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The density and income patterns of metropolitan Vancouver Wiebe, Gary Bernard 1988

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THE DENSITY AND INCOME PATTERNS OF METROPOLITAN VANCOUVER by GARY BERNARD WIEBE B.Sc, The University of British Columbia, 1982 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in Urban Land Economics THE FACULTY OF GRADUATE STUDIES Commerce We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1988 © Gary Bernard Wiebe, 1988 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. Commerce The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: October 1988 ABSTRACT It is the belief in the discipline of Urban Land Economics that as one moves further from the city center population density decreases and average household income increases. These two hypotheses have shown to be accurate in describing cities in the United States, but few studies have been done to see if the two hypotheses are also true for Canadian cities. The general intent of the thesis, therefore, was to properly model the population density pattern and income pattern of Metropolitan Vancouver to see how well they could be explained and to see if they followed the patterns of American cities. In order to address the general intent, several specific issues dealing with density and income studies had to be examined: the functional form of the models, the best proxy of access (straight-line distance or time spent in travel to the city center), whether determinants other than distance should be used in the density equation, and whether Metropolitan Vancouver should be modelled as a monocentric or multi-centric city. The techniques applied to answer these questions and fulfil the general intent included reviewing the literature, applying theory to develop models and then using ordinary least squares to test the models. The results were very good. Although no functional form could be derived for the income pattern, the negative exponential form proved, theoretically and practically, to work well for the density pattern. The distance variable was a better determinant of density than the travel time variable. Two variables, income and ii distance, proved to be the best determinants of population density by explaining almost half of the variation in population density. Finally, Metropolitan Vancouver was shown to be a multi-centric region but added effects of the extra center did little to help explain the density patterns. The results also showed that population densitj' in Metropolitan Vancouver does decrease and, although not conclusive, income does generally increase with distance from the city center. These facts support the hypotheses and suggest that the density and income patterns are much like those of major U.S. cities. m TABLE OF CONTENTS Abstract ii List of Tables vi List of Figures vii Acknowledgement , viii Chapter I. INTRODUCTION 1 Chapter II. LITERATURE REVIEW 6 1. INTRODUCTION 6 2. EARLY OBSERVATION 6 3. THEORY (MUTH) 7 4. THEORY (MILLS) 8 5. FUNCTIONAL FORM AND BIAS 11 6. USE OF NEGATIVE EXPONENTIAL FORM 13 7. CONCLUSION 16 Chapter III. THEORY 17 1. INTRODUCTION 17 2. ASSUMPTIONS 17 3. SINGLE INCOME RENT GRADIENT 18 4. VARIABLE INCOME GRADIENT 20 5. DENSITY GRADIENT 21 6. FUNCTIONAL FORM 24 7. CONCLUSION 31 Chapter IV. THE NEGATIVE EXPONENTIAL DENSITY MODEL 32 1. INTRODUCTION 32 2. DENSITY MODEL 32 3. DATA 33 4. TESTING 34 a. Autocorrelation 35 b. Heteroscedasticity 41 c. Normality 44 5. CONCLUSION 49 Chapter V. BASIC MODEL USING TRAVEL TIME 51 1. INTRODUCTION 51 2. THEORY 51 3. DATA 52 4. MODEL AND TESTING 53 5. ROTHENBURG MODEL 55 6. CONCLUSION 58 Chapter VI. THE INCOME GRADIENT AND ADDED DETERMINANTS 60 1. INTRODUCTION 60 IV 2. INCOME MODEL 61 3. DATA 61 4. TESTING 63 5. EXTENSION OF BASIC MODEL 64 6. ADLER MODEL 65 7. ADLER ESTIMATION 66 8. DATA 67 9. ESTIMATION 67 10. REVISED MODEL 69 a. Multicollinearity 71 11. REVISED REGRESSION 73 12. CONCLUSION 77 Chapter VII. DENSITY PATTERNS IN A MULTI-CENTRIC CITY 80 1. INTRODUCTION 80 2. IDENTIFICATION OF SUBCENTERS 80 a. Introduction 80 b. Theory 81 c. Identification 85 3. THEORY AND MODEL 90 4. ESTIMATION USING DISTANCE 91 a. Data 91 b. Testing 91 5. ESTIMATION USING TIME 93 6. CONCLUSION 95 Chapter VIII. CONCLUSION 97 1. INTRODUCTION 97 2. SUMMARY 97 3. FINAL CONCLUSIONS 102 BIBLIOGRAPHY 104 v List of Tables Table 1: OLS Regression Results 35 Table 2: Autocorrelation Corrected Regression Results 41 Table 3: Diagonal Elements of the Projection Matrix 48 Table 4: OLS Regression Results 54 Table 5: Autocorrelation Corrected Regression Results 55 Table 6: Autocorrelation Corrected Regression Results 63 Table 7: Autocorrelation Corrected Regression Results 68 Table 8: Autocorrelation Corrected Regression Results 70 Table 9: Autocorrelation Corrected Regression Results 73 Table 10: Diagonal Elements of the Projection Matrix 74 Table 11: Autocorrelation Corrected Regression Results 75 Table 12: Autocorrelation Corrected Regression Results 92 Table 13: Autocorrelation Corrected Regression Results 94 v i List of Figures Figure 1: Squared Residuals vs. Predicted Values Plot 43 Figure 2: Residual Histogram 46 Figure 3: Income vs. Distance Plot 62 Figure 4: Density vs. Distance Plot 82 Figure 5: Density vs. Distance Plot 83 Figure 6: Employment Density vs. Distance Plot 86 Figure 7: Employment Density vs. Distance Plot 87 Figure 8: Employment/Residential Ratio vs. Distance Plot 88 Figure 9: Emploj^ment/Residential Ratio vs. Distance Plot 89 vn A C K N O W L E D G E M E N T I would like to express my sincere gratitude to the members of my thesis committee, Dr. Robert Helsley, Dr. Dennis Capozza and Dr. Walter Hardwick. Special thanks goes to Dr. Robert Helsley whose patience, assistance and feedback allowed me to complete this project. Finalty, the completion of my degree and this thesis would not have been possible without the financial support of The Canadian Mortgage and Housing Corporation and The Real Estate Council of British Columbia. I am deeply obliged to these organizations. viii CHAPTER I. INTRODUCTION Almost from the beginning of the discipline of Urban Land Economics two hypotheses have been suggested. These two hypotheses state that as one moves further from the city center urban population density decreases and average household income increases. Over the many years a multitude of studies have been done on population density but relatively few have been done on the income gradient. These studies for the most part have supported the two hypotheses so that now the hypotheses are basically taken as fact. The problem is that almost all of these studies have been done using data from cities in the United States while virtually none have used data from Canadian cities. Yet, when the two hypotheses are stated they are said to hold true for major North American cities in general. It is not practical to believe that all hypotheses which describe cities in the United States also describe Canadian cities. This fact is brought out by Goldberg and Mercer1 who challenge the assumption of a typical North American city. They suggest that besides differences in values, culture, and social and demographic structure of the populations in Canada and America, there are differences in the way cities in the two countries developed. The differences are due to a variety of factors. These factors include the aforementioned differences in population, the differences in political systems and government intervention in 1 Michael A. Goldberg and John Mercer, The Myth of the North American City (Vancouver: University of British Columbia Press, 1986). 1 2 the development of cities, the difference in the production of freeways and transportation systems which allow access to parts of the city, the difference in economies and the difference in the historical development of the two countries. The general intent, therefore, of this thesis is to properly model not only the density pattern, but also the income pattern of Metropolitan Vancouver. These patterns would then be examined in order to refute or support two hypotheses which hold true for many major cities in the United States but have not yet really been tested for Canadian cities. Besides the general issues, there are several specific issues dealing with density and income studies that must be addressed. One of these issues is the identification of the proper functional form of the models. Most densitj' studies have used the negative exponential specification and it has performed remarkably well. The reason why the negative exponential specification should be used must be examined theoretically, not used simply because it is the one that works best. Another important issue pertains to identifying the proper determinants of density. Most studies in their modelling have used a single determinant or variable, namely, straight-line distance to the city's core. Few researchers have allowed for the use of two determinants, while even fewer have used three or more. These determinants could include income, tenure choice, age, etc. Which determinants to use and how many are to be in the gradient must, therefore, be resolved through testing. 3 One determinant already mentioned, which is always included, is distance to the city center. This variable is to be a proxy for access from one's place of residence to one's place of employment, the central business district (CBD). The question must be asked, however, whether distance is the best proxy of access. It may be that travel time, the actual amount of time spent in getting to the CBD is a better proxy for access. This issue must also be addressed and resolved using a combination of theory and testing. A final issue of major importance is whether Metropolitan Vancouver can be best described as a monocentered or multi-centered city. Most density studies have considered cities which are monocentric and have observed the denshty as one moves away from this city center. Vancouver, however, may be best described as having multiple centers. These centers would then interact giving density patterns which are difficult to explain using a single center model. This issue must be addressed using mainly observation and testing. The Second Chapter reviews a significant portion of the literature associated with description of urban population density and income patterns. A complete examination of all the relevant literature would be virtually impossible but several benchmark articles are reviewed in order that the major issues can be addressed and the pattern of progression can be observed. Most of the articles reviewed are empirical in nature but several major works are examined in order to provide the theory that is necessary. In Chapter Three, the theory drawn from these works is presented and a model showing 4 why population density should decrease while residential income should increase with distance from the city center is developed. The chapter goes on to further develop the model in order to theoretically explain why the negative exponential specification should be used to describe urban population density. Chapter Four takes the model developed in Chapter Three and applies it to Metropolitan Vancouver. The model is tested using 1981 data in order to see whether it can successfullj' describe the population density pattern of the area. The use of distance as the proxy for the disutility of commuting is examined in Chapter Five. Time spent in travel would seem to be the better measure and, therefore, replaces the distance variable in the basic model. The model is then tested and the results are compared to those received in Chapter Four where straight-line distance is used. Chapter Six attempts to develop a model which describes the income pattern of Metropolitan Vancouver. This description is not possible and, therefore, the income pattern is deduced by observing how income relates to several key variables. This task is accomplished by adding income and tenure choice to the basic model as determinants of population density. The new model is then tested and the results not only allow the income pattern to be indirectly observed, but show what happens when besides distance, the standard single determinant, more determinants are used to explain the variation in population density. The assumption of a monocentric city is abandoned in Chapter Seven. 5 Metropolitan Vancouver is examined to see if and where urban subcenters are located. A model is then developed and tested to see if including subcenters can better explain the density pattern. The issue of travel time versus distance as the proper proxy for access is re-examined by using both variables separately in the testing process. Finally, Chapter Eight summarizes the findings and concludes the study. CHAPTER II. LITERATURE REVIEW 1. INTRODUCTION The geographical distribution of urban population densities and income groups has been an area of study for many years and has been considered by man}' authors. Because of so many studies, a complete literature review dealing with all articles relating to these topics would be virtually impossible. Several studies remain, however, as benchmarks in this area and a clear pattern of progression evolves. 2. EARLY OBSERVATION Colin Clark 2 in 1951 was the first to provide a methodical empirical investigation of urban population densities. He used census tract data for a number of years in the nineteenth and twentieth centurj' which was available for cities in Europe, Australia and the United States. Taking the central business district as the center, Clark drew concentric rings around the core of each city at intervals of one mile. Clark then took the average density in each ring, excluding the CBD, and regressed the natural log of this density on distance from the core. Although Clark did conclude that density declined exponentially with distance from the CBD, he gave no theory to suggest why this occurred. * C o l i n Clark, "Urban Population Densities," Journal of Economic Theory 114 (1951): 375-386. 6 7 3. THEORY (MUTH) It was not until 1961 that a model was developed giving theoretical justification to Clark's observations. Richard Muth 3 showed that if land rents would decline exponentially then population densities will follow suit. This result, however, depended upon the imposition of several restrictive assumptions. Muth's model assumes a single central market surrounded by nothing but housing. All transactions take place in this market and travel expense is equal in any direction from the market. By assuming transport costs increase at a decreasing rate as one moves from the central market, Muth stated that housing prices must decline. He demonstrated that for all the identical residents to be in equilibrium, the extra travel costs must be exactly offset by the savings due to lower housing prices. The analysis was then further expanded to include the production sector. Virtually the same argument applies, namely that the extra costs of transporting goods to the market must be offset by lower land rents as one moves further from the central market. Since land rents are more expensive near the central market, producers substitute non-land factors for land to increase their profits. This action in turn increases densities. Muth tested these theories using a simple negative exponential model D(k) = D 0 exp(-gk). In this model D is gross population density, D 0 is central density, g is the density gradient and k is the distance to the city center. Using 1950 data, ~* Richard Muth, "The Spatial Structure of the Housing Market Papers," Regional Science Association 7 (1961): 207-220. 8 the average density of twenty-five census tracts for each of the forty-six U.S. cities was used as the sample and applied in the regressions. Results showed that a little less than half of the variation in density could be explained by distance from the city center alone. In all but six of the forty-six cities, the distance parameter or the density gradient, g, was significantly greater than zero at the 0.01 level. In 1969, Muth " again addressed the issue of population density gradients. He drew upon the results of his 1961 study in order to compare and stud}7 patterns of density gradients. Soon after this date other studies began to reinforce the idea that population densities could be approximated by the negative exponential functions. These studies included those by Niedercorn 5 and Mills 6 7 8 9 whose economic model of urban structure gave a theoretical justification to the negative exponential density function. 4. THEORY (MILLS) Mills improves greatly on the Muth model in that no assumption of price declining exponentially with distance is necessary. Mills considered explicit production functions for three industries, the export sector, housing, and " Richard Muth, Cities and Housing (Chicago: U of Chicago P, 1969). 5 John H. Niedercorn, "A Negative Exponential Model of Urban Land Use Densities and Its Implications for Metropolitan Development," Journal of Regional  Science 11 (1971): 317-326. 6 Edwin S. Mills, "Urban Density Functions," Urban Studies 7 (1970): 5-20. 7 —, "The Value of Urban Land, "The Quality of the Urban Environment, ed. Harvey S. Perloff (Baltimore: Johns Hopkins UP, 1969) 231-253. 8 —, Urban Economics (Glenview, Illinois: Scott, 1972). 9 —, Studies in the Structure of the Urban Economy (Baltimore: Johns Hopkins UP, 1972). 9 intra-urban transportation. Mills develops a model taking into account the substitution between land, labour, and capital and the competition for the limited land supply between these three industries. The main result of this model is that land rents decline exponentially with distance from the city center. This result, however, hinges on the assumption that the price elasticity of the demand for housing is negative one. Much has been written on this topic alone. Manjr authors disagree with the assumption and values from -1.28 1 0 to -0.22 1 1 have been reported. On the other hand, many authors do agree that although an exact negative one elasticity is highlj'- improbable, it is a very good approximation and can be used. 1 2 1 3 1 * In Mills' model it is easy to move from land rents to densities and, therefore, Mills suggests that by showing that land rents are approximated by the negative exponential function, population densities will follow suit. If, however, the price elasticity is not equal to negative one, the rent function as well as the density function is approximated by the binomial function. 1 5 Few authors have tested densities using the binomial function, perhaps because C. Fenton, "The Permanent Income Hypothesis Source of Income and the Demand for Rental Housing Analysis of Selected Census and Welfare Program Data to Determine Relations of Household Characteristics and Administrative Welfare Policies to a Direct Housing Assistance Program," (Cambridge, Mass.: Joint Center for Urban Studies, 1974) 1-52. 1 1 J. Friedman and D.H. Weinberg, "Demand for Rental Housing: Evidence From the Housing Allowance Demand Experiment," (Cambridge, Mass.: Abt Associates Inc. 1974). 1 2 Vernon J. Henderson, Economic Theor;- and the Cities (Toronto: Academic Press, 1985). 1 3 J. Maisel, J.B. Burnham and J.S. Austin, "The Demand For Housing: A Comment," Review of Economics and Statistics 53 (1971): 410-413. 1 4 Mills, Urban Density. 1 5 shown in detail in Chapter Three 10 those few have found it inferior to other functions. Using a test suggested by Clark, McDonald and Bowman 1 6 tested several alternative density functions by using them "to predict total population by computing the integral of the function within the appropriate limits". They found that the binomial was completely inferior to all other functions tested. They suggested that the high standard of error was due to the fact that four coefficients were estimated via non-linear least squares. Virtually none of the coefficients estimated were significant due to the high level of collinearity between the variables. Mills goes on in his model by discarding the assumption of uniform income for all residents. By assuming two levels of income, the effects of income differences on the household location pattern can be studied. Again Mills, using his superior model, confirms what Muth concluded, that is, "under realistic conditions high income households live further from the city center than do lower income households". 1 7 The pattern will be the reciprocal of land rents and densities, therefore, a positive function should approximate incomes. Like land rents the result of a positive income gradient also hinges on a crucial assumption of elasticity. For the result to hold true, the income elasticity of demand for housing must be greater than the income elasticity of commuting costs. Again, whether this condition holds true is one of debate with authors such as Mills, Muth and Beckman 1 8 suggesting it does hold true and other ~ r s John P\ McDonald and Woods H. Bowman, "Some Tests of Alternative Urban Population Density Functions," Journal of Urban Economics 3 (1976): 242-259. 1 7 Mills, Urban Economics 85. 1 8 Martin J. Beckman, "On the Distribution of Urban Rent and Residential Density," Journal of Urban Economics 5 (1974): 99-107. 11 authors, such as Wheaton, 1 9 suggesting it does not. 5. FUNCTIONAL FORM AND BIAS Several authors in the mid 1970's studied the validity of the negative exponential function as the proper tool to investigate population density functions. As mentioned, McDonald and Bowman tested several functional forms and concluded that although the negative exponential was adequate, several other forms, such as the general normal, gamma, and standardized normal, performed just as well. Kau and Lee 2 0 in 1976 further examined the functional form of the relationship between population densitj' and distance. Using the Mills model, the generalized functional form was derived. The negative exponential function is a special case of the generalized form and occurs when the price elasticity of demand for housing is negative one. Kau and Lee used the Box and Cox method to determine the optimum value of the functional form parameters. Box and Cox, 2 1 in accordance with the maximum likelihood method derived a maximum logarithmic likelihood for determining the functional form parameter. For the negative exponential form to be the optimal form, the functional form parameter should not have been ~ r 5 William C. Wheaton, "Income an Urban Residence: An Analysis of Consumer Demand for Location," The American Economic Review 67 (1977): 620-631. 2 0 James B. Lee and Cheng F. Lee, "Functional Form, Density Gradient and Price Elasticity of Demand for Housing," Urban Studies 13 (1976): 193-200. 2 1 G.E.P. Box and D.R. Cox, "An Analysis of Transformation," Journal of the Royal Statistical Society 114 (1964): 211-243. 12 significantly different from zero. Kau and Lee tested their theory by running twenty regressions for each of fifty U.S. cities. Each regression used a different functional form parameter ranging from -0.5 to 1.5. By plotting the twenty log maximum likelihood values received from each regression, the true functional form parameter can be determined by observing from the graph which gave the highest maximum likelihood value. After going through this procedure for each of fifty U.S. cities, it was found that slightly more than fifty percent of the cities had optimal parameters which were not significantly different from zero. This study suggests that for these cities the negative exponential functions were optimal in describing density patterns. For those cities with optimal parameters significantly different from zero, Kau and Lee suggest an equation based on the generalized functional form which should be used to obtain unbiased density gradients. Another form of bias was investigated by Frankena 2 2 in 1978. He suggested a problem many studies may incur is one of sampling bias. Frankena states that most neighborhoods are grouped to ensure census tracts have approximately the same population and/or density. This grouping results in an inverse relationship between area of tracts and density at any specific distance to the city's center. Because of this fact, the sample data will tend to over-represent high density areas and under-represent low density areas. Frankena tested for this inverse relationship and found it was significant and ~2_5 Mark Frankena, "Bias iii Estimating Urban Population Density' Functions," Journal of Urban Economics 5 (1978): 35-45. 13 positive for approximately ninety percent of the thirty U.S. cities sampled. He further went on to suggest that in order to correct for the sampling bias, the square root of the census tract area should be used to weight the observations. He showed that these weighted least squares estimates were unbiased when compared to the ordinar}' least squares estimates. Frankena also addressed the problem of heteroscedasticity as did Anderson.2 3 They suggested that in the suburbs, the densities are more homogeneous and, therefore, show less variance than inner city census tracts. This fact will give a variance in density which is inversely related to distance. To correct for heteroscedasticity one must determine how serious a problem it is and adjust it if it's form can be determined. Frankena simply corrected the problem of heteroscedasticity by multiplying his model by the square root of distance. Anderson went further by testing specifically for the pattern of heteroscedasticity and estimated after correcting for it. 6. USE OF NEGATIVE EXPONENTIAL FORM While the above authors and others continued to examine the accuracj' of the negative exponential function to describe urban population density, many authors have assumed that it is a correct approximation and have used it in various studies. John Anderson, "Estimating Generalized Urban Density Functions," Journal of Urban Economics 18 (1985): 1-10. 14 Couch,2" using a "base price" model in which the consumer can purchase non-housing goods and provide employment in various locations in a city, found that the negative exponential function was consistent with his findings in describing residential densities. Macauley 2 5 used the negative exponential function to update and expand the population and employment density gradients Mills 2 6 estimated for 1948 to 1963 in order to study recent patterns. Alperovich 2 7 2 8 applied the negative exponential function to densities for cities in Israel. Comparing the results to results achieved with various other functional forms, Alperovich concluded that the negative exponential function best fit the data. Alperovich continued his studies, and those started by Mills, 2 9 by trying to use variables such as city size, age of citj 7, transportation costs and income level to explain the density gradient once it had been estimated using the negative exponential functions. By doing this estimation, the effect of these variables on the density gradient can be studied. 2 "J.D. Couch "Residential Density Functions: An Alternative to Muth's Negative Exponential Model," Journal of Urban Economics 8 (1978): 16-31. 2 5 Molly Macauley, "Estimation and Recent Behaviour of Urban Population and Employment Density Gradients," Journal of Urban Economics 18 (1985): 251-260. 2 6 Mills, Studies. 2 7 Gershon Alperovich, "Determinants of Urban Population Density Functions: A Procedure for Efficient Estimates," Regional Science and Urban Economics (1983): 287-295. 2 8 —, "An Empirical Study of Population Density Gradients and Their Determinants," Journal of Regional Science 23 (1983): 529-540. 2 9 Mills, Studies. 15 Adler 3 0 uses an approach similar to Alperovich by expanding on the standard Mills negative exponential density equation with the use of several more determinants. Instead of a two step approach, however, Adler assumes the constant (CBD density) and gradient are determined by income and housing tenure choice. Adler's main results show that renters live closer to the city center than owner-occupiers but within each of these groups wealthier households live nearer to the center than lower income ones, a result which goes against prior wisdom by authors such as Mills 3 1 and Muth.3 2 Griffith 3 3 also expanded on the use of negative exponential function by applying it to population densities in a multi-centered city. While almost all of the above studies and theories dealt with a monocentric city, the presence of multiple centers suggests that each location is influenced simultaneously by all centers and the densities which result are affected by all centers. The model j = m k=j , Griffith estimated, therefore, had the form D. = L A. exp(-Z b., dv. ). In this 1 j = i J k = l J k 1J model D. is the population density at location i , d.^  is the distance separating i and center j and A. and b.^  are parameters associated with center j. Griffith found that for Toronto, the city he tested, the CBD swamped all other centers. Once the population density pattern with the CBD was accounted for, the four remaining centers failed to give any additional explanation. Moshe Adler, "The Location of Owners and Renters in the City," Journal of Urban Economics 21 (1987): 347-363. '3~~' Mills, Urban Economics.  3 2 Muth, Cities. 3 3 Daniel Griffith, "Modelling Urban Population Density in a Multi-Centered City," Journal of Urban Economics 9 (1981): 298-310. 16 7. CONCLUSION Even though many authors have found problems with the negative exponential population density function, it is widely used. Its use may be due to the fact that it has relatively good theoretical foundation, is easy to apply and interpret, and has shown, in many studies and cities, to be excellent in the approximation of population densities. Income patterns, on the other hand, have not been studied as extensively. Conventional wisdom and theory agree that distance from the city center and income are positively related, but few authors have bothered to test this relation and Adlers study suggests that this positive correlation ma}' not exist. CHAPTER III. THEORY 1. INTRODUCTION This chapter, develops a model that can explain why density and distance are inversely related and why income and distance are positively related. It is necessary to develop such a model to give theoretical basis and credit to the assumptions. This treatment follows Henderson's book entitled "Economic Theory and the Cities". Once the relationships between density, distance and income are established the model will be re-examined so the proper specification might be used in the testing process. Many density studies have assumed the negative exponential specification because it has given good results and is easy to use. The model will be developed to show theoretically why the negative exponential specification is appropriate for an urban densitj' gradient. This treatment will follow that of Mil ls ' book entitled "Urban Economics". 2. ASSUMPTIONS Man3' standard assumptions must be made. The city is monocentric, located on a flat, featureless plane with all commercial activity taking place in the one center, the central business district. The CBD comes about as a result of the exploitation of scale economies and the cost savings of shipping goods to the very center where they are sold or exported. Al l inhabitants of the city live in 17 18 the residential sector surrounding the CBD. From their home the residents commute daily to the CBD, their place of employment. For the sake of simplicity, it is assumed that there are only two goods produced, h, the housing good and x, the non-housing good. The price of x, Px, does not vary since it is all purchased in the same market place, the CBD. The price of housing, Ph(u), is the rental cost of housing and represents a unit of housing at a particular distance, u, from the city center. Ph(u) prices both housing services and access and, therefore, varies spatially. Only one amenity exists in the city, that being leisure e(u), which also varies with distance u through commuting time to the CBD. Leisure can be defined as the number of non-working hours, T, less the time spent commuting. If t is time spent in commuting, the unit distance to and from work, and t is the same everywhere, then the amount of leisure a consumer at distance u has is given by (1). e(u) = T-tu (1) 3. S I N G L E I N C O M E R E N T G R A D I E N T Considering that all consumers earn an income y, the consumer optimization problem can be expressed by (2), subject to the budget constraint (3) and the leisure constraint (3.1) max V(u) = V {x(u),h(u),e(u)l (2) w.r.t.x,h,e,u 19 y - Px x(u) - Ph(u) h(u) = 0 (3) T - e(u) - tu = 0 (3.1) The consumer must, therefore, choose a location u, and at the same time, a consumption bundle which is optimal. Maximizing equation (2) with respect to e(u) and u via the Lagrange method gives equation (4). To solve equation (4), the first order conditions, equations (5) and (6), are used with X and j being the Lagrange multipliers representing the marginal utility of leisure and the marginal utilit}' of income respectively. L* = { V' [ x(u),h(u),e(u) ] + X [ y-Px x(u)-Ph(u) h(u) ] + y [ T-e(u)-tu ] } (4) 9e(u) 3L* = 3e(u) - 7 = 0 (5) 3 V = 7 3e(u) (5.1) 3L* = -Xh(u) Ph'(u) - 7t = 0 (6) X h(u) Ph'(u) = 7 (6.1) t Combining (5.1) and (6.1) gives the solution to the maximization problem as shown in equation (7). By rearranging equation (7) the bid-rent equation is expressed by equation (8). 20 9 V = - X h(u) Ph'(u) (7) 3e(u) t h(u) Ph'(u) = -Pe(u)t (8) where Pe(u) = 9V" 9e(u)t Pe(u) is the marginal utility of leisure expressed in dollars, therefore, if one moves a distance u away from the center, the money lost due to less leisure time is equal to the money gained by lower housing prices. The variation in housing rents can be described as the rent gradient. As shown in equation (9) Ph' (u) is the slope of this gradient. 4. VARIABLE INCOME GRADIENT The above analysis assumes only one income for all residents. To understand where residents with different incomes locate, the bid-rent curve or rent gradient must be examined when y (income) varies. This analysis can be done by differentiating the slope of the bid-rent curve with respect to y (equation 10). Using an equality shown in equation (11), equation (12) is the result. Ph'(u) = - Pe(u)t h(u) (9) dPh'(u) = _d fPe(u)t dy dy 1-h(u) } (10) 21 using _d_ _u = v(du/dx)-u(dv/dx) dx v v"2 (11) dPh'(u) = -h(u) [ dPe(u)t/dy 3 +Pe(u)t[ dh(u)/dy ] (12) dy = dPe(u)t J_ dy h(u) - Pe(u)t dh(u) hluT^ dy = Pe(u)t r dPe(u)t y_ - j_ dh(u) h ^ y 1 dy Pe(u)t h(u) dy dPh'(u) = Pe(u)t dy h(u)y { \ y • ^Pe.y } (13) Equation (13) is the final result with 77, being the income elastic^ of demand result is one of the hypotheses that this thesis will examine for Vancouver. 5. D E N S I T Y G R A D I E N T Considering densities, one must start with the housing production function (equation 14) where housing at a distance, u, is produced by combining capital, k(u), and land, l(u), at distance u. Producers will want to maximize their profits as shown in equation (15), where h(u) = h[k(u),l(u)] (14) 22 Pk is the price of capital, which does not vary with distance, and Pl(u) is the price of land varying with distance u. maxTr(u) = Ph(u) h(u) - Pk k(u) - Pl(u) l(u) (15) The location u, where profits are maximized, occurs when dff/du = 0 , therefore, taking the derivative of equation (15) with respect to u and setting the result equal to zero gives equation (16) and (16.1). d7T = h(u) Ph'(u) - l(u) Pl'(u) = 0 (16) du h(u) Ph'(u) = l(u) Pl'(u) (16.1) Equation (16.1) suggests that as a producer moves a small distance u from his optimal location, the change in land costs is exactly offset by the changes in housing revenue that he receives. Rearranging equation (16.1) and making use of the equality shown in equation (17), the land's factor share in output revenue, results in equation (18). Pl'(u) = h(u) Ph'(u) (16.2) Ku) Pl'(u)= h(u) Ph(u) Ph'(u) (16.3) Pl(u) l(u) Pl(u) Ph(u) using p, = PUu) Ku) (17) Ph(u) h(u) 23 pr(u)= P PKu) -1 Ph'(u) Ph(u) (18) Equation (18) states that the percentage change in a housing unit's rent is equal to the percentage change in a housing unit's price, multiplied by the inverse of the land's factor share. Since the cost of capital is constant everywhere, changes in housing prices are caused directly and only by changes in residential land rents. Equation (18), therefore, suggests that as one moves towards the city center, land rents should increase more quickly than housing rents. Henderson states that p^  is usually estimated at 0.1 and, therefore, a one percent rise in housing rents induces a ten percent rise in residential land rents. This analysis obviously indicates that densities will increase as one moves to the city center. The producers of housing that face a ten percent rise in land costs compared with a one percent rise in units rent, will increase the intensity of land use in order to remain at profit maximizing position. They will produce more units on equivalent parcels of land as one moves towards the city center, resulting in an increase in density. Henderson's model has given a city with interesting characteristics. Because it is assumed that the income elasticity of the demand for housing is greater than that for leisure, as income goes up the bid-rent curve gets flatter. This result suggests that as one moves from the city center the average income of the residents will increase. Another important characteristic involves densities. Because of profit maximization 24 by the producers of housing, the intensity of land use will increase as one moves closer to the city center. This result suggests that as one moves closer to the city center population densities will increase. The implication of these two results is that in this model city the wealthier residents live on larger lots on the outskirts of the cit3', while lower income residents live on small lots in the city center. 6. FUNCTIONAL FORM In the previous section a model was developed from which a downward sloping density gradient and an upward sloping income gradient were theoretically derived. The functional form which these gradients take, especially density gradients, has been the subject of serious debate as outlined in Chapter Two. While forms such as the binomial and normal have been utilized, the functional form most often used to describe urban population densities has been the negative exponential. Many authors have used this form because it approximates the data well, but a model developed by Mills 3 " theoretically suggests why the negative exponential form is most appropriate. Mills begins with a monocentric city of the same characteristics outlined in Chapter Three. The city is circular in nature with 0 radians of land available for urban uses and 2TT - 0 radians unavailable due to parks, natural causes, roads, etc. As suggested in equation (14), h(u), the output of housing services at 25 a distance u from the city center depends on the inputs of land, l(u), and capital, k(u), at the distance u. Assuming the Cobb-Douglas production function can correctly approximate the linking of land and capital to produce housing, equation (19) can be introduced in the following way: h(u) = Al(u) G k(u) 1 _ a (19) where A and a are constants. The properties of this Cobb-Douglas production function include constant returns to scale, unit elasticity of substitution between any pair of inputs, and an elasticity of output with respect to each input equal to the exponent of that input. Assuming now that the inputs and outputs are bought and sold in perfectly competitive markets, producers will use amounts of inputs until the value of the marginal product of each factor is equal to it's price. Differentiating equation (19), with respect to k and 1, gives the marginal products of capital and land as shown in (20) and (21). MP = (l-q)Al(u) Q = (l-q)h(u) (20) ktul*1 k(u) MP. = aAKu) 0" 1 k(u) X" a = ah(u) (21) Ku) Multiplying each marginal product by Ph(u), the rental rate for housing services, gives the value of marginal product which can in turn be equated to the respective rental rates, -Pl(u) for land and Pk for capital, as seen in equations (22) and (23). 26 gPh(u)h(u) = Pl(u) Ku) (22) (l-g)Ph(u)h(u) = Pk k(u) (23) On the demand side of the model it is assumed that all workers have the same tastes and an equal income y, thereby giving a demand function per worker living at u as the following: where B is a scale parameter and depends on the units in which the housing commodity is measured. The terms 0, and 6 2 are, respectively, the income and price elasticities of the demand for housing. The actual values of 6, and 8 2, as mentioned in Chapter Two, have been a subject of great debate for many years. Mills suggests that values of 1.5 for 0, and -1.0 for 6 2 are reasonable and can be used as approximations for the true figures. The total housing demand at u, D(u), is obviously individual demand multiplied by the number of workers at u as shown in equation (25). d(u) = By 5 7 1 P h ( u r 2 (24) D(u) = d(u) N(u) (25) A few equilibrium conditions complete the model suggested by Mills. First, as outlined in Chapter Three, the location equlibrium in housing requires that equation (8) be satisfied. That equation can be written as 27 Ph'(u)d(u) + c = 0 (26) where c = Pe(u)t is the monetary loss of commuting the unit distance. Equation (26) suggests that in equilibrium families are not able to increase their utility because a change in housing prices gained by a move are exactly offset by an increase in commuting costs. Also in equilibrium, total housing supply and demand must be equal at each u (equation 27). Furthermore, the land used for housing can not exceed the total land available giving equation (28). where u is the distance from the city center to the edge of the urban area. At the distance u, households can no longer outbid nonurban land uses. Assuming Pi is the rent that nonurban land commands, equation (29) must also hold. D(u) = h(u) (27) l(u) = 0U (28) Pl(u) = Pi (29) Finally, all N workers must live in the urban area from the city center at u 0 to the urban fringe at u. (equation 30) 28 u / N(u) du = N (30) u 0 To solve the model for land rents, from which densities can be derived, equations (22) and (23) are rearranged as shown. l(u) = qPh(u)h(u) (22.1) Pl(u) k(u) = (l-q)Ph(u)h(u) (23.1) Pk Substituting for l(u) and k(u) in the production function (19) and rearranging terms gives Ph(u) = [ A q a ( l - q ) ( 1 - a ) ] -1 Pk ( 1- a ) Pl(u) a (31) Taking the derivative of (31) with respect to u gives Ph'(u) = A'^aPk n ( 1 " a ) Pl(u)" ( 1" a )Pl'(u) (32) 1-q Substituting equation (24) for d(u) in equation (26) gives Ph'(u) By^ 1 Ph(u)^ 2 + c = 0 (33) Substituting equation (31) for Ph(u), (32) for Ph'(u) and collecting terms gives E ^ P i a O ^ P l ' O i ) + c = 0 (34) where E and /3 stand for the collection of constants E"1 • = a B y e U A a a (1-a) 1" 0] + F k ^ ^ + ^ ) and 0 = a(l + 62) (34b) The equation (34) represents the locational equilibrium, (26), in terms of land rents Pl(u). Using the initial condition described by (29), the solution to (34) is Pl(u) = [Pl^ + 0cE(U-u)]1//3 if pVO (35a) and Pl(u) = Pi e c E ( u " u ) if /3 = 0 (35b) If /3 does not equal zero, then, as equation (35a) suggests, the binomial function would be a good approximation of the urban land rent function. If, however, j3 equals zero which, as seen in equation (34b), occurs when when 82, the price elasticity of housing, equals negative one, the exponential function would be a good approximation of the urban land rent function. Since it is assumed that 6 2 is approximated negative one, Mills suggests that (35b), the exponential function, should be used to approximate the land rent function. The task now is to relate the land rent function to the density function. Using (27), (28) can be written: 29 (34a) 30 N(u) = h(u)/d(u) (36) Rearranging and equating equations (22) and (23) gives k(u) = 1^ Pl(u) l(u) (37) aPk Substituting (37) for k(u) in (19) results in h(u) = Arl-a,1'0- PKu)1"*1 l(u) (38) laPk J Substituting (22) for Ph(u) in equation (24) results in d(u) = B y 5 l [ A a a ( l - a ) ( 1 " a ] ' 1 P k 1 " a Pl(u) a (39) Finally, substituting (39) for d(u), (38) for h(u) in (36) and rearranging terms results in N(u) = EPKu)1"'3 (40) Ku) Equation (40) shows how workers divided by area varies with distance from the city center. Essentially this ratio is the same as population density and would be so if the labour participation rate were 100 percent. Population densitj' according to equation (40), therefore, is proportional to land rent raised to the 1-/3 power. Since 62 is negative, 1-/3 must be positive, suggesting that densities are high when land rents are high. Moreover, if j3 equals zero, which occurs when 6 2 is 31 negative one, population densit3' is proportional to land rents and, therefore, can be described by the exponential function. Equation (40), therefore, links theory to the number of studies that have used the negative exponential function as a means to describe urban population densities. The equation also provides the necessary theory to begin the testing section of this thesis with the hypothesis that Vancouver's urban population densit3' follows a negative exponential pattern. 7. CONCLUSION In conclusion, it has been shown using a monocentric city model that density and distance from the city center are inversely related while income and distance are positively related. What pattern these relationships took was not known and, therefore, the monocentric city model was re-examined to determine the proper functional forms. Assuming that the price elasticity of housing is equal to negative one, it was shown that the urban density function will follow the negative exponential form. Many density studies have begun with assumptions of how distance, density and income are related and which functional forms should be used. The treatment, however, has provided the theory necessary to allow modelling and testing of certain hypotheses with respect to Metropolitan Vancouver to begin. CHAPTER IV. THE NEGATIVE EXPONENTIAL DENSITY MODEL 1. INTRODUCTION Chapter Four begins the hypothesis, modelling and testing. Because of the examination in Chapter Three the hypothesis will be that distance from the city center and urban population density are negatively related. Furthermore, true to the theories previously examined, this chapter will test a model having a negative exponential form with distance from the city center being the only determinant of density. The purpose of this testing is to see how well Metropolitan Vancouver adheres to the hypothesis developed from a monocentric city model and to aid in the formation of a model which best describes Vancouver. 2. DENSITY MODEL The model to be tested, therefore, is described by equation (41) D(u) = D 0 e' 7 U (41) where; D(u)= population density at a distance u D 0 = constant or density of the citj' center where u = 0 7 = density gradient (the ratio of decline in density with distance from the city center) and u = distance from the citj' center. 32 33 3. DATA Metropolitan Vancouver was used as the area to which this density equation was applied. Included in the sample were Vancouver City, North Vancouver, West Vancouver, Richmond, North and South Delta, White Rock, North and South Surrey, Burnaby, New Westminster, and the North East Sector (Coquitlam and Port Coquitlam). The census tracts in these areas were taken as the observations. The census tracts are good sources of data since besides other characteristics, the census tracts are shaped as compact as possible and are as homogeneous as possible with respect to economic status and living conditions. The densitj' for each census tract was taken as the population of that tract divided by the area of that tract. The population was divided by 1000 giving a density measured in thousands of people per square kilometer. The population statistic was taken from the 1981 Canadian Census.35 The area statistic was, for the most part, taken as the area of the whole census tract. For some census tracts, however, large areas of land were obviously not used for housing purposes, these included government land, parks, golf courses, etc. In order to get closer to the desired net density, areas were adjusted to compensate for these non-residential lands. The adjustment was done by looking at each tract on a map to see whether large areas were devoted to "3~5 Statistics-Canada, Canadian Census (Ottawa: 1981). 34 non-housing purposes. If so, the fraction of non-housing land for that particular tract was approximated by eye and the total area for the tract reduced by that fraction. Of course, pure net density can not be achieved but it is reasonable to assume that streets, corner stores, commercial space, etc. averages out through all tracts so as not to affect the overall results of the study. A few census tracts had so much non-residential land that thej' were deleted. These included tracts in the Central Business District (which has an abundance of commercial, retail, and office space), the airport, the Univers^ of British Columbia Endowment Lands, Stanley Park, and tracts completely enclosed in the agricultural land reserve (land set aside by the provincial government for agricultural use). This left a large group 218 observations to be used for testing purposes. Because of the chance of sampling bias Frankena refers to, all of these observations were used. This eliminated the problem of over-representation of high density areas when a sample is used. The distance variable was taken to be the straight-line distance from the center of each census tract to the center of the CBD. Distances were calculated using a map with a scale of 1 to 50,000. The center of the tracts was eyed and the distances measured using a tape. The distance measured was taken as u. 4. TESTING Using the log of equation (41) gives the equation to be estimated, (42). 35 In D(u) = In D 0 +711 + e (42) The above equation was regressed by SHAZAM using ordinary least squares. The results were encouraging with the gradient being negative and significant and the R 2 being 0.34, as shown in Table One. Variable Name Distance Constant TABLE ONE OLS Estimation Dependent Variable: Density 218 Observations Estimated Coefficient -0.059 1.62 Standard Error 0.006 0.088 T-Ratio 216 DF, 0.05 Level -10.5 18.3 R 2 Adjusted = 0.336 Durbin-Watson Statistic = 1.36 a. Autocorrelation In this regression, however, the assumption of no autocorrelation is violated. The violation occurs when the disturbance term of one observation is related to the disturbance term of the other observations. In cross-sectional studies autocorrelation is usually referred to as spatial autocorrelation. Since the observations are ordered by census tract and these tracts follow a distinct pattern throughout the metropolitan area, spatial autocorrelation is a problem. The detection and correction of spatial autocorrelation is the same as that of 36 times series autocorrelation and, therefore, standard procedures can be used. The preceding regression, as already mentioned, has autocorrelation which is evident when one looks at the Durbin-Watson statistic generated for the regression. The Durbin Watson d Test involves the calculation of a test statistic based on the residuals from the ordinary least-squares regression procedure.3 6 The d-statistic is the ratio of the sum of square differences in successive residuals to the residual sum of squares (RRS). Since the probability distribution of the statistic is obtained from the sequence of residuals and the sequence of all the independent variables, it is difficult to obtain exact figures. This fact leads to the problem that there is no unique critical value that allows rejection or acceptance of the null hyopothesis. For each regression, however, upper and lower limits of the d-statistic can be obtained depending on the number of observations and the number of independent variables. For positive autocorrelation, if the calculated d-statistic is greater than d-upper, then the null hypothesis of no positive autocorrelation can be accepted. If the calculated d-statistic is below d-lower, then the the null hypothesis must be rejected and positive autocorrelation exists. If the calculated d-statistic is between d-upper and d-lower, then no accurate statement about existence of autocorrelation can be given. The d-statistic in this situation is in the zone of indecision. The existence of this zone is due to the fact that in this region, because the "3_E R. Pindyck and J5! Rubinfeld, Econometric Models and Economic Forecasts (New York: McGraw, 1978). sequence of residuals is influenced by the movement of the independent variable, it may be possible that the correlation is not due to the correlation of the errors but the autocorrelation of the independent variables. The lower and upper bounds for the d-statistic to the one percent significance level using two hundred observations are 1.66 and 1.68. The calculated d-statistic for the regression is 1.36 (see Table One), far lower than the lower bound. This fact suggests autocorrelation exists. One cause of autocorrelation could be an incorrect functional form. Another cause may be data manipulation occuring when data is smoothed by taking averages. Both these problems may impose a systematic pattern upon the disturbances which will lead to autocorrelation and both may be a problem in this study. The functional form as mentioned may not be proper, although most prior evidence suggests otherwise. Data manipulation may, however, be the problem in that densities of census tracts and average distances of these tracts to the downtown were taken as observations. The aforementioned would have a smoothing effect considering densities and distances are continuous and not discreet points averaged over large areas of land. Another cause of autocorrelation is again a specification bias, that is, the dropping of a variable. It may be the case that the model specified does not include all the independent variables in the correct or true model. If this is the case, then to the extent that the missing variable affects the dependent variable, 38 the disturbance term will reflect the systematic pattern and thereby create false autocorrelation. Specification bias could be a problem in this study since it is doubtful that distance alone uniquely explains the changes in densities. This subject will be addressed in Chapter Six. When autocorrelation exists, the estimates generated by OLS are still unbiased and consistent but they are no longer efficient. This fact means that the estimates no longer attain the minimum variance, leading to standard errors of the estimates being biased downwards. The final result is that the variance and standard error will be underestimated. This causes the assumption that the estimates are more precise than they actually are, leading to the tendency to reject the null hypothesis when sometimes it should be accepted. The presence of autocorrelation renders the standard t and F tests virtually useless. When autocorrelation exists, most regression computer packages can correct for the effects of autocorrelation and still give good results. Using SHAZAM the autocorrelation problem is rectified via the Cochrane Orcutt Iterative Least Squares Technique. The technique basically estimates p, the coefficient of autocorrelation, and then uses that to transform the equation to account for the autocorrelation. Taking a simple example to be equation (43), it is easy to show how the transformation takes place. Y t = B 0 + B l X t + U t (43) 39 Taking (43) at observation t-1 and multiplying by p gives (44) p Y t l =pB 0 + pB 1X t. 1 + p U ^ (44) Subtracting (44) from (43) gives (45) ( Y t " 'YM> = Bo ( 1-^ + ( B l X t " " B 1 X H > + ( U t " ^  ( 4 5 ) Since the disturbances can not be explicitly observed, it is assumed they follow the first order autoregressive scheme shown in (46), where the absolute value of p is less than one and the errors e follow the classic OLS assumptions. U t = pU t. 1 + e t (46) Substituting (46) into (45) finally gives (47) ( Y t " ' Y n> = Bo ( 1-^ + B i ( x t - " XH> + e t ( 4 7 ) Since e^  is assumed to adhere to the classic assumptions, then the autocorrelation problem is solved and the estimates will be BLUE (Best Linear Unbiased Estimates). This procedure is basically what is performed by the Cochrane-Orcutt Iterative procedure. The original residuals are obtained and are used to perform the 40 regression e^. = pe^ ^  + V . This equation gives an estimate of p which is used to run the regression as in equation (44). The B's from this regression are then used in the initial regression and new residuals are obtained. These residuals are again used to perform a regression to obtain another estimate of p. The procedure or iteration continues until the new estimates of p are less than .01 or .005 apart or after 10 or 20 runs. Generally, this procedure is a good way to get an estimate of p and obtain new parameter estimates which are no longer plagued by autocorrelation or inefficiency. The regression was, therefore, rerun to account for autocorrelation and the results were very good. The problem of autocorrelation seems to have been rectified as a Durbin-Watson statistic of 2.06 suggests. The gradient remained highly significant and negative as theory suggests, while the R 2 increased to 0.404 (see Table Two). An R 2 of this value is excellent when using cross-sectional data and is comparable to the R2's achieved by Mills 3 7 in his studies of gradients for forty-seven U.S. cities and by Alperovich.3 8 Mills, Studies. Alperovich, An Empirical Study. 41 TABLE TWO Least Squares Estimation Dependent Variable: Density Cochrane-Orcutt Type Procedure (convergence = 0.001) 218 Observations Log L.F. = -223.647 at p = 0.320 Variable Name Estimated Coefficient Standard Error T-Ratio 216 DF, 0.05 Level Distance Constant -0.057 1.60 0.007 0.121 -7.61 13.3 R 2 Adjusted = 0.404 Durbin-Watson Statistic = 2.06 What these regression results suggest is that one determinant alone, distance from the city center, can explain 40 percent of the variation in Metropolitan Vancouver population density. These results also give good support to the models developed by Henderson and Mills and the theory that population densities can be described b3' a negative exponential specification. b. Heteroscedasticity Besides the assumption of no autocorrelation, a second assumption, that being homoscedasticity, must be checked for the regression results to be valid. Homoscedasticity assumes that the variances of the disturbances are constant. This assumption is crucial since the presence of heteroscedasticity will lead to estimates which are biased and inefficient, therebj' rendering the usual t and F 4 2 tests useless. The assumption is especially important since authors such as Frankena and Anderson suggest that heteroscadasticity is a likely problem in density studies. One quick way to detect heteroscadasticity is via the graphical method. This method involves plotting the squared residuals of a regression against the predicted values of the same regression to see if some type of pattern exists. Since the residuals are proxies for the disturbances, a systematic pattern will suggest heteroscadasticity and the possible form it takes. This plot was done and is shown in Figure One. There is no obvious pattern to this plot and this randomness suggests that heteroscedasticity is not present. To confirm this finding another test was performed, the Spearman's rank correlation test. The Spearman's rank correlation coefficient is shown as equation ( 4 8 ) . Sr = 1 - 6 r Zd.2 , ( 4 8 ) 1 NTTT 2"-!) where; d. = difference in the ranks assigned to two specific characteristics of the ith observation and N = the number of observations. When testing for heteroscedasticit}', the characteristics to be ranked are the residual and the predicted value. The difference in the rankings give d. which in turn gives Sr. If we assume the rank correlation coefficient Pr to be zero, the 43 FIGURE ONE Residuals Squared Versus Predicted Values Residual Squared 2.2500 2.0962 1 .9423 1 .7885 1 .6346 1 .4808 1 .3269 1 . 1731 1.0192 0.86538 0.71 154 0.55769 0.40385 O. 25000 .96154E-01 .57692E-01 .21154 .36538 .51923 -0.67308 -0.82692 -O.98077 -1 . 1346 -1 .2885 - 1 .4423 -1 .5962 -1.7500 -1.9038 -2.0577 -2.21 15 .3654 .5192 .6731 .8269 -2.9808 -3. 1346 -3.2885 -3.4423 -3.5962 -3.7500 218 OBSERVATIONS * = RES M=MULTIPLE POINT O. -O. -O. -O. -0. * * * * * * * * « M * * * * * « * M * M » . * * * * * * * * * * * » **MMMM* M**M *M*MMMMM* M* M* * M* ****M MM * * * * * * *M M*MM**«** * * * ** MM* *** M** * * * * ******** M * *M *M* ***M* * * * * * M * MM M -2. -2  -2. -2  -1.500 -1.000 -0.500 0.0 0.50O Predicted Value 1.ooo 1.500 2.ooo 44 t-statistic testing for the presence of heteroscedasticity can be calculated as follows: t = Sr/N-2 (49) / l - S r z A calculated t-statistic which is greater than the critical t value suggests heteroscedasticity is present in the regression. The test was performed on our regression and calculated t-statistic is 1.28, far below the critical t value of 1.65 at the 0.05 significance level. This test suggests that heteroscedasticity is not a problem and confirms the results obtained by the residual plot. c. Normality To have credible regression results a final assumption must be considered, the normality of disturbances. It is assumed that in a linear regression model the population disturbances are distributed normally. This assumption is not needed to obtain estimates that are BLUE but is needed for inference and testing of results. The disturbance term represents the combined effect of all independent variables which have not been included in the regression model. If the specification is correct, the influence of these neglected variables is hopefully minor and random. Via the central limit theorem of statistics it can be shown that the sum of a 45 large number of independent and identically distributed random variables tends to go to a normal distribution as the number of these variables increase. With the assumption that the disturbances are normal the distribution of the OLS estimates is easily obtained. Any linear function of normally distributed variables will itself be normal and since the estimates are linear functions of the assumed normal disturbance term, they will also be normal. This fact simplifies the task of establishing confidence intervals and hypothesis testing. One way to check the normalitj' assumption is to make a histogram plot of the residuals, which are proxies for the disturbances. The histogram should appear to have a normal distribution for the assumption to hold. This plot was done and is shown in Figure Two. The mean is shown as 0.00024, virtually zero, and the histogram extends three standard deviations in both directions. It is encouraging that all the residuals are contained in these three standard deviations suggesting that no outliers are present. The plot, although close to normal, appears to be skewed slightly. Seber 3 9 suggests that the fact that the plot may indicate non-normalitj' of residuals can be overlooked if two criteria are met. These criteria are a large number of variables and a projection matrix with roughly equal or similar diagonal elements. George Seber, Linear Regression Anatysis (New York: Wilej', 1977). FIGURE TWO Residual Histogram PCT . N O 454 99 I 0 445 97 I 0 436 95 I O 427 93 I 0 417 91 I 0 408 89 I xxxxxxxxxx 0 399 87 I xxxxxxxxxx 0 390 85 i xxxxxxxxxx 0 38 1 83 I xxxxxxxxxx 0 372 81 I xxxxxxxxxx 0 362 79 I xxxxxxxxxx 0 353 77 I xxxxxxxxxx 0 344 75 I xxxxxxxxxx 0 335 73 I xxxxxxxxxx o 326 7 1 I xxxxxxxxxx 0 317 69 I xxxxxxxxxxxxxxxxxxxx 0 307 67 I xxxxxxxxxxxxxxxxxxxx 0 298 65 I xxxxxxxxxxxxxxxxxxxx 0 289 63 i xxxxxxxxxxxxxxxxxxxx 0 280 61 i xxxxxxxxxxxxxxxxxxxx 0 27 1 59 i xxxxxxxxxxxxxxxxxxxx o 26 1 57 i xxxxxxxxxxxxxxxxxxxx o 252 55 I xxxxxxxxxxxxxxxxxxxx 0 243 53 i xxxxxxxxxxxxxxxxxxxx 0 234 51 i xxxxxxxxxxxxxxxxxxxx 0 225 49 i xxxxxxxxxxxxxxxxxxxx 0 216 47 I xxxxxxxxxxxxxxxxxxxx 0 206 45 I xxxxxxxxxxxxxxxxxxxx 0 197 43 I xxxxxxxxxxxxxxxxxxxx 0 188 4 1 I xxxxxxxxxxxxxxxxxxxx 0 179 39 I xxxxxxxxxxxxxxxxxxxx 0 170 37 I xxxxxxxxxxxxxxxxxxxx 0 16 1 35 I xxxxxxxxxxxxxxxxxxxx 0 151 33 I xxxxxxxxxxxxxxxxxxxx 0 142 31 I xxxxxxxxxxxxxxxxxxxx 0 133 29 i xxxxxxxxxxxxxxxxxxxx o 124 27 i xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 0 115 25 i xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 0 106 23 i xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 0 096 21 i xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 0 087 19 i xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 0 078 17 i xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 0 069 15 i xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 0 060 13 I xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx o 050 1 1 i xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 0 041 9 I xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 0 032 7 ixxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 0 023 5 ixxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx o 014 3 ixxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 0 005 1 ixxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx j j j ! ! i x -2.20 -1.47 -0.735 0.222E-04 0.735 1.47 2.20 Residuals 47 In matrix algebra the regression equation can be expressed as equation (50). Y = XB + E (50) With n being the number of observations and k being the number of parameters, Y is a nxl matrix of the dependent variable, X is a nxk matrix of independent variables, B is a kxl matrix of the parameters and E is a nxl matrix of errors. The estimates of the parameters, B hat, is described in equation (51). B = (X'X^X'Y (51) Inserting (51) into the estimate equation, (52), gives (53). Y = XB (52) Y = X(X'X)' 1X'Y (53) Taking X(X'X) *X'Y to be the hat or projection matrix P (53) can be described as (53.1). Y = PY (53.1) It is easj' to understand why P is described as the projection matrix since it projects or converts the observed Y's to the estimated Y's. 48 The diagonal elements of the P matrix measure the importance or influence of each observation. If, therefore, n is large and the diagonal elements of P are similar, there are a large number of equally influential observations doing the same amount of work. Seber suggests that under these conditions the normalit}' assumption is not crucial as a central limit effect takes over. The projection matrix was calculated for our regression and the diagonal elements are shown in Table Three. The elements do appear to be similar implying that all observations have roughly the same influence in the regression. This fact plus the large number of observations suggests that the skewness observed in the normality plot is not a crucial problem and valid testing can take place. TABLE THREE Diagonal Elements of the Projection Matrix 0.5060082E 0.5798240E-0.7586383E-0.6951279E 0.5907487E-0.7026540E 0.9083004E 0.8021648E 0.9832705E 0.8204009E 0.6263532E 0.1125234E-0.8021648E 0.1202584E 0.9186567E 0.7259397E 0.1088151E 0.1399576E 0.1460104E 0.8680547E. 0.5241810E 02 02 02 02 02 02 02 02 02 02 02 01 02 01 02 02 01 01 01 02 02 0.5498799E. 0.6391648E-0.7420532E-0.6457474E-0.5745385E-0.7586383E. 0.7932236E. 0.8879417E-0.8879417E. 0.7586383E. 0.9186567E. 0.1076026E-0.9832705E. 0.9397229E 0.8021648E-0.8296958E-0.1125234E. 0.1370019E 0.7844004E-0.7586383E. 0.4965197E-02 02 02 02 02 02 02 02 02 02 02 01 02 02 02 02 01 01 02 02 02 0.5907487E-0.6592665E. 0.6804296E. 0.6327000E. 0.5907487E-0.8204009E-0.7844004E-0.8980621E. 0.9722067E-0.7339375E-0.1017169E-0.9397229E. 0.1088151E. 0.1229310E-0.6951279E. 0.8680547E-0.1326568E-0.1312320E-0.8021648E. 0.7102980E 0.6080203E 02 02 02 02 02 02 02 02 02 02 01 02 01 01 02 02 01 01 02 02 02 0.6140134E. 0.6457474E-0.7502868E-0.5545757E-0.6457474E. 0.8486394E. 0.7259397E-0.9832705E 0.8779393E-0.6732573E. 0.1100394E. 0.7932236E-0.1137831E-0.1040358E-0.7102980E-0.9612609E-0.1370019E. 0.1399576E-0.8391086E-0.4593659E-0.7026540E-02 02 02 02 02 02 02 02 02 02 01 02 01 01 02 02 01 01 02 02 02 49 0.6140134E-02 0.6804296E-02 0.6732573E-02 0.8112239E-02 0.6732573E-02 0.5038878E-02 0.4890417E-02 0.4603851E-02 0.2299244E-01 0.9329071E-02 0.9223908E-02 0.7787963E-02 0.3382406E-01 0.3408724E-01 0.1155124E-01 0.9017118E-02 0.1404913E-01 0.1056370E-01 0.5424241E-02 0.5296250E-02 0.5712063E-02 0.4880924E-02 0.4654396E-02 0.5027274E-02 0.4807265E-02 0.4975956E-02 0.4621923E-02 0.5408419E-02 0.4697481E-02 0.8237130E-02 0.5871894E-02 0.1068302E-01 0.1404913E-01 0.1797400E-01 0.5322756E-02 0.9397229E-02 0.5798240E-02 0.5798240E-02 0.5712063E-02 0.4732101E-02 0.4714202E-02 0.4613458E-02 0.2176359E-01 0.9435414E-02 0.1021283E-01 0.3596254E-01 0.2833240E-01 0.1887717E-01 0.1375272E-01 0.1056370E-01 0.1375272E-01 0.9542936E-02 0.4946264E-02 0.5424241E-02 0.5661145E-02 0.4744218E-02 0.4593659E-02 0.5165581E-02 0.4830638E-02 0.5072107E-02 0.4593659E-02 0.6080203E-02 0.5817438E-02 0.6830245E-02 0.6903569E-02 0.8425052E-02 0.1346102E-01 0.8879417E-02 0.5745385E-02 0.8021648E-02 0.7844004E-02 0.4676666E-02 0.5337734E-02 0.5006S27E-02 0.4732101E-02 0.4613458E-02 0.2057718E-01 0.8520781E-02 0.7532614E-02 0.4075739E-01 0.2762776E-01 0.1528195E-01 0.7532614E-02 0.1609078E-01 0.1234141E-01 0.8815046E-02 0.5424241E-02 0.5984344E-02 0.6101511E-02 0.4606171E-02 0.4807265E-02 0.5408419E-02 0.4587428E-02 0.5296250E-02 0.4855192E-02 0.6140134E-02 0.8425052E-02 0.6042338E-02 0.6758100E-02 0.1142367E-01 0.1303233E-01 0.6080203E-02 0.6804296E-02 0.9291308E-02 0.6804296E-02 0.4917751E-02 0.5106516E-02 0.4591387E-02 0.4880924E-02 0.1815227E-01 0.1220677E-01 0.8715778E-02 0.4715095E-01 0.3125710E-01 0.1609078E-01 0.8425052E-02 0.1924670E-01 0.1080351E-01 0.7449856E-02 0.5006827E-02 0.5927529E-02 0.4642392E-02 0.4792876E-02 0.4935926E-02 0.5693709E-02 0.4667578E-02 0.5178872E-02 0.4907836E-02 0.4995646E-02 0.9435414E-02 0.5927529E-02 0.8330501E-02 0.1727268E-01 0.1481080E-01 5. CONCLUSION Mills' simple model was applied to Metropolitan Vancouver and it was discovered that about 40 percent of the variation in density could be explained by a single determinant, straight-line distance from the city center. This Figure compares ver}' well with comparative density studies done by Mills, Niedercorn, Alperovich and others. The parameter of -0.057 suggests that as one moves out from the city 50 center density will decrease exponentially. The testing in this chapter supports the hypothesis that distance from the city center and urban population density is negatively related. Furthermore, the testing in the chapter suggests that population densities in Metropolitan Vancouver can be described very well by the negative exponential function. The results also suggest that Vancouver as a Canadian city follows the density trends of other North American cities. The success of the testing also gives support for the model developed in Chapter Three and provides an excellent base for a model which describes the density and income patterns of Vancouver. C H A P T E R V . B A S I C M O D E L USING T R A V E L T I M E 1. I N T R O D U C T I O N Chapter Five will again use the basic model and the negative exponential function developed in Chapter Three and tested in Chapter Four. Instead of straight-line distance to the city center, however, Chapter Five will use time spent to travel from one's home to the city center as the single determinant of population density. Most studies have simpty used straight-line distance as the proxj' used to measure the access to the city center, which in turn is a measure of the disutility of commuting. It is used because it has been proven successful and is easily available. It may be the case that distance is not the best proxy available. This chapter will attempt to resolve whether travel times are a better measure of the disutilty of commuting than straight-line distance. 2. T H E O R Y One would think intuitively that travel time, not straight-line distance, would be a better measure of access. This thinking comes from the fact that when the housing equilibrium was established it was the time in travel, measured in dollars, that was considererd as the cost endured when moving further from the downtown. ' 0 Time also seems more closely related to the disutility of " 0 see equation 8 and 26 51 52 commuting than distance. Assumptions, such as a flat homogeneous plane and equal access in all directions, have allowed staight-line distance to be used as an accurate proxy for access. Many cities, however, do not fit the assumptions. Access is not equal in all directions from the downtown due to features such as bridges, tunnels and traffic congestion. This fact would suggest that travel time and straight-line distance from the CBD are not simple linear functions of each other. Vancouver is a unique city in that it's downtown core is located on a peninsula. This circumstance translates into several extra hours spent on the road by commuters who must travel out of their way to get to a bridge and then deal with the congestion once there. Clearly time spent in travel, not straight-line distance, would seem to be a far better measure of access for a city like Vancouver. 3. D A T A Accurate travel times from the center of the census tract to the downtown are available for Metropolitan Vancouver. The travel time matrix has been produced by the Greater Vancouver Regional District development services department. The matrix gives length of time, in minutes, to travel during rush hour via a convenient model split, from the center of one transit zone to the center of any other transit zone in the metropolitan area. 53 For the most part one transit zone corresponds directly to one census tract. In this case the observation is simply the given travel time from that zone to the zone encompassing the CBD. For the rest, two to four transit zones map to one census tract. In this case an average of the travel times from those zones to the CBD were taken as the census tract's travel time. 4. MODEL AND TESTING Using travel times, the density regression * 1 has travel time, denoted by the variable t, as opposed to distance as the lone explanatory variable (equation 54). In D(t) = In D 0 + jt + e (54) Equation (54) was regressed, via the SHAZAM regression package using ordinary least squares, and the results were not as expected. Instead of explaining more, the travel time explained less of the variance in density than did straight-line distance. The parameter or density gradient, 7, had the proper sign and was significant but the regression only showed an R 2 of 0.33 (see Table Four). 54 TABLE FOUR OLS Estimation Dependent Variable: Density 218 Observations Variable Name Estimated Coefficient Standard Error T-Ratio 216 DF, 0.05 Level Travel Time Constant -0.026 1.74 0.003 0.099 -10.4 17.5 R 2 Adjusted = 0.330 Durbin-Watson Statistic = 1.50 Tests for heteroscedasticity and normality 4 2 were run and neither assumption seemed to be violated. The regression, however, was plagued with autocorrelation as a Durbin-Watson statistic of 1.50 suggests. Because of this problem the regression was rerun taking into account the autocorrelation." 3 Results of the corrected model were encouraging and are shown in Table Five. 4 2 t e s t s explained in detail in Chapter Four * 3 done via the Cochrane-Orcutt iterative procedure discussed in Chapter Four 55 TABLE FIVE Least Squares Estimation Dependent Variable: Density Cochrane-Orcutt Type Procedure (convergence = 0.001) 218 Observations Log L.F. = -229.070 at p = 0.269 Variable Name Estimated Coefficient Standard Error T-Ratio 216 DF. 0.05 Level Travel Time Constant -0.022 1.63 0.003 0.126 -7.32 12.9 R 2 Adjusted = 0.374 Durbin-Watson Statistic = 2.05 The parameter remained negative and significant and the R 2 increased to 0.374. This figure, although very good, is a bit lower than that for the regression using distance as the explanatorj' variable. This result is somewhat disturbing and suggests that access may not be the measure for which distance is a proxy. 5. ROTHENBURG MODEL In the model developed b}' Mills and used in this paper, the travel time measure should have performed better than the distance measure since it is a truer measure of access. Jerome Rothenburg"' implies that the problem is because sraight-line distance is a proxy for building age not access. Rothenburg suggests that the Mills model is incorrect and that the downward sloping density Jerome Rothenburg, "Heterogeneity and Durability of Housing: A Model of Stratified Urban Housing Markets," working paper. Massachusetts Institute of Technology, 1985. 56 gradient comes not from a trade off between access and location but by the historical nature of downward filtering. In the Mills model, housing and land are constantly related, one is a proxy for the other. Mills assumes that housing is a homogeneous good, bought and sold as units of housing and being easily constructed anywhere in the city. In a model such as this, a household of given income trades off travel time and access to the city center with the cost of housing. Rothenburg and others * 5 disagrees with this type of model. They argue that housing should be treated as a heterogeneous commodity. Being heterogeneous, at any moment of time, houses throughout the urban area are most likely very poor substitutes and thus, the location of one type of house can not be shifted around. Gonsumers, therefore, are looking for a particular t}rpe of housing, not land, and they must locate where that housing is found. Rothenburg sites four dimensions in a housing package. These dimensions include the structural characteristics of the house, the lot characteristics, the neighbourhood and finally the accessibility of the property to various locations. Rothenburg, however, states that distance to the downtown is an oversimplication of accessibility since there is more than one destination and frequency. The accessibility of a specific location will differ for each household living there because of their different desires. 4 5 Timothy Cooke and Bruce Hamilton, "Evolution of Urban Housing Stocks: A Model Applied to Baltimore and Houston," Journal of Urban Economics 16 (1984): 317-338. 57 Where a consumer decides to locate is based upon the desired variations of these dimensions in the housing package and where this housing unit is found. This thinking stems from the belief that a consumer purchases a total housing package and not only a location. Rothenburg suggests that certain types of consumers desire certain types of housing packages and because of where these units are located, the downward sloping density gradient and upward sloping income gradient are found in most North American cities. For a monocentric city most of the oldest housing is naturally found in the center of the city. Because of durability, much of the housing is first generation and costly to demolish. This housing, however, does have a comparative advantage to downgrade. It will be those of low income who are most likely the consumers of this type of housing. Poor people will locate where the low quality housing exists. Around this core of low quality housing, built densely because of transportation problems, comes housing built more recently and because of technology and age, of better quality. Those of higher means will most likely locate in this housing type. This progression of housing will continue towards the fringe where housing of high quality is built. The highest quality of housing is most often provided by new housing because of the advantage it has in providing the latest amenities and utilizing the newest techniques. The desire for space has been facilitated by advancements in transportation allowing densities to decrease as one moves to the fringe. Those of high income will for the most part occupy those new houses of high quality being built on the fringe. 58 This pattern of housing occupancy leads to the low income people dwelling in high density housing in the central city while high income people dwell in low density housing on the fringe. These are the same patterns evident in North America today and the same suggested by the Mills model. Rothenburg suggests that the destruction and replacement with luxurious condominiums of the lowest quality housing in the inner city is perhaps the beginning of a modified second cycle. Obviously Rothenburg does not believe there can be no exceptions to these generalities, but it would seem to be a reasonable theory. In the case of this study the argument for a continuous Mills model is hampered by the fact that travel time does not explain the variations in density as well as straight-line distance. It may be that instead of access, the distance measure is a proxy for the age of construction of the building, which being connected to housing qualuty could support Rothenburg's theories. It was not possible to obtain accurate average age of construction data, but a regression using that as a determinant of density would help to further explore the Rothenburg model. 6. CONCLUSION This chapter has explored the use of travel times instead of straight-line distance as the lone explanatory variable in the density regression. After correcting for autocorrelation it was ascertained that travel time to the downtown core can explain about 37 percent of the variation in Metropolitan Vancouver density. The R 2 obtained, although not as high as that obtained when distance was the lone 59 determinant, is quite good when compared with other density studies."6 * 7 Theoretically the travel time variable should have performed better than the distance variable since intuitively it is a better measure of the disutilty of commuting. The fact that it did not perform better is troublesome but can be explained by Rothenburg who suggests that the Mills model is incorrect. Following Rothenburg's model it may be that the distance variable is a proxy for building age, not access, and thus outperforms the travel time variable. Considering Metropolitan Vancouver this may be the case but it was impossible to test since accurate data on average building age for each census tract was not attainable. The travel time variable, although not performing as well as the sraight-line distance variable, did come very close. A change of only 0.03 in the R 2s suggests that there is not much difference between the two measures of access when they are used to explain Metropolitan Vancouver population density. Because of this fact both measures will again be considered in Chapter Seven when the assumption of the monocentric city is relaxed. Mills, Urban Economics.  4 7 Alperovich, An Empirical Study. CHAPTER VI. THE INCOME GRADIENT AND ADDED DETERMINANTS 1. INTRODUCTION Chapter Six will attempt to describe the income gradient for Metropolitan Vancouver. Unlike the density gradient there is no theory to suggest a functional form or specification. The model developed in Chapter Three does, however, suggest that income should increase with distance from the city center. Because of this hypothesis, the first action will be to tr}' and estimate the income gradient directly via a regression that has income as the dependent variable and straight-line distance from the city center as the independent variable. If the direct method proves unsuccessful it may be possible to analyse the income gradient indirectly. By using income as a determinant in the density equation it will be possible to examine how income relates to population density and distance from the city center. The use of more determinants in the density equation will also aid in the effort to build a model which best describes Metropolitan Vancouver urban population density. Although the R 2 received when straight-line distance was used as the single determinant is very good, it would be foolish to believe there are no other major determinants of population density. 60 61 2. INCOME MODEL Since theory supports an income gradient that is upward sloping but suggests no specification, the first model to be tested is described by (55) Y(u) = Y 0 + Xu + 0u 2 + e (55) where; Y(u) = average income at distance u Y 0 = constant or income at city center where u = 0 u = distance from city center X,/3 = parameters associated with explanatory variables and e = error term. This model is a result of observing the income versus distance plot as shown in Figure Three. Besides a fairly large aberration, the incomes generally appear to be increasing at a decreasing rate with distance from the CBD. 3. DATA The distance measure is the same as was used previously. The income measure comes from the 1981 census and is the average income, including wages, transfer payments, investments etc., of the members of a census family or household. A census household refers to a husband and wife, a single person or a couple living common law, all with or without single children living in the same dwelling. INC 90000 + LEGEND: A = 1 OBS, B = 2 OBS, ETC, 80000 + 70000 + 60000 + 50000 + 40000 + FIGURE THREE Income Versus Distance A A B A A B A AA A A A A A A A AA A A A A AA A A AAA A B A A A 30000 .+ A A A A B A AA A A B A A A A A A A ABBAABBAB A B A D AA AACA A A A A A A A 20000 + B AA AAA A A A A A A A A A AAAA A A AA A A 10000 + A A A A A A A A A A A A A A A A B A ' B A A B A A A A A A A A A A A A A AA AA A A A A A A A A A B A B A A A A A A A A A A A A A AA A A A 0 + - + + -- -- - + + -- -- - + + ..... + + + + + + + + + + + + + + .....+ 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 DIST 63 4. TESTING Regressing equation (55) using OLS yielded disturbing results. The regression was riddled with autocorrelation as suggested by the low Durbin-Watson statistic of 0.97 and the R 2 was virtually zero. The regression was rerun correcting for autocorrelation as described earlier. The residual plots and tests for heteroscedasticity and normality suggest that the results can be taken as accurate.' 8 With the autocorrelation taken care of and no other apparent problems the R 2 jumped to 0.28, and although both parameters had the correct sign, both were insignificant at the 0.05 level (see Table Six). Both parameters were, however, significant at the 0.1 level. This fact suggests that the model and results may still be useful in describing Metropolitan Vancouver income patterns. TABLE SIX Least Squares Estimation Dependent Variable: Income Cochrane-Orcutt Type Procedure (convergence = 0.001) 218 Observations Log L.F. 762.049 at p = 0.518 Variable Name Estimated Coefficient Standard Error T-Ratio 216 DF, 0.05 Level Distance Distance 2 Constant 0.576 -0.014 25.3 0.376 -0.010 2.89 1.53 -1.37 8.75 R 2 Adjusted = 0.282 see Chapter Four for details of tests done 64 Durbin-Watson Statistic = 2.14 The parameters suggest that average income increases with distance until about 21 kilometers from the city center at which point it tends to decline. This fact goes against conventional wisdom but may be explained using equation (13). This equation suggests that higher income residents will live further from the city center because the income elasticity of the demand for housing is greater than the income elasticity of the demand for leisure. It may be the case that this greater demand for housing over leisure reaches a critical point where the time spent in commuting to and from work is too great and can no longer be compensated for by more housing. At this critical point the income elasticity of the demand for leisure is greater than that for housing and the higher income residents no longer outbid lower income residents for land. If this hypothesis is true the critical point for Metropolitan Vancouver would be 21 kilometers. The model described by equation (55), although interesting, did not prove conclusive and, therefore, further attempts at a better model and specification followed. These attempts included a positive exponential and linear function, but none proved adequate. This fact suggests that a simple model relating income and distance may not be applicable for Metropolitan Vancouver. 5. EXTENSION OF BASIC MODEL Although describing the income pattern for Metropolitan Vancouver was somewhat successful, the low significance of the parameters suggest that the indirect 65 method should also be explored. As stated in the introduction this will involve using income as a determinant in the basic density regression and then observing how the income variable relates to density and the other determinants. Virtually all urban population density studies use distance as the primary or only determinant of density. When other determinants are used, measures such as cost of transportation, age of buildings, income and tenure supplement the regression by acting as explanatory variables. Because one of the goals of this chapter is to describe the income gradient for Vancouver, a model developed by Moshe Adler in a recent study entitled, "The Location of Owners and Renters in the City," will be used as the starting point. This will be done since Adler uses income as well as tenure choice as his added determinants. 6. ADLER MODEL Adler develops his model on the lines of the Mill 's model except for one important feature, the demand for privacy. Adler believes that all things being equal, owner-occupiers, because of their higher demand for privacy, will choose to have fewer neighbours than renters. V ia the method utilized in Chapter Three when the location of lower income residents was sought, Adler determines that the land use demand price curve of renters is steeper than that of owners. This fact suggests that renters will live closer to the city center than will owner-occupiers. 66 7. ADLER ESTIMATION Adler in his first estimation assumes that central density, D0, and the density gradient, 7, as shown in equations (56) and (57), are determined by tenure choice, C, and income, Y, onty. D 0 = a 0 + a,C + a 2Y (56) 7 = b 0 + b,C + b 2Y (57) Substituting equations (56) and (57) into (42) gives the equation Adler estimated for the 76 community areas of the city of Chicago. In D(u) = a 0 + a,C + a 2Y - b 0u + b,uC + b 2uY + e (58) where; C = the rate of owner occupancy in each area a T ,a 2 ,b0 ,b T ,b2 = regression parameters a 0 = constant and all other variables as described previously. All the variables were significant at the 0.01 level and the R 2 was 0.934. While the sign on C and u were negative as expected, the sign on Y was positive. The results suggest that renters live closer to the center than owner-occupiers and that within each of these two groups wealthier families locate closer to the center than lower income ones. Adler suggests the reason income decreases with distance from the city center, a fact that goes against conventional wisdom, is that onlj' data from the city proper is used. The suburbs enveloped in the metropolitan area are not included 67 in the observations, thereby foregoing a number of potentially high income areas. 8. DATA Income and distance data are as before. The tenure values came from 1981 GVRD data. The number of owner-occupied dwellings in each census tract was divided by the total number of dwellings giving the percentage of owner-occupied housing in each census tract. 9. ESTIMATION Equation (58) was estimated by OLS using data describing Metropolitan Vancouver. The resulting R 2 of 0.44 was encouraging but several variables were insignificant and a Durbin-Watson statistic of 1.55 suggested that autocorrelation was a problem. The regression was rerun correcting for autocorrelation via the Cochrane-Orcutt iterative technique. The results of this regression are shown in Table Seven. Tests for homoscedasticity and normality again suggest no other problems with the regression." 9 details of tests run are given in Chapter Four 68 TABLE SEVEN Least Squares Estimation Dependent Variable: Density Cochrane-Orcutt Type Procedure (convergence = 0.001) 218 Observations Log L.F. = -207.899 at p = 0.241 Variable Name Estimated Coefficient Standard Error T-Ratio 216 DF, 0.05 Level Distance Income Ownership Dis.*Own. Dis.*Inc. Constant -0.056 -0.010 -0.0009 0.00006 -0.001 2.42 0.021 0.0120 0.0005 0.001 0.260 0.00004 -2.69 -0.841 -2.03 1.59 -1.15 9.32 R 2 Adjusted = 0.475 Durbin-Watson Statistic = 2.05 An R 2 of 0.475, although not as high as Adler obtained, is very good for population density studies and is higher than that obtained with distance as the only explanatory variable. The problem with the regression is that only two variables, distance and ownership, are significantly different from zero at the 0.05 level and only distance is significant at the 0.01 level. The high R 2 and the number of insignificant parameters suggests that the regression had too many explanatorjr variables. The addition of independent variables, in this case determinants of poplation density, will alwaj^s increase the R 2 and, therefore, it is important to have only the correct or most powerful variables in the regression. One reason the results obtained for Vancouver are not as good as those obtained by Adler for Chicago may be the fact that metropolitan data was used for 69 Vancouver while Adler truncated his observations at the city border. While 218 observations are used in the Vancouver regression, Adler only used 76. Suburb locations general^ will have lower densities and less renters than inner citj^ locations and the ommission of these observations in this type of study will obviously bias the results. The suburbs included in this study are very much a part of Vancouver and must be used if proper conclusions are to be drawn. 10. REVISED MODEL In Adler's model he assumed that the density gradient is a function of income and tenure choice. This led to a regression equation that had two interaction terms due to the gradient being multiplied by distance. Both these interaction terms were insignificant at the 0.05 level and, therefore, are candidates to be deleted from the regression. Since Adler gave no theory to suggest the density gradient is determined by income and tenure choice, a simpler way to include them in the densitj' equation is to add them with distance as exponentials (equation 59). By adding income and tenure choice as determinants the basic density equation estimated in Chapter Four is built upon and the interaction terms are dropped. D(u) = D 0 exp(Tju + XY + 0C) (59) Taking logs of equation (59) gives the equation to be regressed, equation (60). 70 In D(u) = In D 0 + TJU + \Y + )3C + (60) Regressing equation (60) via • OLS gave results that again were riddled with autocorrelation as the Durbin-Watson statistic of 1.57 suggests. The regression was rerun correcting for autocorrelation, the results of which are shown in are shown in Table Eight.5 0 T A B L E E I G H T Least Squares Estimation Dependent Variable: Density Cochrane-Orcutt Type Procedure (convergence = 0.001) 218 Observations Log L.F. • Variable Name Distance Income Ownership Constant -209.150 at p = Estimated Coefficient -0.052 -0.022 -0.004 4.71 0.221 Standard Error 0.007 0.007 0.003 0.176 T-Ratio 216 DF, 0.05 Level -7.86 -2.96 -1.29 26.8 R 2 Adjusted = 0.473 Durbin-Watson Statistic = 2.04 The R 2 of 0.473 is virtually the same as when Adler's full model was run, therefore, little explanatory power was lost by deleting the interaction terms. Theory suggests that as density decreases, distance, average income and "5_c tests for normality and homoscedasticity, as detailed in Chapter Four, were again performed and no problems were detected 71 owner-occupancy increases. The regression results support this theory since all independent variables have the expected negative sign. Although signed properly, the tenure variable is not significantly different from zero at the 0.05 level and may be due to the existance of multicollinearity. a. Multicollinearity A linear relationship between two independent variables violates an assumption of the classical linear regression and is labeled as multicollinearity. Intuitively it would seem that income and owner-occupancy are highly related and may be proxies for the same independent measure. If multicollinearity exists the t-ratio significance test is useless and may be the reason that the tenure variable is insignificant. Perfect multicollinearity exists when there is an exact linear relationship between the explanatory variables. In a k variable regression involving X^, X2,...,Xk independent variables, an exact relationship occurs when equation (61) is satisfied, where A^X^, A^X^,...,A^X^ are constants not being simultaneously equal to zero. A 1 X 1 + A 2 X 2 + ... + A k X k = 0 (61) When perfect multicollinearity is present the regression coefficients are indeterminate and their standard errors are infinite. Multicollinearity also defines a condition in which the independent variables are intercorrelated, but not perfectly. This situation is described in equation (62), 72 where Vi is a stochastic error term. A ] [X 1 + A 2 X 2 + ... + A k X k + V. = 0 (62) When this less than perfect multicollinearity exists, the regression coefficients can be estimated but their standard errors are larger than should be. The presence of large standard errors suggests there is a lack of precision or accuracy associated with the coefficients and renders the usual t significance test useless. In the regression run earlier multicollinearity may be present because of the probable linear relationship between tenure and income. As average income goes up one would think that owner-occupancy would go up in proportion. If income and tenure choice are related and measuring the same phenomenon, then it is possible to drop one from the regression equation and not lose any explanatory power. Dropping a variable from a multicollinear model often causes a specification error. If theory states that a certain variable should be included, dropping that variable causes a specification bias leading to coefficients that are over or underestimated. Since there is no concrete theory stating both income and tenure choice should be determinants in the density equation, there should be no problem in dropping one. 73 11. REVISED REGRESSION Deleting tenure choice from the regression equation leaves equation (63) as the one to be estimated. In D(u) = D 0 + rju + j3Y + e (63) This regression was run and as with all other regressions, autocorrelation is present as suggested by a Durbin-Watson statistic of 1.57. The regression was rerun correcting for the autocorrelation and the results, as seen in Table Nine, were very good. TABLE NINE Least Squares Estimation Dependent Variable: Density Cochrane-Orcutt Type Procedure (convergence = 0.001) 218 Observations Log L.F. = -209.989 at p = 0.227 Variable Estimated Standard T-Ratio 216 DF, Name Coefficient Error 0.05 Level Distance -0.055 0.006 -8.84 Income -0.028 0.005 -5.57 Constant 2.41 0.176 13.7 R 2 Adjusted = 0.471 Durbin-Watson Statistic = 2.04 The variables, distance and income, explained over 47 percent in the variation of densit}'. This R 2 is virtual!}' the same as the regression performed with Adler's 74 model and that performed on the equation (60), which included the tenure variable. The equality achieved in the R 2 statistic suggests that the independent variables that were dropped had little or no explanatory power and their expulsion simplifies the model considerably. To make sure these results are valid two of the regression assumptions must be verified. The tests checking for heteroscedasticity suggests that this is not a problem. Tests for normality of the error term did, however, show that non-normality may be a problem. 5 1 The diagonal elements of the projection matrix were examined to see if normality is a major concern. As stated in Chapter Four, if there are a large number of observations and the diagonal elements of the projection matrix are similar, then the normality assumption is not crucial since there are marry observations doing an equal amount of work in the regression. Table Ten shows a portion of the diagonal 218 elements and it is quite evident that one element is considerably larger than all the others. 5 2 The observation corresponding with this element has a larger than average influence on the regression and must be dealt with for the regression results to be valid. TABLE TEN Diagonal Elements of the Projection Matrix details of tests performed are given in Chapter Four 5 2 all other elements were similar 75 0.5162204E-02 0.7587231E-02 0.7273768E-02 0.9910100E-02 0.8403142E-02 0.1081361 0.5526733E-02 0.1653731E-01 0.7016889E-02 0.9567796E-02 0.1111047E-01 0.1885481E-01 0.3264661E-01 0.9395033E-02 0.6797068E-02 0.7076313E-02 0.7242628E-02 0.1128605E-01 0.1378555E-01 0.8556459E-02 0.2412840E-01 0.7179187E-02 0.6713957E-02 0.7771404E-02 0.6937103E-02 0.1066369E-01 0.6662359E-02 0.7534279E-02 0.2301454E-01 0.1235475E-01 0.2311637E-01 0.8682656E-02 The observation in question is census tract number 21. This tract corresponds loosely to an area in Vancouver known as Shaughnessy. This area houses high income residents in old estate homes on relatively large lots. Since this observation was not highly influential in the previous regression when income was not used, it would seem to be the unusually high income figure that makes this observation an outlier. With a large number of observations, the easiest way to deal with an outlier is to drop it from the data. The observation was, therefore, dropped and the regression rerun. The results of this regression, which also corrected for autocorrelation, are shown in Table Eleven TABLE ELEVEN Least Squares Estimation Dependent Variable: Density Cochrane-Orcutt Type Procedure (convergence = 0.001) 217 Observations Log L.F. = -209.227 at p = 0.221 Variable Name Estimated Coefficient Standard Error T-Ratio 216 DF, 0.05 Level Distance Income Constant -0.055 -0.030 2.45 0.006 0.005 0.180 -8.80 -5.58 13.6 R 2 Adjusted = 0.473 76 Durbin-Watson Statistic = 2.04 Comparing the results of this regression with those received in the previous regression (see Table Nine), little difference is observed. The R 2 has increased minutely from 0.472 to 0.473 while the distance coefficient has remained the same at -0.055 and the income coefficient fell from -0.028 to -0.030. The dropped observation, although highly influential in comparison to the other observations, had little effect on the overall regression results. When the results are examined several facts are evident. The negative signs associated with the income variable and the distance variable suggest that for Metropolitan Vancouver as income goes up population density goes down and that in general income and distance from the city center move together. These results support the theory and hypopthesis that as one moves further from the city center income density decreases and income increases. The increase in R 2 from 0.404 when straight-line distance was the only determinant to 0.473 is very encouraging. An increase of almost 7 percentage points in the explanatory power makes a very good model even better. The increase again reinforces the idea that there is more than one determinant of urban population density. 77 12. CONCLUSION This chapter has tried to describe the income gradient for Metropolitan Vancouver. This was first attempted through direct estimation. Since Mills proposed that income should increase with distance from the city center but no functional form was given, a model was developed by looking at an income distance plot for Vancouver. Although this model explained about 28 percent in the variation of income, the parameters were insignificant at the 0.05 level. They were, however, significant at the 0.1 level and if taken as accurate suggest that income increases with distance until 21 kilometers from the city center at which point it declines. This goes against conventional wisdom but can be explained if, at a certain point, the income elasticity of the demand for housing becomes less than the income elasticity of the demand for leisure. At this point higher income residents will no longer outbid lower income residents and average income will decline. Because this model was not entirely successful, other functional forms were tried but none were any better, suggesting that a simple model relating income and distance ma}' not be applicable for Metropolitan Vancouver. A second, more indirect way to examine the income gradient is to use income as a determinant in the density regression and observe how income relates to density and the other determinants. This method also allows for the expansion of the density regression in order to observe whether there are other determinants besides distance that can explain some of the variation in population densities. The adding of more determinants was facilitated by applying to Metropolitan Vancouver a model developed by Moshe Adler in which average income and 78 tenure choice -were assumed to determine central city density and the denshy gradient. Although the model explained over 47 percent of the variation in density, several of the independent variables were insignificant. The reason the model did not work as well for Vancouver as it did for Chicago may be the fact that Adler used observations that were truncated at the city border, while Vancouver data included all the major suburbs. It is felt that the suburbs are very much a part of a study dealing with urban population densities and, therefore, should be included. Adler's model was revised b3' assuming tenure and average income are, along with distance from the city center, determinants of density. This assumption did away with the interaction terms which were not significant. This model also explained over 47 percent of the variation in density. All coefficients had the anticipated sign but the tenure variable was insignificant at the 0.05 level. It was thought that the insignificance of the tenure variable may be due to its collinearity with the income variable. The tenure variable was, therefore, dropped from the model. After correcting for autocorrelation and dropping an outlier, the regression yielded very good results. The distance and income variables alone explained 47.3 percent of the variation in density. The dropping of several variables from Adler's model, therefore, had no effect on the explanatory power of the regression. The negative signs on the distance and income parameters give support to Mills proposal that income tends to increase as one moves from the city center. The 79 increase in R 2 from 0.404 to 0.473 suggests that for Metropolitan Vancouver income is one determinant besides distance that has significant explanatory power for the variation in population density. CHAPTER VII. DENSITY PATTERNS IN A MULTI-CENTRIC CITY 1. INTRODUCTION This study's assumption of a monocentric city is common with virtually all other density studies. The assumption that all commercial activity and employment takes place in one central location makes modelling that city easy, but the problem is that few cities are truly monocentric. In Chapter Seven the assumption of the monocentric city will be eliminated. Metropolitan Vancouver area will be examined to see if and where subcenters exist. Having established the presence of subcenters, a model which takes their influence into account will be presented and tested. 2. IDENTIFICATION OF SUBCENTERS a. Introduction A subcenter is assumed to be a location where a higher than average employment concentration exists. In the classical monocentric model all employment takes place in the one center. In a multi-centric model employment takes place in the main center and also in several smaller centers. As with the monocentric model, population densities should be higher around the subcenters due to the desire to be close to one's place of emplo3?ment. 80 81 Looking at Figure Four, which simply plots population density against distance from the city center, there are three slight increases in density occurring at approximately 11, 17 and 26 kilometers from the city center. These increases may be associated with subcenters. By noting which census tracts at these distances have unusually large population densities, it was determined that the potential subcenters are located at the Central Park-Metrotown area in Burnaby, the commercial build-up along the waterfront in New Westminster and the build-up in North Surrey. It may be that all three are subcenters or it may be that the Burnaby and North Surrey density increases are due to the one New Westminster subcenter. Both these cases are illustrated in Figure Five. b. Theory The uncertainties cited suggest that other criteria, besides population densities, must be used to identhy the employment subcenters. Several methods and criteria are reviewed in a paper by John F. McDonald entitled "The Identification of Urban Employment Subcenters". McDonald suggests that there are five reasonable definitions of an employment subcenter, these being a secondary peak in gross employment density, net employment densitj7, employment-population ratio, gross population density and net population densit}'. An emploj'ment subcenter may show one or all of these characteristics. The studj' of peaks in population densities has given three possible subcenters and although some authors have used this as their only criteria, McDonald LEGEND: A = 1 OBS, B = 2 OBS, ETC. 275 + FIGURE FOUR Density Versus Distance 250 +A 225 + 200 + A A A 175 + A 150 + A 125 +. A 100 + A A A A 75 + A A A A A B A A A A B A B A A 50 + AA B A A A A AA A AB AAB ABA A A A A A ABABA B A A B B A A A A B B B A B A A A AC A A A A A 25 + A A A B AABA A A AAAA A A A ABAA B A B A A A A A A A A AA AA A B A A A A A AB A A A AAA C A A A A A AAA A BA A B A A A AAA B A A AA AB A A A A A A A A A AB A A A A A B AA B AA A A A 0 + A A A A A / . + + + + + + + + + + + + + + + + + + + + + . 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 DIST DEN 275 + 250 +' 225 + 200 + 175 + 150 + 125 + 100 • + 75 + 50 25 LEGEND: A = 1 OBS, B = 2 OBS, ETC. F I G U R E F I V E Density Versus Distance 8 4 suggests this practice may cause problems. Several factors besides employment centers could lead to higher than average population densities. An abundance of amenities, access to radial transportation lines, low crime rates or a good view will generally lead to relatively high land prices which in turn will lead to relatively high population densities. The use of employment densities would seem to be a direct way to identify employment subcenters. The usefulness of net employment densities is questionable since it is easy to imagine a census tract with a very high net employment densit}' with virtually no employment. This tract, being desirable for residential use, would have very little land set aside for the employment sector and, therefore, even with a limited amount of employment, would have a verj' large net employment density. The use of gross employment density would avoid such problems and should be used in the establishment of employment subcenters. The final definition McDonald gives as an indicator of an employment subcenter is the ratio of employment to residential population in each zone. This ratio is useful since a local peak in the ratio suggests the demand for labour at the competitive wage rate is not satisfied by the residents in the zone. This fact implies that commuting into the zone must take place in order to satisfy the labour demands. The ratio is also useful in that it avoids the problems associated with net employment densitj'. A high ratio suggests there are enough jobs to have an impact on the other zones while a high net employment densitj' may not. 85 c. Identification Combined employment data was obtained for 1981 from the G.V.R.D.. The total number of jobs for each census tract is divided by the area of that tract to give the employment density of each tract. Figure Six shows this employment density plotted against distance from the city center. By deleting a few downtown tracts the plot is enlarged (see Figure Seven). At first glance there appears to be many census tracts that have higher than average employment density. Most of these tracts, however, are within five kilometers of the city center and must be considered as part of the downtown center and not a separate subcenter. Excluding census tracts within this five kilometer limit, there are two tracts with high employment densities. The large densities correspond to the two census tracts 201 and 206 which are labeled on the plots and are both in the New Westminster business sector. The employment data was also used to calculate the ratio of employment to residential population for each census tract. Figure Eight shows the ratio plotted against distance from the city center. As with the employment density plot the ratio plot was enlarged by deleting a few central city census tracts (see Figure Nine). CD 00 250 200 150 100 50 EMPDEN 550 500 450 400 + 350 + 300 LEGEND: A = 1 OBS, B = 2 OBS, ETC. + A FIGURE SIX Employment Density Versus Distance 201 A 206 A \ A AA A A \ A A A A AA A AAA C A BA BBBC BADCDAACBAB A A CA A A BA GAAABEC BACABB BCAA A AA A AA A AB A A B A A ABAAC BBCAAAABA B AA AA A AC BEBBB AC BA C AAB A BAA A AA AA 8 10 12 14 IB 18 20 22 DIST 24 26 28 30 32 34 36 38 40 EMPDEN 150 140 LEGEND: A = 1 OBS, B = 2 OBS, ETC. FIGURE SEVEN Employment Density Versus Distance 130 + A 120 1 10 100 206 A 90 80 70 60 50 40 + A A A A 201 A 30 20 10 AA AA A A AA A A BA B A AAA A AA AAA C A CAABB ABB A BA EAAABCC AB + - + -0 B B AAAA 8 10 A A A B A B A A A C A AA B AA A A A A AAAA BAAA BA B B A B B A A AA A AA A AA BCABA AB AA A A A A AA B A A AA A A AA A A A AA AA 34 B 36 12 14 16 18 20 22 24 26 28 30 32 38 40 DIST A LEGEND: A = 1 OBS, B = 2 OBS, ETC. FIGURE EIGHT Employment/Residential Ratio Versus Distance A A 201 A 206 AA A A AA A AABABABB A CAB A A BB A AAAA C AA A A B AA A A BAA A BA CADD IDEDFGDCCBDABB CCBA A C AB BCABCCAC CBB C C BEBBC AC AAA C AAB A BA A AA A A B A A . . + + + ^ ^ + ^ + i + + + h + 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 RATIO 18 + 12 10 LEGEND: A = 1 OBS, B = 2 OBS, ETC. 1G + A 14 -t-FIGURE NINE Employment/Residential Ratio Versus Distance 201 A 206 A A A A A + . AAA A -+-- + . 0 2 BA A AA AA AAAAA A B A AA A BAB AAA A A B A BA BAAC CCDCAB BAAB B B A A A A A A B A A CA FAAAEEDABABABA AC A C AA AC BBCAB ABA A A A B A B BABAA AA A A AA A A C D AB AC A B AAA A A A A A 34 A A A A 8 10 12 14 16 18 20 22 24 26 28 30 32 36 38 40 DIST 90 As with the previous test, if the inner city tracts are disregarded there are two census tracts that have higher than average employment to residential population ratios. These peaks, as before, correspond to census tracts 201 and 206 which are in New Westminster. Using tests of employment densities and employment residential ratios, it appears that only one of the three possible subcenters identified via population densities is actually a subcenter. The identification of a subcenter located in downtown New Westminster is not surprising since it was a dominant city before Vancouver was and, although it has suffered recently, it appears to be resurging and has much history associated with it. 3. THEORY AND MODEL When considering multiple centers the urban population densit}7 is influenced by all centers. How each location is affected depends on the relative power of the centers and the distance the location is from each center. The standard density function 5 3 according to Griffith, therefore, takes the form of equation (64). j = m k = j , D.= I A. exp(-I b.,cQ (64) 1 j = l J k = l J k l J where; D. = the population density at location i d..= the distance separating location i and j, and Aj,bjk = parameters associated with centre j. Having established that there is only one subcenter, for this studj7 m is two. Taking logs of both sides the equation to be estimated becomes equation (65) see equation 41 91 InD. = lnD 0 + TjUy + 0 U n + e (65) where; D. = the population density at census tract i , D 0 = a constant, U = the distance from census tract i to downtown Vancouver, v U = the distance from census tract i to downtown New Westminster, 17^ = the parameters associated with the respective centers, and e = error term. 4. ESTIMATION USING DISTANCE a. Data The only additional data needed to run the regression was the New Westminster distance measures. As in Chapter Four the center of each census tract was eyed and the distances to downtown New Westminster were measured on a map which had a scale of 1 to 50,000. Bj' using a metric measuring tape the distances were easily converted to kilometers and recorded. b. Testing Regressing equation (65) via OLS and the SHAZAM regression package yielded results that were again riddled with autocorrelation as the Durbin-Watson statistic of 1.43 suggests. Estimating rho via the Cochrane-Orcutt iterative technique, the equation was once again estimated correcting for the autocorrelation problem. The results of this regression are shown in Table Twelve. 92 TABLE TWELVE Least Squares Estimation Dependent Variable: Density Cochrane-Orcutt Type Procedure (convergence = 0.001) 218 Observations Log L.F. = -222.533 at p = 0.293 Variable Name Estimated Coefficient Standard Error T-Ratio 216 DF, 0.05 Level Distance N.W. Distance Constant -0.060 -0.016 1.87 0.007 0.010 0.192 -8.08 -1.54 9.58 R 2 Adjusted = 0.407 Durbin-Watson Statistic = 2.05 The Vancouver distance variable is negative and significant, but after correcting for autocorrelation the New Westminster distance variable, while negative, is insignificant at the 0.05 level. The added subcenter also adds little to the explanator}' power of the equation. With the added New Westminster subcenter the R 2 is 0.407 while without it the R 2 is 0.404.5 * The results show that as in Chapter Four straight-line distance from Vancouver center can explain over 40 percent in the variation of population denshty. Straight-line distance from downtown New Westminster, however, does not seem to be a major determinant of Metropolitan Vancouver population denshty and, therefore, should not be included in the model. The new Durbin-Watson statistic of 2.05 suggests that autocorrelation is no 5 " see Table Two 93 longer a problem, and tests 5 5 of the assumptions of homoscedasticity and normality suggest the regression results are valid. 5. ESTIMATION USING TIME The addition of the subcenter had little effect on the regression when distance was used. As discussed in Chapter Five it would seem that time spent in travel is a better measure of access than distance. Chapter Five concluded that even though travel time did not explain as much of the variation in population density as did distance, it explained enough to warrant its use in a multi-centric model. The two center equation (equation 65) was, therefore, modified so that time spent in travel, rather than distance, from the center of each census tract to downtown Vancouver (T ) and downtown New Westminster (T ) is the measure v n of access. The new regression equation is given by equation (66). InD. = lnD 0 + nT^ + j3Tn + e (66) The travel time data comes from the G.V.R.D. travel time matrix which has already been described. Equation (66) was regressed by OLS but again the Durbin-Watson statistic of 1.57 suggests that autocorrelation is a problem. details of tests are given in Chapter Four 94 The regression was rerun correcting for autocorrelation using the Cochrane-Orcutt iterative technique. The results of this regression are shown in Table Thirteen. Both parameters remain negative and significantly different from zero at the 0.05 level. This result is in contrast to that received when distance was used as the access measure. TABLE THIRTEEN Least Squares Estimation Dependent Variable: Density Cochrane-Orcutt Type Procedure (convergence = 0.001) 218 Observations Log L.F. = -222.533 at p = 0.293 Variable Name Estimated Coefficient Standard Error T-Ratio 216 DF, 0.05 Level Time N.W. Time Constant -0.0002 -0.016 2.00 0.00003 0.007 0.200 -7.31 -2.21 10.0 R 2 Adjusted = 0.384 Durbin-Watson Statistic = 2.04 The result suggests that the New Westminster employment subcenter does have an effect on urban population densities. This effect is minor, however, when the R2's are examined. In Chapter Five using only one center, the time spent in travel to the Vancouver C.B.D. explained about 37 percent of the variation in population density. Using two centers, an R 2 of 0.384 suggests that the time spent in travel to the two downtowns explains a little more than 38 percent of the variation in population density. The extra center adds a full precentage point 95 to the explanatory power of the equation.5 6 6. CONCLUSION In this chapter the assumption of a monocentric city was dropped. Metropolitan Vancouver was studied and using a variety of criteria it was discovered that one employment subcenter did exist. This subcenter located in New Westminster along with the Vancouver center means that two centers, not one, simultaneously influence the population density at any location. The influence of the subcenter was included in the original regression by adding distance to downtown New Westminster as a determinant. After correcting for autocorrelation, the regression with distance to downtown Vancouver and distance to downtown New Westminster as the two determinants explained 40.7 percent of the variation in population density. This R 2 is only slightly higher than the 0.404 obtained when distance to downtown Vancouver was the only determinant. The negligible increase in the explanatory power is understandable since, although signed correctly, the parameter associated with the New Westminster distance variable is not significantly different from zero at the 0.05 level. The Vancouver distance parameter is also signed negatively but is significant at the 0.05 level. These results suggest that when distances to the two centers are considered in determining population densities, the power or influence of the Vancouver 5 6 tests checking the assumptions of homoscedasticitj- and normality suggest the results are valid 96 downtown swamps the influence of the New Westminster downtown. When considering travel times as the measure of access, a different conclusion is drawn. The two center regression was rerun using travel times instead of distance to the two centers as the determinants. After correcting for autocorrelation the results showed that the travel times explained 38.4 percent of the variation in density. This R 2 is an increase of one from the 0.374 received when travel times to downtown Vancouver were used as the only determinants. Unlike the regression involving distances, both parameters signed correctly as negative and were significantly different from zero at the 0.05 level. This result suggests that when travel times are considered, although dominated by the Vancouver downtown, the New Westminster center is strong enough to influence population densities in the metropolitan area. The apparent contradiction the above regressions give concerning the influence of the New Westminster subcenter can be explained in the same vein as results in Chapter Five were explained. It may be that the distance variable does not measure access but is a proxy for some unknown determinant, such as age of buildings. In this case the New Westminster subcenter may not be a factor. When travel times are used, an intuitively better measure of access, the New Westminster center is influential. This result again suggests that distance and travel times are proxies for different determinants. CHAPTER VIII. CONCLUSION 1. INTRODUCTION This chapter will summarize the major parts and findings of this study. In doing so it will highlight the procedures and give the results necessary to address the issues and answer the questions raised in Chapter One. The chapter will also show that the general and specific intents of the studjr have been met. 2. SUMMARY The study starts by stating the intentions of the thesis. Basically these intentions are to develop models that would describe the population density and income patterns of Metropolitan Vancouver. This task is being done in order to support or refute the hypothesis that for most major North American cities, as one moves further from the city center, population density increases while average income increases. While many studies on density gradients have been done on cities in the United States very few have been done on Canadian cities and, therefore, it would be interesting to see whether a Canadian city will adhere to the North American trends. Besides the major issue there are several minor questions that need to be answered. These questions include what the proper functional form should be for the models, whether straight-line distance is the best proxy for access to the city center, whether more determinants, besides distance, should be used in the 97 98 density equation and whether Vancouver should be modelled as a monocentric or mutli-centered region. Basically the study is structured so that each chapter addresses one of these issues while still addressing the major goal of describing the density and income patterns for Metropolitan Vancouver. Chapter Two reviews a major portion of the literature associated with this topic. This review shows • the progression of population density studies from their start in 1951 to the present and also gives clues as to what to be aware of when doing this type of study. In Chapter Three the urban density model is developed. Theory is developed to show that under certain conditions population density should decrease and income should increase with distance from the city center. Furthermore, it is shown that under these same conditions population density will decline in an exponential pattern and, therefore, the negative exponential specification should be used to describe urban population densities. The chapter gives theoretical support to the hypotheses and answers the question of which functional form should be used for the density model. Chapter Four takes the model developed in Chapter Three and uses Metropolitan Vancouver data to test its explanatory powers on the variation in population density. After correcting for autocorrelation the model performs very well. The parameter associated with the single determinant of straight-line distance is negative and significant at the 0.05 level and the variable explains over 40.4 percent of the variation in density. This value is good when comparing it to 99 those achieved by those doing similar studies. The results of the regression suggest that the density pattern of Metropolitan Vancouver can be modelled very successfully by the classic density equation developed in Chapter Three. For the most part population density falls off exponentially with straight-line distance from downtown Vancouver. This fact supports the idea that density patterns in major Canadian cities do follow those in the United States and those patterns suggested by theory. Chapter Five uses the model developed in Chapter Three but, instead of distance, uses time spent in travel as the determinant of population densit}'. Travel time is used since intuitively it is better than distance as a measure of the disutility of commuting. In the past most density studies have used distance as the proxy for access because the data was easily available. Metropolitan Vancouver does, however, have accurate travel time data and, therefore, it was used in the regression to explain the density pattern. The results of the regression gave, as theory would suggest, a negative and significant parameter. The travel time variable explained 37.4 percent of the variation in population density. This value is not quite as high as that obtained when distance was used as the single determinant, but it is quite close and is good when compared with other density studies. The fact that the travel time variable did not perform as well as the distance variable is disturbing. As a better measure of access the travel time variable 100 should have performed better. To perhaps explain this phenomenon a model developed by Jerome Rothenburg is introduced. If Rothenburg is correct then distance from the city center would be a proxy for a variable such as building age. not access. It would then be impossible to compare results obtained with the two different determinants since it is not clear what each are measuring. Chapter Six attempts to model the income pattern for Metropolitan Vancouver. The Theory of Chapter Three suggested the gradient is upward sloping but gave no model or specification. Several models, therefore, are tried but only one meets with some success. The model suggests that income increases and then decreases with distance from the city center. This pattern is explainable if the income elasticity of the demand for housing at some point changes and becomes less than the income elasticity of the demand for leisure. At this point lower income residents will outbid higher income residents and the average income will decrease. The income pattern is also explored indirectly when income is introduced as a determinant in the density equation. The use of more determinants in the density' equation is also an issue that needed to be addressed and, therefore, following a model by Moshe Adler, income and tenure choice are added as determinants of the density gradient. Through several revisions, which included deleting the tenure choice variable because of multicollinearity, the final model has income and straight-line distance as the two determinants of population density. The results from the regression which tested this model are very good. Both \ 101 income and distance are significant at the 0.05 level and, as theory suggests, both parameters are negative. The signs on the parameters say that, on average, as one moves further from downtown Vancouver not only does population density decrease but income increases. This fact gives added support to the hypotheses stated earlier and suggests that Vancouver is very similar to major cities in the United States when population density and income patterns are considered. The results also state that straight-line distance and income can explain 47.3 percent of the variation in population density. This excellent figure suggests a couple of things. The increase of almost 7 percentage points in the R 2 suggests that there are other major determinants of population density besides distance and if they can be found they should be included in the density equation. The increase is small, however, when compared to how much of the variation in density distance alone can explain. This fact, which theory supports, indicates that straight-line distance from the city center is the major determinant of population density. Chapter Seven relaxes the monocentric assumption and determines that Metropolitan Vancouver can be best described as multi-centered, the second center being downtown New Westminster. A new model is introduced taking into account the second center. Using straight-line distance measured to each of the centers, the model is tested. The results show that including the New Westminster center added little to the basic model. The R 2 is up ever so slightly from 0.404 to 0.407 and the New Westminster distance variable is insignificant at the 0.05 102 level. These results suggest that the effect that distance to downtown Vancouver has on population density by far outways that of distance to downtown New Westminster. Since travel time performed well earlier, the regression is rerun using time spent to travel to the respective centers instead of distance. The results are different than those obtained when distance is used. This time the parameter associated with New Westminster is significant at the 0.05 level. This fact suggests that when using travel times as determinants New Westminster does have a significant effect on density patterns. The R 2 rises from 0.374 when the New Westminster center is not included to 0.384. Although this is an increase of a full percentage point, the time spent in travel to downtown Vancouver is a far greater determinant on density than is the time spent in travel to downtown New Westminster. 3. FINAL CONCLUSIONS The study has basically fulfilled what it set out to do. The specific issues have been addressed and the density and income patterns of Metropolitan Vancouver have been described. Population densitj', as theorj' suggests, does fall off exponentially. The income pattern, although not directly described, does increase as population density decreases and, therefore, generally does increase with distance from downtown Vancouver. These facts give support to the hypotheses and suggest that density and income patterns of Metropolitan Vancouver are much like those of major U.S. cities. 103 The model which had the best results used income and straight-line distance from the city as determinants of density. These two determinants explained almost half of the variation in population density, a figure which is very good when compared with other density studies. This fact also suggests that Metropolitan Vancouver can be modelled successfully as a monocentric city and that classic Urban Economic Theory describes it very well. BIBLIOGRAPHY Adler, Moshe. "The Location of Owners and Renters in the City." Journal of  Urban Economics 21 1987: 347-363. Alperovich, Gershon. "Determinants of Urban Population Density Functions: A Procedure for Efficient Estimates." Regional Science and Urban Economics 13 1983: 287-295. "An Empirical Study of Population Density Gradients and Their Determinants." Journal of Regional Science 23 1983: 529-540. Anderson, John E. "Estimating Generalized Urban Density Functions." Journal of  Urban Economics 18 1985: 1-10. Beckman, Martin J. "Spacial Equilibrium in the Housing Market." Journal of  Urban Economics 5 1974: 99-107. —. "On the Distribution of Urban Rent and Residential Density." Journal of  Economic Theory 1 1969: 60-67. Box, G.E.P. and D.R. Cox. "Functional Form, Density Gradient and Price Elasticity of Demand For Housing." Urban Studies 13 1976: 193-200. Clark, Colin. "Urban Population Densities." Journal of the Royal Statistical  Society 114 1951: 375-386. Cooke, Timothy and Bruce Hamilton. "Evolution of Housing Stocks: A Model Applied to Baltimore and Houston." Journal of Urban Economics 16 1984: 317-338. Couch, J.D. "Residential Density Functions: An Alternative to Muth's Negative Exponential Model." Journal of Urban Economics 8 1978: 16-31. Fenton, C. "The Permanent Income Hypothesis Source of Income and the Demand for Rental Housing Analysis of Selected Census and Welfare Program Data to Determine Relations of Household Characteristics and Administrative Welfare Policies to a Direct Housing Assistance Program." Cambridge, Mass.: Joint Center for Urban Studies, 1974: 1-52. Frankena, M. "A Bias in Estimating Urban " Population Density Functions." Journal of Urban Economics 5 1978: 35-45. Friedman, J. and D.H. Weinberg. "Demand for Rental Housing: Evidence From the Housing Allowance Demand Experiment." Cambridge, Mass.: Abt Associates Inc. 1974. Goldberg, Michael A. and John Mercer. The Myth of the North American City. Vancouver: University of British Columbia Press, 1986. 104 105 Griffith, Daniel A. "Modelling Urban Population Density in a Multi-Centered City." Journal of Urban Economics 9 1981: 298-310. Gujarati, Damodar. Basic Econometrics. New York: McGraw, 1978. Harrison, D. and J. Kain. "Cumulative Urban Growth and Urban Density Functions." Journal of Urban Economics 1 1974: 61-98. Haurin, Donald R. "Urban Density and Income." Journal of Urban Economics 8 1980: 213-221. Henderson, J. Vernon. Economic Theory and the Cities. Toronto: Academic Press, 1985. Johnston, John. Econometric Methods 2nd ed. New York: McGraw, 1972. Kau, James B. and Cheng F. Lee. "Functional Form, Density Gradient and Price Elasticity of Demand for Housing." Urban Studies 13 1976: 193-200. Klein, Lawrence R. A Textbook of Econometrics 2nd ed. Englewood Cliffs, N.J.: Prentice, 1974. McDonald, John F. "The Identification of Urban Subcenters." Journal of Urban  Economics 3 1987: 242-259. McDonald, John F. and Woods H. Bowman. "Some Tests of Alternative Urban Population Density Functions." Journal of Urban Economics 3 1976: 242-259. Macauley, Molly K. "Estimation and Recent Behaviour of Urban Population and Employment Density Gradients." Journal of Urban Economics 18 1985: 251-260. Maisel, J., J.B. Burnham, and J.S. Austin. "The Demand for Housing: A Comment." Review of Economics and Statistics 53 1971: 410-413. Mayo, Stephen K. "Theory and Estimation in the Economics of Housing Demand." Journal of Urban Economics 10 1981: 95-116. Mills, Edwin S. "The Value of Urban Land." The Quality of the Urban  Environment, ed. Harvey S. Perloff, Baltimore: Johns Hopkins UP, 1969. 231-253. "Urban Density Functions." Urban Studies 7 1970: 5-20. —. Urban Economics. Glenview, Illinois: Scott, 1972. —. Studies in the Structure of the Urban Economy. Baltimore: Johns Hopkins UP, 1972. 106 Montesano, Aldo. "A Restatement of Beckman's Model on the Distribution of Urban Rent and Residential Density." Journal of Economic Theory 4 1972: 329-354. Muth, Richard. " The Spatial Structure of the Housing Market Papers." Regional  Science Association 7 1961: 207-220. —. Cities and Housing. Chicago: U of Chicago P, 1969. Niedercorn, John H. "A Negative Exponential Model of Urban Land Use Densities and Its Implications For Metropolitan Development." Journal of  Regional Science 11 1971: 317-326. Pindyck, R.S. and D.L. Rubinfeld. Econometric Models and Economic Forecasts. New York: McGraw, 1976. Rothenburg, Jerome. "Heterogeneity and Durability of Housing: A Model of Stratified Urban Housing Markets." working paper, Massachusetts Institute of Technology, 1985. Seber, George A.F. Linear Regression Analysis. New York: Wiley, 1977. Statistics-Canada. Canadian Census. Ottawa: 1981. Wheaton, William C. "Income and Urban Residence: An Analysis of Consumer Demand for Location." The American Economic Review 67 1977: 620-631. 

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