UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Housing demand : an empirical intertemporal model Schwann, Gregory Michael 1987

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1987_A1 S39.pdf [ 11.66MB ]
Metadata
JSON: 831-1.0097505.json
JSON-LD: 831-1.0097505-ld.json
RDF/XML (Pretty): 831-1.0097505-rdf.xml
RDF/JSON: 831-1.0097505-rdf.json
Turtle: 831-1.0097505-turtle.txt
N-Triples: 831-1.0097505-rdf-ntriples.txt
Original Record: 831-1.0097505-source.json
Full Text
831-1.0097505-fulltext.txt
Citation
831-1.0097505.ris

Full Text

H O U S I N G D E M A N D : A N E M P I R I C A L I N T E R T E M P O R A L M O D E L by GREGORY MICHAEL SCHWANN B.A., Queen's University, 1976 M.A., University of British Columbia, 1979 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF ECONOMICS) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1987 © G R E G O R Y M. SCHWANN, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of fc^/QoMxe^ The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date n>+ <?, 11*7 A B S T R A C T I develop an empirical model of housing demand which is based as closely as possible on a the-oretical intertemporal model of consumer demand. In the empirical model, intertemporal behavior by households is incorporated in two ways. First, a household's expected length of occupancy in a dwelling is a parameter in the model; thus, households are able to choose when to move. Second, a household's decision to move and its choice of dwelling are based on the same intertemporal utility function. The parameters of the utility function are estimated using a switching regresion model in which the decision to move and the choice of housing quantity are jointly determined. The model has four other features: (1) a characteristics approach to housing demand is taken, (2) the transaction costs of changing dwellings are incorporated in the model, (3) sample data on house-hold mortgages are employed in computing the user cost of owned dwellings, and (4) demographic variables are incorporated systematically into the household utility function. Rosen's two step proceedure is used to estimate the model. Cragg's technique for estimating regressions in the presence of heteroscedasticity of unknown form is used to estimate the hedonic regressions in step one of the proceedure. In the second step, the switching regression model, is estimated by maximum likelihood. A micro data set of 2,513 Canadian households is used in the estimations. The stage one hedonic regressions indicate that urban housing markets are not in long run equi-librium, that the errors of the hedonic regressions are heteroscedastic, and that simple functional forms for hedonic regressions may perform as well as more complex forms. The stage two esti-mates establish that a tight link between the theoretical and empirical models of housing demand produces a better model. My results show that conventional static models of housing demand are misspecified. They indicate that households have vastly different planned lengths of dwelling oc-cupancy. They also indicate that housing demand is determined to a great extent by demographic factors. ii Contents A B S T R A C T i i A C K N O W L E D G E M E N T S i x 1 I N T R O D U C T I O N 1 2 A R E V I E W OF T H E L I T E R A T U R E 3 2.1 THE DEMAND FOR AGGREGATE HOUSING 3 2.1.1 Theoretical and Empirical Frameworks 4 2.1.2 A Review of Past Studies 7 2.2 HEDONIC MODELS 17 2.2.1 The Demand for Housing Characteristics: Theory 18 2.2.2 The Demand for Housing Unit Characteristics: Empirical Considerations . . 21 2.2.3 The Demand for Housing Unit Characteristics: Evidence 22 2.3 SUMMARY 25 3 A N I N T E R T E M P O R A L M O D E L O F H O U S I N G D E M A N D 26 3.1 THE THEORETICAL MODEL 27 3.1.1 Household Preferences 27 3.1.2 Utility Maximization '. 28 3.2 THE EMPIRICAL MODEL 32 3.2.1 The Switching Regression Model 33 3.2.2 The Likelihood Function for a Household's Continuous Demands 34 3.2.3 The Likelihood Function for a Household's Discrete Choices 35 3.2.4 The Length of Occupancy 39 3.3 FUNCTIONAL FORMS 39 3.3.1 A Specification for Household Preferences 39 3.3.2 Housing Expenditures 41 3.3.3 Transaction Costs 42 3.4 CONCLUDING REMARKS 44 4 T H E D A T A 45 4.1 THE MICRO DATA 45 4.1.1 Deficiencies in the SHU Data Set 47 4.2 GENERATED AND FORECAST DATA 48 4.2.1 After-Tax Household Income 48 4.2.2 Forecasting Property Taxes 56 4.2.3 Data on Rejected Housing Alternatives 57 4.3 CONCLUSIONS 57 i i i 5 H E D O N I C P R I C E E S T I M A T E S 61 5.1 THE HEDONIC REGRESSION MODEL 61 5.2 HOUSING PRICES 63 5.2.1 Independent Variables 65 5.3 ESTIMATION TECHNIQUES 68 5.4 RESULTS 72 5.4.1 The Price per Room 73 5.4.2 The Price of Newness 74 5.4.3 The Price per Extra Apartment Unit 75 5.5 OTHER DIAGNOSTICS 77 5.5.1 Testing For Long Run Equilibrium Prices 77 5.5.2 Multicollinearity 81 5.5.3 Location — An Omitted Variable 84 5.6 CONCLUDING COMMENTS 87 6 D E M A N D M O D E L E S T I M A T I O N 89 6.1 INTRODUCTION 89 6.2 PARAMETER ESTIMATES 90 6.2.1 Utility Function Parameters . . . 90 6.2.2 Demographic Scaling Parameters 96 6.2.3 Transaction Costs Constants 99 6.2.4 Optimal Length of Occupancy Parameters 100 6.2.5 Generalized Extreme Value Distribution Parameters 100 6.3 OPTIMAL DEMAND AND DEMAND ELASTICITIES 102 6.3.1 Optimal Demands by Type and Tenure 102 6.3.2 Expected Demands 107 6.4 DEMAND ELASTICITIES 110 6.5 SECOND ORDER PROPERTIES OF THE ESTIMATED FUNCTIONAL FORM . 115 6.6 MODELLING EXPERIMENTS 117 6.6.1 The Hedonic Demand Model versus the Aggregate Housing Model 118 6.6.2 To GEV Or Not To GEV? 118 6.6.3 Full Versus Limited Information Estimation (or Do all Roads Lead to Rome?) 119 6.6.4 Is Flexibility Divine? 123 6.7 SUMMARY 123 7 S U M M A R Y A N D C O N C L U S I O N S 125 B I B L I O G R A P H Y 129 SELECTED BIBLIOGRAPHY 129 A D E T A I L E D S U M M A R Y O F E Q U A T I O N S 138 A.l UTILITY FUNCTION 138 A. 1.1 Continuous Utility Function 138 A.1.2 Normalization on Continuous Utility Function 139 A. 1.3 Normalized Discrete Utility Function 139 A.l.4 First Derivatives of the Continuous Utility Function 139 A.1.5 Second Derivatives of the Continuous Utility Function 139 A.2 STRUCTURAL EQUATIONS 140 A.3 ELEMENTS OF THE JACOBIAN MATRIX 141 A.4 CHOICE PROBABILITIES 143 i v B S T A G E 1, 2, A N D 3 ESTIMATES OF T H E D E M A N D MODELS 144 C ESTIMATES OF HEDONIC PRICE EQUATIONS 147 D ESTIMATES OF HEDONIC PRICE EQUATIONS — LINEAR QUADRATIC M O D E L 158 v List of Tables I STUDIES USING CURRENT INCOME AND INDIVIDUAL HOUSEHOLD DATA 7 II STUDIES USING AGGREGATE DATA OR GROUPED DATA 8 III STUDIES USING PERMANENT INCOME AND INDIVIDUAL HOUSEHOLD DATA 9 IV THE DEMAND ELASTICITIES FOR SELECTED DWELLING CHARAC-TERISTICS 23 V DWELLING ALTERNATIVES . . . 37 VI ESTIMATES FROM AGE-INCOME REGRESSIONS 51 VI ESTIMATE FROM AGE-INCOME REGRESSIONS 52 VII SIMULATED INCOME TAX 58 VIII SAMPLE SIZES 62 IX MEANS AND STANDARD ERRORS OF HOUSING PRICES 66 X WHITES'S TEST FOR HETEROSCEDASTICITY OF UNKNOWN FORM . 70 XI THE PRICE PER ROOM 73 XII THE PRICE OF NEWNESS 74 XIII THE PRICE PER UNIT 76 XIV TESTS FOR LONG RUN EQUILIBRIUM PRICES 79 XV TEST FOR NO JOINT COSTS 82 XVI TESTS FOR CONSTANT PRICES 82 XVII THE PRICE PER ROOM 85 XVIII THE PRICE OF NEWNESS 86 XIX THE PRICE PER APARTMENT UNIT 86 XX DISTANCE-TO-WORK LIKELIHOOD RATIO TESTS 87 XXI PARAMETER ESTIMATES 91 XXII HICKS-ALLEN ELASTICITIES OF SUBSTITUTION 92 XXIII REPRESENTATIVE HOUSEHOLDS 92 XXIV EFFECT OF DEMOGRAPHIC VARIABLES ON DWELLING NEWNESS AND SPACE ACROSS THE LIFECYCLE 96 XXV EFFECT OF DEMOGRAPHIC VARIABLES ON DWELLING TYPE AND TENURE ACROSS THE LIFECYCLE 98 XXVI HOUSEHOLD DEMANDS BY TYPE, TENURE, AND LIFECYCLE CO-HORT 103 XXVII CHOICE PROBABILITIES 107 XXVIII EXPECTED DEMANDS 109 XXIX DEMAND ELASTICITIES I l l v i XXX CHECKS OF SECOND ORDER PROPERTIES 116 XXXI RESTRICTED AND UNRESTRICTED ESTIMATES OF THE DISCRETE CHOICE MODEL PARAMETERS 121 XXXII ESTIMATED COEFFICIENTS FOR MODEL 1 145 XXXIII ESTIMATED COEFFICIENTS FOR MODEL 2 146 XXXIV HEDONIC REGRESSION RESULTS 148 XXXV HEDONIC REGRESSION RESULTS — LINEAR QUADRATIC MODEL . . 159 XXXVI WHITES'S TEST FOR HETEROSCEDASTICITY ON LINEAR QUADRATIC MODELS 165 v i i List of Figures 1 Hedonic Market Equilibrium 20 2 Elasticity of Substitution Between Dwelling Newness and Dwelling Space 93 3 Between Dwelling Newness and Other Consumption 93 4 Between Dwelling Space and Other Consumption 93 5 Demographic Scales for Dwelling Newness and Dwelling Space 97 6 Demographic Scales for Dwelling Type and Dwelling Tenure 97 7 Optimal Value of Newness by Dwelling Type and Tenure 104 8 Optimal Value of Space by Dwelling Type and Tenure 104 9 Optimal Value of Other Consumption by Dwelling Type and Tenure 104 10 Choice Probabilities 108 11 Expected Demands 108 12 Income Elasticity of Expected Demands 112 13 Elasticity of Expected Demands with respect to the Pice of Newness 112 14 Elasticity of Expected Demands with respect to the Price of Space 112 15 Household Size Elasticity of Expected Demands 112 16 Age of the Household Head Elasticity of Expected Demands 112 v i i i A C K N O W L E D G E M E N T S Many people have helped in the production of this thesis. To begin, I would like to thank the three members of my committee — Dr. John Cragg, my supervisor, for his time, all his encouragement, and for his n suggestions, n —• oo , and Dr. Debra Glassman and Dr. Hugh Neary for their helpful criticisms and patient reading of several drafts of the thesis. I gratefully acknowledge the contributions of Prof. Chris Archibald, Dr. Gideon Rosenbluth, and Dr. Ron Shearer, as well as the members of the Economics Workshop at the University of Saskatchewan, in particular, Dr. Don Gilchrist, Dr. Peter Dooley, and Prof. Granville Ansong. I would like to thank Dianne Dupont, Dan Parker, Alan Crawford, and Felice Martinello, my fellow graduate students in Economics, for many hours of discussion. I am grateful for the support and encouragement of my friends, Mary Ann Tisdale, Ann Steven-son, Sue and Tim Diewold, Kassie and Tom Ruth, and Helen and Lome Brooks, who never doubted that it could be done. I thank Cheryl Exner for her meticulous editing of my manuscript, as well as her interest and friendship. My greatest debt is to Evelyn Legare. Her contributions to the thesis are numerous: proofreader, typist, and devil's advocate to name just three. Her greatest contribution, however, has been living with this thesis during the years of its production. i x C h a p t e r 1 I N T R O D U C T I O N Even casual observation indicates that most households move infrequently and that adjustments in housing consumption are often related to events such as marriage, the birth of children, or a job change, rather than to short-term fluctuations in income and prices. As a result of such observations, it has been recognized for some time that housing demand should be analyzed in the context of an intertemporal model of consumer demand. Despite this recognition, almost all of the empirical studies of housing demand are based on the static model of consumer demand. The primary method for adding an intertemporal dimension to empirical studies has been to specify housing demand equations in terms of permanent income rather than current income. Muth (1960) first suggested this approach and it has become a standard in empirical studies of housing demand (e.g., Polinsky 1977, King 1980, Henderson and Ioannides 1983). Another approach to intertemporal modelling, used by studies based on time series data, is to employ a partial stock adjustment model to incorporate time depth (e.g., Houthaker and Taylor 1970, Smith 1974). Although these approaches are well motivated, they are ad hoc. Neither approach transforms the static demand model on which it is based into an intertemporal lifecycle model of consumer demand. In this dissertation, I develop an empirical model of housing demand which is based as closely as possible on a theoretical intertemporal model of consumer demand. In the empirical model, intertemporal behavior by households is incorporated in two ways. First, households are permitted to occupy their dwelling units for any number of time periods; that is, they are able to choose the dates of their future moves. Given that households move infrequently, this feature of the model accords well with the observed economic behavior of households. This feature of the model can also be theoretically justified if one assumes that the monetary and psychic costs of moving from one dwelling to another are relatively large. The second way intertemporal demand behavior is included is by estimating a switching regres-sion model in which a household's decision to move and its choice of dwelling are jointly determined. In the model, both decisions are based on the maximization of the same utility function. Since the decision to move and the choice of housing quantity are joint decisions, it is important to understand how each decision affects the other. 1 The above features result in a highly complex empirical model. In particular, allowing for multiperiod dwelling occupancy means that I must allow for variable time depth in the model. Thus, a central question in this dissertation is whether the complexity of an intertemporal model is necessary to obtain good estimates of a household utility function, or whether a single period model of housing demand provides an adequate approximation to reality. This question is important since the analysis of the welfare implications of possible changes in housing policy, particularly tax policy, depends on having accurate estimates of the parameters of a representative household's utility function. One other feature of my model must be mentioned. In order to deal with the inherent nonho-mogeneity of housing units, I take a characteristics approach to housing demand. Households are assumed to have preferences over the characteristics of a dwelling unit and they generate demands for these characteristics. It is the demands for housing characteristics which I refer to as housing demand. Rosen's (1974) two step procedure is used to estimate the model. In the first step of this procedure, estimates of the implicit marginal prices of the housing characteristics are derived from a set of hedonic regressions of dwelling price on dwelling characteristics. Cragg's (1981) technique for estimating regressions in the presence of heteroscedasticity of unknown form is used to estimate these regressions. The second step is to estimate characteristic demand equations, given the implicit marginal prices estimated in step one. It is this step which involves the switching regression model. This model is estimated using maximum likelihood estimation. A micro data set of Canadian households and dwelling units is used in the estimations. The second step, in particular, uses a sample of 2,513 households. The dissertation is organized as follows. In Chapter 2, I provide a review of the pertinent empirical literature on housing demand. This chapter provides the background for my model, which is derived in Chapter 3. My data sources and data construction procedures are examined in Chapter 4. In Chapter 5, I discuss my hedonic regressions and the derived implicit marginal prices of the dwelling characteristics used in this study. Chapter 6 examines the estimates of my characteristics demand model. Finally, in Chapter 7, I present my conclusions from this research. 2 C h a p t e r 2 A R E V I E W O F T H E L I T E R A T U R E This chapter will review pertinent empirical literature on the demand for housing. Two bodies of literature are examined. The first deals with the demand for "aggregate" housing and is by far the largest body of literature concerning housing demand. A basic assumption of this literature is that housing can be represented as a single, homogenous commodity. Although my model is not based on the aggregate housing assumption, I review the literature because the results of these studies provide a useful backdrop against which to evaluate the results of my thesis. The second body of literature reviewed deals with the demand for housing characteristics. This literature views housing units as a collection of characteristics and assumes that household utility depends on the consumption of the services of these characteristics.1 The housing demand model estimated in my thesis is of this type. 2.1 THE DEMAND FOR AGGREGATE HOUSING The demand for aggregate housing has long been of interest to empirical researchers. The first such work was Engel's (1857) classic family budget study. Engel's data indicated a unitary income elasticity of housing demand. His result was disputed by Schwake (1868) who, using better data, obtained an income elasticity less than unity. The question raised by these two early authors — how sensitive housing demand is to changes in household income — continues to be a major research question in the contemporary housing demand literature. Studies focusing on this question differ in a variety of ways, including their definition of income, their inclusion of a price term, their inclusion of demographic variables, their treatment of tenure choice, their choice of a functional form for the demand equation, and the type of data used (particularly, the level of aggregation of the data). These differences make the evidence on income elasticity unclear. Much of this section will examine the accumulated evidence on the income elasticity of housing demand. Other issues examined are: the price elasticity of housing demand, the role of demographic variables, and the importance of tenure choice to housing demand. 'Models which make this assumption are referred to as hedonic demand models. 3 My overview of the housing demand literature relies heavily on two previous reviews, de Leeuw (1971) and Mayo (1981). De Leeuw reviews four cross-section studies of housing demand. His objective is to reduce the "very wide margin of uncertainty" (1971:1) in existing estimates of the income elasticity of housing demand. He does this by making adjustments to the published elasticities so that they reflect a common definition of housing and relate to the same populations of owner and renter households. For owner occupied dwellings, de Leeuw also adjusts the published elasticities so they include the implicit rental income homeowners derive from their homes. The second review, by Mayo, focuses on analyses completed after de Leeuw's review. Mayo's review is massive, with twenty-seven authors and forty sets of estimates. Although he addresses a number of questions, Mayo is primarily concerned with explaining differences in income elasticity estimates obtained by studies using micro (household) data as opposed to studies using aggregate data. 2.1.1 Theoretical and Empirical Frameworks Before discussing the results obtained by studies of the demand for aggregate housing, some gen-eral observations about the theoretical and empirical frameworks are useful. First, almost all the empirical studies are based on a static model of consumer demand. Households are assumed to maximize their utility by choosing their consumption of two goods, housing services and a compos-ite nonhousing good, subject to a budget constraint. Three implications follow from this paradigm. First, the concept of housing refers to a flow of homogeneous units of housing services; therefore, the quantity and price measures used in an empirical analysis should be flows per unit of time and should take into account the inherent nonhomogeneity of dwelling units. The main problem with this is finding a classification of dwelling units which results in homogeneous units. In practice, most studies of aggregate housing have used either housing expenditure or market value for owner-occupied dwellings as a quantity index for housing services. Both measures are inappropriate. They rank as equal two dwellings which cost the same amount, even though the dwellings may embody substantially different quantities of housing services. Moreover, market value is inappropriate be-cause it is not a flow price. Housing expenditure may be redeemed as a dependent variable if the estimated equations are regarded as housing expenditure equations rather than housing demand functions. In this case, however, one encounters the problem of defining a quality-corrected hous-ing price index. Thus, the problems created by the nonhomogeneity of dwelling units cannot be circumvented by estimating an expenditure function. The second implication following from the paradigm is that the demand for owned and rented dwellings should be analysed simultaneously. This implication arises because household preferences are not assumed to be tenure specific.2 As a result, household utility does not depend directly on the mode of tenure chosen by a household and, hence, neither does housing demand. Tenure, however, has an indirect effect on the demand for housing through the rental price of housing 2 Schwann ( 1 977 ) estimates a mode) in which owned and rented dwellings are treated as separate commodities in a household's utility function. 4 services. All other things being equal, a household will choose the tenure which offers the lowest price per unit of service. Consequently, the price for dwelling services paid by a household contains all the information about tenure needed to determine housing demand. Once this price is known, no further consideration need be paid to tenure. There is also a sound empirical reason for estimating a common demand function for owned and rented dwellings. If separate demand equations for owners and renters are estimated, these equations may be misspecified because of the relationship between housing tenure and the quantity of housing services. Owner occupation is usually associated with single detached dwellings, while tenancy is usually associated with apartments. The former are larger and have a greater range of amenities than the latter. The empirical misspecification arises because households often adjust their housing consumption by "moving up" from a rented apartment to an owned single detached dwelling. This transition will not be accurately reflected in tenure specific demand equations. Hence, fitting separate equations by tenure may result in a downward bias in the estimated effects of the determinants of housing demand. Despite the preceeding arguments, most studies estimate separate equations for owned and rented dwellings. The usual reason given for separate estimates (if a reason is given at all) is an inability to compute comparable price and quantity indexes for owned and rented dwellings. The static nature of the paridigm used in most housing demand studies yields a third impli-cation. A static model implies that households move frequently, changing their consumption of housing services in response to fluctuations in household income, prices, or needs. Casual empiri-cism indicates this is not the case. In fact, households move infrequently, with moves often made in response to major events such as career adjustment, marriage, or changes in household size due to births, deaths, or new household formation. This inconsistency between the theoretical model and reality has long been recognized (e.g., Muth 1960, Reid 1962, Maisel and Winnick 1960). The primary response to the infrequency of household moves has been to specify regression equations in terms of "normal" or "permanent" income rather than current income. The hypothesis underlying this response is that housing demand depends on a household's long run consumption opportunities and is not affected by short-term vagaries in current income. Accordingly, equations estimated with permanent income as an independent variable are interpreted as revealing the long parameters of housing demand. This approach is entrenched in the empirical literature and seldom questioned. The use of permanent income, however, is a "quick fix" to the problem posed by the infrequency of house-hold moves. The essential point is that household housing demand is an intertemporal, lifecycle phenomena and should be modeled as such. The use of permanent income in housing demand equations does not address this issue. Although well-motivated, the use of permanent income does not transform the static model into an intertemporal model and may make little sense in a true lifecycle model. 5 I now consider the empirical framework employed in past housing demand studies. Most studies estimate log-linear demand equations. Among the studies examined below, thirty-nine of the forty-six estimates are obtained using this functional form. The popularity of the log-linear form arises, as Mayo (1981:113) observes, because "the field has been overly preoccupied with estimating 'the' elasticity of demand." The log-linear form is convenient since it imposes a constant elasticity of demand for everyone. Nevertheless, two problems exist with the log-linear functional form. First, the assumption of constant elasticities has "rarely been subjected to even the most rudimentary statistical tests" (Mayo 1981:96). Second, it is not derived explicitly from a known utitility function and, therefore, is unlikely to satisfy the theoretical restrictions required by utility maximization. Notably, in the quest for "the" elasticities of housing demand, researchers have bypassed the linear expenditure equation. This functional form can be estimated as easily as the log-linear equation, and it can be related directly (by utility maximization, subject to a budget constraint) to the Stone-Geary direct utility function (Deaton and Muellbauer 1980, Mayo 1981). A second observation about the empirical framework of housing demand studies is that com-paratively few regressions contain a price index of housing services. This reflects deficiencies in housing data as well as difficulties in correcting for housing quality. The omission of a relevant explanatory variable may lead to biased estimates of the other parameters of a regression equation. One might be tempted to assume that the parameters (elasticities in log-linear regressions) from regressions not containing a price variable are biased, while the parameters from regressions which do contain a price variable are not biased or, at least, closer to the true parameters. Polinsky (1977) demonstrates that this assumption is unwarrented. He considers the biases resulting from estimating a log-linear housing expenditure equation under various common forms of misspecifica-tion. His results indicate that what is important is having the correct price variable. For example, he demonstrates that a log-linear equation estimated on individual household data from several metropolitan areas, but using the mean housing price for dwellings in the area in which the house-hold resides as the price variable, yields an estimated income elasticity further from the true income elasticity than if the same equation were estimated without a price term. Consequently, housing demand equations may be quite sensitive to misspecifications of the housing price variable. My final observation concerns the estimation of permanent income. A variety of methods have been used to obtain measures of normal or permanent income in the literature. The most common method has been to use aggregate data (usually for Standard Metropolitan Statistical Areas (SMSA) or Census Tracts) or to group observations according to some other criteria. This procedure has been criticized by a number of authors (e.g., Nelson 1975, Smith and Campbell 1976, Vaughn 1976, Polinsky 1977, Polinsky and Elwood 1977, and Mayo 1981). If the procedures used to group households are not random, the estimated parameter of a housing demand equation may be biased. In particular, many of the aggregate data sets in use result in upwardly-biased price elasticities. Other methods for estimating permanent income include averaging household income over several years (two to five year averages have been used) and using an instrumental variable(s) for permanent income (e.g., lagged income or income predictions from age-income regressions). A comparison of income elasticity estimates obtained from these other methods is presented in the next section. 6 2.1.2 A Review of Past Studies T A B L E I: STUDIES US ING C U R R E N T I N C O M E A N D IND IV IDUAL H O U S E -H O L D D A T A Demand Elasticities2 Renter, Owner No. Author Functional Form1 Income Price Income Price 1. Barton & Olsen3 SG .39 -.66 2. Carliner LL .44 n.s. .50 n.r. 3a. Kain &c Quigley LL .08 .21 b.3 L .30 .45 4a. Lee LL .65 .80 b. LL .29 .34 5. Lee & Kong LL .35 -.56 .61 -.52 6a. Mayo LL .27 -.17 b.4 LL .30 7. Murray GCES .42 n.r. 8. Nelson LL .28 .24 9. Straszheim LL .42 -.53 .42 10. Wilkinson5 LL .81 -.53 1 L — Linear; LL — Log-Linear; SG — Stone-Geary; GCES — Generalized Constant Elasticity of Substitution; TL — Trans-Log 2 n.s. — not significant; n.a. — not reported 3 Evaluated at $10,000 of income. 4 'Transitory' income elasticity. 5 Based on U.K. data. 6 Based on Canadian data. Here I review the empirical evidence on the demand for aggregate housing. Specifically, I examine the following questions: 1. What is known about the current and permanent income elasticities of the demand for aggregate housing? 2. How do the various measures of permanent income compare in terms of the estimated income elasticities? 3. What is known about the price elasticity of demand? 4. Does the omission of a price term affect the estimated income elasticities? 5. What is the role of demographic variables? 6. Does tenure affect housing demand? Evidence on the income and price elasticities of demand is offered in Tables I, II, and III. Table I presents estimates from studies using current income and individual household data. Table II presents estimates using permanent income, where permanent income is derived by using aggregate 7 T A B L E II: S T U D I E S U S I N G A G G R E G A T E D A T A O R G R O U P E D D A T A Demand Elasticities2 Renter Owner Functional Income Grouping No. Author Form Measure Criterion Income Price Income Price 1. de Leeuw LL current SMSA .81 -.71 1.34 2. Maisel & Winnick LL current city .56 .62 3a. Muth L current city 1.68 -1.59 b. LL effective income SMSA .71 -.76 3a. Nelson LL current SMSA .44 -.68 1.32 -.29 b. LL current income class .25 .40 c. LL current income class .31 .51 d. LL current random .35 .36 e. LL current census tract .76 1.16 5. Polinsky &; Elwood LL effective income SMSA .52 -.70 6a. Reid LL current city 1.00 1.70 b. LL current city .80 1.55 c. LL current census tract & quality 1.16 2.05 7. Schwann6 TL current plus imputed rent city .92 -.93 .92 -1.31 8a. Smith & Campbell LL 2 stage i.v. SMSA .65 b. LL current SMSA 1.21 c. LL effective income random .59 d. LL effective income income class .68 e. LL effective income housing value 1.12 9. Vaughn LL current SMSA .32 -.48 1.88 -.33 10. Wilkinson LL current census tract 1.53 11. Winger LL effective income city 1.05 1 L — Linear; LL — Log-Linear; SG — Stone-Geary; GCES — Generalized Constant Elasticity of Substitution; TL — Trans-Log 2 n.s. — not significant; n.a. — not reported 3 Evaluated at $10,000 of income. 4 'Transitory' income elasticity. 5 Based on U.K. data. 6 Based on Canadian data. 8 T A B L E III: STUDIES US ING P E R M A N E N T I N C O M E A N D IND IV IDUAL H O U S E H O L D D A T A Demand Elasticities2 Renter Owner Functional Income No. Author Form Measure Income Price Income Price 1. Carliner LL 4 year avg. .52 n.s. .63 -.80 2. Fenton LL 3 year avg. .41 -1.28 3a. Friedman & Weinberg LL 3 year avg. .36 -.22 b.3 SG 3 year avg. .44 -.35 4. Mayo LL 2 year avg. .37 -.17 5a. Lee LL 2 year lagged .39 .70 b. LL 2 year lagged .46 .71 6. Lee & Kong LL 3 year lagged full income .70 -.56 .87 -.57 7. Maisel, Burnham, & Austin LL effective income .45 -.89 8. Polinsky & Elwood LL effective income .38 -.67 9. Rosen LL effective income plus imputed rent .35 -.67 10. Smith & Campbell LL effective income .51 11. Mayo LL age-income forecast .46 1 L —- Linear; LL — Log-Linear; SG — Stone-Geary; GCES — Generalized Constant Elasticity of Substitution; TL — Trans-Log 2 n.s. — not significant; n.a. — not reported 3 Evaluated at $10,000 of income. 4 'Transitory' income elasticity. 5 Based on U.K. data. 6 Based on Canadian data. 9 or grouped data. Table III also presents estimates based on permanent income, but, in this table, permanent income is derived using a non-grouping technique. The estimates in Table III are obtained using household data. In addition to evidence on the income and price elasticities, the tables contain information on the functional form, the type of income measure, and the grouping criterion used by each study. Current and Permanent Income Elasticities The current income elasticities presented in Table I are all less than unity. They range from .08 to .81. The average income elasticity for renters is .30, and the average income elasticity for owners is .39. Thus, it appears renters are less responsive to changes in income than owners. This is an interesting finding. A priori, one would assume that renters face significantly lower moving costs than owners and, therefore, would move more frequently. Hence, one would expect the income elasticity for renters to be higher than that for owners — not lower as demonstrated. A plausible explanation for this counterintuitive finding is that increases in income induce renters to move up to the ownership sector of the housing market and out of the renter sample. This would bias the income elasticity for renters downward. Eleven of the twenty-one estimates reported in Table I are from regressions without a price term. For renters, the average income elasticity from the regressions without a price term is .22, while the average income elasticity from the regressions with a price term is .37. Similarly, for owners, the average income elasticity from the regressions without a price term is .40, and the average income elasticity from the regressions with a price term is .51. These figures indicate income elasticities from regressions without a price term tend to be smaller than those from regressions with a price term. Three of the studies in Table I use data from the University of Michigan Panel Survey of Income Dynamics (PSID) (Carliner 1973, Lee 1968, and Lee and Kong 1977). De Leeuw, in his review of the early cross-section studies of housing demand, claims that panel surveys are "not very successful in following movers" (1971:5). He contends the inability to follow movers results in a downward bias in estimates of the income elasticity of housing demand because "households with a high income elasticity may well have a greater probability of moving [and omission from the sample] than households with a low income elasticity" (1971:5). The results of Carliner, Lee, and Lee and Kong do not support de Leeuw's argument, however. These three studies present current income elasticity estimates for renters and owners which are above the average current income elasticities obtained using other types of data sets. Tables II and III contain estimates of the permanent income elasticity of housing demand. These estimates are based on aggregate and micro data, respectively. The average permanent income elasticities from aggregate data are .72 for renters and 1.33 for owners. The average elasticities from micro data are .46 for renters and .58 for owners. Clearly, estimates from aggregate data are 10 higher than estimates from micro data. Moreover, the difference between estimates from aggregate and micro data is greater for owned dwellings than rented dwellings. A number of authors (e.g., Nelson 1975, Polinsky and Elwood 1977, Smith and Campbell 1976, and Wilkinson 1973) have shown that differences between aggregate and micro estimates are due to aggregation bias. These authors demonstrate that serious aggregation bias may occur when households are grouped by city, Census Tract, housing value, or housing quality. Thus, studies 1, 2, 3a, 3b, 4a, 4c, 6a, 6b, 6c, 7, 8b, 8e, 9, 10, and 11, in Table II, may be subject to aggregation bias. The potential bias in these studies can range from approximately 40% (Polinsky and Elwood 1977) to over 400% (Nelson 1975). In addition to causing bias, Polinsky and Elwood have shown that improperly-grouped estimates are extremely sensitive to common specification errors; estimates based on micro data are insensitive to the same errors. The above observations do not rule out the use of all types of aggregate data. When households are grouped randomly (e.g., 4d, 7c), grouped by income class (e.g., 4b, 4c, 8d), or when variables are defined in a way that avoids aggregation bias (e.g., 16, 19a), comparable estimates are obtained from grouped and micro data. The average of the income elasticities based on correctly grouped data is .30 for renters and .51 for owners. These averages compare favourably with the average elasticities from micro data, .46 and .58 for owners and renters, respectively. When estimates based on incorrectly grouped data are omitted, the range of uncertainty over the permanent income elasticity of demand is narrowed considerably. The permanent income elasticity from correctly grouped and micro data ranges from .25 to .52, for renters, and .35 to .87, for owners. The average elasticities for the two tenures are .42 and .55, respectively. As expected, these average elasticities are higher than the corresponding average current income elasticities (.30 for renters and .39 for owners). The current and permanent income elasticities share one feature, however; renters are less responsive to fluctuations in income than owners. Again, this may be the result of model misspecification. All non-grouping methods for deriving permanent income yield roughly comparable estimates for the income elasticity. Two interesting patterns are present in the estimates. First, the income elasticity estimates obtained using average income appear to increase with the number of years in the average. In Table III, the estimate based on a two-year average is .37; the estimates based on three-year averages are .41, .36, and .44; while a four-year average yields an elasticity of .52. Second, a similar pattern is observed in the studies using lagged income. Estimates derived from income lagged two years are less than estimates derived from income lagged three years. These patterns are consistent with the permanent income hypothesis. The above observations suggest estimates of the permanent income elasticity are sensitive to the time frame used to "average out" transitory income fluctuations.3 The income measure with the longest conceptual time frame is "effective income". This measure is available in studies using 3 Muth (1974) advocates using the expected present value of a household's income receipts during its occupancy of a dwelling unit as an income variable. 11 data for Federal Housing Administration (FHA) Section 203 homebuyers. It is an FHA estimate of household income over the first one-third of the FHA mortgage period. Because of its long conceptual time frame, one would expect the permanent income elasticity with respect to effective income to be above the other income elasticity estimates. In fact, the four studies in Table III which use effective income have below average estimates of the income elasticity for homeowners. The reason for this unexpected result lies with the special characteristics of FHA mortgages. There is a legal maximum value on FHA mortgages, and, since dwellings in poor condition cannot be financed under Section 203 (nor can they be financed by conventional mortgages), there is also an implicit lower bound on the value of FHA mortgages. Because of these limits, a disproportionate number of low income households opt for larger dwellngs in order to take advantage of the attractive terms on FHA mortgages; similarly, a disproportionate number of high income households will accept smaller dwellings in order to qualify for an FHA mortgage.4 Thus, estimates of the permanent income elasticity, based on effective income, will be biased downward. On theoretical grounds, homeowners' incomes should include the implicit rental value of their homes. The implicit rental value of an owned dwelling is the factor income an owner household derives from renting the services of its dwelling to itself, and, like all factor incomes, it is part of household income. De Leeuw shows that omitting the implicit rental value of owned dwellings biases the income elasticity estimates away from 1.0. Only two of the studies in Tables II and III, Schwann (1977) and Rosen (1977), include the implicit rental income from owner occupation in their income measures. Unfortunately, the Schwann and Rosen studies cannot be used to confirm de Leeuw's result because both studies are potentially subject to other forms of bias (the Schwann study is subject to aggregation bias; the Rosen study uses FHA data). The discussion of current income elasticites showed that the elasticites derived from regressions without a price term fall below the elasticities derived from regressions with a price term. This result is also displayed by the permanent income elasticities for renters. For owners, there is no appreciable difference in the elasticities obtained by the regressions with or without a price term. The previous discussion also showed that studies using PSID data produce above average income elasticities. This result is confirmed by the permanent income elasticity studies III. 1 and III.2, which produce above average permanent elasticities using PSID data. T h e Pr ice Elas t ic i ty of H o u s i n g D e m a n d Less is known about the price elasticity of housing demand than about the income elasticity. As indicated in section 2.1.1, the major problem is determining the unit price of housing services. Among U.S. studies, four types of housing price variables are used. First, some studies use the metropolitan-wide indexes of housing prices derived from the Bureau of Labour Statistics (BLS) Family Budget surveys (1.1, 2, 5, II.l, 2, 5, 6, 7, 8; III.l, 2, 3a, 4a, 5, 20). The price elasticity *This is supported by FHA data. For a discussion of the evidence, see de Leeuw ( 1971 :6 -7 ) . 12 estimates obtained using this price series range from -1.59 to -.29. This range in the estimates is uncomfortably large. Friedman and Weinberg (1978) and Mayo (1981) observe that, although Carliner (III.l), Fenton (III.2), and Lee and Kong (III.5) all use the PSID data and base their price variables on the BLS series, their analyses produce markedly different price elasticity estimates for rental housing. Carliner estimates price elasticities ranging from —.1 to .2, but his estimates are statistically insignificant. On the other hand, the estimates by Fenton, and Lee and Kong are statistically significant. Fenton's estimates range from -.7 to —1.9, and Lee and Kong's estimate is — .6. Mayo (1981:103) concludes "price elasticity estimates (using the BLS series) are very sensitive to both the way in which prices are denned and to model specification." The second type of housing price variable used is calculated from information on factor prices and the parameters of the housing production function (II.3b, 5; III.7, 8). Studies using this approach obtain estimates of the price elasticity of owned housing ranging from -.76 to —.67. The third type of housing price variable is based on the variable rent rebates offered under the U.S. Housing Allowance Demand Experiment (HADE) conducted in Pittsburgh and Phoenix (III.13a, 13b, 14). The estimates obtained using this variable are close to one another and average — .19. Because of the experimental nature of this programme (especially the limited duration of the programme), these estimated elasticities may be biased downward. The final type of price variable used in U.S. studies is a hedonic price index. Such an index explicitly takes into account variations in the characteristics of dwelling units in order to obtain a "quality corrected" price index. The one study which uses this approach, 1.9, obtains a price elasticity of rental housing of —.53. This figure is comparable with the average price elasticity of rental housing obtained using the BLS price series of —.61. There are two non-U.S. studies in the above tables, Schwann (1977) and Wilkinson (1973). These studies use price series for Canada and the U.K., respectively. In both cases, the prices series used are similiar to the BLS price series. Schwann obtains a price elasticity for renters of -.93 and a price elasticity for owners of —1.31. These estimates are larger in absolute terms than those obtained by most of the American studies. Wilkinson estimates a price elasticity for owned dwellings of - .53, which is of the same magnitude as the elasticities obtained by American studies. Although evidence on the price elasticity of housing demand is far from conclusive, published estimates indicate that, on average, housing demand is inelastic. In addition, it appears renters are less responsive to price changes than owners. The latter result may be due to model misspecification however, as increases in rental price units may induce households to switch from rental to owned accomodation. 13 Demograph ic Effects on H o u s i n g D e m a n d Drawing general conclusions about the effects of demographic variables on housing demand is difficult for two reasons. First, studies of housing demand include different demographic variables. Second, the demographic variables included are incorporated in different ways. The demographic variables most commonly used are the sex of the household head, the age of the household head, the size of the household, and the minority status of the household (white/nonwhite). These variables are either "tacked on" to housing demand equations in an additive manner or used as stratifying variables in the analyses. Because of inconsistent handling of the demographic determinants of housing demand, few pervasive results emerge. The following tentative conclusions may be drawn, however, with regard to the four variables listed above. In Tables I, II, and III, the studies which included variables representing minority status of the household found that nonwhites spend less on housing than whites at comparable levels of income and prices (e.g., Carliner, Fenton, Kain and Quigley, and Smith and Campbell). This result has been rationalized in several ways. Some have contended that nonwhites have a lower preference for housing as compared to other goods. Others have asserted lower housing expenditure by nonwhites is due to market failures such as price discrimination (assuming an elastic demand for housing), nonprice rationing (ghettoization), or discriminatory credit restrictions. Those studies which have investigated the impact of the sex of the household head on hous-ing demand have found that female-headed households spend more on housing than male-headed households. No clear reason for this is known. A number of studies have shown that household size exerts a positive effect on housing demand (e.g., Fenton, Kain and Quigley, and Maisel, Burnham, and Austin). Larger households require more space and, therefore, demand larger dwellings. The relationship between household size and housing demand is nonlinear, however, with the effect of household size decreasing as household size increases (e.g., Miasel and Winnick 1960, David 1962). The effect of the age of the household head on housing demand is not clear. Some studies (e.g., Carliner, Fenton, Kain and Quigley, and David) have shown a monotonic positive relationship between age and the quantitiy of housing services demanded. Other studies, particularly those using age of the household head as a stratifying variable, indicate the relationship between age and housing demand does not follow a consistent pattern — the impact of age may even be negative for some age categories and positive in others. These mixed results are not surprising since this variable is a proxy for a number of factors affecting housing demand. Two factors come to mind. First, the age of the household head summarizes the position of the household in its intertemporal housing consumption plan. Because of the transaction costs involved in moving, housing adjustments will be intermittent; hence, there is no reason to believe the effect of age is monotonic across age. Second, the age of the household head may be a proxy for intergenerational shifts in housing preference. 14 Tenure Choice and the D e m a n d for H o u s i n g All the studies reviewed above estimate tenure-specific demand equations. In section 2.1.1, I ar-gued that this approach may entail a misspecification of the housing demand equations. Specif-ically, misspecification will occur if a household's tenure choice equation is related to its housing demand equation, either through the covariance structure of the equation errors or through com-mon parameters in the tenure choice and demand equations. To avoid potential misspecification when estimating the demand equations, one should take into account the simultaneity between a household's tenure choice and housing demand decisions. In this section, I examine whether one needs to treat tenure choice and housing demand as simultaneous decisions when attempting to obtain reasonable estimates of the demand equation parameters. To answer this question, four studies are considered: Lee and Trost (1978), Gillingham and Hageman (1983), King (1980), and Henderson and Ioannides (1983). These studies were not included in Tables I, II, or III as they represent significant departures from the previous studies. I provide a brief description of each study before reviewing their results. This extended discussion of these studies is warranted because the model of housing demand estimated in this thesis treats tenure choice and the demand for housing services as joint decisions. Hence, the results of these studies are directly applicable to this thesis. The study by Lee and Trost is the simplest of the four. They use a log-linear specification for both their tenure choice equation and their housing demand equation. The income variable they employ is a five-year average of real family income plus real imputed rental income for homeowners. A housing price index is developed from the BLS price data. The dependent variable of the housing demand equation is derived by dividing annual housing expenditure by the housing price index. Data from the PSID are used to estimate the model. Gillingham and Hageman, like Lee and Trost, estimate ad hoc functional forms for their tenure choice and housing demand equations; in their case, the Trans-Log functional form is used. Also, as in Lee and Trost, the authors use PSID data and base their housing price measures on the BLS price indexes. Gillingham and Hageman differ from Lee and Trost, however, in their treatment of the impact of the U.S. Federal Tax System on housing demand. They introduce the tax system in two ways. First, they use after-tax measured income as their income variable. Second, they take account of the mortgage interest and property tax deductability provisions of U.S. income tax in computing the implicit annual rental cost of owned dwellings. The two remaining studies also pay attention to the impact of the tax system on tenure choice and the demand for housing services. King's study of the U.K. housing market has become a seminal piece in housing demand liter-ature. The major innovation in the study is to treat a household's choice of tenure and its demand for housing services as joint decisions based on the maximization of the same utility function. This approach, as King (1980:137) notes, "implies cross-equation constraints on the parameters and the functional form of the equations determining tenure choice and the demand for housing services." 15 Another of King's innovations is incorporating the rationing of dwellings by tenure directly into his model. This feature is made necessary by unique constraints on tenure choice in the United Kingdom. King based his estimating equations on the Trans-Log reciprocal indirect utility func-tion. As an index of housing consumption, he uses a dwelling's "gross rateable value". This value is contained in the U.K. Family Expenditure Survey (FES) — the micro data base for the study; it estimates the rental value of a dwelling assigned by an official dwelling assessor. King's income variable is "normal" gross household income from all sources (defined in the FES), plus the im-puted rent of owner occupiers, less U.K. income taxes and national insurance contributions. Each household in the sample faces individual prices which depend on the tax treatment of housing in the United Kingdom. The model estimated by Henderson and Ioannides is similar in spirit to King's. Their model bases tenure choice and housing demand functions on the same household preference ordering. This preference ordering is given by a variant of the two good Stone-Geary indirect utility function. The model also incorporates rationing by tenure, accomplished by incorporating credit rationing in the mortgage market, since, as Henderson and Ioannides note, whether a household owns or rents in the U.S. depends primarily upon a household's access to mortgage credit. Unlike King, who assumes that rationing is purely random, Henderson and Ioannides include a rationing function which depends on household characteristics, specifically, the marital status, age, and race of the household head, as well as after-tax household income. The model is estimated using U.S. Annual Housing Survey (AHS) data for households which were recent movers in 1975 and which stayed in their dwellings throughout 1976 and 1977. The income variable equals one-third of the present value of household income from all sources, less U.S. federal income taxes and real estate taxes for the years 1975, 1976, and 1977. The price of housing services is based on BLS price data but incorporates the tax position of individual households. The studies by Lee and Trost, and Gillingham and Hageman allow for the interdependence of tenure choice and housing demand by permitting disturbances in the housing demand functions and the tenure choice equation to be correlated. Both sets of authors estimate their models using a two-stage procedure developed by Lee and Trost. The procedure yields direct estimates of the relevant correlations. Lee and Trost estimate a correlation between disturbances of the housing demand function for owners and the tenure choice function of —.2163 and a correlation between the housing demand function for renters and the tenure choice function of —.0047. The analagous estimates by Gillingham and Hageman are .1056 and .1570, respectively. The test of the null hypothesis that the correlations are zero is rejected in both studies; that is, there is simultaneity between tenure choice and housing demand. In addition to considering whether or not simultaneity exists, Lee and Trost examine the impor-tance of simultaneity to the parameter estimates of the housing demand equations. They do this by comparing their two-step maximum likelihood estimates (2SML) of the tenure specific demand equations to OLS estimates of the same equations. They find some differences in the parameters 16 — the value of one parameter obtained using 2SML is five times greater than the corresponding OLS estimate. On the whole, however, they find the two sets of estimates are quite close. In the studies by King and Henderson and Ioannides, the tenure choice and housing demand equations are linked by the parameters of the household utility function and not by a correlation between disturbances of the housing demand and tenure choice equations. To assess the impor-tance of simultaneity in these models, one must compare the maximum likelihood estimates of the parameters from the joint tenure choice and housing demand model with the maximum likelihood estimates of the parameter from the housing demand equation alone. Both King and Henderson and Ioannides do this. For King, the two sets of parameter estimates differ only in the fifth decimal place. Henderson and Ioannides also find a negligible difference in the parameters. In conclusion, the four studies indicate that, while simultaneity exists between tenure choice and housing demand, the parameters of the housing demand function are not sensitive to this simultaneity. Therefore, the common procedure of estimating demand equations without reference to tenure choice may give reliable estimates of the relevant demand parameters. 2.2 HEDONIC MODELS The analyses of housing demand reviewed in the preceeding section are based on an assumption of the existence of a homogeneous unit of aggregate housing services. A large number of authors have questioned the existence of such an aggregate. For example, Ellickson (1981:56) argues that "housing is not a homogeneous commodity, but a collection of commodities that are all distinct to some degree." Many of these authors have adopted the hedonic hypothesis that household preferences depend on the characteristics of a dwelling unit, such as size or age of the dwelling, levels of local public goods (e.g., parkland and police protection), or location-specific levels of air pollution. That is, households do not value dwelling units in themselves; rather, they value dwelling units for their utility bearing characteristics. It follows from the hedonic hypothesis that, since each housing unit embodies a different collection of characteristics, each unit is distinct, but housing units with similar characteristics will be close substitutes for each other. This section decribes a hedonic model of housing demand. It is based on the model of the de-mand for differentiated commodities set out in Rosen (1974). Rosen's paper made two important contributions to the hedonic demand literature. First, the hedonic or implicit prices of the char-acteristics of the differentiated product are the result of a competitive equilibrium in the demand and supply of the differentiated product. This represents a departure from earlier hedonic analyses which regarded housing characteristics as inputs in the household production of aggregate housing. Second, Rosen's paper develops a two-step procedure for estimating the market hedonic demand and supply functions. This procedure has been used by an increasing number of authors; for exam-ple, Harrison and Rubinfeld (1978) and Nelson (1978) use it to estimate the benefits of clean air; Linneman (1981) uses it to analyze residential site choice; and Witte, Sumka, and Erekson (1979) and Blomquist and Worley (1981) consider the demand for a number of housing characteristics. 17 Since the housing demand model estimated in this thesis is based on the Rosen two-step proce-dure, a detailed specification of the Rosen model is presented in the following section. The section is divided into three parts. Part 1 presents the theoretical structure of Rosen's hedonic demand model as applied to housing characteristics. Part 2 examines Rosen's two-step estimation procedure. In part 3, I review the results obtained by five studies of the demand for housing characteristics. 2.2.1 T h e D e m a n d for Hous ing Characterist ics: Theory It is assumed that a household has preferences over both nonhousing goods and the characteristics of the dwelling unit in which it will reside. The nonhousing goods are given by the m dimen-sional vector x = (xi,...,xm) and the dwelling unit characteristics by the n dimensional vector z = (z\,..., zn). The household utility function is given by u(x, z) and is assumed to be strictly quasiconcave and twice continuously differentiable. Each nonhousing good z, can be purchased separately at a fixed (parametric) price pi. The housing characteristics z,-, on the other hand, can seldom be purchased individually; instead, a wide variety of dwelling units are offered for sale in the housing market. The hedonic price function q(z) relates the market price of a dwelling unit to the fixed vector of dwelling characteristics. In a competitive market where single agents treat q(z) as parametric in their decisions, this function gives the minimum price of a dwelling unit with characteristics z (Rosen 1974:37). The function q(z) will, in general, be increasing in its arguments, since firms require additional resources to increase each characteristic z,-, and it will be nonlinear, reflecting the fact that housing units are indivisible and their characteristics not easily modified. The assumption of indivisible housing units is referred to as the "no arbitrage" assumption by some authors. In addition to these properties, q(z) is assumed to be twice continuously differentiable. The household budget constraint is given by y = p'x + q(z) where y is household income. The household maximizes its utility function u(x, z), subject to this nonlinear budget constraint. It is instructive to consider this operation in two stages. In the first stage, the household maximizes u(x, z) with respect to the vector of nonhousing goods x, subject to p'x = y — q(z) for a fixed vector of dwelling characteristics. This yields the household's conditional indirect utility function v(p, z,y- q(z)).5 The second-stage maximization problem for the household is to choose the vector of characteristics z which maximizes v(p, z, y—q(z)). Since each vector of characteristics z represents a different dwelling unit, the household's second-stage problem is equivalent to choosing its utility maximizing dwelling unit. The first order conditions for the household's second-stage maximization problem are dv/dzi = dv/d(y - q(z)) • dq/dz{ z' = l,...,n (2.1) SA conditional indirect utility function has the following properties: (i) if the direct utility function u is continuous, u will be jointly continuous in (p, z, y — ?(z)); (ii) it is nonincreasing and quasiconvex in p for p > 0m, fixed z, and fixed y — q(z); (iii) it is nondecreasing and quasiconvex in (z,y — q[z)) for fixed p; and (iv) it is homogeneous of degree zero in (p, y — q{z)) for fixed z. An excellent discussion of the conditional indirect utility function is given in Diewert (1974). 18 or ~ dv/dzi 3v/c9(y - q(z)) These conditions are not particularly revealing. Rosen (1974:38-41) shows that greater insight into a household's consumption decision is gained when these optimality conditions are expressed in terms of the household's bid function b(p, z, y, u), which is the solution to v(p, z, y- 6) = u. This function gives the maximum amount that a household would willingly pay for z when it has income y, faces prices p, and wishes to maintain the fixed utility level u. By implicit differentiation, it is possible to show that b is (i) decreasing in p, (ii) increasing in z, (iii) increasing in y, and (iv) decreasing in u. The most important result, however, is that db/3zi= , ^ J92', „=dq/dzt i = l,...,n (2.3) Thus, a household's consumption optimum is characterized by the tangency between the household's bid function and the market hedonic price function. In order for this tangency to be a household equilibrium, the bid function must be evaluated at the household's maximum utility level. Figure 1 illustrates the household's tangency condition. In the figure, a household's bid rent curve, 6, is tangent to the market hedonic price schedule at the point A. At the point of tangency, db/dzi > 0, indicating the household has a positive marginal evaluation for the characteristic (i.e., it finds the characteristic desirable), and, therefore, it would pay a positive price for more of it. Only one bid rent curve is shown in Figure 1. In reality, variations in household incomes and preferences lead to a family of bid rent curves. Since dwellings go to the highest bidder, the hedonic price function q(z) must be the upper envelope of this family of bid rent functions. The development of the production side of Rosen's model parallels the consumption side. Pro-ducers are assumed to maximize profits subject to available technology. They have control over the design of each dwelling unit (i.e., the package of characteristics z), the number of units of each design they produce, and the levels of the factors they use. Each producer is a perfect competi-tor; hence, the hedonic price function q(z) is regarded by the firm as parametric to its production decisions. Given this structure on the production side of the model, it is possible to define a producer's offer function <f>(z, 7 r ) , which indicates the minimum price a firm would charge for a dwelling with characteristics z at the fixed profit level n. It is possible to show that d<f>/dzi > 0, and d<f>/dn > 0. A producer's equilibrium is characterized by the tangency of the firm's offer curve and the market hedonic price schedule; that is, d<f>/dz{ = dq/dzi t'=l,...,n (2.4) where <f> is evaluated at the producer's maximum profit level. One such tangency is illustrated in Figure 1. In the figure, the firm's offer function <j> is tangent to the market price schedule at point B. Only one offer curve is shown in the figure. If one allows for variation in the conditions 19 Figure 1 Hedonic Equilibrium Dwelling Characteristic — i z of production, there will be a family of offer functions, and the lower envelope of this family of functions will be q(z). A competitive equilibrium is achieved when the quality demanded of each of the housing char-acteristics equals the quantity supplied. Thus, in equilibrium, the bid and offer functions must have a common gradient in order for market clearing to take place. It follows that observed prices for dwellings, q(z), must represent "a joint envelope of a family of value [bid] functions and another family of offer functions" (Rosen 1974:44). 2.2.2 The Demand for Housing Unit Characteristics: Empirical Considera-tions Rosen (1974:50-51) suggests a two-step method for estimating the demand and supply relationships for dwelling characteristics. The first step is to estimate the market clearing hedonic price function q(z) by regressing the observed market price of a set of dwelling units on their characteristics. Such an equation is, of course, misspecified since it omits a wide range of variables affecting the clearing price schedule. These omitted variables include the prices of nonhousing goods p, factor prices tu, parameters reflecting the distribution of income and tastes, and parameters reflecting technological diversity. Rosen justifies the omission of these variables by arguing that the suggested regression "econometrically duplicates the information acquired by agents in the market, on the basis of which they make their decisions" (1974:50). That is, the agent's expected clearing price schedule is important and this schedule is unlikely to depend upon factors other than housing characteristics.6 Once the hedonic price function has been estimated, the estimated implicit, marginal hedonic prices can be computed as = dq(z)/dzi i = 1,..., n, where q(z) is the estimated hedonic price function. In the second step of Rosen's method, the estimated marginal prices are used as the dependent variables in the inverse demand and supply functions q,\ = D*(z,p, y), i =l,...,n (2.5) qi = Sl(z,p,y), i =l,...,n (2.6) These equations form a system of 2n simultaneous equations in the 2n endogenous variables and Zi. Rosen asserts that, in general, the identification of equations (2.5) and (2.6) involves nothing more than the usual rank and order conditions. This assertion has been challenged by both Brown and Rosen (1982) and Murray (1983). Brown and Rosen show that, contrary to Rosen's claim that "estimation of marginal prices plays the same role here as do direct observations on prices in the standard theory" (1974:50), the estimated marginal prices contain all the information in the observed sample, and therefore, estimating the demand and supply functions above will only 6One possible problem with relying on expected clearing prices is that they must be the product of some form of search process, yet the process of search is exogenous to Rosen's model. A more complete model would treat the amount of search (i.e., knowledge of the hedonic price schedule) and housing consumption as simultaneously determined. 21 reformat the information garnered in estimating q(z). The authors argue that the only way to identify the parameters of the demand and supply functions is to place restrictions (possibly ar-bitrary) on the functional forms of the equations. One type of restriction which can be applied in most situations is to estimate separate hedonic price equations for distinct housing markets7 and then estimate a common set of structural demand and supply equations for all submarkets. Identification is achieved by this procedure because estimated hedonic prices from multiple housing markets will not be exact combinations of the arguments in structural equations. Murray (1983) challenges Rosen's interpretation of equation (2.5) as a demand equation. Con-ventional demand equations, Murray notes, give a household's response to changes in exogenous prices and income. In the hedonic model, however, prices are endogenous because of the nonlin-ear budget constraint. Thus, the equations (2.5) will not have the same behavioral properties as conventional demand equations, and they should not be treated as such. What Rosen's "demand" equations identify are the marginal rates of substitution between housing characteristics and non-housing consumption, as indicated by equation (2.1). There is nothing wrong with estimating marginal rates of substitution, but the estimated equations must be interpreted accordingly. To retain the familiar behavioral interpretations of demand equations, Murray suggests using the linearization approach of Hall (1973), Wales and Woodland (1979b), Hausman (1980), and Wales (1982) . In this approach, one approximates the nonlinear budget constraint by a linear constraint having prices g, = dq/dzi and passing through the household's demand point. The household income consistent with this linearized budget constraint is p/x + q'z, where q = (qi,..., qn)- Given this budget constraint, consumers will purchase the same bundle of nonhousing goods and the same dwelling unit as they did when facing the nonlinear constraint. Murray is careful to point out, however, that the demand functions derived from the linearized constraint exist only in theory; "They refer to the demand behaviour which would be observed if attributes were sold in classical markets" (1983:330). I turn now to the estimates obtained by several hedonic studies of housing demand. 2.2.3 The Demand for Housing Unit Characteristics: Evidence Many authors have used Rosen's two-step proceedure to estimate the demand for various housing characteristics. There is little accumulated evidence on any one housing characteristic, however, because different authors have focused on different characteristics.8 In this section I draw together the scattered evidence on the demand for dwelling size, lot size, accessibility to the workplace, and dwelling age. The evidence on these characteristics comes from studies by Straszheim (1975), Witte, Sumka, and Erekson (1979), Blomquist and Worley (1981), Linneman (1981), and Bajic (1983) . All use Rosen's two-step estimation procedure. Table IV summarizes their results. 7These housing markets may be separated spatially — for example, a hedonic model fitted to data from separate cities or local housing markets. Alternatively, housing markets may be defined in terms of some discrete unit characteristic such as tenure or type. 8The use of different characteristics by different authors is also due to a lack of consistency in the dwelling unit descriptions in different housing data sets. 22 T A B L E I V : T H E D E M A N D E L A S T I C I T I E S F O R S E L E C T E D D W E L L I N G C H A R -A C T E R I S T I C S Renter Owner Income Price Income Price A . Dwe l l ing Size 1. Bajic .08 -8.33 2. Blomquist and Worley n.s. -.26 3. Linneman .03 -.77 4. Straszheim .063 0.00 .12 -.081 5. Witte, et. al. n.s.2 -.183 B . Lot Size 1. Bajic .06 -.94 2. Straszheim .35 -1.02 3. Witte, et. al. .95 -1.91 C . Workplace Accessibi l i ty 1. Bajic .11 -.90 2. Linneman n.s. -.76 3. Straszheim 0.0 n.a. D . Dwe l l ing A g e 1. Straszheim -.19 1.26 -.17 -1.44 1 Computed for a 1200 square foot dwelling, valued at $15.00 per square foot. 2 Computed for a 3500 square foot lot, priced at $2.00 per square foot. 3 Computed for a 3500 square foot lot and an income of $10,000. 23 Each study considers the demand for dwelling size. Three measure dwelling size by the number of rooms in the dwelling (Blomquist and Worley, Linneman, and Straszheim). The two remaining studies, Bajic, and Witte, et. al., measure dwelling size by square footage. The studies show that the demand for dwelling size is extremely income inelastic; the largest income elasticity is .12. The demand for dwelling size also tends to be price inelastic. Except for Bajic's outlying estimate of —8.33, the price elasticity estimates fall in the range 0.0 to —.77. The demand for dwelling size is more responsive to changes in demographic variables than to changes in income and price. The studies show that the demand for dwelling space increases if household size or the age of the household head increases or if the househead is female. Bajic, Straszheim, and Witte, et. al. examine the demand for lot size. In these studies, income and price affect lot size in the expected manner. The income elasticities range from .06 to .95, while the price elasticities range from .94 to —1.91. The demand for lot size is clearly more responsive to income and price than is the demand for dwelling size. Notably, the income and price elasticities obtained by Witte, et. al. for renter households are substantially larger than the corresponding elasticities obtained by Bajic and Straszheim for owner households. Whether this is a general result remains to be seen. While the demand for lot size is more responsive to price and income, it is less responsive to demographic variables. Only the age of the head of the household appears to have an impact. As the househead ages, the household's demand for lot size increases. The demand for accessibility to the workplace is examined by Bajic, Linneman, and Straszheim. Straszheim measures accessibility by the travel time to work of the household head; Linneman measures accessibility by the distance to the central business district (CBD). Bajic mea-sures accessibility using three measures: the walking plus waiting time to the nearest subway or bus stop, the driving time to the CBD weighted by the probability of employment in the CBD, and a probability-weighted average of the driving time to employment centres outside the CBD. The income elasticities reported by these authors are either small or insignificant. Bajic and Linne-man estimate price elasticities, but Straszheim does not. The estimates for Bajic's three measures are —.90, —1.02, and —.86 respectively. Linneman obtains an elasticity of -.76. Thus, while the demand for accessibility responds very little to changes in income, it is reasonably responsive to changes in the marginal price of accessibility associated with a dwelling. Demographic variables have no appreciable effect on the demand for accessibility. Only one of the studies, Straszheim, considers the demand for dwelling age. He reports an income elasticity of —.17 for age of owner households, and —.19 for renter households. The price elasticity of demand is given only for owners; it is 1.44. These elasticities tend to be lower for married households than for unmarried households. Four general findings emerge from the studies examined above. First, the demands for housing characteristics tend to be both income and price inelastic. Second, the demands are substantially more responsive to fluctuations in prices than to fluctuations in income. Third, by comparing the elasticities in Table IV with those in Tables I, II, and III, one sees the income elasticites for individual characteristics are much lower than those obtained for a housing aggregate. Fourth, demographic variables affect the demand for housing characteristics but not in any systematic fashion. Different demographic varaibles affect different characteristics. 24 2.3 SUMMARY This chapter has examined two approaches to estimating the demand for housing. The first and most common approach is based on the assumption that "housing" can be represented by a single aggregate index. Studies employing this approach have obtained the following general results. First, housing is income inelastic. This result holds for both renters and owners, although owners tend to be more responsive to changes in income than renters. Second, the income elasticity depends critically on the income measure used. In particular, studies using permanent income get higher income elasticities than studies using current income. Third, housing is, on average, price inelastic. It is against these general results that all current and future housing demand estimates (including the present work) must be evaluated. Most of the aggregate housing demand studies estimate tenure specific demand equations. This may involve a simultaneity bias since adjustments in housing demand often involve tenure switching. A branch of the aggregate housing demand literature has analyzed whether such a simultaneity bias exists. This literature shows that, while the presence of simultaneity cannot be rejected statistically, it has only a minor effect on the parameters of the demand equations. This result is important for two reasons. First, it validates the existing estimates of the price and income elasticities of housing demand. Second, it indicates two-stage procedures can be used to obtain reasonable estimates of the parameters of the housing demand equation(s) and the tenure choice equation. This is especially important in the present study since I estimate a joint type/tenure choice model and a housing demand model in which there are a fairly large number of parameters. The two-stage estimates should provide a good starting point for the full model. The second approach to estimating the demand for housing examined above uses the demand for housing characteristics. This approach is followed in this thesis. Because the hedonic approach is less popular, the accumulted evidence is fragmentary. The general results appear to be similar to those obtained for aggregate housing — housing characteristics, like aggregate housing, are income and price inelastic. I expect similar results from the present study. One further area of agreement between the two approaches to housing demand is the impact of demographic variables. Demographic variables have been shown to have a significant impact on housing demand. Despite consensus regarding the importance of demographic variables, an understanding of their role has been impeded by the ad hoc ways in which these variables have been included in analyses. Mayo (1981) has suggested a more productive line of investigation may be found in including demographic parameters in the household utility function. He regards the demographic translating techniques of Pollack and Wales (1981) as the most promising way to do this. One such technique is used in this thesis. The studies reviewed above provide a basis for my model of housing demand, as well as a gauge with which to measure my empirical results. There is, however, one crucial difference between the studies reviewed in this chapter and the model developed in the next. The studies reviewed here are all static models of demand, while my model is fundamentally an intertemporal model of housing demand. I turn now to the development of the model. C h a p t e r 3 A N I N T E R T E M P O R A L M O D E L O F H O U S I N G D E M A N D In this chapter, I construct an empirical, dynamic model of housing demand. The central feature of the model is that households are permitted to occupy their dwelling units for any number of time periods. Since most households move infrequently, approximately every five and one-half years on average, this feature of the model accords well with observed economic behavior. Multiperiod dwelling occupancy can also be justified theoretically. The costs of moving from one dwelling to another are relatively large, so it is in the interest of a household to economize on its lifetime moving costs by moving infrequently. I allow for multiperiod dwelling occupancy in my statistical model by including a household's expected length of occupancy as a parameter in my estimating equations. The model has several other features which should be recognized at the outset. First, I jointly estimate the housing demands of households which have recently moved to new dwellings and households which have chosen to stay in their current dwellings. Both sets of households are assumed to have common preferences and moves are treated as endogenous, utility maximizing decisions. The model can also accomodate moves resulting from exogenous events such as a job change or transfer or changes in family composition, provided that these changes are perfectly anticipated. The model cannot handle moves produced by random events, however.1 Second, the model is based on a characteristics approach to housing demand. This approach is taken in order to deal with the inherent nonhomogeneity of housing units. The dwelling char-acteristics I examine are dwelling size, age, type, and tenure. The latter two characteristics are included to test whether households have type and tenure specific preferences. The demands for dwelling type and tenure are regarded as jointly determined with the other demands in the model. To incorporate these discrete demands, a switching regression model is specified. The third feature of the model is that the transaction costs of changing dwellings are intro-duced directly into both the theoretical and statistical models of housing demand. This is done to strengthen the assumption of multiperiod dwelling occupancy. 1A model which does handle moves from random events is constructed by Muth (1974). 26 Finally, the model makes extensive use of information on mortgages held by owner occupiers in computing the housing expenditures of households. Previously, most studies of housing demand have assumed that owner occupied dwellings are purchased outright. In fact, most Canadian; households use some form of mortgage financing to purchase a dwelling. The chapter is organized into four major sections. In the first, I present the dynamic model of housing demand which forms the basis for my empirical model. This dynamic model is tailored into an empirical model of housing demand in the second section. Both the theoretical model in section one and the empirical model in section two are presented in general terms. The third section gives the specific functional forms I use for functions in the empirical model. The final section offers concluding remarks concerning the model. 3.1 THE THEORETICAL MODEL My objective in this section is to construct a dynamic model of housing demand in which households choose their length of occupancy in a dwelling. The basic model is an intertemporal model of consumer demand which has been adapted to take into account some peculiarities of the housing market. Throughout the section, my focus is on the housing choices faced by households and on characterizing the optimal solutions to these choices. The section is divided into three parts. First, I specify household preferences. Second, I describe the constraints on household choice. Third, I characterize the solution to a household's intertemporal utility maximization problem. 3.1.1 Household Preferences Household preferences are represented in this study by the additively separable utility function U = J2u{t)pt (3.1) t-i where u(t) is the household's period t utility index, p > 0 is the household's (constant) rate of time preference, and T is the time remaining in the household's economic life. Each of the utility indexes u(t),t = 1,...,T, is defined over a composite nonhousing good x(t) > 0 and the services from n housing characteristics z(t) = (zi(t),..., zn(i)). The service flow from each housing characteristic is assumed to be proportional to the stock level of the characteristic. Unfortunately, defining household utility in terms of housing characteristic services leaves un-specified the number of housing units occupied by a household. This problem has received no attention in the housing characteristics demand literature.2 I handle this problem by assuming the characteristics z(t) refer to a household's principle residence. The consumption of services from other dwellings, such as a summer cottage, are included in the composite good x(i). 2It has been considered in a production setting by Rosen (1974) and Diewert (1980). 27 The vector z(t) contains two types of housing characteristics, those which are available in any nonnegative quantity and those which are available in discrete units. Accordingly, z(t) can be partitioned into (z 1(i), 2 2(t)), where z1(t) > 0 n i is an nj vector of continuous characteristics and z2(t) is an 712 = n — n i vector of discrete characteristics. In this thesis, dwellings are described by two continuous characteristics and two discrete charac-teristics. The continuous characteristics are: z\ — the size of the dwelling and z\ — the "newness" of the dwelling unit. The two discrete housing characteristics are: z\ — the type of dwelling unit and z\ — the dwelling unit tenure. The discrete characteristics are categorical variables. They des-ignate three types of dwelling units — single dwellings, multiple dwellings, and apartment dwellings — and two modes of tenure — owned dwellings and rented dwellings. These type and tenure cat-egories generate six mutually exclusive varieties of housing. This set of varieties is denoted by D. The period t utility function can now be written as u(t) = u(x(t), zl(t), z2(t)). I assume house-holds have a positive preference for larger and newer dwellings, but I make no specific assumptions regarding the importance of dwelling type and tenure. Two frequently advanced hypotheses can be tested, however: (i) households have a preference for single houses, and (ii) households experience a "pride of ownership", that is, they have a preference for owning their own dwelling. To the best of my knowledge, these hypotheses have never been tested formally. Finally, I assume the function u is twice continuously differentiable, increasing, and strongly quasiconcave in x(t) and z*(t), for x(t) > 0, 2 x(t) > 0ni, and z2(t) G D. 3.1.2 Ut i l i ty M a x i m i z a t i o n Constraints on Household Choice At the beginning of every time period, households maximize their remaining lifetime utility (3.1) by choosing x(t) and z(t),t = 1,...,T, subject to five constraints. The first four constraints are dynamic; they describe the time paths of the housing characteristics during a household's occupancy of a dwelling unit. These constraints are *i(0 = *}(*?) t = taV--;t° (3.2) 4(t) = * = «?,..., t) (3.3) *?(«) = *?(*?) t = (3.4) *!(*) = r = *?,..., (3.5) where ta- is the household's first period of occupancy in its jth dwelling unit (i.e., the acquisition date of the dwelling) and t" is the date on which the household moves from this unit (i.e., its sale or disposal date). The interpretation of these constraints is straightforward. The constraints (3.2), (3.4), and (3.5) state that size of the dwelling, dwelling type, and dwelling tenure do not change between ta- and t*. Constraint (3.3) states that a dwelling ages at the constant rate of one year 28 per year of occupancy. Constraints (3.2) and (3.3) are somewhat simplistic. Constraint (3.2) does not take into account renovations or additions to the dwelling which would alter the effective living space within the unit, while constraint (3.3) imposes a restrictive notion of newness. Nevertheless, these constraints adequately approximate the situation of most households. Constraints (3.4) and (3.5), on the other hand, are quite reasonable. Note that constraints (3.2) to (3.5) transform a household's dynamic optimization problem into a static optimization problem. The whole time profile of the characteristics during the occupancy of a dwelling unit is known once the initial values of the housing characteristics are known. That is, a household only needs to choose the initial value z{t^). The fifth constraint confronting a household is its intertemporal wealth constraint. Without loss of generality, this constraint may be stated in terms of the sequence of dwelling units occupied by the household over the remainder of its economic life. When a household occupies J dwelling units between t = 0 and t — T, its wealth constraint can be written as where tt i ]T \I(t) - x{t) - h[t)\ 6l - [Ha{t°) + TCa{t^)] 6'° - [H'{ty + TC3{t*)] 6*' i 0 (3.6) W— the household's wealth at the beginning of period 1, I{t)= the household's expected after-tax income in period t, h(t)— h(z{t)), the household's expected recurrent housing expenditure in period t, 6= the discount rate in the capital market, Ha{ta-)= Ha(z(t'j)), the capital expenditure required to acquire the jth dwelling unit, Ha(tj) — Hs(z(tj)), the capital payment required on the sale of the jth dwelling unit, TCa{t^)— TCa{z(t^)), the transaction costs associated with acquiring and moving to the jth dwelling unit, TCs(t"j)= TC(z(tj)), the transaction costs assoicated with selling the jth dwelling unit, £}= the dates on which the household moves into its J dwellings, tj,..., t"j= the dates on which the household moves out of its J dwellings. The two sets of dates in equation (3.6) are linked by the J + 1 equations: if = 0; tf+l = t*, j' = 1,..., J — 1; and t8 = T. The first and last of these equations are boundary conditions; the remaining J - I equations simply state that a household moves from its jth dwelling directly into its j + l'st dwelling (i.e., the household occupies one principle dwelling at a time). The functions h, Ha, Hs, TCa, and TCS in equation (3.6) all depend on whether the household chooses to own or rent the dwelling it selects. These functions will be examined later in this chapter. 29 Utility Maximization Households determine their demands by maximizing (3.1) with respect to the arguments of its function, x(t) and z(t), and with respect to the J -1 dates of acquisition and sale ta-,j = 1,..., J -1, subject to conditions (3.2), (3.3), (3.4), (3.5), and (3.6). To begin, I consider a household's demands for x(i) and zl(t). Since these are continuous variables, their demands can be characterized by the first order conditions ux{t)p% = XS* t = l,...,T (3.7) t=t t=t". j = 1,...,J; i = l,2 (3.8) where A is the Lagrange multiplier associated with the wealth constraint. Both the Lagrange multiplier A and the household's subjective rate of time discount p can be eliminated from the first order conditions for dwelling characteristics by substituting the first order conditions for other consumption, (3.7), into (3.8). The transformed first order conditions for dwelling space and dwelling newness are t: E t=t* uZi{t)/ux(t) 6* - ST hZt(t)6< -t=t« H°t(t°) + TC»(t?) 1,...,J; i = l,2 0 i (3.9) The term on the left hand side of (3.9) is the discounted value of the marginal rates of substitution between z}(t) and other consumption x(t). This value is equated to the capitalized marginal cost of the housing characteristic, given by the right hand side of (3.9). The latter term may be viewed as the multiperiod user cost of the housing characteristic, including transaction costs. If a household occupies a dwelling for only one period, equation (3.9) collapses to the single period optimality condition uZt(t)/ux{t) = hZi{t) + HXtp + TCZW) + Hlt(t))+TClt{t°) (3.10) where the right hand side of this condition is the conventional single period user cost of the housing characteristic (including transaction costs). I now examine a household's choices of the J - 1 dates t3- on which it moves to a new dwelling. These choices determine the household's optimal length of occupancy in each dwelling. Since the model is formulated in discrete time, the dates on which a household moves are discrete variables. 30 Because of this, increasing or decreasing tj by one time period causes both marginal and infra-marginal changes in a household's lifetime utility. The necessary conditions for the J — 1 dates, r?, to be optimal are Uj(t? + l)p t" + 1-Ah j(t? + l ) ^ " + 1 (3.11) - A [[(HJ(tJ + 1) + TCJ(tJ + 1)) - (H?(t?) + TCJ(tJ))] + [HJ(tJ) + Tq(t?)] - <$'!")] > u j + 1(t? + l)p<?+1 - Ahj+1(t? + 1)*'J,+1 - A J(H?+1(tJ + 1) + TC?+1(tJ + 1)) - (H?+1(tJ) + TCf+1(tJ))] S^1 + [Hf+1(tJ) + TC?+1(tJ)] -j = l , . . . , J - l The subscripts on the functions in (3.11) indicate the functions are evaluated with respect to the characteristics of the jth and j + l'st dwellings, respectively. The left hand side of equation (3.11) can be broken into two parts. The first part (the part not in square brackets) gives the discounted utility of staying in the jth dwelling one more period, less the discounted recurrent cost of doing so, valued in terms of utility. Hence, this part of (3.11) gives the flow benefits from staying in the unit one additional period. The second part of (3.11) gives the change in the value of the capital expenditures and transaction costs incurred when selling the jth unit. Again, this term is valued in units of utility. The latter term may be either positive or negative, depending on whether the capital and transaction costs increase or decrease over time. The right hand side of (3.11) may be given an analagous interpretation to that of the left hand side of (3.11), except that the second part refers to the change in the capital expenditures and transaction costs from buying the j + l'st unit. Thus, equation (3.11) states that the optimal time to move is when the benefits from staying in the jth unit an additional period, including increased capital gains, are balanced by the opportunity cost of postponing occupancy of the j + l'st unit an additional period. Equation (3.11) highlights the importance of transaction costs in determining the optimal length of stay in a dwelling. It shows clearly that one benefit from staying in a dwelling an extra period is that transaction costs associated with the dwelling disposition or sale are put off to the future. Similarly, the household benefits by postponing the occupancy of the j + l'st dwelling unit because the transaction costs of acquisition are deferred. The magnitude of the benefits from postponing occupancy depend upon the discount rate 6. The smaller the discount rate, or equivalently, the higher the interest rate, the greater the benefits. Of course, the effect of these terms is to increase the optimal length of occupancy in a dwelling unit. Lastly, I consider a household's demands for the discrete housing characteristics z2(t); that is, I consider its choice of dwelling unit type and tenure. The objective of a utility maximizing household is to select the type and tenure of dwelling unit which yields the household the largest intertemporal utility. Given the dynamic nature of the household's maximization problem, this selection is a complex process. 31 A household has seven dwelling alternatives. The first is to stay in its current dwelling. In this case, the household views the characteristics of its current dwelling as optimal (utility maximizing) although the continuous characteristics will not satisfy the first order conditions (3.7) and (3.8). The six remaining alternatives involve moving to a new dwelling. Each alternative corresponds to one of six feasible type and tenure combinations indexed by the set D. In these alternatives, a household can vary its demands for the continuous housing characteristics so as to satisfy the first order conditions (3.7) and (3.8). In fact, in order to evaluate its utility level under each of these "new" alternatives, the household must solve the first order conditions (3.7) and (3.8) to determine optimal levels for dwelling space and dwelling newness. Whether a household stays or moves, it selects a new set of dates ts- for its future moves. There is a different set of dates for each of the seven alternatives. So far, I have examined a household's choice of its next dwelling unit. While the process of selecting this unit appears to be complex, it is only part of the solution to the household's utility maximization problem. The selection of a particular dwelling alternative conditions all the household's future choices. Hence, in order to determine which alternative is best for its next dwelling, a household must solve its complete dynamic program for each alternative and then compare the different lifetime utility values. Specifically, the optimal housing alternative is the solution to maxk{U{k) : k = 1,...,7} (3.12) where U(k) denotes a household's maximal lifetime utility given that it selects alternative k as the alternative for its next dwelling. 3.2 THE EMPIR ICAL MODEL My task in this section is to develop a feasible empirical model of housing demand based as closely as possible on the theoretical model presented above. In moving from the theoretical to the empirical model, one encounters several constraints, since the data include information on a household's current dwelling but not on a household's expected future dwellings. This data limitation occurs in all housing data sets, including panel data sets. The first and most obvious constraint on the empirical model is that the model can be used only to examine a household's current dwelling unit choice. This is not an onerous restriction. The lack of data on a household's expected future dwellings has the greatest impact on the empirical modelling of a household's optimal length of dwelling occupancy and the modelling of a household's discrete choice problem. The optimal solutions to these problems are characterized by equations (3.11) and (3.12), both involving data on a household's future dwellings. Therefore, neither of these equations can be used directly in the empirical model. As a result, ad hoc approaches must be used to model these aspects of dwelling choice. The approaches I have adopted are presented in parts two and three of this section. Briefly, I introduce the optimal length of occupancy into the empirical 32 model by making it a parameter in the estimating equations. To obtain a feasible discrete choice model, I assume a household bases its selection of a current dwelling on the discounted utility it obtains during its tenure in the dwelling. That is, households ignore the discounted utility received from future dwellings. The lack of future dwelling data does not directly affect empirical modelling of household de-mands for dwelling space, dwelling newness, and other consumption. Specifically, future dwelling data do not enter the first order conditions for these demands, except through the length of occu-pancy parameter. I take advantage of this by using the transformed first order conditions (3.9) as the structural equations in my model. The benefits of using the equations (3.9) as structural equa-tions, rather than specifying a set of characteristic demand equations (as is customarily done) are that the parameters of the household's flow utility function are estimated directly and the length of occupancy parameter enters the equations in a straightforward fashion. Murray (1979) follows a similar approach to modelling housing demand in a single period model. To summarize, because of the lack of future dwelling data, the empirical model has two com-ponents: a discrete choice model, concerned with a household's choice of dwelling unit type and tenure, as well as a household's decision to move or not to move, and a continuous demand model, concerned with a household's choice of dwelling size and dwelling newness. A household's optimal length of occupancy, a choice component in the theoretical model, is relegated to a parametric role in the empirical model. 3.2.1 The Switching Regression Model To capture the simultaneity between a household's discrete dwelling choices and its continuous demands, I use a switching regression model. It is helpful to examine the basic form of the switching regression model before considering the detailed specifications of the discrete choice and demand models. The discrete choice component of the switching regression model is based on the probability that household m chooses dwelling alternative j . This probability is denoted by pmj. The continuous component of the switching regression model deals with a household's demands for dwelling space and newness. The joint density of household m's observed demand vector, conditional on its choice of dwelling alternative j», is given by 4>mj. When a household moves, <pmj is well defined. When a household chooses to remain in its existing dwelling, its demand vector is predetermined, hence, nonstochastic. Therefore, the joint density function 4>mi is degenerate with a mass point at the household's current demands. (Recall that the choice not to move is designated as alternative one.) The log likelihood function for household m's demands is L m = Yl Smjl*1 Pmj + Yl ^rnjin <f>mj (3.13) i J*I where fr3 equals one if household t is observed in housing alternative j and is zero otherwise. The first summation in (3.13) is the log likelihood function for household I'S discrete choice problem. 33 The second summation in (3.13) is the log likelihood function for household I'S demands for dwelling space and dwelling newness. The log likelihood function for a random sample of households is given by L = J2Li = £ £ fti/n PH + Y, Y $Hln (3-14) = Ld + Lc where cmj equals one if household m is observed in housing alternative j and is zero otherwise. The first summation in (3.14) is the log likelihood function for household m's discrete choice problem. The second summation in (3.14) is the log likelihood function for household m's demands for dwelling space and newness. In the remaining parts of this section, I specify the discrete and continuous sample log likelihood functions Ld and U. Since the joint density function of household demands follows directly from the first order conditions (3.9), I consider the log likelihood function Lc first, in part two of this section. The choice probabilities underlying Ld are examined in part three. Finally, in part four, I discuss the parametric specification of the optimal length of dwelling occupancy. 3.2.2 T h e Like l ihood Funct ion for a Household's Continuous Demands To obtain a stochastic model from equations (3.9), I assume that households make errors in opti-mization. These errors take the form of random deviations from the first order conditions. I let 9m — (?mi)Sm2)' be the vector valued function of first order conditions for household m. The cor-responding vector of optimization errors for household m is denoted by ojm = (wmi,um2)'. Then, my structural model for a household's demands for dwelling space and dwelling newness can be written as 9m = w m m=l,...,M (3.15) The structural equations are given in detail in Appendix A.3 Each vector o;m is assumed to be normally distributed with mean 0 and nonsingular covariance matrix H. In addition, the error vectors are assumed to be uncorrelated across households — that is, E(u)mu>'n) = 02x2, for all m ^ n, m, n = 1,..., M. Thus, the log likelihood function of the error vectors (ignoring constants) is the multivariate normal density function M 1 M L(u,u...,u>M) = - y/n||n|| - - ]T Jmn-lum (3.16) m=l This log likelihood function does not include the structural parameters, and it is defined in terms of unobservable quantities. To obtain the likelihood function of a household's characteristics demands, one performs a change of variables from the optimization errors wm to the vector of the mth household's demands 3ln order to retain clarity in the exposition of a rather complex model, I have relegated the more difficult equations to Appendix A. The equations in this appendix indicate clearly where all the parameters enter the model. 34 for dwelling space and dwelling newness, denoted by zm(t) — (•jf1, z™)'. The equations (3.15) provide the necessary transformation between the two sets of variables. The Jacobian of this transformation is given by H^ mll demands is given by L(z\...,zM) Because the Jacobian matrices Gm do not depend on the covariance parameters, the log like-lihood function (3.17) can be easily concentrated with respect to the covariance parameters. The concentrated log likelihood function is M M L* = -— ln\\n\\ + ln\\Gm\\ (3.18) m=l where Cl is the residual moment matrix. Both fi and the Jacobian matrices Gm are functions of the structural parameters. The elements of the Jacobian matrix are presented in Appendix A. 3.2.3 The Likelihood Function for a Household's Discrete Choices In the preceeding section, I dealt with a household's demands for continuous quantities. These demands are contingent upon a household's choice of type and tenure of dwelling unit and upon a household's decision to move to a new dwelling. In this section, I set up an empirical discrete choice model which deals with these household choices. Equation (3.12) characterizes a household's decision regarding a move to a new dwelling and its choice of type and tenure if it decides to move. This equation involves a comparison of the lifetime utility levels in each of a household's dwelling alternatives. Since data on a household's future dwellings are not available, equation (3.12) cannot be used as the basis for an empirical discrete choice model. The approach I follow is to compare the discounted utility streams for each housing alternative over the time interval [0, t\] during which the household occupies its current dwelling. I assume households choose the housing alternative with the largest discounted utility; that is, a household will choose the jth housing alternative if U{j;t{)>U(i;t{), i # j; ij= 1,...,12 (3.19) where U(i\t{) denotes the discounted utility over the time interval [0,t{] when alternative i is selected. This approach allows the period of occupancy to vary from household to household as U(i;t[) varies from household to household (this is explained in the next part). All conventional studies are based on single period utility maximization, i.e., t[ = 1 for all households. Hence, my approach represents an advance on current practices. It should be noted, however, that my approach departs from theory in two respects. First, it ignores the discounted future utility of households over the period [if, T]. Since the choice of a particular dwelling alterna-tive conditions a household's dynamic maximization problem for all future dwellings, a household's 35 \dgm/dzm\\. Thus, the log likelihood function of housing • r M M , •—In n - > m=l l-g'mVrlgm + ln\\G7 (3.17) discounted future utility may vary markedly from dwelling alternative to dwelling alternative. Sec-ond, the parameter t\ is different for different households but is constant across the alternatives a household may select. In theory, t\ will vary across both households and alternatives. To obtain an econometric model, I assume households make errors in evaluating their utility levels in the various alternatives. These errors are assumed to be randomly distributed about the discounted utility levels U(i;t\). Thus, the utility index used by a household to make its choice among housing alternatives can be written as Vim = U{i;t[)m + vim i = l , . . . , 7 (3.20) where u, m is the unobserved error. In addition to representing household errors in evaluating utility, the errrors u t m may represent household specific idiosyncrasies in tastes or random behavior on the part of households (e.g., Domencich and McFadden 1975; King 1980). The probability that the mth household will choose the jth housing alternative is prob [Vjm > Vim | i ^ j; i, j = 1,..., 12] = (3.21) prob [U(j; t[)m - U(t; t{)m > vim - vjm | i ^ j, i , j = 1,..., 12] This probability can be determined after a probability distribution has been specified for the random errors. The two most widely used distributions in discrete choice models are the multivariate normal distribution and the multivariate logistic distribution. Neither of these can or should be used in the present context. When the number of alternatives is large, as it is here, the calculation of choice probabilities from a multivariate normal distribution is prohibitively expensive. The multivariate logistic distribution allows for the easy calculation of the choice probabilities, but these probabilities are sensitive to the inclusion or exclusion of roughly similar alternatives from the choice set. This is known as the red bus/blue bus problem. This problem may be important in the present study because many of the housing alternatives are similar and because the choice set varies across households. In this thesis, I assume errors follow a generalized extreme value (GEV) distribution (McFadden 1978, 1981; Manski and McFadden 1981). The GEV distribution allows for a general pattern of dependence among housing alternatives, avoids the red bus/blue bus problem, and permits the easy computation of choice probabilities. The GEV distribution is defined by T(u l m,...,u J m) = exp(-F(e—,...,e-^)) (3.22) where F(w\m,..., wjm) is a nonnegative homogeneous of degree one function of (wim,... ,Wjm) with the following properties 1. l i m ^ - , 0 0 F[wlm,.. .,Wjm) = oo for j'=l,...,J, and 2. for any distinct set of indexes, ..., jk) C {1,..., J}, 1 is positive it k is even and negative it A; is odd. dwh,..., dwjk 36 Using the GEV distribution, the probability that household m will choose housing alternative j is given simply by wjmdF(wlm,...,wJm)/dwjm Pjm - w r — [6.16) F[wim,...,wJm) The F function used in this study is K k=l ieDk i m (3.24) where 0k > 0 and 0 < 77^  < 1 for all A: = 1,..., k. The sets Dk in equation (3.24) are subsets of D, the set of housing alternatives available to a household, where (JjfeLi Dk = D. The sets Dk are arbitrary and do not have to be disjoint. The coefficients r)k in (3.24) measure the similarity of the alternatives in the sets Dk. When rjk is close to zero, the alternatives in the set Dk are almost identical; when r\k approaches unity, the alternatives are distinct (independent). The coefficients dk give the importance of the alternatives in Dk in determining the choice probabilities. As 0k tends to zero, the set of alternatives Dk becomes less and less important. To specify completely the function (3.24), one must describe the set of alternatives D available to any household and the partition structure Dk, k = 1,..., K. I begin by considering the constitution of the set D. In this study, the set D is the set of all dwelling alternatives potentially available to a household. In my preceeding description of household preferences in section 2.1.1, I distinguished three types of dwelling units: single units, multiple units, and apartments. I also distinguished two modes of tenure: owning and renting. As a result, six possible combinations of type and tenure are open to a household. In additon, a household has the choice of staying in its current dwelling or moving to a new dwelling. Combining the choice about moving with the six type and tenure options leads to a set of twelve "elemental" alternatives. These alternatives are presented in Table V, together with the arbitrary choice indexes and acronyms associated with them in this study. The twelve T A B L E V : D W E L L I N G A L T E R N A T I V E S Index Acronym Index Acronym Type Tenure 1 CSO 7 NSO Single Owned 2 CSR 8 NSR Single Rented 3 CMO 9 NMO Multiple Owned 4 CMR 10 NMR Multiple Rented 5 CAO 11 NAO Apartment Owned 6 CAR 12 NAR Apartment Rented alternatives listed in Table V constitute the set D. Not all of the elemental alternatives in the set D are available to a given household. The type and tenure of a household's current dwelling is predetermined by the household's past choice of a 37 dwelling, and, therefore, only one of the six current unit alternatives is open to a household. That is, each household must select its dwelling from a subset of D. The six possible subsets of D are D1 = {CSO, D N e w } D2 = {CSR, D N e w } Dz = {CMO, D N e w ) D4 = {CMR,DNew} D5 = {CAO,DNew} D6 = {CAR,DNew} In defining the function F, I treat each of these choice sets symmetrically; in particular, I use the same partition structure for all sets. In defining a partition structure for the choice sets D*, i = 1,.. .,6, one should group together alternatives which are similar in some respect. What constitutes similarity, however, is an open question. In this thesis, I partition each of the choice sets into four overlapping subsets. The first subset is a singleton consisting of a household's current unit alternative. For example, for the choice set D1, this subset is D\ = {CSO}. The second subset in the partition structure contains two elements: the current dwelling unit and a new dwelling unit of the same type and tenure as the current unit. For the choice set D1, this subset is D\ = {CSO,NSO}. The definition of this subset is empirically motivated. Over 40 percent of moves recorded in my sample take place between dwelling units of the same type and tenure. I regard this as a priori evidence indicating households are better able to evaluate the benefits of a dwelling which is the same type and tenure as the one they currently occupy. The third and fourth subsets in the partition structure contain the sets of owned and rented alternatives, respectively. The similarity among the alternatives in these subsets is due to their similar mode of tenure. The third subset for the choice set D1 is D\ = {CSO, NSO, NMO, NAO). The fourth subset is D\ = {NSR, NMR, NAR}. Having completed the description of the choice set D and the partition structure of this set, it is now possible to specify the F function used in this study. The F function for the choice subset /> is F = 0iwi+ 62 1/-72 1/12 Ji+6 12 + 03 + 04 12 L; = l j = 7 i* 1/173 1 13 .j-4 .7 = 10 (3.25) where, in the third and fourth terms, Wj = 0, if j ^  D\, k = 3,4. The dwelling choice probabilities are generated by applying equation (3.23) to this function. The exact forms of the probabilities are given in Appendix A. The function F has seven estimatable parameters: 0,, i = 1,...,4, and 774, i = 1,2,3. The choice probabilities are homogeneous of degree zero in the 0s; therefore, a restriction must be imposed on the 0s such that the choice probabilities are not homogeneous of degree zero. I impose the restriction 0\ = .02. This provides the necessary normalization and reduces the number of free parameters by one. 38 3.2.4 The Length of Occupancy The optimal date for the household's next move is regarded as an estimatable parameter of the model. Different households, however, plan to occupy their dwellings for different lengths of time. Thus, the households in the sample are broken into three groups, each group having a different, but constant, expected date for its next move. Households were allocated to these groups based on their responses to the survey question (Canada Mortgage and Housing Corporation 1974:15): Which of the following best indicates the chance that this household will move from this dwelling within the next three years? (1) Definitely, (2) Probably, (3) Undecided, (4) Probably not, (5) Definitely not Households replying they would definitely or probably move — categories (1) and (2) — were placed in group one. Households identified as undecided or probably not moving — categories (3) and (4) — were placed in group two. Households stating they would definitely not move — category (5) — were placed in group three. The assumption underlying this grouping of households is that those households with a higher probability of moving within the next three years will move sooner that those with a lower probability of moving. Therefore, I expect tf x < t{2 < t\2, where the superscript indicates the household group, rather than being an index of a household's dwelling units, as was the case above. 3.3 FUNCTIONAL FORMS Thus far in this chapter, I have developed the theoretical and empirical models without specifying functional forms for many functions in the models. Here, I present the various functional forms needed to complete the empirical model. These functions deal with specifications of household preferences, housing expenditures, and transaction costs associated with moving from dwelling to dwelling. 3.3.1 A Specification for Household Preferences The utility indexes u(t), t = 1,.. .,T, are assumed to be the same for all time periods. They are given by the additively separable form u{x{t), z'it), z2{t)) = Kuc{x{t), z'it)) + ud{z2{t)) (3.26) The "continuous" part of the utility function is given by the Generalized Leontief form uc{x{t), z\t)) = 2 a 0 i i ( t ) 1 / 2 + 2 a o 2 2 f ( t ) 1 / 2 + 2aQZz\{t)ll2 + anx{t) + 2a12x{t)1/2zl{t)1/2 + 2a13x(t)1/2z12(t)^2 + a22zl(t) (3.27) + 2a 2 3z 1 1(0 1 / 2^W 1 / 2 + «33^21W 39 where symmetry of the cross-partial derivatives is an imposed restriction. The "discrete" part of the utility function is given by ud(z2{t)) = bndn + 6 i 2 r f i 2 + W i s + &21<*21 + b22d22 (3.28) where the variables d{j are binary variables defined by: d\j = 1, if z\ indicates a dwelling unit of type j and is zero otherwise, d2j = 1, if z\ designates an owned dwelling and is zero otherwise. Finally, K is an arbitrary preset scaling constant. It is needed in the probability calculations to ensure that the value of u remains in the range of computational feasibility (a value of 10 - 5 is used). A scale for utility is determined by the normalizations u(s, z\ ,z\,z\,z\) = uc (3.29) b\i — &21 = 0 where the values x, z\, z\, and uc may be chosen arbitrarily. The values used here are: x = 10,000, ~z\ = 74, z\ — 10, and u° = 1,000. (Appendix A contains the exact form of the normalization on the continuous utility function.) In addition to cardinalizing utility, these normalizations reduce the number of free parameters in the utility function from fourteen to eleven and serve as identification restrictions in the empirical model. Demograph ic Determinants of Household U t i l i t y To this point, I have neglected the role of demographic variables as determinants of household utility, and, therefore, determinants of a household's demand for housing. That demographic variables play a major role has been amply documented (e.g., David 1962, Kain and Quigley 1972, Li 1977, and Steele 1979). The variables used in previous studies have been discussed in Chapter Two. In this study, two demographic variables are incorporated into the household utility function: (i) the age of the household head, and (ii) household size. These variables are denoted by si and s2, respectively. The procedure used to incorporate the demographic variables is a variant of the demographic scaling procedure developed by Barten (1964) and Pollack and Wales (1980, 1981). Mayo (1981) advocates using this procedure to include demographic variables in housing demand analysis. In demographic scaling, the primary arguments of the utility function are replaced by arguments measured in "efficiency units". The efficiency units of goods in the continuous utility function are given by x(t)/fi, z\(t)/ f2, a n d z\(t)/ fz where the /'s are scaling functions which depend on the two demographic variables. The arguments in the discrete utility function can also be scaled. In this thesis, I apply separate scaling functions to housing type and housing tenure. This is accomplished by replacing du, d\2, and d\z by du/f^, d\2/fi, and d\z/fi', and by replacing d2i and ^22 with ^2i//5 a n d d22/f5. This is shown clearly in the equation given in Appendix A. 40 I use the log-linear scaling functions fi = s\il8lia i'=l,...,5 (3.30) These functions generate only positive scaling values, as required on theoretical grounds. They are multivariate generalizations of Pollack's and Wales' (1981) univariate log-linear dynamic scaling functions. While I have interpreted the fs as "efficiency scales", this interpretation is a loose one. The exact role of the scaling functions, as Pollack and Wales point out, is to reallocate expenditure among the goods while leaving total expenditure unchanged. 3.3.2 Hous ing Expenditures In this section, I define the housing expenditure functions h(t), Ha(t), H3(t) used in the estimating equations for a household's demands for dwelling space and dwelling newness. The housing expenditures of owner and renter households are quite different. Therefore, the housing expenditures of the two tenure modes are considered separately. I begin by examining the expenditures of households which own dwellings. Most households in Canada use some form of mortgage financing to purchase a dwelling. The situation I consider here is that of a household which purchases a dwelling unit at time period t± = 0. The dwelling is valued at V{t\). I assume the household makes a downpayment E(t\) > 0 and obtains a single mortgage of value M[t\) to cover the balance of the purchase price; that is V{t1) — E(ti) + M{t\). The outright purchase of a dwelling is a special case of this financing plan in which M{t\) = 0. After the date of purchase, the household makes recurrent payments h(t) which are the sum of its mortgage payments m(t) and its property taxes PT(t). The mortgage payments m(t) consist of an interest payment on the outstanding balance of the mortgage plus some payment toward the outstanding principle. Most mortgages available at the time my sample was collected (1974) were conventional level payment mortgages.4 For these mortgages, m(t) is constant throughout the life of the mortgage and equals * = i - d + r " ) W " ' ' » (3-31> where rm is the rate of mortgage interest and TTM is the amortization period or term to maturity of the mortgage. I assume all mortgages recorded in the sample are constant level payment mortgages; this assumption greatly simplifies many of the calculations below. When a household sells a dwelling, it receives the sale value of the dwelling V(t{) less the outstanding balance of the mortgage on the date of sale M(t[). Thus, the household recoups its equity in the mortgage, plus all capital gains (losses) which have accrued from the time of purchase. *The exotic mortgage forms available today are largely a product of the high nominal interest rates during the late 1970's and early 1980's. 41 The housing expenditures of the owner household described above are summarized by H*{t\) = E{t\), h(t) = fh+PT{t), H°(t[) = M(t\)-V(t[). (3.32) Although these equations are based on a specific financing plan, analagous equations can be easily derived for the situations of outright purchase and multiple mortgages. One cost element which is not included in the above equations is the cost of maintaining an owned dwelling. These costs have been omitted because I have no information on the flow rates of maintenance expenditures by owner households. The expenditure stream of renter households is uncomplicated when compared to that of an owner household. The primary expenditure for renters is their monthly rental payment. In Canada, it is also common for renters to pay a damage deposit when taking up tenancy. The size of the damage deposit is subject to negotiation between the landlord and the tenant; it is usually one-half of one month's rent or less. Providing no damage is done to the unit during a houeshold's tenancy, the damage deposit must be returned with interest (in Ontario) when the household vacates the dwelling. Unfortunately, my sample does not contain information on damage deposits paid by households. It is assumed, therefore, that renter households pay a damage deposit equal to one-quarter of one month's rent — approximately $50. Hence, the housing expenditures of renter households are given by where R(t) denotes the houshold's monthly rental payment and r^d is the rate of interest on the damage deposit. 3.3.3 Transact ion Costs This section explores the transaction costs applicable in the housing market; that is, I define the functions TC a(fj) and TC(t{). As with housing expenditures, transaction costs depend on whether a household is in the rental or ownership sector of the housing market. In the rental sector, the transaction costs of acquiring a dwelling include the time cost of searching for a new dwelling and the direct and time costs of moving to a new dwelling. Households in the ownership sector of the housing market also face these costs. In addition, they have the appraisal cost of securing a mortgage, the legal cost of obtaining clear title to the property, land transfer taxes, and, if a household is having difficulty finding a lender, it may also pay for the services of a mortgage broker. H*{t<{) h(t) H°(t{) .25i?(i?), R(t), -.25i?(*?)(l + r d d) (3.33) 42 There are no obvious monetary costs connected with vacating a rented dwelling. The costs of selling an owned dwelling, however, are significant. These costs are in the form of real estate brokerage fees, typically 5% to 7% of the sale value of the property. In this study, transaction costs are modelled in as parsimonious a manner as possible. The transaction costs of acquiring a dwelling are given by J*1, if the household rents the dwelling, f 2 + (T-! + r2)V(q), if the household purch ases the TCa(t\) = { dwelling and V(t\) < $35,000, and (3.34) 455 + T 2 if the household purchases the +{n + T2) [V (if) - 35000 ) , dwelling and V(i?) > $35,000. T 1 and T2 are constants to be estimated along with the other parameters of the model. These constants are meant to reflect the direct and time costs of search and moving. Separate constants are estimated for owners and renters because owners tend to be older, have larger households, and have more possessions to move than renters. In addition, separate constants are employed because homeowners have the costs of mortgage appraisal which renters do not.5 T\ is the marginal land transfer tax rate. Land transfer taxes in Ontario in 1974 were based on a two-part tariff set out in the Ontario Land Transfer Act 1974- Property valued at less than $35,000 was taxed at .3%, while property valued at over $35,000 was taxed at .3% on the first $35,000 and .6% on the value over $35,000. r2 is the rate of tariff for legal services. The fee charged for legal services varies from firm to firm. A fee of 1% of the property value is employed here. The rate of 1% on the property value was obtained from several sources. First, The Report of the Federal Task Forces on Housing and Urban Development indicates 1% as a customary fee. Second, Hatch (1975:106) states the rate of 1 1/4% on the mortgage value is a reasonable estimate of the legal tariff rate. This rate is equivalent to 1% on the property value when a household's downpayment equals 20% of the property value. Finally, my discussions with several lawyers have indicated that legal fees in the range of 3/4% to 1 1/2% of the property value are common within the legal profession. The transaction costs of moving from a dwelling are !0, if the household rents the dwelling, (3.35) r3Vr(tf), if the household owns the dwelling, where r 3 is the real estate brokerage rate. Real estate brokerage rates were obtained from the Real Estate Boards of the four cities from which the data were collected. These rates are: Hamilton — 5.5%, London — 5.25%, Toronto — 5.5%, and Windsor — 5.4%. It is apparent from these rates that realtors' fees are the largest single transaction cost. For example, at 5.5%, a realtor would charge $2,530 to sell the average home in the sample. s In 1974, C M H C charged a flat fee of $35 for mortgage appraisal. The appraisal fee charged by banks, trust companies, and life insurance companies (the other primary lenders in Canada) was higher, ranging from $50 to over $100 (Hatch 1975:106). 43 3.4 CONCLUDING REMARKS The empirical model in this paper contains thirty-two estimatable parameters. There are eight parameters in the continuous part of the utility function, three parameters in the discrete part of the utility function, ten demographic scaling parameters (six applied to a household's continuous demands and four applied to a household's demands for housing type and tenure), two parameters in the transaction cost functions, three length of occupancy parameters, and six parameters in the GEV probabilities. The equation in Appendix A indicates where the parameters enter the various equations in the model. This is a large nonlinear model. Nevertheless, the model is not complete as it stands. The estimating functions for dwelling space and dwelling age contain the first derivatives of the housing expenditure and transaction cost functions, which, in turn, contain the marginal prices of dwelling newness and dwelling space. These latter quantities are not part of any data set, and, therefore, they must be estimated from data on dwelling prices and dwelling characteristics. There are two methods for approaching this problem. First, the marginal prices may be estimated along with the rest of the model. This approach is followed by Wales (1979) in a much smaller model than that presented here. Second, one may estimate the marginal prices of the housing characteristics separately and then estimate the main model, conditional upon the estimates of the marginal prices. The latter approach is known as Rosen's two-step method (see the preceeding chapter). Given the size of the main model, I have opted to follow Rosen's method. The application of this method to the present study is the topic of Chapter 5. 44 C h a p t e r 4 T H E D A T A The data used in my thesis are examined in the three sections of this chapter. In the first section, I discuss the attributes of the micro data set employed in the thesis. I also discuss my sample selection procedure, as well as two deficiencies of the micro data set. In the second section, I consider the data which must be generated or forecast in order to impliment the intertemporal model presented in the previous chapter. In the final section, I briefly discuss the procedure I follow to obtain data on the housing alternatives rejected by a household. 4.1 THE MICRO DATA The primary data set used in this thesis is drawn from the 1974 Survey of Housing Units (SHU). The SHU was conducted by Statistics Canada in the fall of 1974 for the Central Mortgage and Housing Corporation (CMHC).1 The survey was meant to be the first in a series of surveys which would track a set of dwelling units through time. In this respect, it is similar to the U.S. Annual Housing Survey. Unfortunately, the SHU was discontinued after the first survey. The target population of the survey was the set of private dwellings in existence during the sur-vey period (September-October, 1974) in each of the twenty-twol97I Census of Canada metropoli-tan areas, as well as the set of private dwellings in the city of Charlottetown, Prince Edward Island. A stratified random sample was drawn from this target population.2 The sample contains infor-mation on approximately 74,000 dwellings and their occupants. The SHU data were gathered in personal interviews. Information on the following items was collected from the occupants: 1. the characteristics of the current dwelling, 2. the rental terms for tenant occupied dwellings, 3. the expected market value of owner occupied dwellings and information on the carrying charges for these dwellings; the latter information includes: 1The Central Mortgage and Housing Corporation has since been renamed the Canada Mortgage and Housing Corporation. 2For a description of the sampling strata, see the 1974 Survey of Housing Units: Background Information on the 1974 Survey of Housing Units, or the 1974 Survey of Housing Units: A Report on Sample Design. 45 a. the principal outstanding on a household's mortgages (up to three mortgages are con-sidered), b. the regular yearly mortgage payment, c. the interest rate on the mortgage, and d. the yearly property tax payments on the dwelling, 4. the household composition (i.e., the relationships among household members), 5. the demographic characteristics of each household member, 6. the gross income of each household member,3 and, 7. mobility data for the household, including: a. the dates of the household head's five most recent moves, and b. the household's subjective probability of moving within the next three years. Information on the household's previous dwelling was also collected; specifically, items (1) to (6) were collected as they pertained to the household when it occupied its previous dwelling. A subset of households from the SHU sample is used to estimate my model. The subset consists of primary nuclear families and single member households living in unsubsidized dwellings in Hamilton, London, Toronto, and Windsor, Ontario. In the SHU survey, a nuclear family is defined as a husband and wife with or without never married children (regardless of age) or a parent with one or move never married children living in the same dwelling. A nuclear family may also consist of a man or a woman living with a guardianship child or ward under 21 years of age for whom no pay was received (CMHC 1974:5). and a primary nuclear family is defined as a family which contains the head of the household as one of its members4 (CMHC 1974:6). By restricting the sample to primary nuclear familes and single member households, two cate-gories of households are excluded from the sample: 1. households containing a primary nuclear family and additional members who do not belong to the primary nuclear family; for example, a daughter-in-law, son-in-law, or a boarder. 2. households containing more than one member, none of whom belong to a primary nuclear family, for example, shared or "communal" dwellings. By excluding these two categories of households, I hoped to ensure a stable, short run household composition. A stable household composition is necessary in order to forecast future household income (this will be explained in the next section of this chapter). Subsidized dwellings are omitted from the sample because information on the subsidy is not available, and hence, the effective prices of subsidized dwellings cannot be computed. 3The income of each household member was collected by source (where possible), but the confidentiality restrictions of Statistics Canada prohibit the release of this detailed income data. 4In the SHU survey, the head of the household is defined as the primary wage earner; hence, the head of the household may be male or female even in husband-wife families. 46 The four Ontario cities were selected for two reasons. First, I hoped that by choosing cities in the same province and within the same geographical region, I would control for the institutional and legal arrangements surrounding the renting and buying (selling) of dwellings. Second, the cities chosen are the only cities in Southwestern Ontario for which I could obtain 1974 real estate brokerage rates from the local real estate boards.5 After imposing the above sample selection criteria, my sample contains approximately 12,000 households. 1,893 of these households are recent movers; that is, they had moved to their current dwelling within the past eight months. A sample of 12,000 households is still unmanagable; there-fore, a subsample had to be drawn. Since most of the information on a household's continuous demands is derived from the first order conditions for these demands and because these first order conditions can be applied only to mover households, the entire sample of movers was kept. Thus, the entire reduction in the sample is borne by the set of nonmover households. I did not expect this procedure to impart a sample selectivity bias to the model. Nonmovers are important only to the discrete part of the likelihood function; sample selectivity bias will arise in this part of the likelihood function only if errors made by nonmovers in evaluating their utility in the different hous-ing alternatives are systematically different from errors made by movers. Since both movers and nonmovers are performing the same calculations (it is the outcome that differs), there is no reason for assuming that the two classes of households make different errors. To generate a subsample of nonmovers, I drew a one-in-ten random sample from the set of nonmovers. After imposing the sample selection criteria on the one-in-ten sample, 661 nonmover households remained. Thus, the total sample contains 2554 households — 1893 movers and 661 nonmovers. 4.1.1 Deficiencies in the S H U D a t a Set The SHU data set contains a wealth of information on dwellings and households. Unfortunately, it lacks one piece of data central to this study — the real value of nonhousing consumption, x(t). In principle, the value of nonhousing consumption can be computed by: q(t)x(t) — [current after-tax income + current housing expenditures — housing transaction costs — current saving]. The term q(t) in this expression denotes a price index for nonhousing consumption. This calculation is impossible, however, because the SHU data set does not include current savings.6 There is no entirely satisfactory method for circumventing the problem. The course followed here is to redefine x(t) as nonhousing consumption plus household saving, thereby obviating the need for the savings data. The redefined variable can be computed from: sPrior to 1975, local real estate boards fixed commission rates. This practice was stopped in 1975 by the passage of an extension to the Federal Combines Investigation Act. In principle, real estate commissions are now subject to negotiation between the real estate firm and its client. Some real estate boards, however, continue to issue "suggested" commission rates, so that, despite the extension of the Act, there is a high degree of uniformity in real estate commission rates. 6I made several attempts at imputing a savings rate for the households in the sample; these attempts yielded unreasonable values. (4.1) 47 There are two side effect of defining x(t) in this manner. First, it implicitly introduces a household preference for saving into the analysis. Second, it implies x(t) is nonlinearly dependent upon z\ and z\. This dependence must be taken into account when computing the Jacobain matrixes Gm. To obtain the real value of nonhousing consumption, one needs the price index q(t). In Canada, there is no published price index for nonhousing consumption at the metropolitian level. Further-more, there is no price index which relates prices in different metropolitian areas. As a result, I had to use an ad hoc approach in constructing the price index q(t). I have set the price of nonhousing consumption for the four cities equal to unity in 1974. For time periods after 1974, the price of nonhousing consumption in the four cities is (1 + tfx)', where zx is the rate of inflation for non-housing consumption. For 7rx I have used the average rate of increase in the nonhousing components of the Consumer Price Index for Ontario for the years 1973, 1974, and 1975. The average rate for these years is 9.8 percent. 4.2 GENERATED AND FORECAST DATA The model described in Chapter Three contains data which either cannot be observed directly or were not recorded in the SHU. For the current period, the marginal prices of the dwelling characteristics, dwelling space and dwelling newness, are unobservable. This is also true of the income tax liabilities of the individuals in the SHU sample. The latter values are necessary for the calculation of after-tax household incomes. For future periods, an even larger body of data cannot be observed directly or is missing from the SHU. Included in these data are: (i) rental values of tenant occupied dwellings and the sale values of owner occupied dwellings, (ii) marginal prices of the dwelling characteristics of rented and owned dwellings, (iii) before-tax incomes and income tax liabilities of the individuals in the sample, and (iv) property taxes on owned dwellings. To implement my model, these values must be specified in terms of observed market data. In this section, I describe the methods employed to estimate some of these values. Specifically, part 1 examines the procedure used to generate present and future after-tax household incomes. Part 2 describes the manner in which I have forecast the property taxes on owned dwellings. I discuss the procedures used to forecast the rental and sale values of tenant and owner occupied dwellings and the procedures used to estimate the marginal prices of the dwelling characteristics in the next chapter. 4.2.1 Af ter -Tax Household Income Real after-tax household income is the basic income variable used in this study. It is calculated, for each time period, by summing the after-tax incomes of the economically active members of each household and then deflating this value by a price index. An individual is assumed to be economically active if he/she is 18 years of age or older. As noted above, I have restricted the 48 sample to households consisting of primary nuclear families and single member households in an attempt to ensure short run stability in the number of economically active members. The after-tax incomes of individuals are calculated in two stages. First, an estimate of each individual's income is calculated from male and female age-income regressions. Second, hypothetical income tax liabilities are simulated for each individual, based on his or her estimated income, and subtracted from the individual's estimated income. Each of these stages is discussed below. Forecasting Individual Incomes Individual incomes are subject to two time processes. First, it is well known that as individuals age, their incomes tend to follow an inverted U-shaped time profile. This time profile of individual incomes is explained by the human capital model specified by Becker (1964) and subsequently developed by Ben-Porath (1967), Mincer (1974), Heckman (1976), and many others. The time profiles of Canadians have been examined by Podoluk (1968), Holmes (1976), Robb (1978), and Kuch and Haessel (1979). The second time process affecting individual incomes is the general rate of wage inflation. My approach to forecasting individual incomes takes both processes into account. In the first part of this section, I specify and give the results for the size income regressions used to capture the relationship between an individual's age and his/her income. This is followed by a discussion of the results. Finally, the procedures used to forecast individual incomes and adjust for wage inflation are presented. The relationship between age and income is accounted for by a simple age-income regression with the following independent variables: • A G E — the individual's age. • SINC — the spouse's income, if the individual is married, and the spouse is present. • two dummy variables which indicate the employment status of the household head: — ESI = 1, if the household head is unemployed, and —- ES2 = 1, if the household head's employment status is unknown. • HEAD= 1, if the individual is the household head. • two marital status dummy variables: — MSI = 1, if the individual is married, and — MS2 = 1, if the individual is widowed, separated, or divorced. • three dummy variables indicating the individual's city of residence: — Hamilton= 1, if the individual lives in Hamilton, — London= 1, if the individual lives in London, and — Windsor= 1, if the individual lives in Windsor. Since major differences exist between the incomes of males and females, separate equations are es-timated for each sex. In addition, to capture differences in the effects of the independent variables across an individual's lifetime, separate coefficients are estimated for each of the independent vari-ables for the age groups: 18-24 years, 25-34 years, 35-44 years, 45-54 years, 55-64 years, and 65 49 years and over. The estimated equations are constrained to be piece-wise continuous across these age groups in terms of the variable AGE. More elaborate earnings functions cannot be estimated because the SHU data set does not contain the requisite human capital variables. The results of the two age-income regressions are reported in Table VI. The influence of AGE on male income is as expected. Male incomes rise from age 18 to age 34, decline slightly between ages 35 and 44, are relatively constant between ages 45 and 64, and decline steadily after age 65. The age-income profile of females does not exhibit the inverted U-shape present for males. Female incomes rise sharply between the ages of 18 and 24, are relatively constant from age 25 to age 65, but show a slight rise after age 65. The differences between male and female age-income profiles may be the result of various phenomena. To begin, many women, particularly married women, acquire less labour force experience than do males of comparable age because women tend to drop out of the labour force to raise children. Even if women do not withdraw completely from the labour force after bearing children, they tend to work only part-time. This is particularly true for mothers of younger children. The difference between male and female incomes also reflects wage discrimination against women. Lower wage rates for females lower their incomes directly and provide less incentive for women to increase their hours of work and gain labour force experience. The employment status variables, ESI and ES2, are included in the regressions to capture the effect of unemployment on individual income. The variables do not refer, however, to the employment status of individuals but to the employment status of the household head. Hence, the estimated coefficients will reflect both the direct effect of unemployment on the household head's income and the indirect effect of the household head's unemployment on the incomes of other household members. For males, the coefficients of ESI and ES2 are negative for all age categories (as expected), but only the coefficients for ESI for prime age males (ages 25 to 64) are significant. The effect of the household head's unemployment on the incomes of female household members is not as clear as the effect on male household members' incomes. Female incomes tend to decline if the household head is unemployed (i.e., ESI = 1), but this effect is statistically significant for only two of the six age categories. If the employment status of the household head is unknown (i.e., ES2 = 1), female incomes tend to increase, counter to expectation. Moreover, for females aged 55 to 64, this increase is statistically significant. I do not known the causes of the unexpected results for variable ES2, but the vague definition of ES2 may be a contributing factor. The variable SINC (spouse's income, if spouse is present) applies to married individuals living in the same dwelling as their spouses. This variable is included in the regression in an attempt to incorporate the income effect of a spouse's income on the partner's hours of work and, thus, on the partner's income. If the variable SINC captures the income effect, and if one assumes leisure is a normal good, the coefficient of SINC will have a negative sign. The regression results do not bear out this expectation. The coefficients of SINC are negative for half of the age groups and positive 50 T A B L E VI: ESTIMATES F R O M AGE-INCOME REGRESSIONS Males Age Groups 18-24 25-34 35-44 45-54 55-64 65+ Age 509.13* (185.86) 156.62* (52.405) -140.23* (46.209) 12.273 (50.102) 1.1117 (50.733) -87.582* (42.627) Household Head is Unemployed (ESl) -2351.4 (2063.4) -3392.2* (1526.2) -5076.0* (2424.1) -4288.0* (2079.3) -3834.4* (1848.8) -2471.7 (1238.7) Household Head's Employment Status is Unknown (ES2) -619.57 (820.13) -770.25 (651.70) -204.94 (818.96) -644.51 (801.28) -1044.1 (1083.6) -371.77 (1157.4) Spouse's Income (SINC) .14868 (.23852) -.16775 (.14315) .13431 (.23306) -.19208 (.20678) -.05985 (.38006) 1.2268* (.31069) Marital Status: — Married (MSI) 928.59 (925.14) -566.24 (882.62) 1656.3 (1339.4) -381.89 (1701.0) 4746.0* (1699.4) 2377.3 (1852.9) — Widowed, Separated, or Divorced (MS2) 358.14 (2696.1) -95.089 (1529.4) -2116.7 (2368.4) 1157.7 (2171.0) 3989.0 (2451.2) 2905.6 (1990.2) Household Head (HEAD) 1879.3* (894.37) 1645.8 (917.78) 4422.8* (1452.8) 6843.5* (1893.1) 422.89 (1796.4) 3551.7* (1157.4) City of Residence: — Hamilton 775.97 (980.38) -335.40 (730.60) 499.87 (966.12) -1170.2 (957.87) -1089.2 (1222.2) 1624.6 (953.82) — London 444.88 (924.30) -2258.5* (754.34) -844.00 (996.64) -1036.1 (991.27) 630.86 (1174.8) 1065.3 (940.89) — Windsor 934.23 (935.23) -686.16 (748.42) 1208.7 (949.66) 150.71 (1003.1) -465.84 (1320.0) 565.81 (926.84) Constant -6828.0 (4044.8) R2 = .2548 a = 5160 Equation F = 8.234 51 T A B L E VI: E S T I M A T E F R O M AGE-INCOME REGRESSIONS Females Age Groups 18-24 25-34 35-44 45-54 55-64 65+ Age 380.13* (113.39) 45.773 (34.585) 10.384 (34.850) -21.711 (35.154) -33.785 (37.203) 50.455 (29.106) Household Head is Unemployed (ESl) -578.02 (1388.7) -955.88 (1164.1) -1082.4 (1492.1) -3231.3* (1383.0) 1760.9 (1316.3) -1818.3* (907.48) Household Head's Employment Status is Unknown (ES2) 652.52 (542.15) 493.27 (504.28) -242.69 (688.27) 1021.7 (700.68) 2069.1* (901.35) -1369.8 (827.35) Spouse's Income (SINC) .45425* (.11425) .45963* (.0605) .37366* (.08450) .54670* (.09157) .43675* (.18424) .64304* (.19480) Marital Status: — Married (MSI) -149.37 (504.70) -1170.2* (571.56) -773.44 (1008.9) -2649.6* (972.99) -1746.4 (1616.2) -3252.4* (810.46) — Widowed, Separated, or Divorced (MS2) 400.99 (1301.5) -2435.9* (889.18) -3439.6* (1100.2) -842.76 (1165.4) 68.485 (1593.5) -3732.1* (764.11) Household Head (HEAD) 942.02 (560.52) 2495.6* (683.01) 4652.8* (930.47) 2302.7* (907.87) 2196.2* (894.52) 2848.9* (602.89) City of Residence: — Hamilton -180.10 (562.15) -1095.3* (529.18) -2203.8* (688.76) 481.57 (794.39) 1311.2 (927.51) -266.70 (556.91) — London -46.444 (530.71) -160.51 (496.01) 545.48 (635.90) 1010.8 (774.74) -252.16 (950.79) -230.41 (535.27) — Windsor -88.372 (592.93) -131.23 (518.77) -896.30 (798.89) -122.15 (748.61) 14.589 (1016.8) -482.42 (565.71) Constant -5746.8* (2435.5) R2 = .3452 a = 3008.1 Equation F = 9.330 52 for the other half. Moreover, the only statistically significant coefficient is for males 65 and over and it is positive. Since males over 65 are generally retired, this positive effect probably reflects a positive correlation with the spouse's wealth holdings at retirement. The coefficients of SINC for females are all positive and statistically significant. Again, these results are in strong opposition to the effect predicted by economic theory. It is therefore likely that the variable SINC does not measure the income effect of additional income from one's spouse. The variable HEAD is included to capture the motivational aspects associated with household headship. All other things being equal, I expect that being a household head would induce both males and females to increase their earnings in order to meet the economic responsibilities associated with headship. This expectation is borne out by the estimated coefficients for HEAD. For both sexes, the coefficients of the variable HEAD are uniformly positive and, in most cases, statistically significant. The marital status variables, MSI (married) and MS2 (widowed, separated, or divorced), are used to catch the motivational effects of marital status on income. The motivational effect of marriage (i.e., MSI = 1) depends on one's view of the family. At the time of the SHU survey, the dominant view of the family was probably that of a primary nuclear family with the husband as sole "breadwinner" and his spouse and children as economic dependents. Therefore, married males may have been motivated to increase their earnings relative to single males in order to fulfill their perceived roles. Alternately, married females may have reduced earnings relative to single females in the degree to which they assumed the role of dependent. The motivational aspects associated with being widowed, separated, or divorced (i.e., MS2 = 1) are not clear. The key factors are the presence of children on one hand and child support and/or alimony payments on the other. For males, the regression coefficients of MSI and MS2 do not meet with the expectations voiced above. They are mixed in sign (i.e., they are positive for some age categories and negative for others), and, except for the effect of marriage on males 55 to 64, they are statistically insignificant. Since the motivations ascribed to males, particularly married males, are virtually identical to those ascribed to household heads, it seems likely that the variable HEAD is absorbing all of the "breadwinner" effects. Therefore, it is not clear what the marital status variables are reflecting. As expected, marriage reduces female incomes. Coefficients for the variable MSI are negative for all age categories. These effects are only statistically significant, however, for three of the six age categories. The effects of MS2 on female incomes also tend to be negative, but they are, for the most part, statistically insignificant. The motivational effects associated with being widowed, separated, or divorced are also related to household headship, making unclear what the coefficients of the variable MS2 are capturing. The dummy variables for city of residence are meant to catch inter-city differences in the average wage rate, caused by differences in the demand and supply of labour in the cities. There is no specific pattern among the estimated coefficients for these variables and the coefficients are 53 generally insignificant. This would seem to imply no disparity in the wage rates across the four cities. Age-income regressions are used to forecast the incomes of individuals in the sample. This may be done in two ways. The first and most common procedure is to use the age-income regressions to generate expected incomes for each individual. However, if an individual's actual income in the current period differs markedly from his/her expected income in the next period, there is a problem with this approach. Specifically, there will be a discontinuity predicted income stream for a household. To avoid such a discontinuity, one should use the expected current period income for an individual in place of her/his actual current period income. The second way to generate estimates of an individual's future incomes is to use the individual's reported actual income as an estimate of his/her current income and to forecast future incomes by adding the expected change in his/her income to actual current period income. The expected changes in individual incomes may be derived from the age-income regressions. The major problem with this approach is that it builds current period transitory fluctuations in the individual's income into the estimates of future period income. In deciding which approach was best, I calculated the expected income streams for a number of individuals in my sample using both approaches. After comparing the resulting estimates, I chose the second approach. The primary reason for this choice was that the first approach yielded extreme housing expenditure to income ratios. In some cases, the first approach predicted that a household would spend over 70 percent of its income on housing. Using the second approach, an individual's income in period t is given by /(0) + £ (1 + njY (4.2) »=i where 1(0) is the individual's actual (observed) current period income, A.I(s) is the expected change in the individual's income between periods s — 1 and s, and x/ is the rate of inflation expected by the individual. In this equation, the term in square brackets is the individual's income in period t, valued in 1974 dollars. This value is converted to the individual's income valued in period t dollars by multiplying it by the rate of inflation since 1974: (1 + 7T/)'. The value T:J is set equal to the average (compound) rate of increase in the Consumer Price Index for Toronto for the years 1973, 1974, and 1975. This average rate is 9.8 percent. The expected yearly changes in individual incomes are given by the coefficients of the variable AGE in the age-income regressions. In addition, I assume that individuals regard unemployment as temporary. Therefore, if an individual was unemployed at the time of the SHU survey, the individual's income in the next period is adjusted (upward) by subtracting the coefficient for ESI from the next period's expected change in income. Furthermore, for husband/wife families where the spouse was present, the expected changes in the incomes of the husband and wife are calculated 54 simultaneously. The direct effect of time on the husband's and wife's incomes operates through the variable AGE, while the husband and wife affect each other's incomes through the variable SINC. The Consumer Price Index is not an ideal index for the rate of income increase. Most of the published wage indexes, however, deal only with industrial wages and salaries and are therefore not representative of wage and salary increases for individuals in the sample. I have circumvented this problem by assuming individuals expect their incomes to rise at the prevailing rate of inflation, making their real purchasing power constant. The second step in generating an estimate of the real after-tax income for an individual in the sample is to subtract the hypothetical income tax liabilities of the individual from his/her estimated income. The calculation of an individual's hypothetical income tax liabilities is the subject of the next section. Ind iv idua l Income T a x Liabil i t ies A tax simulator program is used to generate the hypothetical income tax liabilities of the individuals in the sample. The program is based on the 1974 federal and provincial income taxes. The elements of the 1974 tax structure used in the simulation are presented in Table VII, located at the end of this chapter. To simulate individual expected tax liabilities after 1974, I assume that all individuals in the sample expected the tax structure to remain substantially unchanged, in real terms, after 1974. Consequently, the tax rules described in Table VII are used for time periods after 1974, but the nominal values in the table are inflated to reflect either the expected level of price inflation after 1974 or statutory increases in values. The tax rules in Table VII are, for the most part, self-explanatory. The procedures used to generate several of the variables need explanation, however. In Table VII, total income (derived from equation (4.2) is divided into two components: employment income and other income. The division is based on the individual's age and on the average percentage of employment income in total income, by total income class. For individuals under age 65, employment income equals the individual's total income multiplied by the average percentage of employment income in total income for the individual's total income class. Other income equals total income less employment income. Individuals aged 65 and over are assumed to be retired and therefore have no employment income; that is, all income is other income. The average percentages of employment income in total income were obtained from taxation statistics.7 The percentages used in the tax calculations are simple three-year averages of the published percentages for the tax years 1973, 1974, and 1975. I have included deductions in Table VII for contributions to a Registered Pension Plan, a Registered Savings Plan, and a Registered Home Ownership Saving Plan, as well as a deduction 7 The source for these percentages and all other tax data employed is Canada, Revenue Canada. Taxation Statistics. Ottawa: Information Canada. 55 for interest income. These deductions are included because they were major deductions during the study period. Hence, individual income taxes would be inflated if these deductions are ignored. All these deductions are based on imputed values for individual contributions to the three savings plans and the amount of interest income. These values are imputed in the same manner as employment income. Specifically, the average contribution to each of the plans per dollar of total income and the average amount of interest income per dollar of total income, by income class, were culled from the 1973, 1974 and 1975 income tax statistics. The three-year average contribution rates to the plan and the three-year average rates of interest income were calculated. Then the imputed values for an individual's contribution to the three plans, as well as the amount of interest income, are calculated by multiplying the individual's total income by the three-year average contribution rate for his/her income class. To compute the deduction for Registered Savings Plan contributions, one needs to know whether the individual's employer contributed on his/her behalf to a Registered Pension Plan. Since I do not have this information, I have computed each individual's Registered Savings Plan deduction twice: once assuming that the employer contributed to a Registered Pension Plan and once assuming that the employer did not contribute to a Registered Pension Plan. I then averaged the two. The deduction for Registered Home Ownership Savings Plan contributions depends on whether the individual owns the dwelling in which he/she resides. The SHU data set does contain an individual's ownership status for 1974, but this value is unknown after 1974. To calculate the deduction, I have assumed that if an individual owns his/her current dwelling, then he/she will continue to own a dwelling in the future (i.e., no deduction is claimed). On the other hand, an individual who rents his/her current dwelling is likely to "move up" to an owned dwelling in the future. To account for this possibility, I have multiplied the Registered Home Ownership Savings Plan deduction for individuals who rent in 1974 by the ratio of renter households to total households, for households in the same income class as the household in question. This ratio may be roughly interpreted as the income specific probability of renting. 4.2.2 Forecasting Proper ty Taxes In order to calculate future property taxes for owner occupied dwellings, I assume households have static expectations about the rate of property tax increases. As a result, the property taxes on owner occupied dwellings are given by PT(t) = PT(0)(l + nPTy (4.3) where irPT is the household's expected rate of property tax increase. The rate of property tax increase used in this study is 3.2% per annum. This is the average rate of increase for 1973, 1974, and 1975, for the property tax component of the Consumer Price Index for Canada.8 I have resorted to a property tax index for Canada as a whole, since information on the property tax rates for the four cities in my sample and, indeed, for the province of Ontario, is too fragmentary for use in this study. 8 The source for this index is Canada, Central Mortgage and Housing Corporation. Canadian Housing Statistics, 1975. Ottawa: Information Canada. 56 4.2.3 D a t a on Rejected Hous ing Alternatives One set of data, the data pertaining to the housing alternatives rejected by households, remains for discussion. These data are required in order to compute the household's utility levels in the various alternatives which, in turn, are needed to compute the choice probabilities in the discrete part of the log likelihood function. These data, by their very nature, are not included in the SHU data set. The absence of information on rejected alternatives is not particular to this study. Other studies (e.g., King 1980) have dealt with this problem by using average values for the required variables. The averages are usually calculated from the set of households which chose the alternative rejected by the household in question. I also follow this course of action, using the type and tenure stratified averages of the required variables, calculated from the sample of 1893 mover households. 4.3 CONCLUSIONS I have two concluding comments regarding this chapter. First, while the SHU data set has defi-ciencies as outlined in this chapter, such deficiencies are not unique to this data set. For example, in selecting a data set for use in this thesis, I considered the U.S. Annual Housing Survey (AHS). It was not chosen because of deficiencies in its household income data and housing mortgage data. Second, most of this chapter is concerned with forecasting after-tax household incomes and property taxes. These problems are not confronted in single period studies, since the data are taken as given. For multiperiod models, however, forecasting problems will always be a major cost. 57 T A B L E VII: S I M U L A T E D I N C O M E T A X Calculation of Total Income Employment Income a) less: • Employment Expense Deduction whichever is less. b) plus: • Other Income Calculation of Net Income Total Income a) less: • Canada Pension Plan Contribution — 1.8% of an individual's Maximum Contrib-utory Earnings or $106.20, whichever is less. An individual's Maximum Contribu-tory Earnings equal his/her Employment Income less $700 or $5,900, whichever is less. • Unemployment Insurance Premiums — 1.2% of an individual's Insurable Earn-ings, where Insurable Earnings equal his/her Employment Income or $8,840, whichever is less. Net Income Calculation of Taxable Income (=Net Income less Deductions) Net Income a) less: • Basic Personal Exemption — claim $1, 706. • Age Exemption — If the individual is 65 years of age or older, claim $1,066. • Married Exemption — 1) If spouse's income is less than $314, claim $1,492. 2) If spouse's income is between $314 and $1,806, claim $1,806 less the spouse's income. 3) If spouse's income is greater than $1,806, no exemption may be claimed. • Exemption for Wholly-Dependent Children — If the dependent is sixteen years of age or less, then: 1) If the dependent's income is less than $1,166, claim $320. 2) If the dependent's income is between $1,166 and $1,806, claim $320 less one-half of the dependent's income over $1,166. 3) If the dependent's income is greater than $1,806, no exemption may be claimed. — Claim 3% of Employment Income or $150, Total Income 58 If the dependent is over sixteen years of age, then: 1) if the dependent's income is less than $1, 220, claim $556. 2) If the dependent's income is between $1,220 and $1,806, claim $586 less the dependent's income over $1,220. 3) If the dependent's income is over $1,806, no exemption may be claimed. • Medical Exemption — claim the standard medical exemption of $100. • Reduction for Registered Pension Plan Contribution — An individual may claim his/her contribution to a Registered Pension Plan to a maximum of $2, 500. • Deduction for Registered Savings Plan Contribution — One may claim the small-est of the following amounts: 1) The individual's actual contribution to a Registered Savings Plan 2) 20% of Employment Income 3) If the individual's employer does not contribute to a Registered Pen-sion Plan on the employee's behalf, there is a Registered Savings Plan maximum of $4,000. 4) If the individual's employer contributes to a Registered Pension Plan on the employee's behalf, there is a Registered Savings Plan maximum of $2,500. Also, if the individual is married, he/she may transfer the unused portion of his/her spouse's Registered Savings Plan Deduction (i.e., the spouse's maximum deduction less the amount claimed for the spouse's income tax). The sum of an individual's deductions for Registered Pen-sion Plan Contributions and Registered Savings Plan Contributions is limited to $2, 500 or 20% of Employment Income, whichever is less. • Deduction for Registered Home Ownership Savings Plan Contributions — To claim this deduction, the individual must be seventeen years of age or over and must not own a dwelling. An individual may claim his/her actual contribution to a Registered Home Ownership Savings Plan or $1,000., whichever is less. • Interest Income Deduction — An individual may claim his/her actual amount of interest income or $1,000, whichever is less. Taxable Income 59 Calcu la t ion of Taxes and Credi t s Basic Federal Tax — Calculated using the 1974 detailed tax calculation formula. a) less: • Federal Income Tax Reduction — 1) If Basic Federal Tax is less than $100, the reduction equals the Basic Federal Tax. 2) If the Basic Federal Tax is between $100 and $2,000, the reduction equals $100. 3) If the Basic Federal Tax is between $2,000 and $10,000, the reduction equals 5% of the Basic Federal Tax. 4) If the Basic Federal Tax exceeds $10,000, no reduction is available. Federal T a x b) plus: Ontario Provincial Tax — 30.5% of the Basic Federal Tax. c) less: • Ontario Tax Credit — This credit consists of: 1) Property Tax Credit — The "Occupancy Cost" for the individual or $90, whichever is less, plus 10% of the occupancy cost. The "occupancy cost" for an individual equals 20% of the individual's total rental pay-ments, plus all property taxes, plus $25. for students living in a student residence. 2) Sales Tax Credit — 1% of Total Personal Exemptions. 3) Pensioner Tax Credit — 1) If the individual is unmarried, then the tax credit is $110. 2) If the individual is married and lives with his/her spouse, then the spouse with the greater income claims $110. The Ontario Tax Credit equals the Property Tax Credit plus the Sales Tax Credit plus the Pensioner Tax Credit less 1% of Taxable Income. If it is less than zero, a credit of zero is claimed. If it exceeds $400, then $400 is claimed. T o t a l Tax L iab i l i t y 60 C h a p t e r 5 H E D O N I C P R I C E E S T I M A T E S 5.1 THE HEDONIC REGRESSION MODEL This chapter deals with the technique I have used to forecast the rental and sale values of tenant and owner occupied dwellings, as well as the procedure used to estimate the marginal prices of the dwelling characteristics, dwelling size and dwelling newness. In both applications, the basic approach was to fit hedonic regression equations to data on housing prices and characteristics and then use the fitted equations to obtain estimates of the necessary prices. Accordingly, I begin this chapter with three sections describing the specification, data, and estimation of my hedonic regression model. The fourth section discusses my estimates of the hedonic prices of the characteristics in the hedonic regressions. In the last section, I consider three possible sources of misspecification in my hedonic models. Hedonic regressions on the value or cost of dwellings have been estimated for over twenty years.1 During this time, a variety of functional forms have been employed, including the linear, semi-logarithamic, and logarithmic forms and, more recently, Box-Cox regressions and flexible functional forms. Despite a large number of empirical studies, no functional form has been shown to be clearly superior. Hence, the question of what is a "good" functional form for hedonic regressions is as open today as it was twenty years ago. The theoretical literature on hedonic pricing also has little to say on this question. The only author known to have addressed this question is Rosen (1974). As was noted in my review of the hedonic literature (Chapter 2), Rosen argues that because housing characteristics are purchased as a bundle and because this bundle cannot be untied and repackaged, the hedonic price function will be nonlinear in its arguments. The functional form I have chosen to estimate is the modified quadratic2 M j M M N M N Vi = /3oo + Y foizH + o Y Y PikZijZik + Y tojdij + Y Y 1jk2ijdik (5.1) j = l j = lk = l j=l > = 1A;=1 'For example, see: Martin J. Bailey, Richard F. Muth, and Hugh O. Nourse (1963), "A Regression Method for Real Estate Price Index Construction," Journal of the American Statistical Association, 58:933-42. 2The Translog, Generalized Leontief and Generalized Cobb-Douglas forms were also estimated in preliminary ex-periments. These functional forms did not yield appreciably different results from the modified Quadratic. 61 where t>,- is the price of a dwelling unit, z, = (z,i,..., z,^) are housing characteristics which can be measured continuously, and d, = (du,..., d,-jv) are housing characteristics which take binary values. In addition, the symmetry restrictions /3ij = /3Jt-, i,j — 1,...,M are required for identification. This functional form is quadratic in the continuous characteristics z,-y, and it can therefore capture increasing, constant, and decreasing prices for these characteristics. The functional form (5.1) is not wholly flexible as it does not provide a quadratic approximation to the true hedonic price equation. To be flexible, the cross terms dijdik of the discrete characteristics should be included to capture the interaction effects of these variables. They are dropped from (5.1) because their inclusion in preliminary regressions resulted in multicollinearity. The problem of multicollinearity is discussed in the last section of this chapter. The total sample of dwelling units is divided into twenty-four subsamples which are defined by the type, tenure, and location of the dwelling units. A separate version of equation (5.1) is estimated for twenty-one of these samples. The three remaining samples are too small to yield meaningful parameter estimates. Table VIII gives the number of dwelling units in each sample. T A B L E VIII : S A M P L E S I Z E S H a m i l t o n L o n d o n Toronto W i n d s o r A . O w n e r Occupied Dwell ings: 1. Singles 978 2. Multiples 100 3. Apartments 151 B . Tenant Occupied Dwell ings: 1. Singles 100 2. Multiples 153 3. Apartments 852 1 A regression is not estimated for this sample. There are several reasons for dividing the sample in the manner indicated in Table VIII. The primary reason is that more than one hedonic price equation must be estimated in order to identify the parameters of the household demand equations for housing characterstics. This was illustrated by Brown and Rosen (1982) who suggest estimating separate equations for each distinct housing market. The key question in applying this approach is: What is a distinct housing market? I have chosen to partition dwelling units into housing markets according to their type, tenure, and the city in which they are located. The cities in the sample (Hamilton, London, Toronto, and Windsor) are geographically separate. Thus, it is quite natural to regard them as distinct housing markets. The division of dwelling units into distinct housing markets by type and tenure is less easily justified, even though the estimation of separate hedonic regressions by type or tenure is common within the hedonic literature. 1130 779 1217 63 265 64 31 51 291 142 75 175 311 79 218 879 949 659 62 My first reason for estimating different regressions for each type and tenure of dwelling unit is to allow for hedonic prices which may not be long run equilibrium supply prices. That is, I enable the hedonic prices to reflect the short run composition of the existing stock of dwelling units in each city. My second reason for choosing this approach is to be consistent with the household preference ordering I have assumed in Chapter 3. This preference ordering is type and tenure specific, implying some form of separation or distinction between units of differing type and tenure in the housing market, although not necessarily separate markets. A number of authors have stated that the relative immobility of households and the nonmalleability of residential capital due to high conversion costs violate the assumptions of long run equilibrium. They advise taking a short run equilibrium approach. For example, see Ingram, Kain and Ginn (1974), Straszheim (1975), Kain and Quigley (1975), or Goodman (1978). It should be noted that the estimation of separate regressions by type and tenure does not preclude the existence of uniform long run equilibrium prices for the housing characteristics. Indeed, long run equilibrium prices become a special case, subject to empirical testing. I test the hypotheses that the local housing markets are in long run equilibrium in Section 5.1. These tests do not support the hypothesis of long run equilibrium. 5.2 HOUSING PRICES Because different regressions are estimated for owned and rented dwellings, the value of a dwelling can be used as the dependent variable for regressions on the samples of owned dwellings, while the rent of a dwelling can be used as the dependent variable for regressions on the samples of rented dwellings. If the samples could not be separated by tenure, then the two sets of samples would need a comparable dependent variable.3 The value of a dwelling unit is obtained from an owner occupier's response to the SHU survey question, "If you are selling this dwelling now, for how much would you expect to sell it?" 4 Although owner evaluations such as this have long been a means of obtaining information on housing values, one wonders how accurate these evaluations are. Several studies have pursued this question. In the United States, there have been two major studies, Kish and Lansing (1954), and Kain and Quigley (1972). These studies found that while the errors of estimate for individual properties can be quite large, the average discrepancy between the owner's evaluation and the sale value is small. For example, Kain and Quigley indicate a discrepancy of .17% of the appraised value of the dwelling. Two Canadian studies also exist. The primary study was done by Census statisticians Priest, Alford, and Bailey (1973). The authors tried to evaluate the accuracy of the owner's expected selling price in the 1971 Census of Canada by matching the sale values of Multiple Listing Service (MLS) single detached unit sales for August and September 1971 to the expected selling prices 3This could be done by computing the user cost of owner-occupied dwellings, as done by Linneman (1981). *Canada Mortgage and Housing Corporation, Survey of Housing Units, Questionaire, p.5. 63 contained in census records taken on June 1, 1971. Their micro-match yielded a sample of 1,140 dwellings located in Montreal, Hamilton, Toronto, Windsor, Saskatoon, Edmonton, Vancouver, and Victoria. The results of this analysis are reported in the second Canadian study by Steele and Buckley (1976). These authors observe that the average error of the owner estimates is quite small, but owners do overestimate the value of their dwellings. On average, owner estimates exceed the MLS sale value by 5.8%. Steele and Buckley also report a tendency for owners of less expensive dwellings to overvalue their units by a greater percentage than owners of more expensive dwellings. For example, they estimate that a house with an MLS value of $20,000 would be overvalued by 11.2% while a house with an MLS value of $40,000 would be overvalued by a mere 0.6%. While Steele and Buckley explain the differences between the owner estimates and the MLS values in terms of owner-estimation errors, it is also possible that housing prices fell during the summer of 1971, with the prices of cheaper dwellings falling more rapidly than the prices of expen-sive dwellings. The validity of this latter explanation is impossible to establish as price statistics on existing dwellings are not collected by Statistics Canada. Nonetheless, there are two pieces of correlative data which shed some light on this matter. First, the price index for newly constructed homes rose steadily throughout the summer of 1971. Second, the housing component of the Con-sumer Price Index also rose steadily throughout this period. Combined, these series indicate a positive secular trend in housing prices in Canada during the summer of 1971. Hence, it is unlikely that falling prices generated the differences between the owner estimates and the MLS prices. The four studies just discussed imply owners are fairly accurate in their evaluations of dwelling value. The potential bias indicated by the Canadian studies is a matter of some concern, however, since the SHU data set used in this study is closely linked to the 1971 Census of Canada. Should the estimated bias reported in Steele and Buckley be used to adjust the housing values in the SHU? I have chosen not to do so. The results presented in Steele and Buckley apply only to single detached dwellings sold through the MLS, not to owned multiple dwellings, owned apartments, single dwellings sold privately, or single dwellings sold through an exclusive realtor listing. In my opinion, to adjust for the estimated bias from the two Canadian studies would be premature. For rented dwellings, the dependent variable is the net rent of the dwelling. Net rent equals the contract cost less a valuation for all utilities and services provided by the landlord. It may be regarded as the pure rental payment for a dwelling unit, and, like the value of a dwelling unit, it is not encumbered by utility payments. The net rent for a dwelling cannot be computed directly, since the value of the utilities and services provided by the landlord is unknown. To adjust for these utilities and services, a dummy variable is added to the hedonic regressions for rented dwellings for each of the following utilities and services: water, electricity, gas, oil, parking, and "other". If the utility is paid for by the landlord, the dummy variable is assigned the value one; otherwise it is zero. A further adjustment is made to the contract rent for dwellings containing rooms used solely for business or professional reasons. If the rent for these business rooms was recorded in the survey, it is subtracted from the contract rent. In cases where this information was not recorded, 64 an additional dummy variable is added to the regressions; it takes the value one if the dwelling contains business rooms for which the rent is unknown and is zero otherwise. Table IX contains the means and standard errors of the housing prices for the twenty-one regressions. Some notable patterns are evident in these prices. First; looking across cities, one sees that Toronto has the highest prices for dwelling units, followed by Hamilton and London, which have approximately the same prices. Windsor tends to have the lowest prices for dwelling units. The pattern of dwelling prices across dwelling types depends on tenure. Among owner occupied units, single dwellings command a higher price than multiple dwellings, and in Toronto, both single and multiple dwellings have higher prices than apartments. The pattern of dwelling prices is not as definite for rented dwellings. There is a tendency for multiples to have the highest prices, followed by apartments, and then single dwellings. In order to compare the prices of dwellings across tenure, the average monthly user costs of the owner occupied dwellings are computed. The estimation of the capitalization rates used in the calculation of these user costs is discussed in section 5.5.1. The user costs are recorded in panel C of Table IX. Interestingly, the user costs are all lower than the net rentals on the same type of dwelling in the rental sector of the housing market. Viewed in isolation, this seems to indicate a financial advantage for owning over renting. Such a conclusion is premature however, as the values in Table IX do not take into account variations in the quantity of dwelling services provided by the average dwellings in the ownership and rental sectors of the market. 5.2.1 I n d e p e n d e n t V a r i a b l e s There are three categories of housing attributes used as explanatory variables in the hedonic regres-sions: dwelling unit size attributes, dwelling unit quality attributes, and structure type attributes. The dwelling unit size category contains one variable — the number of rooms in the dwelling unit. This variable is denned as the total number of living rooms and the number of bedrooms, excluding bathrooms, halls, vestibules, garages, workrooms, and unfinished rooms. The number of bathrooms and the square area of a dwelling have been used as dwelling size attributes in other hedonic price studies and they have been proven to be statistically meaningful measures of dwelling unit size. Unfortunately, neither of these variables can be used in the present study. Information on the number of bathrooms was not collected during the survey of housing units. The SHU data set does contain the square area of the rooms in the dwelling occupied by a household at the time of the SHU survey. This variable was not employed, however, because the square area of a household's previous dwelling must also be known for the discrete choice part of the demand model and this latter value is not known. The second category of housing attributes contains three variables that measure dwelling unit quality. The first is the construction date of the building containing the dwelling unit. This variable is commonly interpreted as a gauge of the net depreciation rate of housing prices. As such, the construction date of the dwelling is thought to reflect the general level of dwelling unit maintenance 65 T A B L E I X : M E A N S A N D S T A N D A R D E R R O R S O F H O U S I N G P R I C E S H a m i l t o n L o n d o n Toronto W i n d s o r A . Owner Occupied Dwell ings: 1. Singles 46056. 41017. 65208. 35090. (18017.) (15495.) (19700.) (14628.) 2. Multiples 39233. 38586. 54967. 33031. (10510.) (9944.) (13631.) (7969.) 3. Apartments 45104. (13606.) Tenant Occupied Dwell ings: 1. Singles 141.87 147.92 213.44 150.60 (53.93) (58.67) (91.46) (66.80) 2. Multiples 166.90 171.00 240.10 144.28 (64.50) (51.12) (86.23) (47.85) 3. Apartments 158.33 155.77 189.24 161.73 (36.43) (42.12) (56.63) (44.37) User Costs of Owner Occupied Dwel l ings: 1. Singles 107.46 107.57 169.80 83.84 2. Multiples 91.54 101.20 143.14 79.92 3. Apartments 117.45 66 and the architectural obsolescence of the building. Respondents in the SHU were asked to classify the construction date of their building into one of five intervals: (i) 1940 or before, (ii) 1941-1950, (iii) 1951-1960, (iv) 1961-1970, or (v) 1971-present. Dwelling units with a construction date in intervals (ii) to (v) were assigned the date at the mid-point of the interval. Buildings with construction dates in the open-ended interval (i) were assigned a date based on the distribution of construction dates in the 1971 Census. This distribution contains three extra pre-1940 construction date intervals: 1920 or before, 1921-1930, and 1931-1940 and is broken down by tenure and city. The median construction date of pre-1940 buildings for owned and rented dwellings was computed for each city and this median construction date was then assigned to the buildings in the SHU construction date interval: 1940 or before. The second measure of dwelling unit quality is an index of the exterior condition of the building containing the dwelling unit. The index is a summary measure constructed by CMHC from eleven characteristics of the building's exterior. The index takes the values "good", "fair", and "poor". Two dummy variables are used to incorporate the index into the hedonic regressions. The first variable has the value one if the building is in fair condition and is zero otherwise; the second dummy variable has the value one if the building exterior is in good condition and is zero otherwise. An index of the interior quality of a dwelling unit would also be desirable, but such an index is not available. One expects, however, that the interior and the exterior conditions of a dwelling unit will correspond closely. Therefore, the index of exterior quality serves as a proxy for the overall quality of the dwelling. The third and last measure of dwelling unit quality is the presence or absence of a private bathroom with hot and cold running water. This variable is of importance only for tenant occupied apartments, as a private bathroom with hot and cold running water is a standard amenity for almost all other types of dwelling units. The final category of explanatory variables contains structure-type variables. These variables are dummy variables used to subdivide the dwelling types (single and multiple) into a finer division of structure types. Dummy variables are created for two types of single dwellings: single attached dwellings and single detached dwellings, and they are created for three types of multiple dwellings: double houses, row houses, and duplexes. These structure dummies are added to the appropriate regressions. In addition to the variables described above, there are two variables which are available for apartments only. These are the number of apartments in the apartment building and the number of stories in the apartment building. Both variables are included in the apartment regressions. The number of dwelling units in an apartment building reflects the externalities associated with communal living; it is an inverse measure of privacy. The hedonic price of this attribute is therefore expected to be negative. The number of stories is also a measure of privacy. In addition, it mirrors the land prices at the time of the apartment's construction, and, therefore, the expected sign of this variable is ambiguous. 67 5.3 ESTIMATION TECHNIQUES The observed price of a dwelling unit equals the dwelling unit's expected price given by equation (5.1), plus a random error. Hence, the observed price of a dwelling unit is given by Vi = /{zitdiYv + ei (5.2) where /(z,-, <i,)'r/ is the matrix representation of the functional form (5.1), rj is the K = 1 + M + N + M(M + 2N + l)/2 vector of identified parameters (i.e., the identified /3's and 7's) and e, is the random error. The error e, reflects observational errors in the recorded price and variations in the price of the dwelling which are not systematically related to the observed characteristics of the dwelling. If the errors {c\,... ,€j) for a sample of dwelling units are independent and identically distributed with a mean of zero and a constant variance, the parameter vector n can be estimated by ordinary least-squares. The use of OLS may not be warranted, however. Previous studies have suggested variances of the errors may be heteroscedastic; these studies often suggest heteroscedasticity is due to the size characteristics of a dwelling (e.g., Kain and Quigley 1975:194).5 It is well known that OLS estimates are unbiased but inefficient when there is heteroscedasticity. In addition, the computed standard errors will be inconsistent. In the present study, a more compelling argument exists for believing variances of the errors are heteroscedastic. This argument is based on two factors. First, the four urban housing markets may not be in long run equilibrium. Second, the hedonic regressions in this study must be estimated on city-wide data. The roles these two factors play in generating heteroscedasticity will be examined at this point. I have asserted that the housing markets in the four cities will not be in long run equilibrium with respect to type and tenure because of the durability and nonmalleability of dwelling units. For the same reasons, the cities are unlikely to be in spatial equilibrium. Thus, the cities will tend to be balkanized into spatial submarkets, and the prices of dwelling unit attributes will vary from submarket to submarket because of location-specific quasi-rents.6 These variations in the hedonic prices, in turn, imply that the coefficients of the hedonic price regressions will be different for each spatial market. This means separate regressions should be estimated for each market. Unfortunately, this cannot be done as the SHU data set contains no location identifier below the level of city; thus, the hedonic regressions are performed on city-wide data. The necessity of using city-wide, multiple-market data to estimate the hedonic equations rather than single market data changes the nature of the estimation problem. The problem becomes one of efficiently estimating an average parameter vector r) for a metropolitan region. The transformation from a model dealing with a single market to one dealing with multiple markets is not trivial. 5 A common pallitive has been to estimate hedonic price equations in semi-log form. 6Straszheim (1975) amply documents the existence of location-specific quasi-rents for the San Francisco Bay area. 68 Equation (5.2) must be restated in terms of the average parameter vector f), rather than the market-specific parameter vector r\. The first step in this restatement involves a change of notation. Since the dwellings in the sample are in different spatial submarkets, the parameter vector applicable to the j'th dwelling unit can be written as rj,. Then, because fj is an average of the parameter vectors for the different submarkets, the vector rj, can be expressed as the sum of fj and a location-specific deviation from the vector, rj,-; that is, r?,- = fj + 77,. Substituting this expression for rji into equation (5.2) yields Vi = f(zudi)'(fj + f)i) + ei = f{zi,di)'fj + {ei + fiz^diYfn) (5.3) which would be the desired equation if it were not for the term f(zi,di)'f)i. Since the vector 77, cannot be determined from the data, I propose to treat it as a random vector. Thus, tj + /(z,, di)'fji becomes the regression error term for the city-wide regression. Given this interpretation of the city-wide regression, the presence of heteroscedasticity becomes obvious. The regression errors depend on the characteristics of the dwelling units, as do the variances of these errors. While I regard the above as a compelling argument for heteroscedasticity, a compelling ar-gument is not a substitute for a sound empirical test. Therefore, I have tested for the presence of heteroscedasticity in the errors of the regression equations for each of the twenty-one markets. Since the presumed heteroscedasticity is of unknown form, White's (1980) chi-squared test for heteroscedasticity is used. The test results are given in Table X. The null hypothesis for these tests is that heteroscedasticity does not exist. This hypothesis is rejected, quite resoundingly, in each market. Indeed, some of the test statistics are so large as to lead one to question the validity of the tests. The key issue is whether White's test is detecting het-eroscedasticity or picking up a misspecification in the form of the regression equations. (The latter possibility was pointed out by White himself.) I believe misspecification can be ruled out for two reasons. First, the functional form of the hedonic regressions (5.1) is "almost flexible''. Therefore, it provides an "almost second order differentiable approximation" to an arbitrary functional form. Put more simply, no matter what the true functional form is, equation (5.1) will mimic its first and second derivatives. The second reason for doubting misspecification is that similar results to those presented in Table X can be obtained using other specifications for the regression equation. One such specification is the additive quadratic model in Appendix D to this thesis. With the spectre of misspecification set aside, one may conclude that the regression errors are heteroscedastic and given the size of the chi-squared test values, the problem is not trivial. To handle this problem, I have estimated the hedonic price equations using Cragg's (1981) technique for more efficient estimation in the presence of heteroscedasticity of unknown form. This technique will be referred to as 'Cragg estimation' or CE in this study. If the covariance matrix of the 69 T A B L E X : W H I T E S ' S T E S T F O R H E T E R O S C E D A S T I C I T Y O F U N K N O W N F O R M Hamilton London Toronto Windsor A. Owner Occupied Dwellings: 1. Singles 4292.6* 4363.6* 63.4* 6127.9* (25) (18) (20) (25) 2. Multiples 3.5 x 106* 134.1* 383.1* 2.6 x 10 1 2* (24) (10) (12) (34) 3. Apartments 1.1 x 10 1 1* (37) B . Tenant Occupied Dwellings: 1.4 x 107* 1. Singles 1.6 x 106* 1.4 x 105* 10908.0* (41) (38) (24) (52) 2. Multiples 9.6 x 10 1 4* 5.0 x 105* 27.4* 1.1 x 106* (80) (55) (15) (34) 4.7 x 105* 3. Apartments 1.6 x 10s* 48145.0* 84036.0* (104) (85) (133) (86) * Statistically significant at the 5% level of significance. 70 regression errors is symmetric, OLS and CE both yield unbiased estimators. If the variance of the dependent variable is heteroscedastic, however, the CE is usually more efficient than OLS and the covariance matrix of the CE estimator is consistent, whereas the covariance matrix of the OLS estimator is not. The basic assumptions underlying the Cragg estimator for the model given by equation (5.3) are that the regression errors et* = e, + /(z,-, d,)'^ , are independent random variables for which E{et) = 0, and E(e?) = <rf; (5.4) thus, the covariance matrix of the errors for a sample of size T is S = diag(tr* 2,.. .,cr*-2). These assumptions are fulfilled when (a) E(ti) = o, (b) E(fji) = o , (c) E(e2) = * 2 , (d) E(mnl) = *,-, («) Bid*]) = o , (/) E{€if,i) = oK> (9) = 0K Only assumption (g) is difficult to justify. It states that the location specific deviations from the parameter vector fj must be uncorrelated across dwelling units. This condition is not in accord with the theoretical argument for heteroscedasticity presented above. It is consistent with a random coefficients model however.7 Since the latter model also captures the most important point of the above argument — that the parameters of the model will vary across dwellings because of spatial quasi-rents — it is an acceptable alternative model. The Cragg estimator is fj" = [F'Q(Q'5Q)-1Q'F]-1F'Q(Q'5g)-1gV (5.5) where F is the T x K matrix of regressors, Q — [F, G] is a T x (K + J) matrix with G being a T x J matrix of auxiliary variables, V is the T x 1 vector of observation on the dwelling unit prices, and S = diag(f2,.. .,ej) where e2 is the square of the z'th residual from an OLS regression of F on V. Roughly speaking, the role of the auxiliary variables in the Cragg estimator is to assist 7See B. Raj and A. Ullah (1981) for a comprehensive summary of random coefficient models. 71 in explaining the heteroscedasticity of the variances a? i = 1,...,7\ In this study, the logs of the housing characteristics Z{j and the squares of the logs are employed as auxiliary variables. A consistent estimate of the covariance matrix of the estimated parameter vector fjce is given by the matrix One general observation concerning the estimation of the hedonic regressions is in order before turning to my discussion of the hedonic prices. Cragg's technique appears to be particularly successful in reducing the standard errors of the regression coefficients. The parameters from the CE estimated equations frequently have standard errors which are 20% to 40% less than the standard errors from the corresponding OLS regressions. In some cases, reductions in the standard errors of more than 100% are obtained. This is an encouraging result, as it points to the propriety of using the CE estimation technique in the present circumstance. It should be recalled however, that the standard errors from the OLS regressions are inconsistent when there is heteroscedasticity, so the indicated reductions in the standard errors may be spurious to some extent. In this section, I discuss the estimates of the hedonic prices of the continuous housing attributes. The estimated prices are examined with respect to three criteria: (i) the sign of the average hedonic price for the sample, (ii) the standard error of the average price, and (iii) the proportion of dwelling units in the sample which have an estimated hedonic price with the wrong (i.e., negative) sign. This section does not contain an equation by equation discussion of the parameter estimates, since these estimates are not readily interpretable. The parameter estimates for each equation are presented in Appendix C to this dissertation. I now examine the hedonic prices for three continuous housing characteristics: the number of rooms, the newness of the dwelling, and the number of units in the building containing the dwelling. The prices of the first two characteristics can be computed for all three types of dwelling units; the price of the last characteristic applies only to apartments. The estimates of the implicit or marginal prices are obtained by differentiating the estimated hedonic price function with respect to the characteristics. Thus, the hedonic price of the jth characteristic for the ith dwelling unit is [F'QiQ'SQyiQ'F] - l (5.6) 5.4 R E S U L T S [df^d^/dz^'f, (5.7) 72 5.4.1 T h e P r i c e p e r R o o m Table XI contains the average price per room for the twenty-one housing markets, together with the standard error of the average prices and the proportion of dwelling units in each market with a negative estimated price per room. This table shows that the estimates of the marginal price per room are exceedingly robust. In all markets, the average price is positive as expected, and the average prices are statistically significant in most of the markets. In addition, few dwelling units have a negative estimated price per room — less than one percent of the total number of dwelling units. T A B L E X I : T H E P R I C E P E R R O O M H a m i l t o n L o n d o n Toronto W i n d s o r A . Owner Occupied Dwell ings: 1. Singles 4985.* (852.2) 0% 5026.* (716.9) 1% 6253* (988.8) 0% 4815* (671.2) 0% 2. Multiples 2947. (1701.) 27% 2148. (989.1) 0% 2838. (1468.) 1% 3165 (1987.) 30% 3. Apartments 6590.* (2763.) 0% Tenant Occupied Dwell ings: 1. Singles 13.49* (6.26) 1% 20.57* (5.71) 0% 29.15* (12.00) 0% 21.90* (10.12) 0% 2. Multiples 16.03* (6.47) 2% 21.20* (5.02) 0% 31.63* (7.41) 1% 23.70* (4.19) 0% 3. Apartments 17.13* (2.53) 0% 17.13* (3.22) 0% 23.26* (4.50) 1% 19.57* (2.84) 0% * Statistically significant at the 5% level of significance. Table XI also displays several type/tenure specific features. The most striking of these is the similarity of the average prices for the four cities within each type-tenure category. The price per room for owner occupied single dwellings ranges from $4815 to $6253, and all prices are statistically significant. For owner occupied multiple dwellings, the price per room varies between $2148 and $3168. The four prices are statistically insignificant however, and the Hamilton and Windsor markets have large proportions of dwelling units with a negative estimated price. The average price of owner occupied apartments in Toronto is $6596, which is statistically significant. Among owner occupied units, multiple dwellings are the best buy in terms of dwelling space. 73 For tenant occupied dwellings, the differences in the average price per room by dwelling type are minor. The average prices vary between $13.49 and $29.15 for single dwellings, between $16.03 and $31.63 for multiple dwellings, and between $17.13 and $23.26 for apartments. All average prices per room are statistically significant. A negligible number of dwellings have a negative estimated price. There is one further regularity in the average prices. The average price per room is highest for dwellings in Toronto. 5.4.2 T h e Price of Newness The estimated prices for a one-year newer dwelling are presented in Table XII. These estimates are less robust than the estimates of the price per room. First, seven of the twenty-one markets have a negative average price of newness, but none of these negative average prices is statistically signifi-cant. Second, while fifteen of the twenty-one markets have a positive average price of newness, in only six of these markets is the positive price statistically significant. Finally, as might be expected, there is a dramatic rise in the proportion of individual dwellings with an estimated negative price of newness. On average, twenty-five percent of the dwellings have a negative estimated price. T A B L E XII: T H E P R I C E O F N E W N E S S Hamilton London Toronto Windsor A . Owner Occupied Dwellings: 1. Singles 441.23* (71.03) 0% 311.81* (70.29) 21% 362.75* (111.8) 21% 329.68* (64.83) 6% 2. Multiples 272.18 (171.6) 0% -20.56 (91.36) 100% 216.56 (130.7) 8% -172.30 (230.3) 60% 3. Apartments 1110. (617.4) 22% Tenant Occupied Dwellings: 1. Singles -.25 (1.00) 64% .11 (.72) 58% -.47 (1.46) 68% -.15 (.82) 67% 2. Multiples 1.24 (.88) 54% 1.50* (.46) 28% .46 (.40) 0% 1.16 (.86) 17% 3. Apartments .26 (.23) 19% .72* (.32) 2% -.15 (.65) 80% .42 (.46) 9% * Statistically significant at the 5% level of significance. 74 The strongest results are for owner occupied single dwellings. The average prices for these dwelling units range from $311.81 in London to $441.28 in Hamilton and these prices are all statistically significant. The next strongest results are for tenant occupied multiple dwellings. The average prices for this category of dwellings are uniformly positive but only one of the four prices is statistically significant. Apart from the eight markets just mentioned, there remain thirteen markets; five of these have positive average prices and eight have negative average prices, but in only one market is the average price statistically significant, albeit in a market with a positive price. These results are not heartening and one wonders why these prices are so poorly defined, when those for the price per room perform so well. One reason which has been suggested8 is that newness as an object of preference is a subjective and hazily defined phenomenon. Some households prefer new homes in new subdivisions while others may prefer older, architecturally interesting homes in established neighbourhoods. One can argue then, that because preferences are so diverse, the estimated price of newness is destined to be fuzzy. This view certainly has some credibility. On the other hand, one cannot deny the priority the real estate profession places on a dwelling's newness when describing a dwelling for sale. This interest manifests itself in their newspaper advertisments and in their professional documentation, for example, the MLS fact sheets on listed dwellings. If the real estate profession is correct in its evaluation of newness as an important dwelling characteristic, the estimated price of newness should be positive and statistically significant. A second factor which may contribute to the poor results in the estimates of the price of newness is the imprecise manner in which "newness" was constructed. Newness is defined as the mid-date of a decennial time interval. Thus, this variable is constructed with some imprecision. The importance of this error in the variable is unknown. If the estimates had been obtained by ordinary least squares, one would conclude that, the estimates of the parameters are inconsistent, in which case, one might suggest reestimating the parameters using instrumental variables. This is not the case, however. The parameters are estimated using Cragg's technique which is itself an instrumental variables technique. Whether Cragg's technique adequately deals with the inconsistency caused by the error in newness, in addition to dealing with the problem of heteroscedasticity, is a topic I have not dealt with, nor has anyone else in the literature done so. 5.4.3 T h e Price per E x t r a A p a r t m e n t U n i t The price per room and the price of newness are the prices of principle interest in this thesis. In addition, the hedonic regressions for apartments yield estimates of the marginal price of an extra apartment in an apartment building. These estimates are given in Table XIII. The number of units in an apartment building is treated as an inverse measure of privacy and is expected to be negative. Given this expectation, the results for this price are surprising. Only two of the five markets have an average price per unit which is negative and both of these prices are statistically insignificant. 75 T A B L E XIII : T H E P R I C E P E R U N I T H a m i l t o n L o n d o n Toronto W i n d s o r A . Owner Occupied Apar tments : -148.4 (90.92) 65 % B . Tenant Occupied Apar tments : .07 .34 -.12 .32* (.06) 19% (.20) 2% (.14) 80% (.16) 9% * Statistically significant at the 5% level of significance. 76 Alternatively, the market for tenant occupied apartments in Windsor has a statistically significant positive average price. There are two possible reasons for these results. First, if it is the case that larger apartment buildings are situated closer to the city centre or other employment nucleii, dwelling units in these buildings will earn positive quasi rents because of their proximity to employment. These positive quasi rents may outweigh the benefits of privacy, rendering the marginal price per unit positive. This argument hinges, of course, on larger apartments being built closer to employment centers. This is a reasonable economic supposition. Land close to employment centres, particularly the central business district of a city, generally commands a higher price than land farther out, therefore, landlords close to such centres will economize on their land costs by constructing more dwellings per unit of land. The exact density of construction will depend on the marginal rate of substitution between time (for transport) and privacy and on the marginal rate of technical substitution between land and other materials. The second possible reason for a positive price per unit is that the number of units in a building may serve as a proxy for building amenities such as access to a swimming pool, sauna, or tennis court. These characteristics would have positive prices and their presence would counteract the loss of privacy. 5.5 OTHER DIAGNOSTICS This final section of the chapter is devoted to a critical assessment of some of the assumptions and limitations of the hedonic pricing model described in the preceeding sections. In particular, I consider three items. The first is the assumption that the housing markets are in short run rather than long run equilibrium. The second item concerns the problem of multicollinearity. This problem vexed a number of the hedonic regressions. Finally, I consider whether the omission of a location variable from the hedonic regressions is likely to cause a significant bias in the results. 5.5.1 Testing For L o n g R u n E q u i l i b r i u m Prices I have argued that the urban housing markets in this study are not in long run equilibrium. While this assumption is not crucial, its violation would mean that the estimated hedonic price equations are not fully efficient, since the estimation did not take into account the extra information conveyed by the existence of long run equilibrium. In view of this consequence the assumption that the housing markets are not in long run equilibrium should be tested. In order to do so, it is necessary to test whether the estimated hedonic equations are the same. Ordinarily, one would test whether the coefficients of the hedonic equations are equal. This is impossible here, as the hedonic regressions do not contain the same set of regressors. This variation 8 I would like to thank Professor Chris Archibald of the Department of Economics, The University of British Columbia, for this suggestion. 77 in the set of regressors exists because several of the regressors are constant in some samples but not in others. Specifically, the number of units and stories in the building containing the dwelling unit and the dummy variables for fair and good quality dwellings and dwellings with a private bath are constant in some of the housing markets. The number of units and the number of stories are constant in the samples of single dwellings by design and in the samples of multiple dwellings because of missing information. The dummy variables are constant because of the lack of sample variability in some samples. Therefore, neither the constant regressors nor their cross-products with the other regressors are present in these samples. As a consequence, to test for long run equilibrium, the various hedonic regressions have to be put on a common base. The base used in the tests presented below is that of a dwelling in a building of one unit, one story in height, in good condition, with a private bath. Having placed the hedonic regressions on a common base, Lagrange multiplier tests are con-structed which test whether: (1) the coefficients are the same in the hedonic price functions for dwelling newness, (2) the coefficents are the same in the hedonic price functions for the number of rooms, and (3) the coefficients are the same across both the hedonic price functions. The results are presented in Table XIV. The figures in parentheses are the degrees of freedom for the tests. The tests are straightforward for the renting equations and owning equations taken separately. It is more difficult to test for equality of the coefficients across the renting and owning equations, since the coefficients in the owning equation should be the capitalized values of the coefficients in the renting equations. This means that the null hypothesis states the coefficients in the owning equation are proportional to those in the renting equation. The factor of proportionality in the test (i.e., the capitalization rate) is unknown however, and should be estimated. This is done using a nonlinear variant of the Cragg estimator9 on pooled samples of owned and rented dwellings. The estimated annual capitalization rates are presented in panel IV of Table XIV. These rates are centered (approximately) at 3% per annum, with the exception of the rate for Toronto, which is 1.3%. All the rates are statistically significant. The chi-squared test values for the Lagrange multiplier tests are presented in panels I—III of Table XIV. The null hypothesis of these tests — that the housing markets are in long run equilibrium — is rejected in 33 of the 45 tests. Underneath the pure numerical superiority of rejected hypotheses lie a number of interesting results. To begin, it is reasonably clear that the hedonic price function for dwelling newness is different for different types of dwellings. This result holds both within and across the two tenure groups. It also holds for all the cities combined. The tests for the city of Hamilton are an exception to this result. For Hamilton, the null hypothesis of long run equilibrium prices cannot be rejected 9This is really an exercise in nonlinear instrumental variables estimation. The minimand is the weighted sum of squares in the metric IT' where T = F'Q{Q'S~1Q)~lQ'S"^. The matrixes F and Q are defined above; 5 — the covariance matrix of equation errors — is obtained from a nonlinear least squares estimation of the equations on the pooled sample. 78 T A B L E X I V : T E S T S F O R L O N G R U N E Q U I L I B R I U M P R I C E S H a m i l t o n L o n d o n Toronto W i n d s o r A l l Cities I. T h e Pr i ce O f Newness A . Across T y p e but W i t h i n Tenure: 1. Owner Occupied 3.225 26.788* 15.614* 34.511* 84.995* (3) (3) (6) (3) (24) 2. Tenant Occupied 7.815 736.234* 190.151* 49.901* 1390.692* (6) (6) (6) (6) (33) Across T y p e and Tenure: 3.343 44.550* 25.781* 35.667* 89.350* (12) (12) (15) (12) (58) II. T h e Pr ice of a R o o m Across T y p e but W i t h i n Tenure: 1. Owner Occupied 15.605* 23.603* 21.775* 39.289* 6.787 (3) (3) (6) (3) (24) 2. Tenant Occupied 1.431 11.406 33.965* 34.182* 5.474 (6) (6) (6) (6) (33) Across T y p e and Tenure: 16.492 43.920* 24.505* 46.830* 6.972 (12) (12) (15) (12) (58) III. Jo int Test of B o t h Prices Across T y p e and Tenure: 1. Owner Occupied 26.042* 79.182* 76.213* 68.068* 89.627* (5) (5) (10) (5) (40) 2. Tenant Occupied 8.544 1026.866* 202.162* 62.360* 1494.690* (10) (10) (10) (10) (55) Across T y p e and Tenure: 27.365 124.027* 55.975* 75.590* 94.120 (20) (20) (25) (20) (97) I V . A n n u a l Capi ta l izat ion Rates 2.836%* 3.193%* 1.317%* 2.905%* 3.633%* (0.577) (0.501) (0.623) (0.815) (0.353) * Statistically significant at the 5% level of significance. 79 for owner and tenant occupied dwellings taken separately or in the composite test across type and tenure. The tests for long run equilibrium in the price per room are largely rejected when one considers each city separately, but cannot be rejected when one considers the cities taken together. This result is disturbing. Normally, one would expect that the test on the cities taken together would reject the null hypothesis if the underlying within city tests rejected it, because the former test involves a greater number of restrictions than do the latter tests. This is not the case with these tests however, because the high correlations in the differences in the parameter vectors across cities greatly lower the discriminating power of the individual restrictions. Succinctly, the hedonic price functions for dwelling space are more similar across cities than across type and tenure, and this leads to the non-rejection of the hypothesis of long run equilibrium when city, type, and tenure differences are combined. Viewed in this way, I believe the test results indicate that the hedonic price functions for dwelling space are not in long run equilibrium. Again, the city of Hamilton is to some extent an exception. The test of long run equilibrium is rejected for owner occupied dwellings. It is not rejected for tenant occupied dwellings nor in the joint test across owner and tenant occupied dwellings. Finally, I consider the joint tests for long run equilibrium in the price of newness and the price per room given in panel III. Full long run equilibrium exists if these tests do not reject the null hypothesis of equality in the parameter vectors. This is not the case. Full long run equilibrium is rejected in all the tests for London, Toronto, and Windsor, but not for Hamilton. It is rejected for owner occupied dwellings in Hamilton and not rejected for tenant occupied dwellings or across owner and tenant occupied dwellings. When the four cities are combined, full long run equilibrium is rejected for owner and tenant occupied dwellings taken separately. On the other hand, the test for full long run equilibrium across all cities, all types, and both tenures cannot be rejected. Nevertheless, since the probability value for this test is approximately 7%, the "acceptance" of the null hypothesis in this case is far from resounding. In conclusion, I believe the bulk of the evidence presented in Table XIV goes against the hypothesis of long run equilibrium. These tests support my claim that the urban housing markets studied in this thesis are in short run, not long run equilibrium. As noted above, however, there are indications that some of the submarkets may be approaching long run equilibrium. Specifically, it appears the hedonic price structure for Hamilton is close to being in full long run equilibrium. In addition, the joint test for long run equilibrium in both prices across type and tenure cannot be rejected. Why Hamilton should exhibit long run equilibrium while the other cities do not is a question I am unable to answer. 80 5.5.2 M u l t i c o l l i n e a r i t y In a number of the hedonic regressions, the functional form (5.1) could not be estimated because of the singularity of the matrix Q'SQ in the CE estimator (5.5). Analysis of this matrix revealed that the submatrix F'SF of Q'SQ was either singular or ill-conditioned in all cases where the problem occured. (The matrix F'SF is the matrix of the weighted cross-products of the regressors.) In principle, no problem should have resulted as both F and S have full column rank. Thus, the problem is computational in nature. To circumvent this problem, I have followed the simple expedient of dropping the most collinear of the weighted regressors in S^F. The chief cause of the problem is the cross-products Zijdik between the continuously measured and discrete characteristics. These regressors are deleted from a number of the hedonic regressions. Unfortunately, this does not wholly eliminate the problem of multicollinearity. In two markets, owned multiples in London and rented multiples in Toronto, removing the cross-product regressors zijdik. was not sufficient to yield an invertable Q'SQ matrix. In these two cases, estimatable regressions are obtained by deleting the cross-products ZijZik of the continuous characteristics. This implies that marginal prices of the characteristics are constant in these latter markets. While the above procedure works in the sense that it results in estimatable models, it was suggested I may have merely reduced the collinearity among the regressors to the point where I could estimate and that further simplifying the estimating equations would improve the precision of the estimates without significantly affecting the explanatory power of the equations.10 I have investigated the possibility of simplifying the estimating equations. Two sets of hypothesis tests shed some light on this matter. These tests are: (1) tests of the null hypothesis of no joint costs and (2) tests of the null hypothesis of constant hedonic prices. The test for no joint costs involves the null hypotheses that the coefficients on the cross-product terms ZijZik, j ^ k, j,k = 1,..., M, and Zijdik, j = 1, • • •, M, k — 1,..., /V are zero. If this hypothesis cannot be rejected, the cross-products contribute little to the explanatory power of the regression and one might consider dropping them from the estimating equation. Table XV contains the chi-squared values for the tests of this hypothesis on the CE estimates of the hedonic price functions. The null hypothesis can be rejected in thirteen of the nineteen tests and cannot be rejected in six. One interesting feature of the tests is that all the failures to reject the null hypothesis occur in the regressions on samples of single dwellings. I am unclear why this should be the case. The test of constant hedonic prices is a test of the null hypothesis that all the cross-products ZijZik and z^dik a r e zero. Hence, this test involves all the restrictions in the test of no joint costs plus the restrictions that zt2- = 0 for all j. Table XVI contains the chi-squared test values for the tests of this hypothesis. The null hypothesis of constant hedonic prices can be rejected in all nineteen tests. l 0 I would like to thank Dr. John Cragg of the Department of Economics, The University of British Columbia, for first voicing this possibility. 81 T A B L E X V : T E S T F O R N O J O I N T C O S T S H a m i l t o n L o n d o n Toronto W i n d s o r A . Owner Occupied Dwel l ings: 1. Singles 9.4 7.3 10.4 33.6* (5) (5) (5) (5) 2. Multiples 11.3* n.a. 22.9* 13.7* (3) (3) (5) 3. Apartments 32.5* (6) Tenant Occupied Dwel l ings: 1. Singles 1.1 29.6* 2.1 19.8* (1) (5) (1) (7) 2. Multiples 10.2 25.2* n.a. 24.2* (7) (7) (1) 3. Apartments 250.6* 48.8* 32.3* 167.3* (12) (12) (14) (12) * Statistically significant at the 5% level of significance. n.a. Not applicable. T A B L E X V I : T E S T S F O R C O N S T A N T P R I C E S H a m i l t o n L o n d o n Toronto W i n d s o r A . Owner Occupied Dwel l ings: 1. Singles 25.5* 44.1* 54.0* 76.7* (7) (7) (7) (7) 2. Multiples 18.7* 27.5* 23.2* (5) (7) (7) 3. Apartments 50.1* (10) B . Tenant Occupied Dwell ings: 1. Singles 21.7* 46.2* 12.7* 46.4* (3) (7) (3) (9) 2. Multiples 25.6* 53.4* 28.8* (9) (9) (3) 3. Apartments 264.8* 80.7* 46.9* 243.8* (16) (16) (18) (16) * Statistically significant at the 5% level of significance. 82 These tests provide a qualified measure of support to the hypothesis that there are joint costs in the pricing of dwelling unit characteristics. They also show that a linear form is too simple a functional form for the hedonic price function. Thus, they indicate that simplifying the functional form of the estimating equations in the direction of either of the tests would injure the overall fit of the hedonic price function. Although these tests imply that functional simplification may not be desirable, I have re-estimated the hedonic equations by dropping the cross-products z,-;-2,jfc, j ^ fc, j , k = 1,...,M, and Zijdik, j = 1,...,M, k = l,...,iV, for all j and k. This was done to learn (1) whether the damage from simplifying the functional form is important (i.e., whether it markedly distorts the marginal prices of the housing characteristics) and (2) whether the damage is compensated for by increased precision resulting from reduced collinearity of the regressors. The hedonic prices computed from the re-estimated equations are presented in Tables XVII, XVIII, and XIX. The parameters from the re-estimated equations are given in Appendix D of this thesis. Two findings are revealed by comparing the hedonic prices from the hedonic equation (5.1), given in Tables XI, XII, and XIII, with the hedonic prices from the simplified form. First, the two sets of hedonic prices are not radically different. With one exception, all the hedonic prices in Tables XVII, XVIII, and XIX have the same sign as those in Tables XI, XII, and XIII and they are of approximately the same magnitude. A close scrutiny of the prices shows a tendency for the price per room in owner occupied dwellings to be lower in the simplified model than in the full model. At the same time, there is a tendency for the price of newness in owner occupied dwellings to be higher in the simplified model than in the full model. The one exception to the preceding trends is the hedonic price of newness of owner occupied apartments in Toronto. In the full model, this price is $1,110 per year and is statistically significant, while in the simplified model, the price drops to S — 178.56 per year and is not statistically significant. The second finding from the comparison of the hedonic prices from the full and simplified models is the considerable reduction in the standard errors of the average prices in the simplified model. The standard errors are smaller for all three prices in every market. Moreover, the decreases are large — 61% on average. The average decrease in the price per room and the price per year of newness are approximately the same, 40% and 42% respectively, while the decrease in the price per extra unit in an apartment building is 73%. The preceeding two findings appear to show that simplifying the functional form of the hedonic price equation does not greatly distort the marginal prices of the housing characteristics and that the distortions that do exist may be more than offset by the decrease in the standard errors in the prices. This observation conflicts with the hypothesis tests in Tables XV and XVI, which imply that a simple specification of the hedonic price equation would entail a serious specification error. When combined, the two sets of results indicate that, in estimating the hedonic prices, there is a trade-off between specification error with the simple model and sampling error with the flexible 83 form (5.1). Rather than err in either direction, I have estimated the second stage of my model (i.e., the demand and choice equations) using both sets of prices. This second stage price experiment yields an interesting result. The demand choice model given in Chapter 3 has a significantly lower log likelihood when the simpler hedonic price equation is used than when the more complex hedonic price equation is used. This result will be examined in more detail in the next chapter. The lower log likelihood with the simpler hedonic equations does point to a problem with the two-stage methodology used in this thesis, which should be noted at this point. The problem is that errors in the estimates of the hedonic prices (independent from their source) become measurement errors in the variables in the second stage. Yet these second stage measurement errors are not taken into account in estimation.11 While I acknowledge the problem, I do not deal with it. No other hedonic price demand study has been concerned with this problem (Indeed, as far as I know, no one else has even raised the issue.) and to deal with it here would be well beyond the scope of this thesis.12 5.5.3 Locat ion — A n Omit ted Variable Urban economic theory provides minimal guidance as to whether a variable should be included in a hedonic regression. As a result, a wide array of housing characteristics has been included in other hedonic price studies, although the characteristics used in any one study are usually dictated by the peculiarities of the data set used. The present study uses comparatively few characteristics to explain the variation in housing prices. The possibility of omitted variable bias is therefore a concern. Of particular concern is the omission of a location variable. A number of studies have indicated dwelling unit attributes are systematically related to dwelling location. Straszheim's (1975) analysis of the San Francisco housing market is of particular note. He examines the spatial variation in the price per room and the price per year of dwelling age — the two prices used in this study. In San Francisco, both prices decrease with distance from the city centre, with the price gradient for age being much steeper than the price gradient for rooms. These findings suggest location variables should have been included in the hedonic regressions. The SHU data set contains two measures of dwelling unit location: the distance to work by the household head and the travel time to work by the household head. Unfortunately, neither variable can be endorsed for inclusion in the hedonic regressions since they do not actually measure the location of the dwelling unit. Rather, they give the position of the dwelling unit relative to I I My estimation of first order conditions rather than the demand equations may partially alleviate the measurement error problem. The hedonic prices are the dependent variables in the first order conditions, therefore the mea-surement errors will be incorporated into the equation disturbance terms. Unfortunately, the hedonic prices also are used to construct the income variable and to construct the Jacobian matrixes, G m , in the likelihood function. Hence, I have not entirely escaped from the measurement error problem. 1 2 The measurement error problem in hedonic demand models is a research problem I am currently examining in post-dissertation research. 84 T A B L E XVII: T H E PRICE PER R O O M Hamilton London Toronto Windsor A. Owner Occupied Dwellings: 1. Singles 4860.84* (363.80) 0% 4999.60* (302.01) 0% 6109.06* (417.85) 0% 4529.07* (282.65) 0% 2. Multiples 3595.27* (935.96) 2 3 % 2148.00 (989.10) 0% 2786.92* (658.81) 3% 3384.61* (492.25) 0% 3. Apartments 5409.30* (1037.21) 0% B. Tenant Occupied Dwellings: 1. Singles 13.31* (2.60) 2% 20.76* (3.12) 0% 30.60* (6.85) 0% 17.61* (2.68) 0% 2. Multiples 14.63* (2.39) 0% 21.55* (2.28) 0% 31.60* (7.41) 0% 22.13* (2.45) 0% 3. Apartments 17.11* (0.94) 0% 16.67* (1.08) 0% 22.23* (1.57) 0% 19.59* (1.08) 0% * Statistically significant at the 5% level of significance. 85 T A B L E XVIII: T H E P R I C E O F N E W N E S S Hamilton London Toronto Windsor Owner Occupied Dwellings: 1. Singles 460.32* 313.50* 386.82* 346.25* (24.21) (21.59) (34.90) (17.86) 0% 28% 30% 0% 2. Multiples 211.23* -20.27 243.90* -239.51 (92.53) (91.36) (29.96) (136.87) 0% 100% 0% 73% 3. Apartments -178.56 (250.12) 100% B . Tenant Occupied Dwellings: 1. Singles -0.24 0.13 -0.31 -0.14 (0.36) (0.24) (0.95) (0.40) 64% 56% 68% 67% 2. Multiples 1.27* 1.42* 0.46 1.24* (0.19) (0.20) (0.40) (0.47) 54% 28% 0% 22% 3. Apartments 0.37* 0.70* -0.11 0.69* (0.12) (0.12) (0.25) (0.11) 0% 18% 100% 0% Statistically significant at the 5% level of significance. T A B L E X I X : T H E P R I C E P E R A P A R T M E N T U N I T Hamilton London Toronto Windsor A . Owner Occupied Dwellings -30.44 (18.00) 92% B . Tenant Occupied Dwellings 0.05' 22.14* -0.04 0.37* (0.02) (0.03) (0.05) (0.05) 14% 3% 9 3 % 5% * Statistically significant at the 5% level of significance. 86 an arbitrary workplace. The measures also involve a degree of sample selectivity, as the measures cannot be applied to dwelling units where the household head is unemployed or does not have a fixed employment location. Moreover, both measures are missing from a substantial number of cases in the data set. The problems of sample selectivity and missing data result in the exclusion of over one-quarter of the dwellings in the sample. This loss of observations has an even more injurious effect on the individual markets. The loss of data eliminates the sample of owner occupied apartments and leaves seven of the remaining twenty samples with fewer than one hundred observations. I believe these problems are significant enough on their own to warrant excluding these variables from the regressions. Nevertheless, I have re-estimated hedonic regressions using the distance-to-work variable in order to learn whether these work-related location variables have a significant impact on the price of a dwelling unit. A set of likelihood ratio statistics is used to test whether the inclusion of the location variables has a statistically significant impact on the performance of the hedonic regressions. The test statistics are % 2 random variables with the number of degrees of freedom equal to the number of distance-to-work related terms in the first regression. The test values and the degrees of freedom for each test are recorded in Table XX. The distance-to-work variable has a statistically insignificant effect in sixteen of the twenty tests and is significant in four tests. Thus, on balance, the statistical power of the hedonic regressions is not improved by including this variable. Because of the problems with the two location variables included in the SHU and because of the poor test results for the distance-to-work variable given in Table XX, neither of the location variables is included in the hedonic regressions. Of course, the possibility of omitted variable bias remains, but I am unable to deal with this possibility due to a lack of data. T A B L E X X : D I S T A N C E - T O - W O R K L I K E L I H O O D R A T I O T E S T S H a m i l t o n L o n d o n Toronto W i n d s o r A . O w n e r Occupied Dwell ings: 1. Singles 2. Multiples 12.0 (5) 18.2 (5)' 6.0 (5) 0.0 (1) 4.0 (5) 10.0 (5) 32.0 (5)* 15.0 (5)' B . Tenant Occupied Dwell ings: 1. Singles 2. Multiples 3. Apartments 1.6 (4) 5.8 (4) 4.0 (6) 11.6 (5) 8.0 (7) 12.0 (6) 4.8 (4) 4.6 (4) 6.0 (6) 82.8 (6)* 2.8 (4) 4.0 (6) * Statistically significant at the 5% level of significance. 5.6 CONCLUDING COMMENTS In this chapter, I have concentrated largely on the mechanics of estimating efficient and robust hedonic price equations as well as discussing my estimation results which use these equations. 87 Since the demand model in the next chapter is based on the estimated hedonic price equations, it is imperative that good estimates be obtained. In this chapter, three general results emerge concerning the specification of hedonic regression equations for urban housing markets. First, the evidence suggests strongly that urban housing markets are not in long run equilibrium. Therefore, separate hedonic price equations should be fitted to each well defined housing submarket. Second, the White's tests for heteroscedasticity presented in Section 5.3 and Appendix D in-dicate my hedonic price equations are fraught with a high degree of heteroscedasticity. The het-eroscedasticity may exist because I was forced to estimate city-wide regresssions. Alternatively, one cannot rule out the possibility that heteroscedasticity is an endemic feature of hedonic equations fitted to urban housing data. Certainly, this has been suspected for some time. If this is the case, hedonic regressions for urban housing markets should always be corrected for heteroscedasticity, or the parameter estimates will be inefficient and their standard errors inconsistent. Third, my results suggest that a simple functional form for the hedonic price equation may perform as well as a more complex functional form, in terms of estimating the marginal prices of housing characteristics. Moreover, the more complex forms may result in multicolinearity, because of the large number of discrete characteristics commonly used to describe dwellings. I began this chapter by stating that little is known about the correct functional form for a hedonic price equation. I believe I have advanced our knowledge about these equations by showing the problems inherent in a more complex functional form. It is clear that research is needed on the theoretical and empirical underpinnings of hedonic equations, if further advances are to be made. Before continuing on to the next chapter, I would like to make some comments about the omitted location variable problem. It is clear to anyone who has gone house/apartment shopping that location plays an important role in determining housing prices. The question is how to model these locational effects. It is argued that in urban markets, distance to work is the paramount factor in determining housing prices. As a consequence, many empirical studies include a variable measuring the distance from the dwelling to the central business district, or if more sophisticated, a weighted average of the distances from the dwelling to various employment foci. It was this set of variables that I attempted to include in Section 5.5.3, and which I ultimately rejected. While the distance to work is an important locational consideration, it may not be the penul-timate factor urban economists believe it to be. At least equally important, in my opinion, is the desire of households to live in a "good neighbourhood". For example, there is the perceived difference between the east side and west side of Vancouver, British Columbia, or, for that matter, the east side and west side of Saskatoon, Saskatchewan. Without a doubt, similar distinctions exist in the cities in my sample. These neighbourhood effects will increase or decrease the price of a dwelling. What is at issue here, however, is whether omitting neighbourhood variables biases the estimates of the hedonic prices. The hedonic price literature provides no evidence to support or refute this. Studies which have included neighbourhood location variables do so by including neighbourhood dummy variables and hence, pick up only the increase or decrease in the level of dwelling prices. I suggest neighbourhood location effects, which are basically ideosyncratic, merely introduce a mild degree of randomness to the parameters of my hedonic equations and do not bias the estimates. 88 C h a p t e r 6 D E M A N D M O D E L E S T I M A T I O N 6.1 INTRODUCTION In this chapter, I discuss the estimation results for the demand model described in Chapter 3. To highlight the central feature of the model — that households may engage in multiperiod dwelling occupancy — I have estimated two forms of the model. The first form of the model is based on two assumptions: (1) households are single period utility maximizers and (2) households disregard transaction costs when choosing a dwelling.1 This form of the model is referred to as Model 1. The second form of the model is the full intertemporal model with transaction costs as described in Chapter 3. I refer to this model as Model 2. The results from these two models are compared to determine whether the theoretically consistent, intertemporal approach to housing demand advo-cated in this thesis produces a better estimated model than the conventional single period housing demand model. Both models have been estimated using Fletcher's algorithm (1970, 1972), with analytic first derivatives used in the calculations. Occassionally, Fletcher's algorithm would stall in a region of parameter space. In these cases, the Gauss-Newton algorithm was used to escape to a well-behaved region of parameter space and the Fletcher algorithm restarted. (The Gauss-Newton algorithm could not be used throughout to estimate the models because it was too computationally expensive.) The models were difficult to estimate. Initial attempts to estimate Model 1 yielded a nonsingular parameter covariance matrix and the parameter estimates themselves were nonsense. This problem was distressing as it indicated the parameters in the model were not identified. Numerical analysis of the parameter covariance matrix showed that the demographic scaling parameters, cu and C12, were approximately colinear with the utility parameters, aoi, 021, and 031, and this colinearity was causing the problem (all of these parameters are associated with other consumption). Since, in principle, these parameters are identified, the close relationship among the five parameters had to be computational in nature. I have circumvented the problem by setting cu and C12 to zero in the estimations. This sets the demographic scale for other consumption equal to unity. 'These are the standard assumptions in the empirical literature on housing demand. 89 My computational experience also showed that the performance of the optimization algorithm was sensitive to the scaling of the parameters and that one initial scaling was not sufficient for all of the regions of parameter space along the trajectory of optimization. Poor scaling manifested itself in a computationally ill-conditioned Hessian matrix. This problem was resolved by periodically stopping the algorithm and rescaling the parameters and Hessian. The most effective scales were the absolute values of the partial derivatives of the log likelihood function with respect to the parameters. That is, the algorithm worked best when the elements of the gradient were close to ±1. 2 There was one further problem encountered in maximizing the log likelihood functions. The log likelihood functions for the two models contained a number of local maxima and saddlepoints. Because of these, I adopted the following approach to convergence. First, whenever apparent convergence was achieved, the estimation was restarted using the Gauss-Newton algorithm, since this algorithm was particularly effective in moving off saddlepoints and in moving away from a "flat'' spot in parameter space. When the Gauss-Newton algorithm also indicated convergence, a variety of starting values were tried in order to detect a local maximum. The models are said to have converged only after both of these procedures failed to yield different parameter estimates. 6.2 P A R A M E T E R ESTIMATES At this point I turn to the discussion of the parameter estimates obtained for the two models. These estimates are presented in Table XXI. For convenience, I have divided the parameters in the table into five sets: utility function parameters, demographic scaling parameters, transaction costs constants, optimal length of occupancy parameters, and generalized extreme value (GEV) distribution parameters. Each set of parameters is discussed in a subsection. Since almost all the parameters are statisti-cally significant, I mention statistical significance only when the null hypothesis that the parameter is zero cannot be rejected. 6.2.1 Utility Function Parameters The first nine parameters are the parameters of the continuous part of the household utility function. It is difficult to interpret these parameters individually as they are part of a quadratic approximation to the true utility function. One way to describe the estimated household utility function is in terms of the elasticities of substitution between z\, z\, and x. The Hicks-Allen elasticities of substitution for Models 1 and 2 are presented in Table XXII. In this table, separate sets of elasticities are given for young, middle-aged, and old households. The necessity for lifecycle categories of households arises because the demographic scaling of the household utility function makes the elasticities of J G i l l et.al. (1981) provides an excellent discussion on the effects of poor scaling in numerical optimization, as well as a discussion on scaling techniques. 90 T A B L E XXI: P A R A M E T E R E S T I M A T E S Mod el 1 Model 2 Parameter Coefficient Std. Error Coefficient Std. Error 168.67967 0.72796 166.30538 0.65938 O-02 363.75807 6.98280 369.62200 4.16490 -287.68552 5.51613 -315.27640 7.67812 a u -2.28365 0.01010 -2.23166 0.00987 a 2i -1.84519 0.03549 -1.84688 0.01965 031 0.95629 0.02053 0.91584 0.06029 a22 10.00848 0.77302 9.83205 0.57625 <*32 -97.15427 2.97252 -103.8660 1.82185 «33 209.73470 6.23115 243.90350 3.68233 611 0.0 norm. 0.0 norm. 612 0.02180 0.00891 0.00301 0.00178 613 0.07838 0.02937 0.00711 0.00421 621 0.0 norm. 0.0 norm. 622 -9.99071 5.29935 -6.42738 0.59272 cu 0.0 fixed 0.0 fixed C12 0.0 fixed 0.0 fixed C21 0.22829 0.01123 0.23900 0.00983 C22 -0.60244 0.01167 -0.60305 0.00649 C31 0.35476 0.01102 0.23459 0.01513 C32 0.076660 0.01093 0.08517 0.00839 C41 -0.36215 0.09978 -0.02012 0.14449 C42 1.14859 0.08907 1.11675 0.12913 C51 -0.05939 0.16713 -0.27756 0.07689 C52 -0.13752 0.21363 -0.46543 0.07712 T 1 -4.25189 6.15460 T 2 -22.86649 5.81692 0.99781 0.00338 '12 3.37214 0.00457 '13 1.34608 0.00820 Oi 0.02 norm. 0.02 norm. 02 0.02570 0.14835 0.02039 0.01350 Oz 0.00450 0.01217 0.00358 0.00123 04 46.37084 286.97491 0.68363 0.29716 m 0.03089 0.02039 0.36364 0.87505 m 0.99901 0.28240 0.99902 0.31911 0.33183 0.10893 0.04216 0.01423 Lc -33221.009 -27845.673 Ld -3536.692 -3506.845 L -36757.701 -31352.517 91 T A B L E X X I I : H I C K S - A L L E N E L A S T I C I T I E S O F S U B S T I T U T I O N Model 1 Model 2 Young Middle-Aged Old Young Middle-Aged Old Elasticity Household Household Household Household Household Household 2.105 1.764 3.590 1.057 0.988 1.017 0~z1z -0.090 -0.103 -0.080 -0.004 -0.005 -0.003 °~z2x -0.470 -0.477 -0.966 -0.007 -0.007 -0.009 substitution dependent upon the age of the household head and the size of the household. I have chosen to represent this dependence in terms of lifecycle differences in age and size. The ages and sizes of my representative young, middle-aged, and old households are given in Table XXIII. A young household has a head of age 30 and a size of three; a middle-aged household has a head of age 50 and a size of five; and an old household has a head of age 70 and a size of two. The ages for the household heads were picked arbitrarily. The household sizes are the average sizes of households with heads aged 30, 50, and 70, respectively. T A B L E XXIII: R E P R E S E N T A T I V E H O U S E H O L D S Age of Household Head Size Young Household Middle-Aged Household Old Household 30 3 50 5 70 2 The elasticities of substitution for Models 1 and 2 basically yield the same qualitative results. The elasticities of substitution between dwelling newness and dwelling space, °~z*zi» are positive for the three categories of households and are (approximately) unity or above. That is, an increase in the relative price of newness to space leads to a substitution of dwelling space for newness and, since o~zi zi is greater than one, it also leads to increased consumer expenditure on dwelling space relative to expenditure on dwelling newness. The elasticities of substitution between dwelling newness and other consumption, czix, and dwelling space and other consumption, o~zix, are all negative. Hence, both dwelling characteristics are net compliments with other consumption. Moreover, the elasticity of substitution between dwelling newness and other consumption for Model 1 is so small that household preferences with respect to these commodities are "Leontief like". Similarly, in Model 2, both the elasticity of substi-tution between dwelling newness and other consumption and the elasticity of substitution between dwelling space and other consumption are very small, again indicating Leontief like preferences between the housing characteristics and other consumption. While Models 1 and 2 share the same qualitative results, in quantitative terms the models are different. The quantitative differences between the models is clarified in Figures 2, 3, and 4, which depict the values of the three elasticities, o~z\z\-> az\xi an(^ az\z-> f°r t n e three lifecycle categories of households for Models 1 and 2. There are two main differences in the elasticities of substitution 92 Figure 2 Newness and Dwelling Space c o 10 in (0 a Young Legend • Model 1 • Model 2 Middle-Aged Lifecycle Cohort Old Figure 3 Dwelling Newness and Other Consumption c o 0-00-. « -0.05--O — -0.10-Young Legend • Model 1 • Model 2 Middle-Aged Lifecycle Cohort Old Figure 4 Dwelling Space and Other Consumption o 3 3 Legend • Model 1 • Model 2 Young Middle-Aged Old Lifecycle Cohort 9 3 for Models 1 and 2. First, all the elasticities of substitution from Model 2 are smaller in absolute value than the analagous elasticities in Model 1. This is shown in Figures 2, 3, and 4 by a shift toward zero by the lifecycle profiles of the elasticities for Model 2 as compared to the profiles for Model 1. Assuming that Model 2, the more general model, is correct, Model 1 overstates the magnitude of the elasticities of substitution. This overstatement is most evident in the elasticities of substitution between dwelling newness and other consumption and dwelling space and other consumption; in both cases, the elasticities for Model 2 have almost vanished. Second, Models 1 and 2 have differently shaped lifecycle profiles. The elasticities of substitution for Model 1 exhibit distinctly bowed profiles across the lifecycle cohorts. The elasticities of.substi-tution between dwelling newness and space and between dwelling newness and other consumption have U-shaped lifecycle profiles. Early in the lifecycle, households are comparatively responsive to changes in relative prices. This responsiveness decreases as the household moves to middle-age, but re-emerges as the household becomes old. The lifecycle profile for the elasticities of substitution between dwelling space and other consumption bends the other way, into an inverted U-shaped profile. In contrast, the elasticities of substitution for Model 2 are virtually constant across the lifecycle. Figures 2, 3, and 4 clearly illustrate these patterns. Since the demographic scaling parameters do not change appreciably between Models 1 and 2, it must be the differences in the utility function parameters which effect the changes in the elasticities of substitution. In other words, it appears the Model 1 estimate of the utility function produces bogus lifecycle effects in the elasticities of substitution. I believe the quantitatiave differences between Models 1 and 2 reflect the different optimal lengths of occupancy for the households in the sample. In Model 1, the length of occupancy is restricted to one year for all households. As a result, the housing consumption changes of long-term occupiers are put on the same basis as year-by-year occupiers. Long-term occupiers probably are sticky in their housing adjustments, making infrequent but large changes in their consumption of housing, while short-term occupiers probably make frequent but smaller changes. By lumping these two groups of households together and forcing year-by-year housing adjustments on all the households, Model 1 obtains inflated estimates of the elasticities of substitution (in absolute values) in order to accomodate the larger housing adjustments of long-term occupiers. The U-shaped lifecycle pattern in the elasticities exhibited by Model 1 may also result from the restriction on the lengths of occupancy in this model. Young and old households are generally more mobile than middle-aged ones, since the middle-aged households are larger (see Table XXIII) and are limited in their mobility by the presence of children. Because of this, the length of occupancy is, in part, a lifecycle phenomenon. Thus, the lifecycle pattern in the elasticities of substitution for Model 1 may occur because of the correlation between a household's lifecycle position and its optimal length of occupancy. 94 There is one problem with the above results. In a three good utility function which satisfies the usual convexity properties, either all three Hicks-Allen elasticities of substitution are positive or one of the elasticities is negative and the other two positive (R.G.P. Allen 1937:502). Clearly, this is not the case with the elasticities here. Therefore, the convexity principles are being violated. I examine the issue of convexity of the utility functions and the second order properties of the model in Section 6.5. The last five utility function parameters to be examined are the parameters of the discrete part of the household utility function. They reflect a household's type and tenure preferences. It should be recalled that the parameters bu and 6 2 1 have been set to zero, and therefore, a household's preferences are measured relative to owned, single dwellings. A household's preference for multiple and apartment dwellings is given by the parameters 6 1 2 and 6 1 3 respectively. These parameters are positive in Model 1. Thus, according to Model 1, house-holds prefer multiple and apartment dwellings — a result that runs strongly against the common presumption that Canadians have a preference for single dwellings. The estimated parameters for Model 2 are also positive, but they are much smaller than those for Model 1 and they are statisti-cally insignificant. Hence, Model 2 agrees with the commonly held view that Canadian households have no statistically distinguishable preference for any dwelling type. The parameter 622 gives a household's preference for rented dwellings as opposed to owned dwellings. The estimated coefficient is negative, large (in absolute value), and statistically signif-icant in both models. This finding shows that households have a distinct preference for owning their own dwelling — a "pride of ownership". Both the type and tenure preference results have an important implication for Canadian housing markets. They indicate that the pre-eminence of the owned single family dwelling in Canada has less to do with the "singleness" of these dwellings than is generally assumed. I believe it may have something to do with establishing legal title. If this is the case, however, one might expect a large market in Canada for owned multiple dwellings and strata-titled apartment dwellings. It is somewhat surprising then, that there are few of these dwellings in the 1974 SHU survey. Owned multiples and apartments comprise just 13% of owned dwellings and 6% of all dwellings. It may be that household preferences in 1974 had shifted over time toward owned dwellings of all types but the stock of owned multiple and apartment dwellings has lagged behind this change in preference. In this situation, the excess demand pressure in these markets should force up the prices of the characteristics for these markets. Such is not the case for owned multiples, however. As noted in Chapter 5, owned multiples are a definite bargain among owned dwellings because of their low characteristic prices. On the other hand, owned apartments in Toronto (the only city with owned apartments) do have higher prices for their characteristics. Thus, the evidence is inconclusive as to whether there is excess demand in these markets. The question of why the markets for owned multiple and apartment dwellings are so thin cannot be answered with the present model and the available evidence. Further investigation using time series data is required. 95 6.2.2 Demographic Scaling Parameters The next set of parameters are the demographic scaling parameters. The first four parameters give the effects of the age of the household head and household size on the effective amounts of dwelling newness and dwelling space, respectively. The latter quantities are the arguments of the continuous part of the household utility function. The coefficients c2\ and c22 are the parameters of the demographic scaling function for dwelling newness. These coefficients are similar for Models 1 and 2. They reveal that effective dwelling newness decreases with the age of the household head and increases with the size of the household. The coefficients C31 and C32 are the parameters of the demographic scaling function for dwelling space. They are also roughly the same for Models 1 and 2. Both coefficients are positive; therefore the effective amount of dwelling space decreases with both the age of the household head and the size of the household. To understand more fully the role played by the demographic variables, I have examined their impact on the effective quantities of newness and space across the household lifecycle. The effective quantities of newness and space for young, middle-aged, and old households are presented in Table XXIV and the lifecycle profile for these quantities are illustrated in Figure 5. This table and figure give the demographically scaled value of one unit of dwelling newness or dwelling space. T A B L E X X I V : E F F E C T O F D E M O G R A P H I C V A R I A B L E S O N D W E L L I N G N E W -N E S S A N D S P A C E A C R O S S T H E L I F E C Y C L E Cohort Demographic Scales Model 1 Model 2 Dwelling Dwelling Newness Space Dwelling Dwelling Newness Space Young Households Middle-aged Households Old Households 0.89 0.28 1.08 0.22 0.58 0.21 0.86 0.41 1.04 0.35 0.55 0.35 The effective quantities of dwelling newness have an inverted U-shaped profile across the lifecycle for both Model 1 and Model 2. This profile arises from the conflicting effects of the demographic variables in the following way. Early in the lifecycle, both the age of the head and the size of the household are increasing. The positive effect of increases in the size of the household dominates the negative effect of increases in the age of the head, however, producing an increase in the effective quantity of dwelling newness. Later in the lifecycle, the age of the head continues to increase but the size of the household decreases. Both of these changes operate to produce a decrease in the effective quantity of dwelling newness. Because of the shape of the lifecycle profiles, if a household wished to maintain a constant level of effective dwelling newness over its lifecycle, it would have to occupy increasingly older dwellings until about age 40, and then increasingly newer dwellings for the remainder of its economic life. The effective amount of dwelling space decreases monotonically over the household lifecycle in both Models. The most rapid decrease occurs early in the life of the household. During this part of 96 Figure 5 Demographic Sales for Dwelling Newness and Dwelling Space •o •a Young Middle-Aged Old Lifecycle Cohort Legend • Model 1: Newness • Model 1: Space • Model 2: Newness • • • • • • • • m m m O Model 2: Space Figure 6 Demographic Sales for Dwelling Type and Dwelling Tenure Legend • Modell^Tyoe^ • Model 1: Tenure • Model 2: Type O Model 2: Tenure Y o u n g M i d d l e - A g e d L i f e c y c l e C o h o r t Old 97 the lifecycle, increases in the age of the household head and the size of the household both decrease effective dwelling space. Later on, when the age of the head is increasing but household size is decreasing, the demographic effects oppose each other. The effect of the age of the household head dominates, however, and effective dwelling space decreases, albeit at a slower rate than before. This means that in order to maintain a constant level of effective dwelling space across its lifecycle, a household would have to occupy increasingly larger dwellings. It should be noted that while Models 1 and 2 share the same lifecycle profile for effective dwelling space, they differ in level. The effective quantity of dwelling space for Model 2 is thirty-six percent higher, on average, than the effective quantity of dwelling space for Model 1. Figure 5 illustrates this clearly. I turn now to the demographic scaling coefficients for dwelling type and tenure. The coefficients C41 and C42 give the effects of the demographic variables on dwelling type. The same demographic scale is applied to all types of dwellings. Thus, the effect of increasing the demographic scale is to deflate or inflate a household's preference for multiple and apartment dwellings (relative to single dwellings), depending on whether the scale is greater or less than unity. The coefficients C51 and C52 give the effects of the demographic variables on dwelling tenure. Again, the same demographic scale is applied to both tenures of dwellings, and therefore, the demographic scale for dwelling tenure deflates or inflates a household's preference for owned dwellings (relative to rented dwellings). The demographic scaling parameters for dwelling type and tenure for Models 1 and 2 yield the same qualitative results, but there are considerable differences in the quantitative magnitudes of the coefficients between the two models. In Model 1, the scaling parameters for dwelling type, C41 and C42, are —.362 and 1.149, while in Model 2 they are —.020 and 1.117. These two sets of coefficients indicate that a household's relative preferences for multiple and apartment dwellings increase with the age of the household head and decrease with household size. Moreover, since the coefficient C42 is much larger than the coefficient C41, the household size effect will predominate and the total demographic effect will follow household size. Like dwelling newness and dwelling space, the effect of the demographic variables on dwelling type is most clearly illustrated in terms of the household lifecycle. Table XXV presents the effective quantities of dwelling type across the lifecycle for Models 1 and 2. These quantities are illustrated T A B L E X X V : E F F E C T O F D E M O G R A P H I C V A R I A B L E S O N D W E L L I N G T Y P E Demographic Scales Mod el 1 Mod el 2 Dwelling Dwelling Dwelling Dwelling Cohort Type Tenure Type Tenure Young Households 0.97 1.42 0.31 4.29 Middle-aged Households 0.65 1.57 0.18 6.26 Old Households 2.10 1.42 0.50 4.49 98 in Figure 6. An examination of the figure indicates that small, young households have a relative preference for multiple and and apartment dwellings, as do small, old households. Bigger, middle-aged households, however, shift their preference toward single dwellings. This lifecycle shift in preferences is more noticable in Model 1 than Model 2. The lifecycle profile for Model 1 is more bowed because the coefficient for the age of the household head is larger (in absolute value) in Model 1 than in Model 2. Thus, as a household moves from middle-age to old age, there is a greater shift in its relative preferences toward multiple and apartment dwellings. The larger lifecycle effect in Model 1 may be spurious, however. In connection with the elasticities of substitution, I argued above that the age of the household head may be correlated with a household's optimal length of occupancy, and that in Model 1, this correlation may upwardly bias the effect of the age of the head. This may be the case here. If so, the large upward bend in the lifecycle profile of effective dwelling type in Model 1 is illusory. The demographic scaling parameters for dwelling tenure, C 5 1 and C 5 2 , are -.059 and -.138 for Model 1 and -.278 and -.465 for Model 2. This means the relative preference for owned dwellings increases with the age of the household head and with household size, but the household effect is larger since C52 is approximately twice the size of C 5 1 . Thus, the effective preference for owned dwellings is similar to the lifecycle profile for dwelling size and has an inverted U-shaped profile across the household lifecycle. Small, young households have a weak relative preference for owned dwellings, large, middle-aged households have a strong preference for owned dwellings, and small, old households have a weak relative preference for owned dwellings. This is illustrated in Table XXV and Figure 6. The difference between Model 1 and Model 2 occurs in the absolute magnitudes of the two coefficients. The coefficients for Model 2 are approximately four times the size of the coefficients in Model 1, leading to a much higher effective preference for owned dwellings in Model 2. The size of the Model 2 coefficients also make the effective preference for owned dwellings more volatile with respect to changes in the demographic variables. Therefore, effective dwelling tenure will react more strongly to lifecycle changes in household size and the lifecycle profile of dwelling tenure has a sharper bow to it. These effects are readily apparent in Figure 6. 6.2.3 Transact ion Costs Constants I turn now to the five parameters specific to Model 2. The first two of these are the constants in the transaction cost functions for owners and renters — T 1 and T 2, respectively. Both of the estimated constants are negative and statistically significant. Their negative values indicate an overstatement of transaction costs by the other parts of the transaction cost functions. In the case of owners, this overstatement is trivial. It is only ^ to ^  of 1% of the total transaction cost. For renters, the overstatement is significant. In constructing the transaction cost function for renters, I assumed that renters pay a damage deposit equal to | of one month's rent or approximately $50. Thus, T 2 implies an overstatement of approximately one half of the damage deposit. This implication is misleading, however. In Ontario, renters get their damage deposit back with interest when they move out, so the discounted value of their transaction costs is approximately zero. As a result, a negative value for T 2 has little to do with transaction costs. It is my opinion that T 2, and possibly T 1, are acting as "curve fitting" parameters in addition to being transaction costs constants. 99 6.2.4 O p t i m a l Length of Occupancy Parameters The last three parameters specific to Model 2 are the estimated lengths of occupancy for households with high, medium, and low subjective probabilities of moving within the next three years. The households with a high probability of moving plan to stay in their dwelling one year; the households with a medium probability of moving plan to stay in their dwelling 3.4 years, and the households with a low probability of moving plan to stay in their dwelling only 1.3 years. In Chapter 3, I speculated that the estimated lengths of occupancy would be ordered as t{j < t\2 < t\3, where the second subscript denotes the probability of moving within the next three years. Obviously, the estimated coefficients do not follow this scheme. The anomoly is that households which state they probably or definitely will not move have an optimal length of occupancy not much greater than households which state that they probably or definitely will move. One possible explanation is that long-term occupiers may regard themselves as being fixed or rationed in their quantities of housing services, even though they themselves pick their length of occupancy. If this is the case, the estimated length of occupancy will reflect the length of the household's planning period for the nonhousing commodities embodied in the aggregate commodity, other consumption. 6.2.5 Generalized Extreme Value Dis tr ibut ion Parameters The seven remaining coefficients in Table XXI are the elasticities of the generalized extreme value distribution. There are two sets of parameters: = 1,2,3,4, and r/,,t = 2,3,4. The parameters 0{ measure the importance of the choice subset D{ in determining a household's choice probabilities. (A copy of the table of alternative choice sets presented in Chapter 3 is recreated here.) As noted Index Acronym Index Acronym Type Tenure 1 CSO 7 NSO Single Owned 2 CSR 8 NSR Single Rented 3 CMO 9 NMO Multiple Owned 4 CMR 10 NMR Multiple Rented 5 CAO 11 NAO Apartment Owned 6 CAR 12 NAR Apartment Rented in Chapter 3, these parameters must be normalized by a transformation which is not homogeneous of degree one in order to identify the parameters. I have normalized them by setting 6\ = .02. Therefore, the free parameters Qi, i = 2, 3, 4, give the importance of the choice subsets D,, j = 2, 3, 4, in determining the choice probabilities relative to the choice subset D\. A value for 0, greater than .02 means that the alternatives in D{ are of greater relative importance than the alternatives in D\, while a value 0, less than .02 means the opposite. Since D\ contains a single alternative — the household's current dwelling — the alternatives in the other choice sets are weighted relative to a household's current dwelling. 100 While the parameters 0, are used to compare choice subsets, the parameters 77,, 1 = 2,3,4, measure the similarity of the alternatives within a choice subset. When rjj is close to zero, the alternatives in the subset are very similar; at the other extreme, when r?,- is close to unity, the alternatives in the subset D{ are distinct or independent. One problem in estimating the 77 coefficients is that if an 77, gets too close to zero, the choice probabilities involve exponentials with exponents ^ , which may be too large to be represented by the computer. To avoid this problem, all the 77s were bounded below by .02. The 77s were also constrained to be less than unity, as required by the GEV distribution. The estimated 9 coefficients for Model 1 and Model 2 are similar. They place the greatest weight in determining a household's choice probabilities on the choice subset D 4 , which consists of all rented dwelling alternatives. Moreover, the sheer magnitude of #4, as compared to the other 9 coefficients, implies that the inclusive value for rented dwellings plays a large role in the calculation of the choice probabilities. Beyond this, the coefficient 92 indicates that the choice subset D2 is of approximately the same importance as a household's current dwelling in determining its choice probabilities. This is not surprising as the set D2 consists of a household's current dwelling and a dwelling of the same type and tenure as the current dwelling, but involves a move. Finally, the coefficient #3 indicates that the choice subset Ds, which consists of all owned dwelling alternatives, is the least important choice subset. Unlike the 9 coefficients, the similarity coefficients 77,-, 1 = 2,3,4, are markedly different in the two models. In Model 1, the estimated value of 772 is .03089 and is statistically significant from zero. This finding indicates that a household's current dwelling and a "new" dwelling of the same type and tenure are virtually perfect substitutes for one another. In Model 2, the estimate of 772 increases to .36364, but this estimate is not statistically significant from zero. Thus, the message from Model 2 is mixed. On the one hand, in comparison to the estimate for Model 1, the increase in the estimated value of 772 indicates a decrease in the estimated similarity between the two dwelling alternatives in D2. On the other hand, the statistical insignificance of the Model 2 estimate of 772 indicates that the Model 1 and 2 estimates are not statistically different. If a conclusion may be drawn from these results, it is that a household's current dwelling and a new dwelling of the same type and tenure are somewhat similar. However, the measured similarity may be sensitive to the time frame over which utility is compared. That is, the alternatives will be measured as more similar, the shorter the time frame of comparison. The estimates of the similarity coefficient 773 for owned dwelling unit alternatives are .99901 for Model 1 and .99902 for Model 2. These estimates are statistically significant. Therefore, the different types of owned units are independent alternatives. On the other side of the market, the estimates of the similarity coefficient 774 for rented dwelling unit alternatives are .33183 for Model 1 and .04216 for Model 2. Both of these estimates are statistically significant. Since the two are comparatively small, the different types of rented units seem to be highly substitutable. Moreover, 101 when multiperiod optimization is taken into account in Model 2, the perceived similarity of rented dwellings increases. The finding that owned and rented dwellings are on opposite ends of the similarity spectrum is an intriguing result. Two factors may contribute to it. First, the hedonic prices for owned dwellings show more variability across dwelling type than the hedonic prices for rented dwellings (see Tables XI, XII). Second, the transaction costs for owned dwellings are higher and more variable than they are for rented dwellings. They are more variable for owned dwellings because realtors' charges depend on dwelling sale values, which differ widely across type (see Table IX), while for rented dwellings, the discounted value of damage deposits do not vary across type. These two factors make the expenditure paths for owned dwellings more diverse than the expenditure paths for rented dwellings. Thus, the difference in the degree of similarity between owned and rented dwellings may be due to the different financial circumstances associated with each type. 6.3 OPTIMAL DEMAND AND DEMAND ELASTICITIES 6.3.1 O p t i m a l Demands by T y p e and Tenure To this point, I have examined the results from the two stages of Rosen's two stage estimation technique in isolation from one another. Chapter 5 dealt with the first stage — the estimation of the hedonic price functions — while the previous section of this chapter dealt with the second stage — the estimation of the hedonic demand model. In this section, I integrate the two sets of results by analyzing the predicted optimal values for dwelling newness, dwelling space, and other consumption. The optimal values for dwelling newness and space are obtained by numerically solving the first order conditions for newness and space given by equations (3.9). The optimal values for other consumption are calculated residually, from the budget constraint given in equation (4.1). The household's planned period of occupancy is assumed to be 3.37 years (the estimated length of occupancy for households uncertain about whether they will move in the next three years). A set of hedonic price functions is needed in order to set the household budget constraint. My calculations are based on the hedonic price functions for Hamilton. Separate sets of optimal values are calculated for young, middle-aged, and old households. The demographic profiles of these households are presented in Table XXIII above. The before-tax incomes of these households are assumed to be $14,500, $17,500, and $11,500, respectively. These figures are the average before-tax income (approximately) of households with heads aged 30, 50, and 70. The calculated optimal values of dwelling newness, dwelling space, and other consumption are tabulated in Table XXVI for each lifecycle cohort of households (by type and tenure). The values in this Table are portrayed in Figures 7, 8, and 9. 102 T A B L E X X V I : H O U S E H O L D D E M A N D S B Y T Y P E , T E N U R E , A N D L I F E C Y C L E C O H O R T Dwelling Dwelling Other Type x Tenure x Lifecycle Cohort Newness Space Consumption Single-Owned •Young Household 5.7 1.9 2696 •Middle-aged Household 4.8 2.2 2747 •Old Household 10.4 2.4 1485 Multiple-Owned •Young Household 11.0 2.5 991 •Middle-aged Household 8.7 2.7 1364 •Old Household 21.2 3.4 369 Single-Rented •Young Household 28.1 6.2 4722 •Middle-aged Household 25.1 7.6 4644 •Old Household 36.3 6.4 3486 Multiple-Rented • Young Household 28.0 6.2 4630 •Middle-aged Household 25.0 7.5 4641 •Old Household 37.5 6.4 3330 Apartment-Rented •Young Household 23.7 13.1 2895 •Middle-aged Household 20.0 9.9 3676 • Old Household 35.0 6.8 2983 103 Figure 7 Optimal Value of Newness 4-0 10 V> C 20 <D Young M idd l a -Aged Old Lifecycle Cohort Figure 8 Optimal Value of Space Legend • Slnnl. - O . n t d Pw.Htnc 0 Multtpl. - 0»n«d Pw.lllno • J J n j ^ . ^ j h M l t ^ ^ w ^ H j l A ^ O MuHI.I. - R.nt.d P..Ulna A Aoartm.nt - R.nt.d P..Ulna 15-10 « o D a . C/1 Young M i d d l s - A g s d Lifecycle Cohort Old Legend • S l n a l . - Q w n i d O w . H l n a a MuH Ip l . - Own.d Dw. l l Inq • S l n a l . - R . n l . d D « . l l l n a « M u l l l a l . - R . n l . d Q . . t l l n a A A n a r t m . n l - R .n t .d O w t l l l n a Figure 9 Optimal Value of Other Consumption c o ^ 4 0 0 0 3 in c o o x: 2 0 0 0 H Young M idd le -Aged Old Lifecycle Cohort Legend • Slnal. - o. n .d O w . H l n a a Mulli.l. - O . n . d O . .M Ina • S l n a l . - R .n t .d P w . l l l n o O y u l l l a l . - R .n t .d O w . H l n a 104 Three points are brought out in the table and figures. First, the optimal values depend critically on the type and tenure of dwelling unit. This is seen primarily in the vertical dispersion of the lifecycle profiles of the optimal values, although the dependence on type and tenure is also made apparent to a lesser degree in the variations in the shapes of the lifecycle profiles. Since each type and tenure is associated with a different hedonic price function, the apparent dependence on type and tenure actually indicates the sensitivity of consumer demands to the level and shape of the hedonic price functions. It is imperative, therefore, to obtain good, robust estimates of the hedonic price functions, such as those used in this thesis. Second, all three optimal demands are smaller for owned dwellings than for rented dwellings. This is because the total user costs3 for owned dwellings in Hamilton are much larger than those for rented ones. Thus, the budget constraint for owners lies below the budget constraint for renters, and consequently, the owners' consumption of dwelling newness, dwelling space, and other consumption is reduced relative to the consumption of renters, provided these are normal goods. The high user costs for owned dwellings in Hamilton arise because of the high prices for owned dwellings in that city. Table IX in Chapter 5 shows that Hamilton has the second highest prices for owned dwellings, while at the same time, it has some of the lowest prices for rented dwellings. Therefore, the difference in the optimal demands between owned and rented dwellings will not be as large in the other cities. Third, the lifecycle profiles for dwelling newness tend to have the same characteristic shape for all types and both tenures of dwellings. The same is true for the lifecycle profiles for other consumption. The optimal demands for dwelling newness (see Figure 7) decline as a household moves from youth to middle-age and then rises as a household moves from middle-age to old age. Interestingly, this is the lifecycle profile one would expect if a household maintains a constant level of effective dwelling newness over its lifecycle, as was discussed in Section 6.2.2. One feature of the optimal demands for newness not readily apparent from the figure is that they are considerably smaller than the average values for Hamilton. The optimal dwellings depicted in Figure 7 were built between 1905 and 1936, while the average construction date for Hamilton is approximately 1950. Thus, for Hamilton, the model underpredicts the demand for dwelling newness. The choice of an older dwelling may be due to the higher price per year of newness for dwellings in Hamilton as compared to the price of newness in London, Toronto, and Windsor (see Table XI). The optimal demands for other consumption (see Figure 9) tend to increase or remain stable as a household goes from youth to middle-age, but fall as a household moves from middle-age to old age. The drop in other consumption in the household's latter years is due to a large drop in household income. On average, the change in income from middle-age to old age is -$6,000 (a 34% decrease), which induces a decrease in other consumption of $1304 (or 38%). 3The total user cost of a dwelling is the yearly cost of using the dwelling. This is to be distinguished from the marginal user costs of dwelling newness and dwelling space, which are the yearly costs of an extra unit of the characteristics. 105 Unlike the lifecycle profiles for dwelling newness and other consumption, the lifecycle profiles for dwelling space are not similar in shape for all types and both tenures. In owned single and multiple dwellings, the optimal demands for space increase slightly across the lifecycle. In rented single and multiple dwellings on the other hand, the optimal demands for space follow an inverted U-shaped profile across the lifecycle cohorts. Finally, in rented apartments, the demand for space declines steadily over the lifecycle. In addition to having differently shaped lifecycle consumption profiles, the actual amounts of dwelling space demanded differ widely in the different dwelling alternatives. The average amounts demanded are: 2.5 rooms in owned single and multiple dwellings, 6.7 rooms in rented single and multiple dwellings, and 9.9 rooms in rented apartments. The corresponding sample averages are: 7.5 rooms, 6.2 rooms, and 4.2 rooms, respectively. A comparison of the predicted amounts and the sample average amounts of space demanded establish that the predictions for rented apartments are significantly above their sample averages and the predictions for owned singles and multiples are below their sample averages, while the predictions for rented singles and multiples come very close to their sample averages. One must immediately ask why the model generates such off-the-mark results. In the case of rented apartments, a glance at the household budget constraint provides the answer. The marginal price of an extra room in a rented apartment is almost zero in the range of values for dwelling newness and space given in Table XXVI and Figures 7 and 8. Therefore, it is optimal to choose a large apartment. The optimal demands drop as one moves across the lifecycle because of changes in the demographic variables. I have already established that as the household ages, the effective amount of dwelling space declines (see Section 6.2.2). The effective price per room increases, since one must purchase more actual rooms to obtain the same amount of effective space. As a result, the optimal demand for space decreases. The small values for owned singles and multiples are not so easily explained. The marginal price per extra room for these two types of dwellings is actually somewhat lower than in rented dwellings of the same type. Yet, the predicted demands for the rented dwellings are close to their sample averages while the predicted demands for owned dwellings is too low. Apparently, it is not a price effect which lowers the predicted demands for owned singles and multiples. Therefore, the low predicted demands must be due solely to the income effect caused by the relatively high total user costs for owned dwellings. I have already dealt with this effect in my second point above, so I do not dwell on it here. The results in this section are some of the most dramatic in the thesis. They demonstrate that the predicted optimal demands are extremely sensitive to both the level and shape of the estimated hedonic price functions. This result is not obvious from either the estimated parameters of the hedonic price functions or the estimated parameters of the demand model. As a result, I believe the most important finding in this section is that the two steps of Rosen's two stage estimation technique must be treated as a unified model (which it is), and that the tendency to treat the two stages as somehow independent must be totally eschewed. 106 6.3.2 Expected Demands One important point should be clarified regarding the optimal lifecycle demand profiles illustrated in Figures 7, 8, and 9: they are conditional on the type and tenure of dwelling. Therefore, a given profile will be realized only if a household stays in the same type and tenure of dwelling throughout its lifecycle. Yet households do move among the various types and tenures of dwelling and this mobility must be taken into account in calculating a household's lifecycle demand profiles. This can be accomplished by calculating a household's expected demands, 7 z, = Y,PiZiJ 1 = dwelling newness, dwelling space 3 = 1 7 3 = 1 where (zij, z2j, Xj) is a household's vector of demands for dwelling newness, dwelling space, and other consumption in dwelling alternative j, pj is the probability that a household selects dwelling alternative j, and (z\,Z2,x) is a household's expected demand vector. 1 have calculated a household's expected demands using probabilities calculated from the GEV distribution. The parameters of the GEV are those estimated in Model 2. The choice probabilities are tabulated in Table XXVII and graphically portrayed in Figure 10. T A B L E X X V I I : C H O I C E P R O B A B I L I T I E S Young Middle-aged Old Type/Tenure of Dwelling Household Household Household Current Dwelling (Single-owned) .047 .048 .048 Single-Owned .021 .022 .022 Multiple-Owned .005 .005 .005 Single-Rented .113 .225 .300 Multiple-Rented .112 .220 .286 Apartment-Rented .702 .480 .338 There are two points to note concerning the choice probabilities. First, the probabilities of choosing an owned dwelling are small in comparison to the probabilities of choosing a rented dwelling. They are also small in comparison to the observed proportion of households in owned dwellings; apparently, the optimal demands obtained from the Hamilton budget constraints for owned dwellings yield less utility than the optimal demands for rented dwellings. This is not surprising since all three demands are smaller in owned dwellings than in rented ones. Moreover, since the household preference coefficient for rented dwellings is large and negative, the difference in the utility levels of owner and tenant households arising from the smaller demands for dwelling newness, space, and other consumption must be quite large. 107 Figure 10 Choice Probabilities Xi a xt o O.8-1 0.6 0.4 0.2 0.0 oo oo o o o o d d — CN <N CN M CM o q o o d d m m ^ o o . ai o o d d CN O Legend Wk Y o u n g H o u s e h o l d Z3 M i d d l e - a g e d H o u s e h o l d C2 O l d H o u s e h o l d . o«« 1 1"" i . - O * " 8 0 , , . - 0 ' < ' ' , A ..-Ren** 4 , . _R«n , , < 1 »_Re Current °* s i " * ' * y.u*«<Pu Sln9 1 4 * wlul^P'* ^ p a r « " « M D w e l l i n g A l t e r n a t i v e n»«d' Figure 11 Expected Demands C o E a> a 50-i 40 30 A 20 10 Legend • D w e l l i n g N e w n e s s • D w e l l i n g S p a c e • Other Consumption f'00) Young Middle-Aged Lifecycle Cohort Old 108 Second, the rented apartments are the most probable choice for all three lifecycle cohorts of households. This probability decreases rapidly across the lifecycle, and correspondingly, the prob-abilities of choosing the other types/tenures of dwellings, particularly rented singles and multiples, increase across the lifecycle. Because of the dominant position of rented dwellings among the choice probabilities, the three expected demands in Table XXVIII (Figure 11) tend to follow the lifecycle profiles of the demands of households occupying rented apartments. In addition, the shift in the choice probabilities toward T A B L E X X V I I I : E X P E C T E D D E M A N D S Dwelling Dwelling Other Lifecycle Cohort Newness Space Consumption Young Household 23.4 10.8 3275 Middle-aged Household 21.1 8.3 4030 Old Household 34.3 6.2 3114 rented single and multiple dwellings in the middle-aged and old household cohorts has a strong effect on the expected demands. The effect of this shift in the choice probabilities depends on the level of demand in rented single and multiple dwellings relative to the level of demand in rented apartments. If the demand is greater in rented single and multiples than rented apartments, the expected demand is pulled up; if the demand is less, the expected demand is pulled down. From an examination of Figure 11, it is apparent that the confluence of the two factors affecting the expected demand produces no surprises. All three have the same basic lifecycle profiles as the underlying type and tenure-specific demands. The expected demand for dwelling newness follows a U-shaped profile, with young households demanding a dwelling built in 1923, middle-aged households demanding a dwelling built in 1921, and old households demanding one built in 1934. These demands are higher than those for households in rented apartments because the demands for households in rented single and multiple dwellings are greater than those for households in rented apartments. The expected demand for dwelling space declines across the lifecycle cohorts. In this, it mimes the profile for rented apartments. Young households demand a dwelling of eleven rooms, middle-aged households demand a dwelling of eight rooms, and old households, a dwelling of six rooms. These demands are less than the demands for households in rented apartments because the demands for space by households in all other types and tenures of dwelling are less than those for households in rented apartments. Finally, the expected demand for other consumption follows an inverted U-shaped profile. Young households consume $3275 worth of other consumption, middle-aged households consume $4030 worth of other consumption, and old households consume $3114 worth of other consumption. This lifecycle profile is more steeply bowed than the type/tenure specific profiles. The extra bowedness is caused by the shift in the probabilities toward rented single and multiple dwellings. The demand for other consumption by households occupying rented single and multiple dwellings is much greater than the demand for other consumption by households 109 occupying rented apartments. This imparts a greater upward thrust to the lifecycle profile between young households and middle-aged ones. While the reason there are no surprises in the expected demands is because there are no truly profound changes in the choice probabilities across the lifecycle, this is only part of the story. The other part explains why the choice probabilities are so stable. The stability of the choice probabilities rests on the fact that the utilities of Hamilton households in owned dwellings never manage to overtake the utilities of households in rented dwellings as households move along their lifecycle. Therefore, there is no shift in the choice probabilities toward owned dwellings, as one might expect. At a more basic level, however, the differences in the utilities for owner and renter households are the consequence of the hedonic price functions for owned dwellings, which yield such small demands by owner households. Thus, the importance of the hedonic price functions is once again highlighted. While the expected demands presented above are useful in measuring the sweeping changes in household demands brought about by changes in household income, household size, and the age of the head across a household's lifecycle, they are of minimal use in gauging the sensitivity of household demands to small variations in income, prices, and the demographic determinants of demand. The latter objective may be accomplished by calculating the elasticities of the demands with respect to these parameters. Table XXIX presents the demand elasticities of the expected demands with respect to: (l) after-tax household income, (2) the marginal price of newness, (3) the marginal price of space, ,(4) household size, and (5) the age of the household head. The various elasticities are illustrated in Figures 12 to 16. The elasticities in Table XXIX are arc elasticities. The base of the arc elasticities are the ex-pected demands presented in the preceeding section. The elasticities are calculated at the expected demands, rather than at sample averages as is commonly done, because the former are on a house-hold's demand curve while the latter are not. Hence, elasticities at sample averages are meaningless in the context of this study. The elasticities are obtained by increasing the variable of interest by 1%, computing the new roots of the first order conditions, and calculating the percentage change in the optimal demands. Although I calculate the elasticities by numerical means, their analytic representation can be easily written as where e is the elasticity of the expected demand with respect to some parameter, ' is the elasticity of the choice probability pj, e; is the demand elasticity for a dwelling of type j, and Sj is dwelling type j's share of the expected value (i.e., pjZ{j/zi or pjXj j i j ) . Thus, the elasticity for each type of 6 . 4 DEMAND ELASTICITIES 7 (6.2) 110 T A B L E X X I X : D E M A N D E L A S T I C I T I E S Elasticity by Lifecycle Cohort Dwelling Newness Dwelling Space Other Consumption Income Elasticity •Young Households •Middle-aged Households •Old Households 2.20 -1.19 0.17 -1.85 -0.60 0.16 3.26 1.62 1.43 Price of Newness * • Young Households •Middle-aged Households •Old Households -0.01 -0.02 -0.01 -0.13 -0.01 -0.00 0.00 -0.01 -0.02 Price of Space * •Young Households •Middle-aged Households •Old Households 0.13 -0.41 -0.3 -0.71 -0.73 0.00 0.46 -0.16 -0.06 Household Size Elasticity •Young Households •Middle-aged Households •Old Households -0.49 -0.83 -0.46 -0.01 -0.00 0.03 0.01 -0.00 -0.00 Age of Head Elasticity • Young Households •Middle-aged Households •Old Households -12.97 -4.54 10.87 -32.90 8.31 27.08 28.68 -2.18 -16.93 own price elasticities for each type/tenure of dwelling 111 Figure 12 Income Elasticity Figure 13 Price of Newness Elasticity a u CD £ o o c 0.00 ~2 ° ^ L i f « c y c l « C o h o r t Figure 14 Price of Space Elasticity 0.5i O CO O -0.5 Legend • D w e l l i n g N e w n e s s • D w e l l i n g S p a c e • O t h e r C o n s u m p t i o n Y o u n g Middle-Agsd Lif«cycls Cohort O l d Figure 15 Household Size Elasticity Figure 16 Age of Head Elasticity 0.5 01 a in o _c <D </) 3 O I -0.5 -1 40-| r.9 uJd*« - A 9 73-o T3 a <D O _ CD < L l f t c y c l * C o h o r t 1 1 2 dwelling is made up of two parts, a probability elasticity and a conventional demand elasticity, and the elasticity of an expected demand is a weighted average of the type/tenure specific elasticities. There are two further points to note concerning the expected demand elasticities. First, since alternative one in the sum (6.2) is a household's current dwelling, its dwelling characteristics cannot be altered; therefore, the conditional demand elasticities 6 j equal zero for dwelling newness and dwelling space. Second, each dwelling alternative has a different hedonic price function; hence, a variation in the marginal price of a dwelling characteristic for dwelling alternative j affects only the demands for that type of dwelling. As a result, the price elasticities can be rewritten as 7 € = 5^ c}p)a,- + €t*t (6.3) J'=I where alternative k is the alternative experiencing the price change. The formula (6.3) indicates the size of the price elasticities will be governed in large part by the size of the probability elasticities, all other things being equal. I turn now to the elasticities themselves. The after-tax income elasticities yield two surprises. First, the dwelling characteristics are neither clearly normal nor inferior goods. Dwelling newness is a normal income elastic good for young households, an inferior income elastic good for middle-aged households, and a normal income inelastic good for old households. I would expect newness to be a normal good for a young household since youth tends to view all things new in a positive light. The fall to inferiority by middle-age is unexpected, however, and I cannot explain it. Second, dwelling space is an inferior good for young and middle-aged households and a normal but income inelastic good for old households. This result is not supported by other empirical studies (see Chapter 2). These other studies have shown that dwelling space is a normal income inelastic good. The income inelasticities for other consumption, on the other hand, are not surprising — other consumption is a normal income elastic good for all lifecycle cohorts. The three sets of income elasticities do have one feature in common. They decline in absolute value as households age. The elasticities with respect to the price of newness are all close to zero. Only the cross-price elasticity on dwelling space has any significant magnitude, and then only for young households. The elasticities with respect to the price of space are not as extreme. In particular, the own-price elasticity for dwelling space is -.71 for young households and -.73 for middle-aged ones. These elasticities are comparable to the elasticities obtained in other studies. While not above one in absolute value, the elasticities indicate that dwelling space is responsive to its marginal price. This responsiveness disappears for old households, however, as do all the price elasticities. It should be noted that price elasticities are not well defined in the present model. Prices are not constant in the model but are linear functions of the dwelling characteristics. I have calculated the price elasticities by adding a one percent change in the price of the characteristic added to the constant term in the hedonic price equation. This corresponds to the conventional notion that 113 price changes alter the slope of the budget line. However, this manner of proceeding leaves the second order properties of the budget line unaltered, and it may be that the curvature of the budget constraint also plays an important role. The last two sets of elasticities give the effects of fluctuations in household size and the age of the household head on the household demands. The elasticities reflect the combined effects of changes in a demographic variable acting through the demographic scales on all the goods. The elasticities of dwelling newness with respect to household size are: -.49 for young households, -.83 for middle-aged households, and -.46 for old ones. Thus, an increase in household size decreases the demand for dwelling newness. Moreover, since middle-aged households are the largest, the effect of household size increases as household size increases. Both of these results are exactly opposite the results found in previous studies of aggregate housing demand. In the earlier studies, household size was found to exert a positive effect on housing demand, with the magnitude of this increase decreasing as household size increases. The argument accompanying this result explains that larger households require more space and hence, demand larger dwellings. Since this argument focuses on dwelling space and not dwelling newness, one could argue that the results of the previous studies of aggregate housing demand are not applicable here. However, the results of previous studies do bear on the demand for household size. They imply that dwelling space increases with household size. This is not the case in this study. The elasticities in Table XXIX (Figure 15) show that household size has a negligible effect on dwelling space. Household size also has a negligible effect on other consumption. If the results of this study are correct, the household size effects observed in previous studies may have been produced by the procedure used to aggregate dwelling characteristics into "aggregate housing". The elasticities with respect to the age of the household head are phenomenally large. They indicate that the age of the head is the single most powerful factor determining a household's demands for dwelling newness, space, and other consumption. In addition, the responsiveness of the demands follow distinct lifecycle paths. For both dwelling characteristics, an increase in the age of the household head decreases consumption early in the lifecycle, but increases consumption later in the lifecycle. The elasticities for other consumption follow the opposite lifecycle pattern, with increases in the age of the head producing increases in consumption early in the lifecycle and decreases later in the lifecycle. The effect of the age of the household head appears to work in the following way. Early in the lifecycle, households have a strong potential demand for other consumption and are willing to reduce their consumption of housing to realize this demand. Since other consumption includes saving in my model, households are probably accumulating assets. Later in the lifecycle, when household wealth is larger, households are willing to consume more housing and less of other consumption. If true, this explanation implies that the age of the household head is acting as a proxy for household wealth. This last result is, perhaps, the most important result of the section; it speaks strongly about the structure of my model. First, it indicates the household wealth, or equivalently, household 114 saving, should enter the model on its own, rather than being lumped in with other consumption. That this is not possible with the data at hand is cold comfort. Second, the strength of the age of the household head variable in determining the probability elasticities, e^p\ indicates that there may be a strong wealth effect missing from the type and tenure choice equations. This deficiency probably emerges because of the manner in which dwellings are compared. In Chapter 3, I explained that the choice probabilities are based on a comparison of a household's discounted utilities in the different dwelling alternatives taken over a common time period — the household's optimal length of occupancy. I also explained that this is incorrect, that the correct basis of comparison is a household's discounted utility over the remainder of its economic life, and that the correct approach is impossible, because no one has data on a household's future dwellings. As a result, the intertemporal wealth and savings effects inherent in a household's choosing a particular type and tenure of dwelling are missing from the empirical analysis. In my opinion, there is a conflict here between theory and empiricism. In theory, a household's discrete dwelling choice and its dwelling demands are derived from the same underlying prefer-ence ordering. In practise, the dwelling choice equation derived from utility maximization is too demanding of the data and cannot be properly implimented. Consequently, it may be preferable to use an ad hoc discrete dwelling choice equation which docs capture a household's intertemporal wealth allocation than to use a theoretically derived choice equation which does not. 6.5 SECOND ORDER PROPERTIES OF THE ESTIMATED FUNCTIONAL FORM In order for the estimated functional form of the household utility function to be theoretically valid, two sets of curvature conditions should be satisfied. First, the Generalized Leontief continuous utility function should be strongly quasiconcave. (This was an assumption made in Chapter 3.) While the Generalized Leontief function will not be globally strongly quasiconcave over its entire domain of definition, it is possible for this function to be locally strongly quasiconcave. Such will be the case if, and only if the bordered Hessian has at least two negative eigen values (Diewert, Avriel, and Zang 1979:400-402) or, equivalently, at least two negative Cholesky values (Lau 1978:409-453). I have checked for local strong quasiconcavity of the estimated Generalized Leontief utility function by computing the Cholesky values of the bordered Hessian of the function at every point at which it is evaluated. The results of these checks are reported in the top panel of Table XXX. The findings indicate that in Model 1, only 6.8 percent of the 1893 points checked violate local strong quasiconcavity, while in Model 2, 14.4 percent of the 4317 points checked violate local strong quasiconcavity. Given the usually poor performance of flexible functional forms in satisfying curvature conditions,4 these results are encouraging. 4See Diewert and Wales (1985). 115 T A B L E X X X : C H E C K S O F S E C O N D O R D E R P R O P E R T I E S Percentage Number of Failures of Checks Strong Quasiconcavity Model 1 6.8 1893 Model 2 14.4 4317 Second Order Conditions Model 1 100.0 1893 Model 2 100.0 1893 116 The second set of curvature conditions which must be satisfied are the second order conditions for a utility maximum. The second order conditions arise because of the nonlinearity of the household's budget constraint. With a nonlinear budget constraint, strong quasiconcavity of the household utility function is neither a necessary nor a sufficient condition for a utility maximum. The second order conditions are satisfied if the Hessian of the Lagrangian function of the utility maximization problem is negative semidefinite; that is, if it has nonpositive eigen values or Cholesky values. Since the utility maximization problem in this thesis is a full intertemporal problem, the full Hessian matrix cannot be computed because it involves the demands for future dwellings, which cannot be observed. However, the submatrix of the Hessian associated with a household's current dwelling can be computed; This submatrix or sub-Hessian must also be negative semidefinite. I have calculated the sub-Hessians for each of the mover households (nonmovers do not optimize with respect to the continuous variables), calculated the Cholesky values for these matrixes, and checked for negative semidefiniteness. The results of these checks are given in the bottom panel of Table XXX. In both Models 1 and 2, 100 percent of the households fail the second order conditions. That a significant number of households should fail to satisfy the second order conditions is not surprising. In the model, households are only required to satisfy the first order conditions on average; that is, they are not exact. Thus, requiring exact satisfaction of the second order conditions may be too demanding. 6.6 M O D E L L I N G E X P E R I M E N T S In the preceding sections of this chapter, I have concentrated on describing the estimation results of the model specified in this thesis. I have also examined the implications of these results, but in doing so I have kept within the analytic structure of my model. In contrast, this section deals directly with questions concerning the appropriateness of the structure specified. Four questions are addressed: (1) whether a hedonic model of housing demand is necessary, or equivalently, whether a model of aggregate housing demand is sufficient for modelling a household's housing choices, (2) whether the generalized extreme value distribution was a good choice for the distribution for the errors from a household's discrete housing choice problem, (3) whether one needs to estimate both the continuous and discrete choice parts of a household's housing choice problem in order to obtain resonable estimates of the household utility function, and (4) whether the complicated flexible functional form used to estimate the hedonic price functions generates better estimates of the demand functions than a simpler functional form. These questions are taken up in turn. 117 6.6.1 T h e Hedonic D e m a n d M o d e l versus the Aggregate Hous ing M o d e l Without a doubt, the most difficult and time consuming problems confronted in this thesis arose from my decision to model the demands for dwelling characteristics, rather than the demand for a housing aggregate. Because of these difficulties, most authors have shunned the hedonic approach. While the aggregate housing demand model is simpler to work with, it has inadequacies. From my perspective, the major one is that a housing aggregate may not exist! For an aggregate housing index to exist, the housing characteristics over which a household has preferences must be separable from the other arguments of the household's utility function. Whether this is the case is an empirical problem. In order to examine the question of whether an aggregate housing index exists, I have tested the proposition that dwelling newness and dwelling space are separable from other consumption in my estimated utility function. The null hypothesis for the test involves the restrictions: A 0 3 A 2 1 — A 0 2 A 3 1 — 0 0 3 2 0 2 1 - 022<231 = 0 (6-4) A 3 3 A 2 1 - G 3 2 A 3 1 = 0 A Wold test of these restrictions yields a x2 value of 3239.2 with three degrees of freedom; hence, the null hypothesis of separability is rejected quite convincingly. This test validates the extra work involved in the hedonic demand model. Moreover, given this result, one wonders exactly what the studies of aggregate housing demand have been estimating. 6.6.2 T o G E V O r Not To G E V ? In the discrete choice part of my likelihood function, the probabilities of type/tenure choice are based on a generalized extreme value (GEV) distribution for the errors made by households in choosing their dwellings. I used the GEV distribution in order to admit a general pattern of dependence among housing alternatives and to avoid the red bus/blue bus problem inherent in the logit distribution. Despite its merits, the GEV distribution has one major disadvantage — the pattern of dependence among alternatives must be specified in advance of estimation. Because of this, the partitioning of choice alternatives into related groups appears to be somewhat arbitrary, no matter how well motivated the specification of the related groups. In order to assess the sensitivity of the results to my partitioning of the alternatives, I have re-estimated my model using a logit distribution to generate the choice probabilities. The logit distribution is particularly suited to the task. It is a special case of the GEV distribution in which 0j = 02 = 03 — 64 and 772 = »?3 = T)A = 1 and therefore, the re-estimated model is nested within my more general model. Consequently, a likelihood ratio test can be used to test the parameter restrictions and, in this sense, test my partitioning structure against a well-known alternative partition structure.5  5 The logit distribution lumps all the alternatives into one set. 118 The x2 value for the likelihood ratio test of the restrictions given above is 1.67 with six degrees of freedom. Thus, the logit specification cannot be rejected. In addition, the parameters change very little as a result of the restrictions (the changes are scarcely noticable). It appears that the GEV distribution was not needed and that the logit distribution would have worked just as well. The results also indicate that the other parameters in Model 1 are robust with respect to changes in the probability distribution of the errors associated with type/tenure choice. It is gratifying to note that while my "arbitrary" partition structure does not lend much power to the model, it also does not distort the other results. 6.6.3 F u l l Versus L i m i t e d Information Es t imat ion (or D o all Roads Lead to Rome?) Table XXI contains the full information estimates of parameters for Model 1 and 2. This is not the way the models were estimated. Both Model 1 and Model 2 were estimated in three stages. In the first stage, the log likelihood function of the demands for dwelling space, dwelling age, and other consumption, Lc, was maximized with respect to the parameters of the continuous part of the household utility function and the parameters of the demographic scaling functions applied to dwelling space and dwelling age. For Model 2, the tenure specific constants in the transaction cost function and the three time horizon parameters were also estimated in the first stage. Under the usual regularity conditions (Amemiya 1977), the estimates of these parameters are consistent and asymptotically normal, but are not full information estimates, since the information conveyed by a household's discrete choices is ignored. In the second stage, the log likelihood function for a household's discrete choices, Ld, was maximized with respect to the parameters of the discrete part of the household utility function, the parameters of the demographic scaling functions for dwelling type and tenure, and the parameters of the GEV probability distribution. During this stage of estimation, the parameters estimated in stage one are held constant. Again, the estimated parameters are consistent and asymptotically normal, but not full information estimates. The combination of stages one and two yields two stage maximum likelihood estimates (2SML) of all the parameters. In the third stage, the joint log likelihood function, L = Lc + Ld, was maximized with respect to all the parameters. The estimates from the third stage are consistent, asymptotically normal, full information estimates (FIML). The parameter estimates for the three stages of estimation for both Model 1 and Model 2 are presented in Tables XXXII and XXXIII of Appendix C. An overview of Table XXXII reveals a remarkable similarity in the parameter estimates from stages one, two, and three of Model 1. This is an important result. It indicates that reasonable estimates of the parameters of the continuous part of the household utility function may be obtained without reference to the household's discrete choice problem, via the stage one estimates, and that all the parameters of a household utility function may be obtained by two-stage maximum likelihood. Both estimation strategies involve 119 relatively few parameters, as opposed to full information maximum likelihood, which involves a large number of parameters. The similarity of the different parameter estimates calls into question the simultaneity of a household's discrete and continuous housing choices. If the two sets of choice really do interact, one might expect the full information estimates, which take this interaction into account, to be further from the two-stage estimates than they are. The issue of whether simultaneity exists can be settled by performing a likelihood ratio test using the log likelihood values from the 2SML estimates of stage two and the FIML estimates of stage three. The test statistic equals 16.114, which is distributed as a x2 random variable with eight degrees of freedom. The null hypothesis that a household's discrete and continuous choices are independent can be rejected at the five percent level of significance. It appears that, while the parameter estimates from stages one, two, and three of Model 1 are not sensitive to the simultaneity between the two sets of household choices, an efficiency gain (as measured by the log likelihood value) may be obtained by allowing for the jointness in household demands. Similar results have been reported by Lee and Trost (1978), Gillingham and Hageman (1983), King (1980), and Henderson and Ioannides (1983). Model 2 differs from Model 1 in that the move from 2SML estimates to FIML estimates produces changes in the values of the estimated coefficients. It should be noted that not all the coefficients change, however. Those parameters most closely associated with a household's continuous demands — the parameters of the continuous part of the household utility function, the parameters of the demographic scales for dwelling newness and dwelling space, the transaction cost constant terms, and the optimal lengths of dwelling occupancy — stay the same in stages two and three and these results are the same as those obtained in Model 1. Model 2 differs from Model 1 in the parameters associated with a household's discrete choices; that is, in the parameters of the discrete part of the household utility function, the parameters of the demographic scales for dwelling type and tenure, and in the GEV distribution parameters. Table XXXIII reveals significant differences between stages two and three with respect to these parameters. The degree of these changes is manifest in the likelihood ratio test for the presence of simultane-ity between a household's discrete and continuous choices. The xi test value is 461.05; therefore, from a statistical perspective, simultaneity clearly exists. Thus, there is simultaneity between a household's discrete and continuous choices, but it appears that it manifests itself only in terms of changes in the parameters connected with a household's discrete choices. The conclusion one is tempted to draw from the above comparisons is that the 2SML estimation is inefficient, relative to the FIML, estimation for the problem at hand. This conclusion is warranted only if the continuous and discrete choice parts of the problem have common parameters, as I have assumed throughout my estimations. Since the interpretation of all the results hinges on the assumption of common parameters, this assumption must be tested. The test used is a likelihood ratio test. The restricted alternative in the test is the model given by the stage two estimates of Model 2. The model is restricted in the sense that the continuous 120 utility function parameters, the demographic scaling parameters for dwelling newness and space, the transaction costs constants, and optimal length of occupancy parameters are all drawn from the stage one estimates of the demand functions. An unrestricted alternative is obtained by estimating all of the parameters in the model, using the discrete choice likelihood function. The likelihood ratio for this test is the ratio of the discrete likelihoods, Ld, from the restricted and unrestricted models. The x2 value for this test is 1283.136 with seventeen degrees of freedom. Thus, the null hypothesis of common parameters is rejected. The rejection has two implications. First, given my data, there is a statistical cost to imposing common parameters and, second, the parts of the household's choice problem are, in a sense, distinct. Interestingly, the restricted parameters do not change much when re-estimated in the unre-stricted model. The largest changes occur in the second significant digit. The parameters which do change are those found only in the discrete choice model. Table XXXI contains the restricted and unrestricted estimates of these parameters. T A B L E X X X I : R E S T R I C T E D A N D U N R E S T R I C T E D E S T I M A T E S O F T H E DIS-C R E T E C H O I C E M O D E L P A R A M E T E R S Restricted Unrestricted Parameter Estimates Estimates 612 0.00296 0.01040 &13 0.00703 0.07816 622 -6.47294 -5.68900 C41 -0.47122 0.77568 c 4 2 1.12188 0.80610 C51 -0.28206 -1.72026 C52 -0.01749 -1.15835 O2 0.17035 0.45787 03 0.02174 0.25836 0.00367 0.87588 m 0.87211 0.96518 m 0.23795 0.56835 14 0.03239 0.43577 The parameters 612, 613, and 622 of the discrete part of the household utility function in the unre-stricted model give increased preference weight to multiple and apartment dwellings and decreased weight to owned dwellings, relative to the restricted parameters. Nevertheless, qualitatively, the results of these two sets of estimates are the same. On the other hand, there are dramatic changes in the demographic scaling parameters for dwelling type and tenure. In the restricted estimates, the effective preference for multiple and apartment dwellings increases with the age of the household head and decreases with the size of the household. In the unrestricted estimates, the effective preference for multiple and apartment dwellings decreases with both the age of the head and with household size. The effect of this change is to negate any preference for multiple or apartment dwellings by middle-aged households. 121 The demographic scaling parameters for dwelling tenure have the same signs in the restricted and unrestricted models, although their magnitude increases considerably in the unrestricted version. The GEV parameters also display noticeable changes. The coefficients of importance — 02, 03, and 04 — all increase relative to 0\, indicating a downweighting of a household's current dwelling in the choice probabilities. In addition, the weight given owned dwellings in the unrestricted model rises relative to the other choice sets. Apparently, the restricted choice model gives too much weight to the current dwelling and too little weight to owned dwellings. Finally, the similarity coefficients, 772, 773, and 774, increase in value in the unrestricted estimates (relative to the restricted estimates). The increase in value implies that the alternatives in each partition of the choice set are less similar than is indicated by the joint discrete choice/demand model. All in all, the parameters from the unrestricted model conform more closely to my expecta-tions concerning discrete housing choice than the parameters from the restricted model. I find the increased importance and decreased similarity of the owned dwelling alternatives particularly believable. Why the changes in the discrete choice parameters found in Table XXXI should follow relatively minor changes in the demand model parameters is a puzzle. Presumably, the changes in the parameters are brought about by the different structures underlying the restricted and unrestricted models. In economic terms, the key difference between the models is that the restricted model forces the utility function to mimic the first order conditions for a utility maximum with respect to the continuous demands, while at the same time, constraining the utility function to meet the budget constraint, while the unrestricted discrete choice model requires only that the utility function meet the budget constraint. As a result, in the unrestricted model, the utility function is allowed to twist to take full advantage of the differences in income across households for the purpose of explaining dwelling type and tenure choice. Perhaps this is why the discrete choice parameters appear more reasonable. The ability of the utility function to twist is limited, however, by the requirement that it meet the budget constraint. Therefore, the changes in the parameters are not likely to be large. The question raised in this section is whether one needs to estimate both the continuous and discrete parts of a household's housing choice problem in order to obtain reasonable estimates of its household utility function. The answer to this question is, it depends on which parameters of the household utility function one is interested in. If it is the parameters of the continuous part of the household utility function, then the maximun likelihood estimates of the household's demand equations provide fairly accurate estimates vis-a-vis 2SML and FIML. In the case of these parameters, all roads lead to Rome. If one is interested in the parameters of the discrete part of the household utility function, then one is well-advised to take the long road and use full information maximum likelihood — provided one believes that a common set of parameters underlies both a household's discrete and continuous choices. Alternatively, if one does not believe in a common set of parameters, there is no efficiency gain to FIML. There is also no efficiency gain to 2SML. The latter estimates, however, provide a good starting point for the maximum likelihood estimations of the discrete choice equations. 122 6.6.4 Is Flexibi l i ty Div ine? One question raised in Chapter 5 was whether a simpler functional form for the hedonic price equations would improve the precision of the estimates of the hedonic prices without significantly affecting the explanatory power of the equations. This question arose because of the high collinearity among the regressors in the flexible quadratic form used to estimate the hedonic price functions. In that chapter, I showed that a simple additive quadratic functional form yielded marginal prices of the housing characteristics which were reasonably close to those generated by the more complex flexible quadratic form and that the marginal prices from the simpler form had smaller standard errors than the prices from the more complex form. This suggests a trade-off between specification error with the simple model and sampling error with the complex model. In order to further examine the nature of the trade-off, I have re-estimated Model 2 using the estimated additive quadratic hedonic price equations to generate the marginal prices of the dwelling characteristics and the expected values of dwellings. If the simpler hedonic model involves large specification errors, the parameters of the demand equations from the re-estimated model should be at some distance fromthe parameters from Model 2. Alternatively, if the sampling errors for the simple model are smaller than those for the more complex hedonic model and the simple model has no substantial specification errors, the parameters from the two models should be reasonably close and the value of the log likelihood function for the simple model should climb relative to that for the complex model, due to the smaller sampling errors. The result of the experiment is quite conclusive. The value of the log likelihood increases by 474.367, from -31352.517 in the model with the flexible quadratic hedonic price function to -30878.150 in the model with the simple additive quadratic hedonic price function. Moreover, the log likelihood function converges after a scant twelve iterations to a set of parameters which are virtually the same as the estimates for Model 2. Thus, the trade-off between specification error and sampling error is somewhat onesided, with sampling error being the dominant factor. The moral of this story is that flexibility is not divine, at least for my data, and parsimony a virtue. 6.7 SUMMARY The question set out in the introduction to this chapter is whether my theoretically consistent, in-tertemporal approach to housing demand produces a better estimated model than the conventional single period model of housing demand. The results indicate that the intertemporal model is bet-ter, although the reason for its superiority is not what I expected when specifying the model. My original expectation was that allowing for multiperiod dwelling occupancy with different lengths of occupancy for different households would significantly alter the dwelling prices faced by different households and that this would be reflected in vastly different parameter estimates of the house-hold utility functions from those obtained from the single period model. Instead, I obtain relatively minor differences in the utility function parameters of the two models, but significant differences in 123 the parameters of the demographic scaling functions, particularly the scaling parameters relating to dwelling type and tenure. These differences in the demographic scaling parameters applied to dwelling newness and space result in smaller elasticities of substitution in Model 1 than in Model 2. As well, the U-shaped profile in the elasticities of substitution evident in Model 1 vanishes in Model 2. The demographic scaling parameters for dwelling type in Model 1 impart a strong U-shaped profile to the effective preference for multiple and apartment dwellings. As with the elasticities of substitution, the move to the Model 2 parameters eliminates this profile; it also basically removes any preference for single and apartment dwellings. Finally, the demographic scaling parameters for dwelling tenure for Model 1 generate an inverted U-shaped lifecycle profile in the effective preference for owned dwellings. Such a profile exists in Model 2 as well, but it is greatly accentuated as compared to the Model 1 profile. This finding suggests that Model 1 understates the relative preference of middle-aged households for owned dwellings. Taken on their own, the above changes in the effective quantities of dwelling newness, dwelling space, dwelling type, and tenure demonstrate that a single period model misrepresents the effects of the demographic determinants of housing demand. However, the further results of this chapter suggest that the demographic variables, the age of the household head in particular, may be, in part, proxies for household wealth, which could not be included in the analysis. If this is true, both the single and multiperiod occupancy models estimated in this thesis are misspecified, although the multiperiod model appears to partially redress the problem. There are two other major results in this chapter. First, the section on optimal values and demand elasticities shows clearly that the performance of the demand model is highly sensitive to the level and shape of the estimated hedonic equations. This finding implies that the two stages of estimation must always be treated as a unified model. In the same vein, it casts some doubt on the efficacy of Rosen's two stage estimation technique. If the stages are so tightly linked, Rosen's two stage methodology is probably throwing away valuable cross-parameter information, which could improve the efficiency of the estimates. Ideally then, one should simultaneously estimate all the parameters. Unfortunately, this leads to intractably large models; therefore, if hedonic demand models are to be explored at all, they must be either very small or some two-stage technique such as Rosen's must be used. Second, the results indicate that a switching regression model is not necessary if one seeks only to model a household's continuous demands. Adequate parameter estimates may be obtained by estimating the conditional demand functions. Similar results have been found in four other studies; thus, this finding is slowly becoming an empirical rule. Modelling a household's discrete choices is more difficult and it appears that two stage maximum likelihood or full information maximum likelihood estimates are needed in this case. 124 C h a p t e r 7 S U M M A R Y A N D C O N C L U S I O N S The objective of this dissertation has been to construct and estimate an empirical model of housing demand which is more closely linked to a theoretical intertemporal model of consumer demand than has been the case in past empirical studies of housing demand. In the dissertation, this close linkage takes the form of a set of estimating equations based directly on a household's first order conditions for the optimal choice of a dwelling. The closeness of the link between the theoretical and empirical models results in unambiguous interpretations of all the parameters in the model. In addition, it permits me to add two innovations to the analysis of housing demand. First, in the model, households are allowed to occupy their dwellings for any length of time. Second, a household's decision to move and its choice of its optimal dwelling are based on the maximization of the same intertemporal utility function. The model in this dissertation has three other features. First, a characteristics approach to housing demand is taken. Households are assumed to have preferences over the dwelling character-istics, dwelling newness and dwelling space, rather than over an aggregate housing good. I take this approach as a means of dealing with the nonhomogeneity of housing units. Second, the transaction costs of changing dwellings are incorporated directly into the empirical model of housing demand. The existence of significant transaction costs provides a justification for multiperiod dwelling oc-cupancy. Third, I make extensive use of the sample data on household mortgages in computing the user costs of the dwelling characteristics for owner-occupiers. The use of this information is an advance over the conventional assumption that owner occupied dwellings are purchased outright. The primary question posed in the introduction to this dissertation is whether the added com-plexity of an intertemporal model of housing demand is necessary in order to obtain good estimates of the parameters of the household utility function, or whether a single period model is sufficient for this task. In order to answer this question, I estimated two forms of the empirical model. The first form of the model is based on the assumptions that households are single period utility maximizers and that they disregard transaction costs when choosing a dwelling. The second form of the model incorporates both multiperiod dwelling occupancy and transaction costs. 125 As noted above, the first question to be answered is whether it is important to allow for mul-tiperiod occupancy. In my opinion, the answer is "yes"; allowing for multiperiod occupancy is important. The inclusion of the length of tenure parameters has two primary effects. First, the elasticities of substitution decrease in magnitude compared to the elasticities in the single period model. In other words, the single period model has downwardly biased elasticities of substitution. Second, allowing for variable lengths of dwelling occupancy brings about changes in the demo-graphic scaling parameters used to capture the effects of the size of the household and the age of the household head on household demands. This is perhaps the most important effect, since the demographic variables play a large role in determining household demands. First, the changes in the demographic scaling coefficients have a profound effect on the elasticities of substitution. In the first model, the elasticities of substitution exhibit a. U-shaped profile across household age groups, beginning high with young households, decreasing for middle-aged households, and increasing for older households. This profile virtually disappears in the second model. Second, the changes in the demographic scaling parameters produce marked changes in the effective demands for the three types of dwellings and the demands of owned versus rented dwellings. In the single period model, the demographic scaling coefficients for dwelling type imply a U-shaped profile in a household's effective preference for multiple and apartment dwellings; i.e., young households and old house-holds have a greater preference for multiples and apartments than middle-ages households. Once multiperiod occupancy is allowed, this lifecycle profile disappears. As well, the demographic scale for dwelling type becomes so large as to effectively remove any preference for single and multiple dwellings. The changes in the demographic scaling parameters for dwelling tenure work in the op-posite way; they accentuate the lifecycle differences in a household's preference for owned dwellings. Specifically, they increase a middle-aged household's preference for owned dwellings relative to the estimated preference in the single period model. It appears that the length of occupancy in a dwelling and the demographic variables, house-hold size and age of the household head, are significantly related. As a result, by restricting the length of occupancy, a single period model introduces biases into its estimated demographic scal-ing parameters. The relationship between the length of occupancy and the demographic variables suggests that, in the future, length of occupancy functions should contain demographic variables such as those used in this dissertation, in order to make explicit the relationship between length of occupancy and the demographic variables. Moreover, since both demographic variables essentially position a household in its lifecycle, there is a hidden relationship between the optimal period of occupancy and household wealth. In future research, the relationship between the length of occupancy and household wealth needs to be explored. The second question posed in this dissertation is whether the decision to move equation and the dwelling demand equations should be jointly estimated. This question is more difficult to answer than the preceeding question. The basic statistical result is that there is simultaneity between 126 these equations. THis is clearly demonstrated by the likelihood ratio tests presented in Chapter 6. In contrast, however, the inclusion of the dwelling choice equation produces few noticable changes in the parameter estimates, and the parameters which do change are associated with the discrete choice problem. I suggest, therefore, that if one seeks to estimate a household's demands for continuous characteristics, one can obtain adequate estimates by estimating the demand functions for the continuous demands without reference to the discrete choice problem. This conclusion is not unqualified, however. The lack of changes in the parameter estimates may be due to a misspecification of the dwelling choice equation. In Chapter 6, I argued that the dwelling choice equation used in this dissertation may be deficient in its ability to capture a household's intertemporal wealth allocation decisions. This deficiency arises because of the lack of data on a household's future dwellings. Without these data, one cannot properly assess a household's discounted future utility and, therefore, one cannot correctly specify the dwelling choice equation. In my opinion, there is a conflict here between theory and empiricism. In theory, a household's discrete dwelling choice and its dwelling demands are derived from the same underlying preference ordering. In practise, the dwelling choice equation derived from utility maximization is too demanding of the data and cannot be properly implimented. Consequently, it may be preferable to use an ad hoc discrete dwelling choice equation which does capture a household's intertemporal wealth allocation than to use a theoretically derived choice equation which does not. My final observations concern the use of Rosen's two step model for the estimation of charac-teristics demands. The model is appealing for two reasons. First, it is popular because it allows one to use conventional demand theory to analyze the demand for packages of characteristics such as houses, cars, trucks, etc. Second, it allows one to break the estimation of the demand model into two estimation steps, each of which has comparatively few parameters. Nevertheless, it has two disadvantages which have been understated in the literature. First, the same data are used to estimate the implicit marginal prices of the characteristics and to estimate the demand equations themselves. Brown and Rosen (1982) and Murray (1983) have shown that this will not work unless identifying restrictions are placed on the functional forms of the demand equations (see Chapter 2). Estimating separate hedonic price equations for distinct housing markets and then estimating a common demand equation for all submarkets provides an easy means of obtaining identifying restrictions. While these restrictions work in theory, they add only a limited amount of information to the model. In particular, when the demand equations contain shift variables for the different housing markets, as is common, the information added by estimating separate hedonic price equations for different housing markets may not be sufficient to enable one to discern the parameters of the housing demand equations. This topic needs further investigation if Rosen's model is to continue being used in econometric research. A second disadvantage of Rosen's model is its focus on continuously measured characteristics. In practise, many characteristics are discrete or qualitative in nature, or they are measured by discrete or qualitative variables. Consequently, the use of Rosen's technique places asymmetric 127 weighting on the demands for the few continuous characteristics in the data set. A richer analysis might be obtained by treating the demands for all dwelling characteristics symmetrically. My results in Chapter 5 and 6 point to a third limitation; namely, that the demand model is extremely sensitive to the form of the estimated hedonic price equations. Because of this, it seems advisable to simultaneously estimate all the parameters of the model in order to take advantage of all the cross-parameter information available. Unfortunately, the hedonic demand models involve a large number of parameters, making simultaneous estimation intractable. In conclusion, I beleive this thesis establishes that tightly linking the structure of an empirical model of housing demand to a theoretical intertemporal model of consumer demand produces a better model. My results show that households have vastly different planned lengths of dwelling occupancy; moreover, for some households, the planned length of occupancy is substantially above one. For this reason alone, one can reject the static, single period approachs which have been used to date. In addition, my results demonstrate that the estimated household utility function from my theoretically consistent model is better able to capture changes in a household's housing demands resulting from variations in the demographic determinants of demand. My model is not without its shortcomings. From my perspective, the major deficiency is its failure to adequately capture the role of housing as an asset. There are two reasons for this. First, household wealth and saving are not handled well due to a deficiency in my data. Therefore, this shortcoming was beyond my control. Second, the discrete choice equations appear not to capture the fact that the own/rent decision is one of a household's most important asset accumulation decisions. As already noted, the problem is one of data; specifically, one does not know how the current housing choice fits into a household's overall asset accumulation plan. I have grappled with both problems in the thesis, but with limited success. Clearly, these problems need to be addressed in future housing research. 128 B I B L I O G R A P H Y Aaron, Henry J. [1972], Shelter and Subsidies: Who Benefits from Federal Housing Policies, The Brookings Institution, Washington, D.C. Allen, R.G.P. [1938], Mathematical Analysis for Economists, Macmillan, New York, N.Y. Alonso, W. [1964], Location and Land Use, Harvard University Press, Cambridge, Mass. Amemiya, Takeshi [1973], "Regression Analysis when the Dependent Variable is Truncated Normal", Econometrica, 41, 997-1016. [1974], "Multivariate Regression and Simultaneous Equation Models when the De-pendent Variables are Truncated Normal", Econometrica, 42(6), 999-1012. [1977], "The Maximum Likelihood and Nonlinear Three-stage Least Squares Esti-mator in the General Nonlinear Simultaneous Equation Model", Econometrica, 45, 955-968. Appelbaum, Elie, and Harris, Richard [1978], "Imperfect Capital Markets and Life-cycle Sav-ing", Canadian Journal of Economics, 11(2), 319-324. Atkinson, A.B., and King, M. A. [1980], "Housing Policy, Taxation and Reform", Midland Bank Review, Spring, 7-15. Atkinson, Anthony B., and Stiglitz, Joseph E. [1980], Lectures on Public Economics, McGraw Hill Book Co., New York, N.Y. Bailey, Martin J., Muth, Richard F., and Nourse, Hugh O. [1963], "A Regression Method for Real Estate Price Index Construction", Journal of the American Statistical Associa-tion, 58, 933-42. Bajic, Vladimir [1983], Housing-Market Segmentation and Demand for Housing Attributes: Some Empirical Findings.M'imeo. Ball, M.J. [1973], "Recent Empirical Work on the Determinants of Relative House Prices", Urban.Studies, 10, 213-233. Barten, A. [1964], "Family Composition, Prices and Expenditure Patterns" in Econometric Analysis for National Economic Planning, P.E. Hart, G. Mills, and J.K. Whitaker (eds.), London: Butterworth. Barton, D.M., and Olsen, E.O. [1976], "The Benefits and Costs of Public Housing in New York City", Institute for Research on Poverty, Madison, Wisconsin. Becker, G.S. [1964], Human Capital.Columbia University Press, New York, N.Y. Ben-Porath, Y. [1967], "The Production of Human Capital and the Life Cycle of Earnings", Journal of Political Economy, 75, 352-65. Blackorby, Charles, Primont, Daniel, and Russel, Robert R. [1978], Duality, Separability, and Functional Structure: Theory and Economic Applications, Elsevier North-Holland, Inc., New York, N.Y. Blomquist, G., and Worley, L. [1981], "Hedonic Prices, Demands for Housing Amenities, and Benefit Estimates", Journal of Urban Economics, 9(1), 125-138. Boadway, Robin W., and Kitchen, Harry M. [1980], Canadian Tax Policy, Canadian Tax Paper No. 63, Canadian Tax Foundation, Toronto, Ont. 129 Bouthelier, F. [1978], An Efficient Methodology to Estimate and Predict with Multinomial Probit Models: Applications to Transportation Problems, Ph.D. dissertation, Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, Mass. Bossons, John [1973], "Credit Rationing, Indivisibilities, Portfolio Balance Effects, and the Wealth Elasticity of the Demand for Housing", Working Paper, Institute for the Quantitative Analysis of Social and Economic Policy, University of Toronto, Toronto, Ont. Brown, J., and Rosen, H. [1982], "On the Estimation of Structural Hedonic Models", Econo-metrica, 50, 765-768. Cameron, Trudy Ann [1982], "Home Weatherization Choices: Alternative Logit Models and Pol-icy Implications", Discussion Paper No. 82-31, Department of Economics, University of British Columbia, Vancouver, B.C. Carliner, Geoffrey [1973], "Income Elasticity of Housing Demand", Review of Economics and Statistics, 55(November), 528-32. [1974], "Determinants of Home Ownership", Land Economics, 50(May), 109-19. [1975], "Determinants of Household Headship", Journal of Marriage and the Family, 37(February), 28-38. Chinloy, Peter T. [1977], "Hedonic Price and Depreciation Indexes for Residential Housing: A Longtitudinal Approach", Journal of Urban Economics, 4, 469-482. [1978], "Depreciation, Adverse Selection and Housing Markets", Journal of Urban Economics, 5(2), 172-89. [1980a], "An Empirical Model of the Market for Resale Homes", Journal of Urban Economics, 7(2), 279-92. [1980b], "The Effect of Maintenance Expenditures on the Measurement of Deprecia-tion in Housing", Journal of Urban Economics, 8, 86-107. Clarke, CE. [1961], "The Greatest of a Finite Set of Random Numbers", Operations Research, 9, 145-162. Clayton, F.A. [1974], "Income Taxes and Subsidies to Homeowners and Renters: A Comparison of U.S. and Canadian Experiences", Canadian Tax Journal, 22(May), pp 295. Cragg, J.G. [1971], "Some Statistical Models for Limited Dependant Variables with Application to the Demand for Durable Goods", Econometrica, 39, 829-844. [1981], "More Efficient Estimation in the Presence of Heteroscedasticity of Unknown Form", Discussion Paper No. 43-81, The University of British Columbia. Cragg, J.G., and Uhler, R.S. [1971], "The Structure of the Asset Portfolio of Households", Review of Economic Studies, 38, 341-57. Cramer, J.S. [1967], "A Dynamic Approach to the Theory of Consumer Demand", Review of Economic Studies, 24, 73-86. Curley, Anthony J., and Guttentag, Jack M. [1974], "The Yield on Insured Residential Mort-gages", Explorations in Economic Research, 1(1), 114-161. Daganzo, Carlos F. [1979], Multinomial Probit: The Theory and Its Application to Demand Forecasting, Academic Press, New York, N.Y. Daly, Andrew J., and Zachary, Stanley [1979], "Improved Multiple Choice Models" in Identifying and Measuring the Determinants of Mode Choice, D. Hensher and Q. Dalvi (eds.), Teakfield, London, 335-357. David, Martin [1962], Family Composition and Consumption, North-Holland Publishing Co., Amsterdam, Holland. Deaton, Angus [1979], "Theoretical and Empirical Approaches to Consumer Demand Under Rationing", Discussion Paper No. 72/79, University of Bristol, Bristol, U.K. [1980], "Demand Analysis" in Handbook of Econometrics, Z. Griliches and M. Intril-ligator (eds.), Forthcoming. 130 Deaton, Angus, and Muellbauer, John [1980], Economics and Consumer Behavior, Cambridge University Press, Cambridge, U.K. Debreu, G. [1954], "Representation of a Preference Ordering by a Numerical Function" in Deci-sion Processes, R.M. Thrall, C.H. Coombs, and R.L. Davis (eds.), Wiley, New York, N.Y., 588-592. de Leeuw, Frank [1971], "The Demand for Housing: A Review of Cross-section Evidence", Review of Economics and Statistics, 53(February), 1-10. de Leeuw, Frank, and Ekanem, Nkant F. [1971], "The Supply of Rental Housing", American Economic Review, 61, 806-17. de Leeuw, Frank, and Struyk, Raymond J. [1975], The Web of Urban Housing: Analyzing Policy with a Market Simulation Model, The Urban Institute, Washington, D.C. De Rosa, Paul [1978], "Mortgage Rationing and Residential Investment", Journal of Money, Credit and Banking, 10(1), 75-87. Diewert, W.E. [1974], "Intertemporal Consumer Theory and the Demand for Durables", Econo-metrica, 42(May), 497-516. [1976], "Exact and Superlative Index Numbers", Journal of Econometrics, 4, 115— 145. [1977], "Walras Theory of Capital Formation and the Existence of a Temporary Equilibrium", in Equilibrium and Disequilibrium in Economic Theory, G. Schwodi-auer (ed.), D. Reidel Publishing Company, Dordrecht, Holland, 73-126. [1978a], "Optimal Tax Perturbations", Journal of Public Economics, 7, 179-202. [1978b], "Duality Approaches to Microeconomic Theory", Discussion Paper 78-09, Department of Economics, University of British Columbia, Vancouver, B.C. [1978c], "Hick's Aggregation Theorem and the Existence of a Real Value-Added Function", in Production Economics: A Dual Approach to Theory and Applications, Melvyn Fuss and Daniel McFadden (eds.), North-Holland Publishing Co., New York, N.Y., 17-51. [1980], "Aggregation Problems in the Measurement of Capital" in The Measurement of Capital, D. Usher (ed.), University of Chicago Press, Chicago, 433-538. Diewert, W.E., Avriel, M., and Zang, I. [1979], "Nine Kinds of Quasiconcavity and Concav-ity", Discussion Paper No.79-29, Department of Economics, University of British Columbia, Vancouver, B.C. Diewert, W.E., and Wales, T.J. [1986], "Semiflexible Functional Forms", Discussion Paper No.86-25, Department of Economics, University of British Columbia, Vancouver, B.C. Dixit, A.K. [1979], "Price Changes and Optimal Taxation in a Many-person Economy", Journal of Public Economics, 11, 143-157. Domencich, Thomas A., and McFadden, Daniel [1975], Urban Travel Demand: A Behavioural Analysis.North Holland American Elsevier, New York, N.Y. Dougherty, Ann, and Van Order, Robert [1982], "Inflation, Housing Costs, and the Consumer Price Index", American Economic Review, 72(1), 154-63. Duesenberry, James S., and Kistin, Helen [1953], "The Role of Demand in Economic Structure", in Studies in the Structure of the American Economy, Wassily Leontieff (ed.), Oxford University Press, New York, N.Y., 451-82. Ellickson, B. [1981], "An Alternatiave Test of the Hedonic Theory of Housing Markets", Journal of Urban Economics, 9, 56-79. Epstein, Larry [1973], Duality and Optimal Consumption Under Uncertainty, Research Projects Group, Department of Manpower and Immigration, Ottawa, Ont. Feldstein, Martin S. [1976], "Personal Taxation and Portfolio Composition: An Econometric Analysis",Econometrica, 44(4), 631-650. 131 Fenton, C. [1974], "The Permanent Income Hypothesis, Source of Income, and the Demand for Rental Housing", Analysis of Selected Census and Welfare Program Data to Deter-mine Relations of Household Characteristics, and Administrative Welfare Policies to a Direct Housing Assistance Program , Joint Center for Urban Studies, Cambridge, Mass., 3-1-3-52. Fisher, F.M., and Shell, K. [1971], "Taste and Quality Change in the Pure Theory of the True-Cost-of-Living Index", in Price Indexes and Quality Change, Z. Griliches (ed.), Harvard University Press, Cambridge, Mass., 16-54. Fletcher, R. [1970], "A New Approach to Variable Metric Algorithms", Computer Journal, 13, 317-322. Fogelman, F., Guesnerie, R., and Quinzii, M. [1978], "Dynamic Process for Tax Reform Theory", Journal of Economic Theory, 17, 200-226. Friedman, J., and Weinberg, D.H. [1978], "The Demand for Rental Housing: Evidence from the Housing Allowance Demand Experiment", Abt Associates Inc., Cambridge, Mass. Gillingham, R., and Hagemann, R. [1983], "Cross-Sectional Estimation of a Simultaneous Model of Tenure Choice and Housing Services Demand", Journal of Urban Economics, 14(1), 16-39. Goodman, Allen C. [1978], "Hedonic Prices, Price Indexes and Housing Markets", Journal of Urban Economics, 5, 471-484. Grebler, L., Blank, D.M., and Winnick, L. [1956], Capital Formation in Residential Real Estate, Princeton. [1959], "Once More: Capital Formation in Residential Real Estate", Journal of Po-litical Economy, 67(Dec), 612-27. Greenway, H.F. [1939], Housing in Canada, Census Monograph No. 8, Seventh Census of Canada, 1931, King's Printer, Ottawa, Ont. Griliches, Z. [1971], Price Indexes and Quality Change, Harvard University Press, Cambridge, Mass. Guesnerie, Roger [1977], "On the Direction of Tax Reform", Journal of Public Economics, 7, 179-202. Hall, R.E. [1973], "Wages, income and hours of work in the U.S. labor force", in Income Main-tenance and Labor Supply , G. Cain and H. Watts (eds.), Markham, Chicago, 111. Harrison, D., and Rubinfeld, D. [1978], The Air Pollution and Property Value Debate: Some Empirical Evidence, Review of Economic Statistics 60, 635-638. Hatch, James E. [1975], The Canadian Mortgage Market, Toronto: Ministry of Treasury, Eco-nomics, and Intergovernmental Affairs. Hausman, J.A. [1980], "The Effect of Wages, Taxes, and Fixed Costs on Women's Labor Force Participation", Journal of Public Economics , 14(2), 161-94. Hausman, Jerry A., and Wise, David A. [1976], "A Conditional Probit Model for Qualitative Choice: Discrete Decisions Recognizing Interdependence and Heterogeneous Prefer-ences", Discussion Paper 173, Massachusetts Institute of Technology, Cambridge, Mass. [1980], "Discontinuous Budget Constraints and Estimation: The Demand for Hous-ing", Review of Economic Studies, 47, 75-96. Hebert, M., Jakubecki, M., Lee, P., and Sheikh, M.A. [1972], "Sector C: Residential Construc-tion", CANDIDE Model 1.1, Economic Council of Canada, Ottawa, Ont., 65-76. Heckman, J.J. [1976], "A Life Cycle Model of Earnings, Learning, and Consumption", Journal of Political Economy , 84, S11-S44 (supplement). Hendershott, Patric, and Hu, Shing [1979], "Inflation and the Benefits from Owner Occupied Housing", National Bureau of Economic Research Working Paper No. 383, Cam-bridge, Mass. 132 Henderson, J. Vernon [1977], Economic Theory and the Cities, Academic Press, New York, N.Y. Henderson, J.V., and Ioannides, Y. [1983], "A Model of Housing Tenure Choice", American Economic Review, 73(1), 98-113. [1983], Tenure Choice and the Demand for Housing, mimeo. Hildreth, C, and Houck, J.P. [1968], "Some Estimators for a Linear Model with Random Coefficients", Journal of the American Statistical Association, 63, 707-724. Holmes, R.A. [1976], "Male-female Earnings Differentials in Canada", Journal of Human Re-sources , 11, 109-17. Houthakker, H.S., and Taylor, L.D. [1970], Consumer Demand in the United States, Harvard University Press, Cambridge, Mass. Ingram, Gregory K., Kain, John F., and Ginn, J. Royce [1972], The Detroit Prototype of the NBER Urban Simulation Model, Urban and Regional Studies No. 1, National Bureau of Economic Research, New York, N.Y. Johnson, Norman L., and Kotz, Samuel [1970], Distributions in Statistics: Continuous Univari? ate Distributions - 1, Houghton Miffin Company, Boston, Mass. [1972], Distributions in Statistics: Continuous Multivariate Distributions, John Wiley & Sons, Inc., New York, N.Y. Kain, John F., and Quigley, John M. [1972], "Note on Owner's Estimate of Housing Value", Journal of the American Statistical Association, 67, No. 340, 803-806. [1975], Housing Markets and Racial Discrimination: A Microeconomic Analysis, Na-tional Bureau of Economic Research, New York, N.Y. Kearl, James [1979], "Inflation, Mortgages, and Housing", Journal of Political Economy, 87(Oc-tober), 1115-1138. King, A. Thomas [1973], Property Taxes, Amenities, and Residential Land Values, Ballinger Publishing Co., Cambridge, Mass. King, Mervyn A. [1980], "An Econometric Model of Tenure Choice and Demand for Housing as A Joint Decision", Journal of Public Economics, 14, 137-59. [1981a], "The Distribution of Gains and Losses from Changes in the Tax Treatment of Housing" in Behavioral Simulation in Tax Policy Analysis, M.S. Feldstein and D. Hartman (eds.), Chicago University Press, Forthcoming. [1981b], "Welfare Analysis of Tax Reforms Using Household Data", SSRC Pro-gramme on Taxation, Incentives and the Distribution of Income Discussion Paper 21, London School of Economics. [1981c], "An Index of Inequality with Applications to Horizontal Equity and Social Mobility",Econometrica , Forthcoming. Kish, L., and Lasing, J.B. [1954], "Response Errors in Estimating the Value of Homes", Journal of the American Statistical Association, 49(Sept.), 520-38. Kitchen, H.M. [1967], "The Imputed Rent on Owner-Occupied Dwellings", Canadian Tax Jour-nal, 15(September), 482-491. Kuch, Peter, and Haessel, Walter [1979], An Analysis of Earnings in Canada, Statistics Canada, Ottawa. Laidler, David [1969], "Income Tax Incentives for Owner Occupied Housing", in The Taxation of Income from Capital , Arnold C. Harberger and Martin T. Bailey (eds.), The Brookings Institution, Washington, D.C, 50-76. Lau, L.J. [1978], "Testing and Imposing Monotonicity, Convexity and Quasiconvexity Con-straints", in Production Economics: A Dual Approach to Theory and Applications, Vol. 1, M. Fuss and D. McFadden (eds.), North Holland, Amsterdam, 409-453. Lee, Lung-Fei, and Trost, Robert P. [1978], "Estimations of Some Limited Dependant Variable Models with Application to Housing Demand", Journal of Econometics, 8, 357-382. 133 Lee, Tong Hun [1963], "Demand for Housing: A Cross Section Analysis", Review of Economics and Statistics, 45, 190-196. [1964], "The Stock Demand Elasticities of Non-Farm Housing", Review of Economics and Statistics, 46(Feb.), 82-89. [1967], "More on the Stock Demand Elasticities of Non-Farm Housing", Review of Economics and Statistics, 49(Nov.), 640-642. [1968], "Housing and Permanent Income: Tests Based on a Three Year Reinterview Survey", Review of Economics and Statistics, 50, 480-490. Lee, T.H., and Kong, CM. [1977], "Elasticities of Housing Demand", Southern Economic Jour-nal , 43,298-305. Li, Mingche M. [1977], "A Logit Model of Home Ownership", Econometrica, 45(5), 1081-1097. Linneman [1981], "The Demand for Residence Site Characterisitcs", Journal of Urban Eco-nomics, 9(2), 129-148. Lloyd, Cliff L. [1980], "Durables and Demand: An Alternative Paradigm for Intertemporal Demand Theory" in The Collected Works of Cliff L. Lloyd, School of Business Ad-ministration and Economics, Simon Fraser University, Burnaby, B.C. MaCurdy, Thomas E. [1981], "An Intertemporal Model of Portfolio Choice and Human Capital Accumulation under Uncertainty with Extensions Incorporating Taxes, Consumer Durables, Imperfections in Capital Markets, and Nonseparable Preferences", Work-ing Papers in Economics No. E-81-18, The Hoover Institution, Stanford University. Maddala, G.S. [1977], "Identification and Estimation Problems in Limited Dependent Vari-ables Models" in Natural Resources, Uncertainty, and General Equilibrium Systems, Alan S. Blinder and Philip Friedman (eds.), Academic Press, New York, N.Y., 219-239. Maisel, Sherman J. [1966], "Rates of Ownership, Mobility and Purchase" in Essays in Ur-ban Land Economics, Real Estate Research Program, University of California, Los Angles, Calif. Maisel, S.J., Burnham, J.B., and Austin, J.S. [1971], "The Demand for Housing: A Comment", Review of Economics and Statistics, 53, 410-413. Maisel, S., and Winnick, L. [1960], "Family Housing Expenditures: Elusive Laws and Intrusive Variances" in Consumption and Savings, Vol. 1, Irwin Friend and Robert Jones (eds.), Wharton School of Finance and Commerce, University of Pennsylvania, Philadelphia, Penn., 359-435. Manski, Charles F., and McFadden, Daniel [1981] Structural Analysis of Discrete Data With Econometric Applications , MIT Press, Cambridge, Mass. Marshall, Alfred [1952], Principles of Economics, 8th ed., Macmillian Co., London, U.K. Mayo, Stephen K. [1981], "Theory and Estimation in the Economics of Housing Demand", Journal of Urban Economics, 10, 98-116. McFadden, D. [1973], "Conditional Logit Analysis of Qualitative Choice Behavior" in Frontiers in Econometrics, P. Zarembka (ed.), Academic Press, New York, N.Y., xxx-xxx. [1978], "Modelling the Choice of Residential Location" in Spatial Interaction Theory and Residential Location, A. Karlqvist et al. (eds.), North Holland, Amsterdam, 75-96. [1981], "Econometric Models of Probabalistic Choice" in Structural Analysis of Dis-crete Data, C.F. Manski and D. McFadden (eds.), Massachusetts Institute of Tech-nology Press, Cambridge, Mass., 198-280. Mincer, J. [1974], Schooling, Experience, and Earnings. Columbia University Press, New York, N.Y. Morgan, James N. [1965], "Housing and the Ability to Pay", Econometrica, 33(April), 289-306. 134 Morton, Walter A. [1955], Housing Taxation, University of Wisconsin Press, Madison, Wiscon-sin. Murray, Michael P. [1978], "Hedonic Prices and Composite Commodities", Journal of Urban Economics, 5, 187-197. Muth, Richard F. [1960], "The Demand for Non-Farm Housing" in The Demand for Durable Goods, Arnold C. Harberger (ed.), University of Chicago Press, Chicago, 111., 29-96. [1965], "The Stock Demand Elasticities of Non-farm Housing: Comment", Review of Economics and Statistics, 47(Nov.), 447-449. [1969], Cities and Housing, University of Chicago Press, Chicago, 111., Ch. 3. [1971], "The Demand for Urban Residential Land", Urban Studies, 8, 243-254. [1973], "A Vintage Model of the Housing Stock", Papers of the Regional Science Association, 30, 141-56. [1974], "Moving Costs and housing Expenditures", Journal of Urban Economics, 1(1), 108-125. [1982], "Effects of the U.S. Tax System on Housing Prices and Consumption", Work-ing Papers in Economics No. E-82-3, The Hoover Institution, Stanford University. Nagatani, K. [1972], "Life Cycle Saving: Theory and Fact", American Economic Review, 62, 344-53. [1981], Macroeconomic Dynamics, Cambridge University Press, Cambridge. Nelson, D. [1975], "Income Elasticity of Housing Demand", Unpublished Ph.D. dissertation, Department of Economics, University of Oregon. Nelson, J. [1978], "Residential Choice, Hedonic Prices and the Demand for Urban Air Quality", Journal of Urban Economics, 5, 357-369. Oksanen, E.H. [1966], "Housing Demand in Canada, 1947 to 1962: Some Preliminary Exper-imentation", The Canadian Journal of Economics and Political Science, 32(Aug.), 302-318. Ontario Economic Council [1976], Housing: Issues and Alternatives - 1976, Toronto, Ont. Orcutt, G.H., et. al. [1961], Microanalysis of Socioeconomic Systems, Harper Bros., New York, N.Y. Ostas, James R., and Zahn, Frank [1975], "Interest and Non-interest Credit Rationing in the Mortgage Market", Journal of Monetary Economics, l(April), 187-201. Palmquist, Raymond B. [1979], "Hedonic Price and Depreciation Indexes for Residential Hous-ing: A Comment", Journal of Urban Economics, 6, 267-271. Parkin, J.M., Cooper R.J., Henderson, J.F., and Danes, M.K. [1975], "An Integrated Model of Consumption, Investment and Portfolio Decisions" in Papers in Monetary Eco-nomics, Volume II, Reserve Bank of Australia, Sydney. Podoluk, J.R. [1968], Incomes of Canadians . Dominion Bureau of Statistics, Ottawa. Polinsky, A. Mitchell [1977], "The Demand for Housing: A Study in Specification and Group-ing", Econometrica, 45(2), 447-462. Polinsky, A.M., and Elwood, D.M. [1977], "An Empirical Reconciliation of Micro and Grouped Estimates of the Demand for Housing", Discussion Paper No. 567, Harvard Institute of Economic Research, Cambridge, Mass. Pollak, R.A., and Wales, T.J. [1980], "Comparison of the Quadratic Expenditure System and Translog Demand Systems with Alternative Specifications of Demographic Effects", Econometrica, 48(April), 595-612. [1981], "Demographic Variables and Demand Analysis", Econometrica, 49(Novem-ber), 1533-1554. Priest, G., Alford, D., and Bailey, M. [1973], Evaluation of 1971 Census Reporting of 'Sell-ing Value' of Owner-Occupied Dwelling Units: Micro-Match with Comparison Sales Catalogues Provided by Members of the Canadian Real Estate Association , (Results 135 Memorandum CDN 71-E-9, Housing and Families Section, Census Field, Statistics Canada), Ottawa. Purvis, J.M. [1975], "Portfolio and Consumption Decisions: Towards a Model of the Trans-mission Process" in Papers in Monetary Economics, Volume II, Reserve Bank of Australia, Sydney. Quigley, J. [1973], Housing Demand in the Short Run: An Analysis of Polytomous Choice, Yale University Press, New Haven, Conn. [1976], "Housing Demand in the Short Run: An Analysis of Polytomous Choice", Explorations in Economic Research, 3(1), 76-102. Raj, Baldev, and Ullah, Aman [1981], Econometrics: A Varying Coefficients Approach, Croom Helm, London. Reid, Margaret G. [1958], "Capital Formation in Residential Real Estate", Journal of Political Economy, 66, 131-153. [1962], Housing and Income, University of Chicago Press, Chicago, 111. Robb, R.E. [1978], "Earnings Differentials Between Males and Females in Ontario, 1971", Cana-dian Journal of Economics , 11, 350-359. Rosen, Harvey S. [1977], Owner-occupied housing and the federal income tax: Estimates and simulations, Princeton University, mimeo. [1979], "Owner Occupied Housing and the Federal Income Tax: Estimates and Sim-ulations", Journal of Urban Economics, 6(2), 247-265. Rosen, Harvey S., and Rosen, Kenneth T. [1980], "Federal Taxes and Home Ownership: Evi-dence from Time Series", Journal of Political Economy, 88(February), 59-75. Rosen, S. [1974], "Hedonic Prices and Implicit Markets: Product Differentiation in Pure Com-petition", Journal of Political Economy, 82, 34-55. Schwab, Robert M. [1982], "Inflation Expectations and the Demand for Housing", American Economic Review, 72(1), 143-53. Schwann, Gregory M. [1977], The Demand for Housing and Related Commodities in Canada: An Application of the Trans-Log Utility Function. Unpublished M.A. paper, Department of Economics, University of British Columbia, Vancouver, B.C. Shaffner, Richard [1975], "Housing Policy in Canada: Learning from Recent Problems", HRI Observations, No. 9, CD. Howe Research Institute, Montreal, P.Q. [1979], "Housing Needs and Economic Policy: The Mortgage Interest and Property-Tax-Deduction Proposal", HRI Observations, No. 19, CD. Howe-Research Institute, Montreal, P.Q. Shelton, John P. [1968], "The Cost of Renting Versus Owning a Home", Land Economics, 44(1), 59-72. Sheppard, N.E. and Baillie, D.C. [i960], Compound Interest, University of Toronto Press, Toronto, Ont. Smith, B., and Campbell, J.M., Jr. [1976], Aggregation bias and the demand for housing, Uni-versity of Houston/University of Chicago, mimeo. Smith, L.B. [1974], The Post-war Canadian Housing and Residential Mortgage Markets and the Role of Government, University of Toronto Press, Toronto, Ont. Steele, Marion [1979], The Demand for Housing in Canada, Census Analytical Study, Statistics Canada, Ottawa, Ont. Steele, Marion, and Buckley, Margaret [1976], "Error in Estimating the Value of Homes: A Source of Downward Bias in Estimated Elasticities of Demand For Housing", Journal of the American Statistical Association, 593-598. Stigler, George J. [1954], "The Early History of Empirical Studies of Consumer Behavior", Journal of Political Economy, 62(April), 95-113. 136 Straszheim, Mahlon R. [1973], "Estimation of the Demand for Urban Housing Services from Household Interview Data", Review of Economics and Statistics, 55, 1-8. [1975], An Econometric Analysis of the Urban Housing Market, Urban and Regional Studies Number 2, National Bureau of Economic Research, New York, N.Y. Struyk, Raymond J., and Marshall, Sue [1974], "The Determinants of Household Home Own-ership", Urban Studies, 11, 289-299. Sweeney, James L. [1974], "A Commodity Heirarchy Model of the Rental Housing Market", Journal of Urban Economics, 1(3), 288-323. Tirole, J., and Guesnerie, R. [1981], "Tax reform from a Gradient Projection Viewpoint", Jour-nal of Public Economics, 15, 275-293. Tobin, James [1958], "Estimation of Relationships for Limited Dependant Variables", Econo-metrica, 26, 24-36. Vaughn, G.A. [1976], "Sources of downward bias in estimating the demand income elasticity for urban housing", Journal of Urban Economics, 3, 45-56. Vousden, Neil [1980], "An Open-City Model with Nonmalleable Housing", Journal of Urban Economics, 7, 248-277. Wales, T.J., and Woodland, A.D. [1979a], "Sample Selectivity and the Estimation of Labour Supply Functions", International Economic Review, 21(2), 437-468. [1979b], "A Random Preferences Model for the Estimation of Consumer Demand Systems with Binding Nonnegativity Constraints", Discussion Paper No. 79-32, De-partment of Economics, University of British Columbia, Vancouver, B.C. Waslander, H.E.L. [1973], CANDIDE Model 1.0: Residential Construction, CANDIDE Project Paper No. 3, Economic Council of Canada, Ottawa, Ont. Weymark, John A. [1978], "On Pareto-improving Price Changes", Journal of Economic Theory, 19, 338-346. [1979], "A Reconciliation of Recent Results in Optimal Taxation Theory", Journal of Public Economics, 12, 171-189. [1980], "Undominated Directions of Tax Reform", Discussion Paper 8007, CORE, Universite Catholique de Louvain, Louvain-La-Neuve, Belgium. Wheaton, William C. [1977], "A Bid Rent Approach to Housing Demand", Journal of Urban Economics, 4, 200-217. White, Michelle J., and White, Lawrence [1977], "The Subsidy to Owner Occupied Housing: Who Benefits?", Journal of Public Economics, 7(1), 111-126. Wilkinson, R.K. [1973], "The Income Elasticity of Demand for Housing", Oxford Economic Papers, 25, 361-377. Winger, Alan R. [1963], "An Approach to Measuring Potential Upgrading Demand in the Hous-ing Market", Review of Economics and Statistics, 45(3), 239-45. [1968], "Housing and Income", Western Economic Journal, 6(June), 226-233. Wingo, L. [1961], "An Empirical Model of the Utilization of Urban Land for Residential Pur-poses", Papers and Proceedings of the Regional Science Association, 191-205. Winnick, L. [1955], "Housing: Has There Been A Downward Shift in Consumer Preferences?", Quarterly Journal of Economics, 69(Feb.), 85-98. Witte, A., Sumka, M., and Erekson, H. [1979], "An Estimate of a Structural Hedonic Price Model of the Housing Market: An Application of Rosen's Theory of Implicit Mar-kets", Econometrica , 47, 1151-1173. Woodland, A.D. [1978], "On Testing Separability", Journal of Econometics, 8, 383-398. 137 A p p e n d i x A D E T A I L E D S U M M A R Y O F E Q U A T I O N S The following sections give in detail a number of the equations used in the empirical model. The equations demonstrate how the parameters to be estimated enter the model. In order to state the equations in a manner which is applicable to both owned and rented dwellings, a number of the equations contain the parameter /z; this parameter takes the value one for owned dwellings and the value zero for rented dwellings. A . l UTILITY FUNCTION A . 1.1 C o n t i n u o u s U t i l i t y F u n c t i o n 2(203 ( S l S 2 ) 2 + a uz(*;(l - M I T 1 + nT2) + (A.1) **(*?)-(*-*;) ) 2 + 138 Normal i za t ion on Continuous Ut i l i ty Funct ion , i _ i i _ i i i i 1 2 2 1 2 2 2 1 2 aoi = 5000x 2 - a o 2 ^ i x - arj32 2 * ~~ n a n z _ a 2 i ^ i -12 1 -1- 2 _12_12_ 2 1 - l - ~ 2 -a 3 1z 2 - 2 ° 2 2 Z i x _ a32*i 2^ 1 ~~ 2 a 3 3 Z 2 x Normal i zed Discrete Ut i l i ty Funct ion d &12^ 12 1^3^ 13 , 62 ^22 « = J ^ T ^ r - + a l *2 1 2 Firs t Derivatives of the Continuous Ut i l i ty Funct ion ^ . M W M I + + ^ ( ^ ) * + . „ § u i 1 x ( t ; ( l - ^ ) T " + ( » t 2 ) 2 1 , , I ^ 0 2 + M 2 x ( t ; ( l - > . ) f 1 + M t 2 ) 2 + a 2 2 ^ - ^ T ^ ) 2 + » 3 2 ( ' ' ^ ,'4^ ) ' I , 1 . 0 8 + . i , x ( t i ( X - M ) * ' + | . t a ) i + *23 ^ / 2 \ % ) 2 + '33 ^ %,' t32 t ? ) y = -d } ( t j ) - < t - t ; » 2 Second Derivatives of the Continuous Ut i l i ty Funct ion c 1 M l - - j ( t ) U. , ( t ) = 1 1 2 x ( t ; ( l - ^ ) t l + | i f 2 ) u £ 2 ( t ) ( » i 2 1 » 2 2 2 ) 2 x ( t ; ( l - ( . ( T l + M 1 - 2 ) 5 1 u < ( t ) = 2*31 "Is*1) ( » J ( t J ) - ( t - t j ) ) 2 ( » i 3 1 > 2 3 2 ) 2 * < t ; ( l - ^ ) t l + M T 2 ) 2 2 2 , ! ( , * ) 5*32 •{ M 2 1 '2") * - <* - 1 ! » ( - c i 3 1 ' 2 3 2 ) * '21 «22 u3> u 3 3 ( ' ' = 2 a » ( t } ) - ( t - t » ) 139 A . 2 STRUCTURAL EQUATIONS £ a02 + .„,(.; (1 - „)* 1 + 2 + a 2 2 [ 'IfQ V + a 3 2 ( a 0 1 + aux(t; (1 - u.)T^ + pt*) 2 + a 2 1 ( ^  ^ »j(t«)-(«-t») + "31 i ,c 3 1 c 3 2 1 2 x (t;(l - ^ IJ-l + M f 2 ) 2 ( i - u ^ ) - ™ ) E''U'*.?>.4<.;> -(1 --M [ t 2 + ( T ! + r 2 ) n ( 4 ( t " ) , » » ( t j ) ) ] -(1 - n) [ f 1 + .2S,1(4(tJ),«J(t«))] (A.13) I - (1 + rm) 1 1 - (1 + r m ) - T ™ »i(4(«?).4c?)) - (i - '3)n(4('?).4('?) - c j -«?)) -(1 - p) ••2*0 +r(W)'i«l<4<«?).4<«?) - ( i j - ( J ) ) - £ »03 + »13*(*i U - M)*1 + /if 2) 2 + a 2 s 4(«?) ,c21,c22 ' l *2 + »33 »l(t»)-(t-tf) ,c31,'32 1 2 "oi + «u*(«, ( i - + M ? 2 ) 2 + a 2 1 C 2 \ | 2 2 . *«?)-(«-««) ^ 2 + "31 | 2 c'31 c 3 2 1 1 2 x ( t ; ( l - n ) ? - 1 + ^ T 2 ) 2 <*{(<?) - (' - '?»2 ' i y]«(4c?)-4('?)- («-'?))<' 1=1 [ T 2 + (n + r2),2(4(tJ),4(«J))] - ( i - M) [ r 1 + .2s,2(4(«J),4(tJ))] ,I!-TTM" 1 - (1 + <-m) 1 - (1 + r m ) " 92(4('?).4({?)) - <' - ' s l B W C f l . ' j l ' l ) " ('I - '?)) -(1 - M) -2S(1 + rdi)'i<,2i,\(t°),z\(,t«) - (tj -«?))j «'l (A.14) 140 A . 3 E L E M E N T S O F T H E J A C O B I A N M A T R I X 8 1 1 • E u | 2 ( t ) » f ( t ) - » f 2 ( t ) u ° ( Q « f 2 ( 0 ) ±> - ( i --/» [ ( M + r 2 ) „ 1 ( » j ( t j ) , 4 ( t j ) ) ] -(l-,»)[-2Sm<«l<«?).»2<«J))] 911 (*?(«?). *1 (<?)) 1 - (1 + 'm) 1 1 - ( l + ' m ) - ™ - • 2 S ( l + '<W)''«ii(*J(«?),4<«l)> (A. IS ) ( t ) ( i - ( l + r » ) - ™ ) £ ' -(1 - „) +'2)»12(»l1('?).4('?»] -(1 - [.2S„ 2(*J(«J),.»(«J))] ^«j(*}(«?).»j(«f) -(«-«?))<' 1 - (1 + 'm) 1 1 - ( l + r m ) - T ™ - ( 1 - M ) --25(1 + r < l l | ) , l n 2 ( , } ( t j ) , ^ ( t f ) ) »12(»}(tJ),x^»J)) - (1 - r s ) , 1 2 ( , J ( « J ) , . * ( « J ) - ( t j - t j ) ) ( A .16) 141 *21 » § S ( ' ) » f (') - » 1 3 ( " » 3 ( ' ) « ? 2 ( « ) - ( i - ^) ^•72l(»l«l).«2('f) -(«-'?))*' - c [ ( r i + r 2 ) , j 1 ( » J ( t J ) , * J ( « J ) ) ] -(1 - „ ) [ . 2 5 , 2 1 ( , J ( t J ) , , 2 « J ) ) ] , f, j _ v t f - T T M 1 - ( l + 'm) 1 . - ' ( H ^ - r m J Mi<«i<«i>.-a<«i)> - ( i - r s ) , a i ( . , 1 ( i ; ) 1 . i ( i j ) - « f - I f ) ) - • 2 S ( l + r ( J < J ) , J 0 2 1 ( , J ( t J ) , , » ( i J ) ) i ' l (A.17) 522 - £ » | » ( ' K C ) - " j 2 ( t ) u | ( t ) -(i - M ) ^ 722 ( » } « ? ) , 4 <«?) - (' - «?))•' + ' 2 ) « 2 ( » l 1 ( ' ? ) . * 2 ( « ? ) ) ] - ( i - „) [.as«22(»}(«f),4(tj))] i - ( 1 + T . ) - r r M ) «M(*i<*i).«2(*r» " <» - '»)»»a(«i,(«?).«»(«i) " «I - •?» (A.18) - ( ! - » . ) - 2 S ( l + r ( l d ) ' l , 2 2 ( z 1 1 ( . J ) , z J ( , J ) ) «'l 142 A . 4 C H O I C E P R O B A B I L I T I E S .02 + 62 [ w."2 + w.^  1_\('>2-1) j _ w 12 i Pi = 1_ \ 12 .02 W i + e2 [ + w.^ 6 12 . \ "»« ,14 ,k = l k = 7 i € { ! , . . . ,8} O.I9) _1_ _1_ \ ( 1 2 - 1 ) J _ i+6 i+6 "*3 |EW" + E" ( 1 3 - 1 ) 13 1 i+6 T k 14 Pi + 6 -vk = 4 ( 1 4 - 1 ) 1 i+6 _1_ _ l _ \ 12 .02 w; +»2 ( " j " 2 + w j + , I + \ k = l k =7 « € {1 « } 13 + »4 E ^ + E' r k 4 U = 4 (.4.20) E^3+EW*3 9 j \ ( 1 3 - 1 ) i V k = l k =7 1 \ 12 + »3 E^ +E' Vk = l k = 7 • S D 3 , j > . + « ,14 k = 10 (/1.21) E-F + E-( 1 4 - D 14 ,''4 a = 4 k = 10 .02 w ; + 92 ,12 + w 1 2 i + 6 + >3 | ^ " k " 3 + k = l k=7 E-k=7 .13 M E -F + E' ,14 Vk=4 (*-22) 143 A p p e n d i x B S T A G E 1, 2, A N D 3 E S T I M A T E S O F T H E D E M A N D M O D E L S 144 T A B L E X X X I I : E S T I M A T E D C O E F F I C I E N T S F O R M O D E L 1 S t a g e 1 S t a g e 2 S t a g e 3  s t a n d a r d s t a n d a r d s t a n d a r d p a r a m e t e r c o e f f i c i e n t e r r o r c o e f f i c i e n t e r r o r c o e f f i c i e n t e r r o r U t i l i t y F u n c t i o n '. P a r a m e t e r s 1 6 6 . 1 4 5 0 7 0 . 3 3 5 7 6 1 6 6 . 1 4 5 0 7 fixed 1 6 8 . 6 7 9 6 7 0 . 7 2 7 9 6 * 0 2 3 6 3 . 7 5 8 3 2 2 . 9 1 2 7 S 3 6 3 . 7 5 8 3 2 fixed 3 6 3 . 7 5 8 0 7 6 . 9 8 2 8 0 a 0 3 - 2 8 7 . 6 8 5 3 6 8 . 5 2 2 3 6 - 2 8 7 . 6 8 5 3 6 fixed - 2 8 7 . 6 8 5 5 2 5 . 5 1 6 1 3 * 1 1 - 2 . 2 3 5 4 0 0 . 0 0 4 8 5 - 2 . 2 3 5 4 0 fixed - 2 . 2 8 3 6 5 0 . 0 1 0 1 0 » 2 1 - 1 . S 2 5 5 1 0 . 0 1 0 7 1 - 1 . 8 2 5 5 1 fixed - 1 . 8 4 5 1 9 0 . 0 3 5 4 8 6 • 3 1 0 . 9 5 6 7 8 0 . 0 2 1 1 6 0 . 0 9 5 6 7 8 fixed 0 . 9 5 6 2 9 0 . 0 2 0 5 3 » 2 2 1 0 . 0 0 9 5 8 0 . 4 7 5 9 7 1 0 . 0 0 9 5 8 fixed 1 0 . 0 0 8 4 8 0 . 7 7 3 0 2 " • 3 2 - 9 7 . 1 5 3 1 6 1 . 7 2 9 4 6 - 9 7 . 1 5 3 1 6 fixed - 9 7 . 1 5 4 2 7 2 . 9 7 2 5 2 a 3 3 2 0 9 . 7 3 4 9 9 5 . 5 9 3 9 3 2 0 9 . 7 3 4 9 9 fixed 2 0 9 . 7 3 4 7 0 6 . 2 3 1 1 4 5 b l l 0 . 0 0 0 0 0 n o r m . 0 . 0 0 0 0 0 n o r m . b 1 2 0 . 0 2 1 4 9 0 . 0 0 8 7 8 0 . 0 2 1 8 0 0 . 0 0 8 9 1 b 1 3 0 . 0 7 5 9 0 0 . 0 2 9 0 6 0 . 0 7 8 3 8 0 . 0 2 9 3 7 b 2 1 0 . 0 0 0 0 0 n o r m . 0 . 0 0 0 0 0 n o r m . b 2 2 - 9 . 7 7 2 0 0 5 . 5 1 6 1 3 - 9 . 9 9 0 7 1 5 . 2 9 9 3 5 D e m o g r a p h i c S c a l i n g P a r a m e t e r s c l l 0 . 0 0 0 0 0 fixed 0 . 0 0 0 0 0 fixed 0 . 0 0 0 0 0 fixed c 1 2 0 . 0 0 0 0 0 fixed 0 . 0 0 0 0 0 fixed 0 . 0 0 0 0 0 fixed c 2 1 0 . 2 3 4 7 2 0 . 0 0 7 4 4 0 . 2 3 4 7 2 fixed 0 . 2 2 8 2 9 0 . 0 1 1 2 3 c 2 2 - 0 . 5 9 4 6 7 0 . 0 0 3 6 9 - 0 . 5 9 4 6 7 fixed - 0 . 6 0 2 4 4 0 . 0 1 1 6 7 0 c 3 l 0 . 3 5 0 0 8 0 . 0 1 3 2 3 0 . 3 5 0 0 8 fixed 0 . 3 5 4 7 6 0 . 0 1 1 0 2 c 3 2 0 . 0 7 6 2 3 0 . 0 0 6 7 4 0 . 0 7 6 2 3 fixed . 0 7 6 6 6 0 1 0 . 0 1 0 9 3 c 4 1 0 . 1 1 8 6 4 0 . 0 9 9 4 5 - 0 . 3 6 2 1 5 0 . 0 9 9 7 8 c 4 2 1 . 1 4 6 0 3 0 . 0 8 6 6 5 1 . 1 4 8 5 9 0 . 0 8 9 0 7 C S 1 - 0 . 1 6 6 9 6 0 . 0 3 6 4 7 - 0 . 0 5 9 3 9 0 . 1 6 7 1 S C S 2 - 0 . 2 2 0 4 3 0 . 0 3 7 8 3 - 0 . 1 3 7 5 2 0 . 2 1 3 6 3 G E V D i s t r i b u t i o n P a r a m e t e r s » 1 0 . 0 2 0 0 0 n o r m . 0 . 0 2 0 0 0 n o r m » 2 0 . 1 8 5 8 7 0 . 1 3 9 2 6 0 . 0 2 5 7 0 0 . 1 4 8 3 5 » 3 0 . 0 1 7 0 1 0 . 0 1 0 7 4 0 . 0 0 4 5 0 0 . 0 1 2 1 7 » 4 4 6 . 3 7 3 0 6 2 8 4 . 6 6 8 6 8 4 6 . 3 7 0 8 4 2 8 6 . 9 7 4 9 1 1 2 0 . 0 2 c . a . o . 0 . 0 3 0 8 9 0 . 0 2 0 3 9 1 3 1 . 0 0 0 0 0 c . a . o . 0 . 9 9 9 0 1 0 . 2 8 2 4 0 1 4 0 . 3 3 1 8 3 0 . 1 0 8 4 8 0 . 3 3 1 8 3 0 . 1 0 8 9 3 L o g l i k e l i h o o d V a l u e s L c - 3 2 9 3 7 . 8 4 6 - 3 2 9 3 7 . 8 4 6 - 3 3 2 2 1 . 0 0 9 - 3 8 2 7 . 9 1 2 1 - 3 5 3 6 . 6 9 2 L = L C + L d - 3 6 7 6 5 . 7 5 8 - 3 6 7 5 7 . 7 0 1 norm. — p a r a i n e t c r h a s b e e n n o r m a l i z e d ( s e e C h a p t e r 3 ) . fixed — p a r a m e t e r h a s b e e n s e t t o a fixed v a l u e . c.a.o. — p a r a m e t e r i s c o n s t r a i n e d b y a b i n d i n g i n e q u a l i t y c o n s t r a i n t a t t h e o p t i i t h e l o g l i k e l i h o o d f u n c t i o n . 145 T A B L E XXXIII: ESTIMATED COEFFICIENTS FOR M O D E L 2 S t a g e 1 S t a g e 2 S t a g e 3 s t a n d a r d s t a n d a r d s t a n d a r d p a r a m e t e r c o e f f i c i e n t e r r o r c o e f f e c i e n t e r r o r c o e f f i c i e n t e r r o r U t i l i t y . F u n c t i o n P a r a m e t e r s <*01 1 6 6 . 2 6 2 0 3 0 . 2 4 7 9 0 1 6 6 . 2 6 2 0 3 f i x e d 1 6 6 . 3 0 5 3 8 0 . 6 5 9 3 8 » 0 2 3 6 9 . 6 2 1 9 8 1 . 8 4 5 7 4 3 6 9 . 6 2 1 9 8 f i x e d 3 6 9 . 6 2 2 0 0 4 . 1 6 4 9 0 • 0 3 - 3 1 S . 2 7 6 3 8 4 . 0 7 7 5 5 - 3 1 5 . 2 7 6 3 8 f i x e d - 3 1 5 . 2 7 6 4 0 7 . 6 7 8 1 2 M l - 2 . 2 3 0 7 2 0 . 0 0 3 5 7 - 2 . 2 3 0 7 2 f i x e d - 2 . 2 3 1 6 6 0 . 0 0 9 8 7 " 2 1 - 1 . 8 4 7 1 4 0 . 0 0 7 4 8 - 1 . 8 4 7 1 4 f i x e d - 1 . 8 4 6 8 8 0 . 0 1 9 6 5 " 3 1 0 . 9 1 S 7 S 0 . 0 1 4 7 2 0 . 0 9 1 S 7 S f i x e d 0 . 9 1 5 8 4 0 . 0 6 0 2 9 » 2 2 9 . 8 3 2 0 4 0 . 2 6 4 9 3 9 . 8 3 2 0 4 f i x e d 9 . 8 3 2 0 5 0 . 5 7 6 2 5 • 3 2 - 1 0 3 . 8 6 5 9 8 0 . 8 4 1 8 0 - 1 0 3 . 8 6 5 9 8 f i x e d - 1 0 3 . 8 6 6 0 0 1 . 8 2 1 8 5 • 3 3 2 4 3 . 9 0 3 5 1 2 . 4 4 7 0 1 2 4 3 . 9 0 3 5 1 fixed 2 4 3 . 9 0 3 5 0 3 . 6 8 2 3 3 fell 0 . 0 0 0 0 0 n o r m . 0 . 0 0 0 0 0 n o r m . b 1 2 0 . 0 0 2 9 6 0 . 0 0 0 7 9 0 . 0 0 3 0 1 0 . 0 0 1 7 8 b 1 3 0 . 0 0 7 0 3 0 . 0 0 1 7 6 0 . 0 0 7 1 1 0 . 0 0 4 2 1 b 2 1 0 . 0 0 0 0 0 n o r m . 0 . 0 0 0 0 0 n o r m . b 2 2 - 6 . 4 7 2 9 4 0 . 5 4 2 9 2 - 6 . 4 2 7 3 8 0 . 5 9 2 7 2 D e m o g r a p h i c S c a l i n g P a r a m e t e r s c l l 0 . 0 0 0 0 0 fixed 0 . 0 0 0 0 0 fixed 0 . 0 0 0 0 0 fixed c 1 2 0 . 0 0 0 0 0 fixed 0 . 0 0 0 0 0 fixed 0 . 0 0 0 0 0 fixed c 2 1 0 . 2 3 8 8 7 0 . 0 0 5 3 5 0 . 2 3 8 8 7 f i x e d 0 . 2 3 9 0 0 0 . 0 0 9 8 3 c 2 2 - 0 . 6 0 3 9 0 0 . 0 0 2 0 8 - 0 . 6 0 3 9 0 f i x e d - 0 . 6 0 3 6 S 0 . 0 0 6 4 9 c 3 1 0 . 2 3 4 6 1 0 . 0 0 8 0 9 0 . 2 3 4 6 1 f i x e d 0 . 2 3 4 5 9 0 . 0 1 5 1 3 c 3 2 0 . 0 8 S 2 4 0 . 0 0 4 2 9 0 . 0 8 5 2 4 f i x e d 0 . 0 8 5 1 7 0 . 0 0 8 3 9 c 4 1 - 0 . 4 7 1 2 2 0 . 0 6 4 0 1 - 0 . 0 2 0 1 2 0 . 1 4 4 4 9 c 4 2 1 . 1 2 1 8 8 0 . 0 9 1 9 5 1 . 1 1 6 7 5 0 . 1 2 9 1 3 c 5 1 - 0 . 2 8 2 0 6 0 . 0 6 8 5 5 - 0 . 2 7 7 5 6 0 . 0 7 6 8 9 C S 2 - 0 . 0 1 7 4 9 0 . 1 0 8 9 7 - 0 . 4 6 5 4 3 0 . 0 7 7 1 2 T r a n c a n c t i o n C o s t s C o n s t a n t T e r m s T l - 4 . 2 5 1 8 9 1 . 3 8 5 7 6 - 4 . 2 5 1 8 9 fixed - 4 . 2 5 1 8 9 6 . 1 5 4 6 0 X 2 - 2 2 . 9 6 6 4 9 1 . 8 0 5 1 4 - 2 2 . 9 6 6 4 9 fixed - 2 2 . 8 6 6 4 9 5 . 8 1 6 9 2 O p t i m a l L e n g t h s o f O c c u p a n c y 1 2 0 . 9 9 9 7 7 1 . 3 X 1 0 _ e 0 . 9 9 9 7 7 fixed 0 . 9 9 7 8 1 0 . 0 0 3 3 8 3 . 3 7 1 9 4 0 . 0 0 1 5 1 3 . 3 7 1 9 4 fixed 3 . 3 7 2 1 4 0 . 0 0 4 5 7 *i» 1 . 3 4 6 0 3 0 . 0 0 0 3 5 1 . 3 4 6 0 3 fixed 1 . 3 4 6 0 8 0 . 0 0 8 2 0 G E V D i s t r i b u t i o n P a r a m e t e r s « 1 0 . 0 2 0 0 0 n o r m . 0 . 0 2 0 0 0 n o r m » 2 0 . 1 7 0 3 S 0 . 0 3 0 6 1 0 . 0 2 0 3 9 0 . 0 1 3 5 0 » 3 0 . 0 2 1 7 4 0 . 0 0 9 5 4 0 . 0 0 3 5 8 0 0 0 1 2 3 U 0 . 0 0 3 6 7 0 . 0 0 0 5 7 0 . 6 8 3 6 3 0 . 2 9 7 1 6 12 0 . 8 7 2 1 1 0 . 2 2 7 9 4 0 . 3 6 3 6 4 0 . 8 7 5 0 5 13 0 . 2 3 7 9 5 0 . 5 3 3 0 8 0 . 9 9 9 0 2 0 . 3 1 9 1 1 n* 0 . 0 3 2 3 9 L o g ! i 0 . 0 0 2 0 2 k e i i h o o d V a 0 . 0 4 2 1 6 l u e s 0 . 0 1 4 2 3 Lc - 2 7 6 6 6 . 1 3 4 - 2 7 6 6 6 . 3 4 - 2 7 8 4 5 . 6 7 3 hd - 3 9 1 6 . 9 0 8 - 3 5 0 6 . 8 4 5 L = L C + L d - 3 1 5 8 3 . 0 4 2 - 3 1 3 5 2 . 5 1 7 norm. — p a r a m e t e r h a s b e e n n o r m a l i s e d ( s e e C h a p t e r 3 ) . fixed - p a r a m e t e r h a s b e e n s e t t o a f i x e d v a l u e . c.a.o. - p a r a m e t e r i s c o n s t r a i n e d b y a b i n d i n g i n e q u a l i t y c o n s t r a i n t a t t h e o p t i m u m o f t h e l o g l i l 146 A p p e n d i x C E S T I M A T E S O F H E D O N I C P R I C E E Q U A T I O N S The following table contains the coefficients and standard errors of the parameters in the twenty-one hedonic regressions estimated for this dissertation. The mnemonics for the variables are: Rooms - number of rooms in dwelling Age - construction date of dwelling Units - number of dwelling units in building (apartments only) Stories - number of stories in building (apartments only) QFAIR - building in fair condition QGOOD - building in good condition Bath - dwelling has a private bath with hot and cold running water Type 1 - single attached dwelling Type 2 - row house Type 3 - duplex Water - water payment included in rent Elec - electricity payment included in rent Gas - gas payment included in rent Oil - oil payment included in rent Parking - parking payment included in rent Condo - condominium charges Other - 'other' payments included in rent BusRooms - business rooms in dwelling 147 T A B L E X X X I V : H E D O N I C R E G R E S S I O N R E S U L T S Constant Rooms Age Units Stories (Rooms x Rooms)/2 Rooms x Age Owner Occupied Dwellings: 1. Singles Hamilton 21801.' -2614.1 -175.49 957.14' 47.030* (10968.) (2787.8) (165.5) (455.55) (22.494) London 46588. -7158.5* -182.39 1404.0* 5.7319 (11483.) (2540.8) (281.07) (3.8802) (16.903) Toronto 42061.' 1859.1 -1208.4* 311.26 31.851 (9455.5) (2412.3) (351.96) (366.04) (24.539) Windsor 18756.* -2154.0 -343.15 630.19 84.577* (8292.9) (2040.1) (183.17) (349.44) (18.159) 2. Multiples Hamilton 70109.' -16489.* 234.68 2625.5* 32.127 (25333.) (6702.2) (312.83) (886.25) (32.600) London 27069.* (7416.5) 1528.7 (1147.4) 3.0721 (72.155) Toronto 31137. -859.78 -182.71 932.28 22.343 (21130.) (5080.6) (260.96) (591.61) (21.927) Windsor -1348.5 -977.88 1357.4* 2295.8* -159.12* (16135.) (5277.1) (324.14) (1056.7) (66.753) 3. Apartments Toronto -16932. 12691. 2048.9* -1111.4* 14360. -1394.5 53.475 (42805.) (10118.) (1034.6) (280.19) (34119.) (1537.4) (102.33) Rooms x Units Rooms x (Age x Stories Age)/2 5.0949 (3.2358) 10.332* (2.5074) 20.482* (4.5521) 8.4460* (2.5620) 1.7523 (5.7933) -2.4837 (4.9535) -12.031 (8.3594) 2.1802 -2397.1 -63.524* (18.907) (2100.9) (19.306) T A B L E XXXIV: HEDONIC REGRESSION RESULTS (continued) Constant Rooms Age Units Stories (Rooms x Rooms x Rooms x Rooms x (Age x Rooms)/2 Age Units Stories Age)/2 enant Occupied Dwellings: Singles Hamilton 110.30* 28.361* -5.5700* -3.9128* .16868 .14325* (36.192) (10.755) (1.4997) (1.8592) (.16165) (.04339) London 50.189 25.647* -4.0338* -1.1376 .07077 .11301* (31.354) (9.6662) (1.2939) (2.1270) (.17716) (.0340) Toronto -33.103 61.193* -3.3251 -2.8398 -.50588 .17877* (108.61) (29.692) (2.6916) (4.9694) (.35319) (.06084) Windsor 129.03* 12.969 -4.6597* .21820 .05496 .13595* (44.753) (10.397) (1.4140) (1.8378) (.13733) (.03379) Multiples Hamilton 37.890 29.019 -1.3481 -4.6484 -.02326 .07396* (34.342) (16.263) (1.3901) (3.5083) (.17961) (.03310) London 53.905 12.333 -2.0108 -.23625 .19701 .06015* (30.484) (9.7023) (1.0501) (2.9936) (.14425) (.01640) Toronto 42.936 31.629* .46166 (45.161) (7.4100) (.40356) Windsor -29.385 45.976* .41959 -7.9514* .49106* -.05157 (33.677) (12.996) (1.4719) (2.9326) (.09978) (.03447) Apartments Hamilton 17.060 19.566* 1.7480* -1.1534* 2.2403 .82391 -.01811 .01509 5.8080* .00873 (14.314) (5.9522) (.63770) (2.4821) (10.193) (1.5272) (.06041) (.01519) (2.2430) (.00792) London 7.4939 18.165* -.77576 1.4941 -90.848* 2.8471 -.06016 .08029* -2.7631 .04126* (15.413) (8.2646) (.41903) (.89942) (41.501) (1.9440) (.06255) (.03175) (3.7121) (.00799) Toronto 43.978 4.9047 2.8423 -.83269* -.16176 8.4395* -.05069 -.03421 .80893 -.01383 (27.907) (7.7657) (2.2703) (.40775) (13.615) (2.0454) (.10257) (.04024) (.63175) (.01513) Windsor 12.072 27.244* -.60021 -.02852 87.733 -.38653 .16057* .15131* -14.866* .00677 (21.538) (6.4303) (.73164) (.47609) (56.952) (1.4665) (.05292) (.03074) (5.0199) (.01377) T A B L E X X X I V : H E D O N I C R E G R E S S I O N R E S U L T S (continued) Age x Age x (Units x Units x (Stories x QFAIR QGOOD Bath Type 1 Type 2 Units Stories Units)/2 Stories Stories)/2  A . Owner Occupied Dwellings: 1. Singles Hamilton London Toronto Windsor 2. Multiples Hamilton Lond on Toronto Windsor 3. Apartments Toronto 14.434* 222.99 .87994 -87.453* 1959.1 (4.8815) (601.37) (.46031) (42.912) (4001.6) 7566.7 (10408.) -9735.1 (12260.) -18542. (23193.) 18215.* (8627.5) 8604.0 (8891.0) -14643. (9548.8) -334.18 (8139.8) 13631. (7450.6) 22253. (17337.) 10.355 (9.1840) -20072. (11703.) 1826.9 (3206.8) -14919. (10495.) - 19954. (10223.) T A B L E X X X I V : H E D O N I C R E G R E S S I O N R E S U L T S (continued) Age x Age x (Units x Units x (Stories x Units Stories Units)/2 Stories Stories)/2 QFAIR QGOOD Bath Type 1 Type 2 B . Tenant Occupied Dwellings: 1. Singles Hamilton London Toronto Windsor 2. Multiples Hamilton London Toronto Windsor 3. Apartments Hamilton London Toronto Windsor -.00347 (.00200) -.01160* (.00244) .00843 (.00569) -.00735* (.00216) -.14116 (.18880) 1.6120* (.55902) -.12030 (.08512) -.39179 (.71666) .00105* (.00043) -.00080 (.00084) .00003 (.00084) .00014 (.00106) .06101 (.03174) .03326 (.10123) -.00053 (.00427) -.21959 (.12710) -20.483* (5.2997) -15.398 (12.646) -.04843 (.10222) 23.991 (12.529) 100.04 (52.986) -12.083 (24.459) -36.938 (21.350) 5.4521 (26.756) 36.712 (21.091) -20.088 (46.421) -19.863 (23.473) 71.793* (28.410) -10.146 (20.392) 12.670 (12.993) 8.5734 (13.658) 13.090 (25.298) -6.1985 (16.319) 46.917* (7.5281) 63.928* (18.170) 59.907' (20.108) 76.052* (9.4513) -24.284 (9.5206) 63.155 (26.332) 127.31 (92.744) -64.854 (46.357) -27.242 (26.573) -5.2815 (23.120) 7.2513 (7.9235) T A B L E X X X I V : H E D O N I C R E G R E S S I O N R E S U L T S (continued) Type 3 Owner Occupied Dwellings: 1. Singles Hamilton London Toronto Windsor 2. Multiples Hamilton London Toronto Windsor Rooms x QFAIR 6859.8 (3565.9) 16058. (13266.) -2099.3 (1706.0) 1179.6 (1991.0) 3453.2 (3726.4) -1620.2 (1176.7) Rooms x Rooms x Rooms x Rooms x Rooms X Age x QGOOD Bath Type 1 Type 2 Type 3 QFAIR -605.64 (1488.6) 3022.3* (1541.8) 851.20 (1355.9) -554.38 (1039.6) -3929.1 (3106.9) Age x Age x QGOOD Bath 4103.4* (1926.2) 850.12 (1802.7) 515.30 -3354.7 (1059.3) (3330.5) -135.95 (153.00) 50.487 (236.39) 142.22 (346.55) -322.06* (148.46) 85.707 (96.239) -27.279 (225.87) 434.00 (296.44) -213.99 (131.07) 400.27* (95.913) T A B L E X X X I V : H E D O N I C R E G R E S S I O N R E S U L T S (continued) Type 3 Rooms x Rooms x Rooms x Rooms x Rooms x Rooms x Age x Age Age X QFAIR QGOOD Bath Type 1 Type 2 Type 3 QFAIR QGOOD Bath B . Tenant Occupied Dwellings: 1. Singles Hamilton London Toronto Windsor 2. Multiples Hamilton London Toronto Windsor 3. Apartments Hamilton London Toronto Windsor -12.869 (80.841) 7.4545 (4.7970) 16.460 (5.1051) 13.661 (23.120) -3.0926 (4.9782) .02589 (5.6785) 5.2062 (6.8964) 5.2224 (4.9556) -13.114 (7.8921) -1.-7832 (3.2184) 2.8311 (4.5627) -4.2226 (6.7727) -1.5954 (3.4724) -7.0699 (4.6023) -16.957* (7.1981) -13.334 (5.7538) -17.032' (3.0331) -18.596* (4.6115) 70.636 (39.136) 10.550 (8.3438) 12.948* (5.1122) .32374 (.46191) -.79375 .15950 (1.0281) (.90822) -.25151 -.20832 (.56124) (.27286) -.64411* -.10266 (.32420) (.32262) -.39981 (.20665) -.18878 (.22132) .97768' (.40111) 1.0437* (.29053) -1.1728* (.44865) .03710 (.22434) -2.8613 (2.1983) -.46956 (.20065) T A B L E X X X I V : H E D O N I C R E G R E S S I O N R E S U L T S (continued) Age x Age x Age x Units x Units x Stories x Stories x Water Type 1 Type 2 Type 3 QGOOD Bath QGOOD Bath  Owner Occupied Dwellings: 1. Singles Hamilton London Toronto Windsor 2. Multiples Hamilton —153.63 (86.114) London Toronto 11.003 (73.160) Windsor 150.18 160.51 (124.93) (148.85) T A B L E X X X I V : H E D O N I C R E G R E S S I O N R E S U L T S (continued) Age X Age X Age x Units x Units x Type 1 Type 2 Type 3 QGOOD Bath Stories x Stories x Water QGOOD Bath  B . Tenant Occupied Dwellings: 1. Singles Hamilton London Toronto Windsor 2. Multiples Hamilton London Toronto Windsor 3. Apartments -.90702 (.52231) -2.2739 (1.3034) .38152 (.42833) -.26444 (.35654) -21.760* (9.2180) -3.6138 (13.428) 70.935 (27.567) -7.7052 (7.8691) 8.1149 (5.4970) -12.484* (4.7259) 20.985 (17.575) 12.058* (4.7798) Hamilton .29703* 1.1538* -10.894* (.06903) (.21155) (2.0337) London -.88938 .06644 10.914* (.85613) (.22817) (3.9147) Toronto .47962 -.05689 -9.6581 17.220 9.5539 (.32390) (.34453) (6.4761) (12.421) (6.4431) Windsor .12305 .08785 15.457* (.48053) (.05031) (2.7384) T A B L E X X X I V : H E D O N I C R E G R E S S I O N R E S U L T S (continued) Elec Gas Oil Parking Condo Other BusRooms R2 A . Owner Occupied Dwell ings: 1. Singles Hamilton .5022 London .4722 Toronto .4789 Windsor .4680 2. Multiples Hamilton .5180 London .0548 Toronto .3968 Windsor -1352.1 .5482 (2740.7) 3. Apartments Toronto -2540.9 .5369 (2289.0) T A B L E X X X I V : H E D O N I C R E G R E S S I O N R E S U L T S (continued) Elec Gas Oi] Parking Condo Other BusRooms P? B . Tenant Occupied Dwellings: Hamilton 3.5382 15.304 -8.1423 28.369 .4427 (16.550) (12.978) (14.549) (14.567) London 18.797 20.290 21.176 -9.9041 .4840 (14.046) (13.289) (13.202) (6.8054) Toronto -41.209 -54.371* 11.257 7.6666 .3500 (26.059) (18.909) (23.168) (37.710) Windsor 11.338 -.81901 -72.368 .4575 (13.092) (11.933) (49.941) Multiples Hamilton 9.5690 4.1778 -7.2017 13.577 .7695 (6.6251) (7.0188) (6.2830) (9.1538) London 10.227* 15.554* 1.4397 .21407 1.4288 .6748 (4.6195) (4.0060) (6.7851) (3.9793) (7.6669) Toronto 22.043 8.0899 -29.072 -2.6725 .4218 (13.992) (20.179) (16.956) (15.064) Windsor -26.398* 39.605* 3.3691 10.702 .5748 (5.6730) (6.9251) (6.0353) (10.557) Apartments Hamilton 7.6825* -1.0253 -3.5950* 7.7285* -.67857 .6249 (2.0726) (1.7156) (1.6545) (2.1069) (1.9894) London 2.2620 2.2277 1.6616 5.2093* -9.1252 .6051 (2.3823) (1.6108) (3.5685) (1.8713) (5.2089) Toronto -3.1507 -12.701* -16.143* 12.930* -1.1319 83.620 .4254 (3.3758) (3.3935) (2.9670) (2.9859) (3.2811) (44.797) Windsor 1.3933 14.380 5.1641* -20.440* -6.3170 .6790 (2.2294) (10.754) (2.6061) (7.3212) (4.3356) A p p e n d i x D E S T I M A T E S O F H E D O N I C P R I C E E Q U A T I O N S — L I N E A R Q U A D R A T I C M O D E L 158 T A B L E X X X V : H E D O N I C R E G R E S S I O N R E S U L T S — L I N E A R Q U A D R A T I C M O D E L Constant Rooms Age U nits Stories i Rooms2 \Age2 ^Units2 ^Stories2 A . Owner Occupied Dwellings: 1. Singles Hamilton 15960. -2520. 108.5 1176.* 7.528* (8978.) (2692.) (128.0) (435.1) (2.988) London 29180. -4290. -178.4 1449.* 10.53* (6811.) (2254.) (105.1) (354.6) (2.515) Toronto 25870. 2069. -704.1* 618.4 23.50* (7094.) (2149.) (180.3) (344.0) (4.185) Windsor 16180. -1936. -132.5 1106.* 10.55* (5891.) (2009.) (114.1) (344.2) (2.573) 2. Multiples Hamilton 72090.* -20450.* 487.6 3941.* -4.991 (22570.) (6953.) (250.8) (1142.) (5.938) London 24310.* (5122.) 2148.* (989.1) -20.27 (91.36) Toronto 46070. -5026. 243.8 1187.* .00140 (13760.) (4031.) (229.5) (574.0) (5.198) Windsor 9 3 1 4 . 2 1 3 7 . 756.6* 211.7 -18.09* ( 1 3 4 6 0 . ) (4799.) (316.7) (808.7) (7.667) 3. Apartments Toronto 37120. 13230. -1128. -66.20* 8241. -1505. 15.34 .2083* -4338. (24340.) (7355.) (778.7) (31.89) (4224.) (1356.) (16.07) (.0942) (2575.) T A B L E X X X V : H E D O N I C R E G R E S S I O N R E S U L T S (continued) Constant Rooms Age U nits Stories | Rooms2 \Age2 \Unils2 1 Stories1 B . Tenant Occupied Dwellings: 1. Singles Hamilton 93.14* (34.63) 31.85* (10.32) -5.116* (1.530) -3.593 (1.917) .1557' (.0400) London 83.43* (32.59) 10.56 (12.27) -3.739* (1.142) 13.90 (2.542) 1.964 (.0305) Toronto 12.22 (116.4) 55.50 (31.58) -5.405 (2.798) -4.436 (4.992) .1601* (.0592) Windsor 102.4* (28.02) 23.99* (5.600) -4.886* (1.430) -1.256 (1.429) .1498* (.0336) 2. Multiples Hamilton . -8.083 (35.36) 39.21* (14.09) -1.992* (.9510) -4.815 (2.799) .08671* (.0239) London 37.77 (24.99) 20.50* (9.043) -2.340* (.6588) .2144 (1.894) .07532* (.0164) Toronto 43.13 (45.16) 31.60* (7.410) .4621 (.4036) Windsor -48.34 (43.59) 38.70* (17.12) 2.841 (1.477) -3.565 (3.699) -.05346 (.0348) 3. Apartments Hamilton 56.45* 8.348 .4399 .0999* 22.05* 2.126 -.0012 .0006 -15.01* (9.956) (5.345) (.3001) (.0431) (3.547) (1.366) (.0070) (.0003) (3.405) London 21.05* 11.59 -.4556 .3067* 6.708 1.278 .0208 -.0012* -7.727 (10.61) (6.606) (.3058) (.0582) (6.307) (1.744) (.0074) (.0006) (7.502) Toronto 94.09* .1548 .0383 -.0682 3.792* 5.435* -.0028 .0002 -.1007 (17.17) (6.556) (.6953) (.0693) (1.422) (1.786) (.0169) (.0002) (.0720) Windsor 3.924 24.06* .6500 .4775* -15.89* -1.245 .0008 -.0024* 6.529 (15.28) (5.028) (.5973) (.0637) (7.439) (1.438) (.0137) (.0004) (3.970) T A B L E X X X V : H E D O N I C R E G R E S S I O N R E S U L T S (continued) QFAIR QGOOD Bath Type 1 Type 2 Type 3 Water Elec Gas A . Owner Occupied Dwellings: 1. Singles Hamilton 133.3 7192.* (2194.) (1475.) London -1832. 2800. (3785.) (3518.) Toronto 5949. 16070* (5297.) (2223.) Windsor -825.2 4091.* (1766.) (5891.) 2. Multiples Hamilton London Toronto Windsor 3. Apartments Toronto -3536.* (1630.) 1784. (2075.) -8709.* (1643.) -7158.* (1315.) 4352. (3702.) -3629. (2090.) T A B L E X X X V : H E D O N I C R E G R E S S I O N R E S U L T S (continued) QFAIR QGOOD Bath Type 1 Type 2 Type 3 Water Elec Gas B . Tenant Occupied Dwell ings: 1. Singles Hamilton 9.782 -22.38* -21.98* 5.252 14.73 (9.031) (9.636) (9.145) (15.93) (12.49) London 13.90 -35.33* -8.885 16.75 24.85 (8.087) (9.946) (14.53) (14.27) (13.85) Toronto 35.62 71.28* -39.07 -54.22* (22.32) (28.83) (26.72) (18.29) Windsor 13.21 11.44 3.871 -11.66 14.30 -1.688 (12.44) (7.166) (30.98) (7.420) (15.38) (14.78) Multiples Hamilton 1.538 11.40 23.28* 6.676 11.08 7.464 (6.695) (11.47) (7.827) (5.133) (6.681) (6.525) London 11.61 32.40* 18.83* -11.73* 10.04* 14.10* (6.524) (5.118) (6.125) (4.580) (4.771) (3.909) Toronto -10.14 -5.256 13.69 20.92 22.04 8.050 (20.39) (14.12) (23.12) (17.57) (13.99) (20.18) Windsor 13.27' 8.817 -3.047 -27.18* 43.56* 2.973 (4.819) (8.374) (5.151) (6.203) (7.133) (6.288) Apartments Hamilton 3.694 13.72* -10.02* 7.250* -1.196 (3.707) (6.988) (2.013) (2.065) (1.751) London 9.017* 27.19* 12.23* .3006 2.199 (3.812) (5.740) (4.085) (2.349) (1.709) Toronto 4.408 17.06 7.326 -2.773 -11.71* (6.965) (9.895) (6.638) (3.462) (3.491) Windsor 14.19* 12.21 -3.672 14.70* 3.158 (3.529) (7.679) (4.457) (2.728) (2.330) T A B L E X X X V : H E D O N I C R E G R E S S I O N R E S U L T S (continued) Oil Parking Condo Other BusRooms R2 A . Owner Occupied Dwell ings: 1. Singles Hamilton .4995 London .4688 Toronto Windsor .4691 .4486 2. Multiples Hamilton .4333 .0550 London Toronto Windsor .3832 -3629. (3152.) .4408 3. Apartments Toronto -3632. (3749.) .3690 T A B L E X X X V : H E D O N I C R E G R E S S I O N R E S U L T S (continued) Oil Parking Condo Other Bus Rooms R2 B. Tenant Occupied Dwellings: 1. Singles Hamilton London Toronto Windsor 2. Multiples Hamilton London Toronto Windsor 3. Apartments Hamilton London Toronto Windsor -6.522 (14.36) 19.33 (15.07) 8.172 (22.82) -8.410 (5.934) -2.229 (7.239) -29.09 (16.96) -4.047* (1.680) 2.337 (3.444) -15.17* (3.111) 10.96 (10.78) 27.10 (14.42) -8.706 (6.687) 15.11 (36.07) -19.89 (25.62) 11.47 (8.997) -.1254 (3.863) -2.681 (15.06) 5.182 (10.95) 7.900* (2.139) 5.121* (2.011) 12.21* (3.063) 7.029* (2.699) .2213 (7.596) -.8115 (2.006) -11.93' (5.165) -1.709 (3.270) -17.10' (6.845) 86.00 (47.16) .4396 .4742 .3431 .3334 .7582 .6607 .4218 .5424 .6100 .5718 .4008 .6447 T A B L E X X X V I : W H I T E S ' S T E S T F O R H E T E R O S C E D A S T I C I T Y O N L I N E A R Q U A D R A T I C M O D E L S H a m i l t o n L o n d o n Toronto W i n d s o r A . O w n e r Occup ied Dwel l ings: 1. Singles 2. Multiples 3. Apartments B . Tenant Occupied Dwel l ings: 1. Singles 2. Multiples 3. Apartments 1024.4* (18) 884.02* (16) 7.7 x 105* 134.13* (17) (10) 34.305* (14) 1283.4* (18) 79.048* 3.0 x 106* (10) (24) 1.1 x 109* (30) 3501.7* 2.1 x 106* (20) (36) 5.2 x 105* 65714.0* 27.393* 6.5 x 105' (64) (40) (15) (29) 21671.0* 21918.0* 13472.0* 91573.0* (68) (63) (88) (61) 4.5 x 105* 68989.0* (37) _ (33) * Statistically significant at the 5% level of significance. 165 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0097505/manifest

Comment

Related Items