UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Valuation theory and real property assessment Rollo, Gordon Paul 1971

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

Item Metadata

Download

Media
831-UBC_1971_A4_5 R64_6.pdf [ 8.87MB ]
Metadata
JSON: 831-1.0101845.json
JSON-LD: 831-1.0101845-ld.json
RDF/XML (Pretty): 831-1.0101845-rdf.xml
RDF/JSON: 831-1.0101845-rdf.json
Turtle: 831-1.0101845-turtle.txt
N-Triples: 831-1.0101845-rdf-ntriples.txt
Original Record: 831-1.0101845-source.json
Full Text
831-1.0101845-fulltext.txt
Citation
831-1.0101845.ris

Full Text

VALUATION THEORY AND REAL PROPERTY ASSESSMENT by Gordon Paul Rollo B.A., University of Br i t i s h Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION in the Department of Commerce and Business Administration We accept this thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA August, 1971 In presenting th is thesis in par t ia l fu l f i lment of the requirements for an advanced degree at the Universi ty of B r i t i s h Columbia, I agree that the Library shal l make i t f reely avai lab le for reference and study. I further agree that permission for extensive copying of th i s thes is for scholar ly purposes may be granted by the Head of my Department or by his representat ives. It i s understood that copying or publ icat ion of th is thesis for f inanc ia l gain shal l not be allowed without my wr i t ten permission. Department of (jyMWUJL 0A\1 QuwwtM iX^mjMj^P^oduek. The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada The real property tax has a major impact on real property owners in a l l Canadian municipalities. As with a l l systems of taxation it is important that the burden of this tax be distributed fairly and equitably. Legislators have attempted to ensure equitable treatment among real property owners by requiring that the basis of assessment should be 'actual value'. However, due to the large numbers of properties to be valued, assessors have not been able to use the market approach to value, a valuation technique known to produce 'actual values'. Rather, they have resorted to the more subjective cost approach to value. While the mechanics of the cost approach lend themselves to the mass valuation problem, they rarely produce values that can be equated with actual market values. The application of multiple regression analysis is presented as a solution to this valuation problem. Multiple regression analysis enables the assessor to produce objectively the 'actual value' of a l l single family homes in a municipality. After presenting multiple regression analysis as a modern application of the market approach to value, the applicability of this valuation technique is tested on actual sales data. A sample of approximately four hundred recently sold single family homes is subjected to valuation by multiple regression analysis. Various experiments, including means of stratifying the data are presented in an attempt to produce high standards of solution. While the statistical results of the experiment are not of sufficient calibre for practical assessment purposes, they do reveal how continued experimentation can improve the applicability of this valuation technique to mass appraisal. Multiple regression analysis is the assessor's tool of the future. It facilitates the application of a valuation technique that will permit the assessor to meet his statutory obligation while s t i l l allowing him to adhere to sound appraisal methodology. The writer wishes to thank Professor Richard U. Ra t c l i f f for his very helpful advice and direction, Mr. Norman Goode and Mr. E. C. Twining of the Burnaby Assessment Office for providing the very valuable sales data, and his wife Geri, for her help i n editing and typing the f i n a l draft of the thesis. I INTRODUCTION . . 1 Statement of the Problem 3 Purpose of the Thesis 3 Importance of the Thesis . U Limitations of the Thesis 9 Organization of the Thesis 10 II ESSENTIAL FEATURES OF THE ASSESSMENT FUNCTION . . . . . . . 13 Purpose and Importance of Assessment 13 Equality of Treatment H Measurement of the Tax Base . . . . . . 17 The Assessor's Dilemma 20 Breakthrough in Assessment Methodology 27 III VALUATION TRENDS AND MULTIPLE REGRESSION ANALYSIS . . . . . 32 Introduction 32 Trends i n Valuation Theory . . . . . 32 New Appraisal Theory and the Assessor . . . . . 37 Recent Trends in Assessment Technique 38 Econometric Models for the Assessor 39 Multiple Regression Analysis and Equality of Treatment . . 43 Current Use of Multiple Regression Analysis in Assessment. 46 A Total Assessment System 51 Limitations i n Using Multiple Regression Analysis . . . . 52 IV TECHNICAL ASPECTS OF REGRESSION ANALYSIS FOR ASSESSORS. . . . 58 Introduction . . . . . . . . . . . . . . . . . . 58 Uses of the Multiple Regression Model 59 Aspects of Multiple Regression Analysis 65 V TESTING MULTIPLE REGRESSION ANALYSIS IN BURNABY 75 Introduction 75 Sample Selection. . . . . . . . . . . . . . . 76 Property Characteristics and Their Coding 83 S t a t i s t i c a l Processing and Data Refinement 90 Analysis of the Tests 101 Implementation of the Technique 106 VI SUMMARY AND CONCLUSIONS 109 Restatement 109 Summary of Findings I l l Conclusions I l l Suggestions for Further Study 112 LITERATURE CITED 115 APPENDIX A COMPONENTS OF A TOTAL ASSESSMENT SYSTEM .... 121 APPENDIX B THE COST OF CONVERTING TO A NEW VALUATION SYSTEM . . . 128 APPENDIX C TYPICAL SEQUENCE OF EVENTS IN THE ASSESSMENT CYCLE. . . 132 APPENDIX D EXAMPLE OF REAL PROPERTY CHARACTERISTICS REQUIRED FOR MULTIPLE REGRESSION ANALYSIS . 134 APPENDIX E THE GEOGRAPHIC AREA UNDER STUDY 135 APPENDIX F DETAILED RESULTS OF EACH STATISTICAL TEST 136 TABLE I Property Taxes in Selected Countries i n Recent Years. . . 5 II Selected Provincial and Real Property Tax Data - 1966 . . 7 III Calculation of Taxable Assessed Values 19 IV Distribution of Differences Between Estimated and Actual Selling Price . 49 V Average Deviations of Estimated Selling Price from Actual Selling Price and Coefficients of Dispersion . . . . 50 VI Distribution of Sample by Property Type and Mean Assessed Values 82 VII Property Characteristics (Variables) 85 VIII Coefficient of Determination and Relative Standard Error of the Estimate for Each Test i n the Study . . 103 ; . INTRODUCTION STATEMENT OF THE PROBLEM At the heart of any system of taxation i s the concept that the taxpayer must bear a f a i r burden of the tax. This concept should also be true of the real property tax. The burden to be borne by each real property owner in B r i t i s h Columbia i s described by provincial statutes, since property taxation i s a provincial function. These statutes require that there be equality of treatment among real property owners for taxation purposes. Equality of treatment means that those individuals who own real property of the same value should be taxed equally. The tax payable arises when the tax rate i s applied against the value of the property derived from the process of assessment. While i t i s not the purposes of this thesis to describe in detail the administration of the assessment function, i t may be worth while to describe b r i e f l y how a tax on real property i s derived i n order to i l l u s t r a t e the importance of the basis of assessment, which i s a major subject of examination i n this thesis. Assessment i s the process of discovering, l i s t i n g and placing a value on real property. The discovering and l i s t i n g of properties provides the tax base of the municipality and the valuation of the tax base deter-mines the share of the tot a l taxation which each real property owner w i l l have to bear. The procedure requires that the assessor record a l l taxable property within a municipality and then calculate the taxable assessed value of each individual property. Other municipal officials calculate a tax rate, which, when applied to the aggregate taxable value will produce the required tax revenue. Each real property owner's share of the total tax burden is the product of the taxable property assessment and the tax rate. Since the property tax is the principal source of municipal revenue in Canada, i t is of paramount importance that its burden be distributed as equitably as possible. The basis of assessment or measurement of the tax base is the means of distributing the real property tax burden and i t i s , therefore, imperative that its use by municipal assessors to generate the tax revenue be as equitable as possible, in order to assure that each real property owner does not bear a disproportionate share of the total tax burden. The problem however, is that present techniques of valuing real property for tax purposes are not capable of providing an objective method of valuation, and hence may create an unfair distribution of the burden of the tax. Provincial statutes require the basis of assessment to be 'actual value', and the courts have interpreted this to mean market value or value in exchange at the time of the assessment.^ But the assessor is unable to f u l f i l l his statutory duty to assess at 'actual value' because of the overwhelming number of properties he must assess within a limited time and with a limited budget. Consequently, the assessor relies on the cost approach to value, an approach long condemned by the appraisal fraternity because of the lack of logic in the approach attributed to the separation of total value into land value and improvement value, and the subjectivity introduced by depreciation estimates. The danger of the approach i s simply that i t i s more subjective than other approaches to value, and hence i s less capable of ensuring equitable treatment among real property owners. Objectivity i s essential i n a tax assessment procedure that i s designed to distribute the tax burden equitably. White states: (Equality of treatment} must be determined by s t a t i s t i c a l methods, and i t i s necessary, there-fore to have data reasonably available which can be used for this purpose. The data must be ob-jective, and the only basis of assessment that w i l l meet this requirement i s current market value.^ It w i l l be shown in this thesis that present techniques of assess-ment u t i l i z i n g the cost approach to value do not create objective measures of value, hence cannot produce a high degree of equality of treatment. PURPOSE OF THESIS If there is a danger that present assessment procedures lack the objectivity needed to ensure an equitable distribution of the tax burden, then there should be enough concern to attempt to develop assessment techniques that w i l l increase this objectivity. It i s believed that the shortcomings of the cost approach to value can be overcome by adopting s t a t i s t i c a l aids that enable the market approach to value to be used in a mass valuation context. In fact, this thesis w i l l explore the use of a s t a t i s t i c a l device that enables assessors to make inferences about the market values of a l l municipal real property on the basis of a s t a t i s t i c a l examination of past market sales. This device i s c a l l e d multiple regres-sion analysis and i s already proving very useful for assessment purposes i n the United States. In addition to enabling assessment to be made by the market approach, i t also generates objective measures of the quality of these assessments. It w i l l be argued in this thesis that traditional assessment techniques employing the cost approach to value should be, and can be, replaced by the more objective market approach as implemented by multiple regression analysis* An effort has been made in this thesis to go beyond an arguement for the adoption of this multiple regression technique. The thesis w i l l test the app l i c a b i l i t y of such a technique i n valuing single family residential properties i n the Corporation of the Munipality of Burnaby, B r i t i s h Columbia, (hereinafter referred to as Burnaby). IMPORTANCE OF THE THESIS The real property tax i s a major source of tax revenue for many countries of the world, including Canada. Table I i l l u s t r a t e s the very high proportion of property tax revenue as a percentage of t o t a l tax revenues for various levels of government i n selected countries. The high rate of property tax revenue for l o c a l government places a re l a t i v e l y large financial burden on real property owners. Table II il l u s t r a t e s some measures of the impact of the tax on individuals i n the Canadian provinces. With a high property tax per capita, the real property owners of Br i t i s h Columbia should have a desire for the tax to be as equitable as possible. It i s important for the individual taxpayer to be assured that the value of his property, and hence the burden of his tax, should be similar to the tax burden of real property owners with similarly valued real property. TABLE I PROPERTY TAXES IN SELECTED COUNTRIES IN RECENT YEARS* Property Tax Revenue aa Percentage of Total Tax Revenue Local Plus Local Country and Year Governments Intermediate Governments A l l Governments National Income United States, 1962c 87.7 45.9 15.4 4.3 Australia, 196ls62 100.0 40.8 6.6 2.0 Canada, 1960-61 90.3 46.3 16.0 5.2 Ireland, 1962-63 100.0 13.5 3.8 New Zealand, 1962-63 96. A - 7.6 2.3 South Africa, 1961-62 84.3 34.3 7.5 1.4 United Kingdom, 1963 100.0 - 11.2 4.2 Austria, 1962 n.a. n.a. 1.6 0.7 Belgium, 1962 f 31.8 32.6 4.4 1.2 Denmark, 1961-62 29.7 - 7.9 2.3 France, 1961-628 20.7 2.2 0.6 Germany, 1963 17.0 n«a© 2.2 0.6 Iceland, 1961 5.5 1.5 0.4 Luxembourg, 1962 22.2 - 2.5 0.8 Netherlands, 1962 60.1 - 2.2 0.6 Norway, 1962 2.1 - 0.6 0.3 Japan, 1963*1 21.1 - 6.3 1.2 n.a. Not available a The selected countries include v i r t u a l l y a l l developed countries in which property taxation appears to be of consequence and for which data can be found. Spain andSwitzerland are the principal exclusions on the grounds of data a v a i l a b i l i t y . For this purpose, property taxation i s defined as ad valorem taxation of tangible assets - usually land and buildings only -either on the basis of capital value or annual rental value; ad valorem taxes on property which form part of broader taxes mainly levied on other bases are excluded. These exclusions include net wealth taxes in a number of European countries, walth transfer taxes in most of the countries l i s t e d , and real property levies which form part of the income tax structure in several countries, notably in Sweden. Except where specific publications are cited, the tax data are based on information supplied by the central s t a t i s t i c a l offices of the countries l i s t e d whose assistance was most helpful. National income data are from United Nations publications and estimates by United Nations personnel. b Includes local governments plus states and provinces. Shown only for federal countries, and for others in which the data are organized to permit separate treatment of the different t i e r s of sub-national governments. Sources: U.S. Bureau of the Census, Governmental Finances in 1962 (1963); U.S. Department of Commerce, Office of Business Economics, Survey of Current Business, July, 1963. ^ Source: Dominion of Canada, Statistics Bureau, A Consolidation of Public Finance S t a t i s t i c s j 1961 (Ottawa, 1964). 6 Source: Department of St a t i s t i c s , New Zealand, Report on National Income and Expenditure, 1962-63 (Wellington, 1963). Source: Statistics Department, Denmark, Statisik Arbog, 1963-64 (St a t i s t i c a l Yearbook), Vol. 68 (Copenhagen, 1964)• Final column shows property tax revenue as a percentage of net national product at factor costs. & Property tax and l o c a l tax data based on information i n O.D. Gorven, Diff e r e n t i a l Taxation of Business by Local Authorities (City Treasurer of Durban, South Africa, May 1964), pp. 92-93. n Sources: Ministry of Finance, Japan, Quarterly Bulletin of Financial S t a t i s t i c s , 4th Quarter 1963 (Tokyo, March 1964) and An Outline of Japanese Tax, 1963 (Tax Bureau, Tokyo, 1963). Source: P.H. White, Land Taxation in Canada, unpublished paper, (Vancouver University of British Columbia, 1969) pp. 41-42. Province Area i n (So.Miles) Population Index of Income Income Real Prop-Per Per Capita erty Tax Capita (Nat.Ave..100) Per Capita Index of Real Property Tax Per Capita (Nat.Ave.«100) Real Prop-erty Tax as % of Muni-cipal Inc. Newfoundland 156,185 493,396 1,287 60.0 8.956 11.98 38 Prince Edward Island 2,186 108,535 1,376 64.2 33.353 44.6 65 Nova Scotia 21,425 756,039 1,575 73,5 55.145 73.8 60 New Brunswick 28,354 616,788 1,475 68.8 46.188 61.8 47 Quebec 594,860 5,780,845 1,885 87.9 61.410 82.2 54 Ontario 412,582 6,960,870 2,454 114.4 133.760 179.0 76 Manitoba 251,000 963,066 2,054 95.8 95.648 128.0 68 Saskatchewan 251,700 955,344 2,238 104.4 112.144 150.0 70 Alberta 255,285 1,463,203 2,215 103.3 95.563 127.9 59 B r i t i s h Columbia 366,255 1,873,674 2,438 113.7 105.216 140.8 65 A l l calculations based on 1966 data. Source: P.H. White, Land Taxation in Canada, unpublished paper, (Vancouver) University of British Columbia, 1969) Table 1, p. 40. In addition to a concern for equality of treatment among individual real property owners, i t i s also important to ensure that there be equality of treatment among municipalities. Equality of treatment among municipal-i t i e s i s known as assessment equalization. This is an important concept, since property assessment i n different municipalities offers a convenient basis for apportionment of provincial grants towards several municipalities that are financing j o i n t l y some undertaking from which each w i l l benefit. If each municipality carries out i t s assessments quite independently, then there i s created a danger that the levels of assessments throughout the province may vary considerably. As a result, municipal contributions or provincial grants which are apportioned according to t o t a l assessment i n each area, w i l l be unequally shared. Advantages of having low assessment (indicating greater need) w i l l encourage under assessing for those purposes. With low assessments, i t follows that required tax revenue from re a l property owners to provide adequate municipal services w i l l necessitate a correspondingly high tax rate. For p o l i t i c a l reasons, an elected council i s reluctant to raise the tax rate and so as a result of low assessments and low rates, a municipality may be unable to provide adequate services. Not only i s this undesirable i n a general way, but i t increases demands upon the provincial government for financial aid. An attempt has been made i n this thesis to show that the applica-tion of multiple regression to mass valuations w i l l ensure equality of treatment, both between individuals withiia municipality, and between municipalities. The revenue requirements of municipalities are met by taxes on the value of real property, and large grants are received from the provincial government on the basis of the aggregate of rea l property values i n municipalities. It i s essential to apply valuation techniques that provide equality of treatment i n the carrying out of these two important financial functions. LIMITATIONS OP THESIS This thesis w i l l not be concerned with the many details of real property tax administration, as there are many excellent sources which describe i t . It i s f e l t that i t s inclusion at this point would be unnecessary and detract from the fundamental issue of the basis of assessment.^ The thesis w i l l be limited to the consideration of taxes that provide revenue for purposes commonly associated with l o c a l government, for example, education, f i r e and police protection, refuse collection, water and drainage. Excluded are levies on a particular group to pay for services which are deemed to be of special benefit to them, such as charges for street lighting or sewers (known as l o c a l improvement taxes). The thesis w i l l also be restricted to considering single family residential properties i n Burnaby, as they provide the best data base for generating a sample of sales that can be used to derive a market oriented basis of assessment. The problem of the de s i r a b i l i t y of the institution of the real property tax w i l l not be considered here. It has been discussed at length 5 elsewhere. The real property tax has been maintained by the legislators largely as a matter of expediency. However, no matter how good or how bad the institution i s , assessors should direct their efforts to ensuring that the methods they use to implement the real property tax distribute the tax burden f a i r l y . The adoption of multiple regression analysis as an aid in assessment w i l l provide the most objective means of ensuring that the tax burden i s distributed equitably. ORGANIZATION OF THE THESIS The thesis w i l l be organized into chapters as follows: Chapter Two w i l l discuss the assessor's statutory requirement to assess at 'actual value'. It w i l l i l l u s t r a t e the problems of mass appraising that have led the assessor to adopt an approach to value that i s known to be i l l o g i c a l but that has had to be adopted as a matter of expediency. The chapter w i l l point out that there i s room for reform of the basis of assessment and w i l l suggest the direction that this reform should take. Chapter Three w i l l investigate i n depth the nature of the reform to valuation theory that i s currently being advocated by leading academics i n the f i e l d of valuation. It w i l l also investigate how new computer technology and s t a t i s t i c a l aids are helping to automate much of appraisal. It w i l l explore various aids to the assessor and w i l l conclude that multiple regression analysis i s the device that w i l l equip the assessor with the best means for deriving a market oriented basis of assessment. Chapter Four w i l l describe some of the technical features of multiple regression analysis that are important to the assessor. The chapter w i l l pose the basic technical problems of the technique as they confront the assessor, and w i l l i l l u s t r a t e how the assessor must react to some of the peculiar features of valuation by multiple regression analysis. Chapter Five w i l l discuss i n detail the methodology followed i n attempting to implement the market value basis of assessment with multiple regression analysis in Burnaby. It w i l l also present the results of the study and w i l l measure the quality of the valuation produced by multiple regression analysis. Chapter Six w i l l present the conclusions of the thesis and w i l l make specific recommendations for further research aimed at developing the use of multiple regression analysis i n an assessment context. Finally, the Appendices w i l l contain an i l l u s t r a t i o n of how a t o t a l valuation system based on multiple regression analysis could work in Burnaby. Included w i l l be a description of the costs of such a system; the sequence of events in a hypothetical assessment cycle; examples of the data required by other studies; a description of the geographic area of Burnaby being studied and the detailed s t a t i s t i c a l results of this thesis. REFERENCES CITED CHAPTER I Real Estate Institute of B r i t i s h Columbia, Local Assessment and  Taxation, unpublished course material (Vancouver: The University of B r i t i s h Columbia, 1966), lesson Ut p. 3. Phil i p H. White, Land Taxation i n Canada, unpublished paper (Vancouver: Faculty of Commerce and Business Administration, The University of Bri t i s h Columbia, 1969), p. 12. Real Estate Insitute of Br i t i s h Columbia, Local Assessment and Taxation, lesson 1, p.6. The topic of real property tax administration i s thoroughly explained i n F.H. Finnis' Property Assessment l n Canada, Canadian Tax Paper No. 50 (Toronto: Canadian Tax Foundation, 1970), 1L4 pages. Dick Netzer, Economics of the Property Tax (Washington: The Brookings Institution, 1966), 326 pages. ESSENTIAL FEATURES OF THE ASSESSMENT FUNCTION PURPOSE AND IMPORTANCE OF ASSESSMENT The assessment of single family residential property i s the process by which the municipal tax base i n the form of real property i s valued for purposes of municipal taxation. It i s t y p i c a l l y described as the act of discovering, l i s t i n g and valuing property by the municipal assessment o f f i c e . 1 Two major purposes are served by the process of assessing. These are: determining those properties i n the municipality that are subject to taxation and thereby the t o t a l tax base against which the tax rate i s applied to raise the required municipal taxes, and determining what 2 proportion of the t o t a l tax burden each taxpayer w i l l be required to pay. It i s essential that the assessment process provides an equitable means of valuing property so that the burden of the property tax may be distributed as f a i r l y as possible. In order to achieve equality of treatment, the value of a l l r e a l property should be determined by the application of uniform principles and up-to-date accurate methods that can provide a reasonably equitable relationship between and within a l l groups of real 3 property. The achievement of an equitable distribution of the tax burden within a municipality by methods that enable assessors to adhere to sound appraisal methodology i s a key concern i n this thesis. But before investigating the system of assessment further, i t i s essential that the concept of equality of treatment be further explored. EQUALITY OF TREATMENT The real property tax i s levied on individuals i n respect of the property that they own. The term 'property' usually refers to the physical aspects of real estate and other tangibles, but i t more correctly refers to the 'bundle of rights' possessed by individuals i n those assets. Thus, the real property tax i s levied on individuals according to their rights of ownership of real assets. Since the real property tax i s a tax on individuals there must exist 5 equality of treatment between individuals. As White says: Equality of treatment within the framework of the taxing statute i s at the root of the concept of individual freedom and the institution of private property rights." When equality of treatment i s lacking, the individual taxpayer may bear a disproportionate share of the t o t a l tax. There must also be equality of treatment among municipalities. Where grants to municipalities are calculated with reference to the t o t a l assessed value within the municip-a l i t y , i t i s important for assessments to be at the same l e v e l i n each municipality i n order that the municipality's taxable capacity be equitably assessed. White identifies two objectives founded in equity: 1 Equality of assessments within the same municipality - called uniformity of assessment; and 2 Equality of the average level of assessments between municipalities - referred to as equalization of assessments.? A very important part of real property taxation i s concerned with devising methods that ensure equality of treatment i s bu i l t into the basis of assessment. Unfortunately, the principles of taxation do not provide the means for ensuring an equitable distribution of the burden of the real property tax. Much of the discussion of the real property tax centers on the 8 principles of a b i l i t y to pay and benefits received. The a b i l i t y to pay principle states that those individuals with equal wealth should bear the same burden of taxation. Wealth i s considered i n terms of annual income or assets possessed. Originally, real property holdings were a reliable indicator of wealth, and the efficiency of the real property tax could be judged with respect to this cr i t e r i o n . In today's world however, in view of the number of forms i n which assets can be held, including bonds, stocks, and personal tangible property, and i n view of the essential nature of the shelter provided by houses and other accommodation, real property i s no longer a good indicator of wealth, though some relationship may exist between annual income and the value of the premises occupied. Bonbright says of the real property tax: Certainly the a b i l i t y to pay doctrine can no longer be adduced i n support of a discriminatory burden on realty. Whatever force there may have once been i n the argument that the tot a l wealth of individuals corresponds roughly to the values of their houses and l o t s , has disappeared with changing economic conditions. If a real estate tax can be defended, i t must be on other grounds.^ The principle of the benefits received requires that the taxpayer pay taxes that bear some relationship to the benefits received from the services provided out of those taxes. But, the benefits received from services provided out of the real property tax are not clearly traceable to the taxpayer in question. The extent to which one benefits from municipal services i s indeterminate. The value of real estate does not provide a measure of those benefits. Only in one way can the value of property be used to estimate the benefits accruing to i t , and that i s by considering the differences i n value that would exist i f the public services were not provided. This would be as hypothetical and subjective an evaluation as an attempt to estimate directly the benefits received. The use of the real property tax i s founded neither on the principle of a b i l i t y to pay nor on that of benefits received, but depends upon expediency and practical considerations."^ It provides municipalities with a form of revenue that i s particularly local in character, easy to administer (once assessments have been made), relatively stable and 11 d i f f i c u l t to evade. Since the problem of equality of treatment cannot be solved by resorting to the principles of taxation, i t must be determined by provincial legislators. Under this system, equals can only be discovered by reference to the intentions of the legislators. Assessors must adopt, therefore, that criterion of equality that i s specified in the legislation. The basis of assessment of real property i n B r i t i s h Columbia i s 'actual 12 value' Individuals with real estate of similar 'actual value' should bear similar taxes, and the l i a b i l i t y to tax should be i n direct proportion to the actual value of one's real property. A concern of the legislators should be to construct a basis of assessment that i s based on objectivity rather than subjectivity. When subjectivity i s introduced into the valuation process, equality of treatment may be endangered. The basis of assessment must be easily identified and measured. It w i l l be shown next that while legislators have striven for objectivity i n determining the tax base, they have i n s t i l l e d into the taxing statutes a degree of subjectiveness that makes i t d i f f i c u l t for assessors to apply correct appraisal methodology and techniques. This has created the danger that an equitable distribution of the tax might not be assured. MEASUREMENT OF THE TAX BASE In Canada, re a l property forms the principal municipal tax base. In most provinces real property i s assessed at i t s capital value rather than rental value. Rental value reflects only the value of a property in i t s present use while capital value includes any latent value which could 13 be realized by development schemes. A principle concern i n assessment legislation i s the measuring of the tax base in such a way that equality of treatment results. Legislation in B r i t i s h Columbia requires property to be assessed at 'actual value' with the value of the land being estimated separately from the value of the building and other improvements. The term 'actual value' i s not defined by statute, though the Municipal Act (Section 330 (1)), as well as the Assessment Equalization Act (Section 37 (1)) give some guidance as to the factors which the assessor may consider in arriving at 'actual values'. These Statutes provide thati The assessor may give consideration to present use, location, original cost, cost of replacement, revenue or rental value, and the price which such land and improvements might be reasonably expected to bring i f offered for sale i n the open market by a solvent owner, and any other circumstances affecting v a l u e . ^ Legislation has required assessments for general and for school purposes to be made at 'actual value'. While the statutes do not define 'actual value', the courts have decided i t s h a l l mean exchangeable value -the price which a property shall bring when exposed to the test of competition. 1-' A famous case concerning the definition of value i s the Sun L i f e Assurance Company of Canada versus the City of Montreal. 1^ In this case Taschereau, J . said the following with respect to the use of replacement costs and market data: Although this method of valuation [cost approach) for municipal purposes i s of frequent use, there are cases where i t would be dangerous to attach to i t too much importance...It i s not always, although i t might happen that the market or ex-changeable value of a building i s represented by the amount of the costs less depreciation. The test [the market approach] i s an objective one which i n many cases may be applied by seeking the exchange value or the value i n a competitive market. If there i s no such market, then one may ask what would a prudent investor pay for the subject of taxation, bearing i n mind the return._ that might be expected upon the money invested. This clearly identifies market value and endorses the use of the market (or comparative) approach to value, f a i l i n g which the income .approach may be acceptable, using a reasonable capitalization rate. The important factor to note from these court decisions i s that the courts uphold the market value approach as the best objective standard which can be applied with f a i r l y reasonable uniformity to a l l classes of real property. But as shown earlier, the provincial statutes that estab-l i s h the basis of assessment are not as specific as j u d i c i a l decisions and leave the assessor considerable freedom in the use of the cost, income or market approach to valuation. The simplicity of the basis of assessment of residential property in B r i t i s h Columbia, (by being 'actual value 1) i s complicated by the method of calculating taxable assessed value for general and school purposes. The assessor, to derive taxable assessed value, must f i r s t calculate 'actual value' from which i s derived assessed value. Assessed values for general and school purposes in Burnaby are 10055 of 'actual value', and taxable assessed value for land and improvements i s calculated as il l u s t r a t e d i n Table III. TABLE III - CALCULATION OF TAXABLE ASSESSED VALUES General Purposes School Purposes Land 10056 Assessed Value 100^ Assessed Value Improvements up to 7556 Assessed Value Always 75% Assessed Value Source t F.H. Finnis, Property Assessment in Canada, Canadian Tax Paper No. 50 (Toronto: Canadian Tax Foundation, 1970), p.84. Burnaby i s one of the few municipalities that generate an assessed value of IOO56 of 'actual value'. Other municipalities create further complexities by generating an assessed value of 50% of 'actual value'. As this i s not a problem in Burnaby, i t s further consideration at this point i s not relevant. What i s important to note in Burnaby however, i s that taxable assessed value of improvements for school and general purposes i s not based directly on actual value, while land i s . This introduces an unnecessary element of complexity into the taxation process. The simplicity of the tax system i s lost when the calculation of taxable assessed values for improvements i s not based directly on 'actual value'. THE ASSESSOR'S DILEMMA According to provincial statutes, the assessor i s required to value separately a l l land and improvements in the municipality once a year as of an appointed day, and to use 'actual value' as the basis of assess-ment. With limited staff and budget available to accomplish this task, s t r i c t adherence to the statutes i s impossible and the assessor i s forced to use the freedom given to him by the statutes in an attempt to u t i l i z e mass valuation methods and compromise with his instructions. Assessors are forced to resort to the cost approach to valuation as i t f a c i l i t a t e s the separate valuation of land and improvements required by the statutes. By using this method R a t c l i f f says the assessor . . . i s able to reconcile this obligation [that i s , his statutory obligation to value at actual value] with the requirement for equity and for a rational relationship with market value by judicious fudging of cost estimates and depreciation allowances.^ The literature i s f u l l of comments which describe the inaccuracy and discrimination in the assessment process. Finnis makes the following statements! . . . i t may be gathered that reasonably accurate valuations for assessment purposes may be determined only by the application of a rather complex and time consuming procedure. Valuation for assessment purposes results from a series of judgements based on fact. 20 (It a l i c s mine.) On the complexity of the cost approachi ...for each individual property he [[the assessor] must decide among other things, i n which cla s s i f i c a t i o n to put the building, what cost schedule to apply, what environmental or i n t r i n s i c features cause loss in value and to what extent. A l l these matters require the exercise of judgement, with one assessor being influenced one way and another assessor i n some other way. Accordingly no two assessors are l i k e l y to come up with exactly the same value for a given property even though they may start with the same facts. 2*-The B r i t i s h Columbia Assessment Commissioner supplies the municipal assessor with a comprehensive manual to assist in carrying out this cost approach to valuation. Building costs i n the manual are factored annually to reflect changes i n current construction costs and municipal assessors often adjust these for differences i n lo c a l labor rates and material costs. The Provincial Assessment Commissioner also constructs assessment sales ratios for each area i n the province as an aid to municipal assessors. Armed with the knowledge of how assessment sales ratios for certain classifications of properties in the municipality have changed since the last assessment, the assessor i s i n a position to 'judiciously fudge' his land and improvement values by changing the values on the r o l l proportionately to the change in assessment sales ratios. The c a l l for annual revaluation of properties puts an immeasurable strain on the assessor. Finnis states: The practical impossibility of annually reflecting a l l changes in value i n times when the economic situation i s changing rapidly i s f a i r l y generally conceded. It seems certain that even i f assess-ments are made at market values, the assessments cannot possibly reflect current market values but must always be one or two years behind current values. Valuation i s not an exact science. At best, assessments of properties en masse can only reflect established trends within each cl a s s i f i c a t i o n of property with modifications made for individual properties at the discretion of the assessor. A change of values that seems to be appearing i n one year should not be reflected i n revised assessments u n t i l the assessor can be sure that the change i s real l y an established trend. This may take two years to establish. Re-assessment of a l l properties in large assessment jurisdictions usually takes two or more years to accomplish by which time the assessed values w i l l be representative of the le v e l pertaining to some four or even five years prior to the completion of the r o l l . To this extent therefore, assessment levels after a careful re-assessment based on market values must always lag behind current market values. Once an assessment close to current values has been achieved however, established trends i n changes of value could start to be reflected i n the r o l l , and over a period of perhaps two or three years assessments would approach very close to current market values. As sales and rental analyses should cover at least a two year period i f reliable trends are to be established, i t would probably be undesirable as well as unlikely that assessments _ would be made at so-called current market values. This viewpoint seems to indicate that there i s l i t t l e likelihood that the assessor w i l l ever be able to assess property at 100$ of actual market value. The assessor, i n short, appears to be in a position of never being able to assess in the manner required by provincial statutes. Assessors have had to compromise with the wishes of the statutes i n order to assess so many thousands of properties. It i s important to examine the implications of this compromise. 1 The assessor i s using the cost approach to value where cost of replacement new less depreciation purports to measure the current market value. This approach i s condemned in the literature for i t s subjectiveness. The cost approach i s very seldom, i f ever, equal to market value and therefore w i l l only permit the assessor to estimate actual value by even more subjective methods of 'fudging'. 2 The cost approach implies that real property has an i n t r i n s i c value. Current appraisal literature vehemently denies this proposition and advocates that the value of property is determined solely i n the market by buyers and sellers. Craig argues that the value for assessment purposes should be 'most probable s e l l i n g price', a view that i s gaining widespread acceptance i n the appraisal of real property i n a non assessment 23 context. If i t is accepted that the cost approach i s identified with i n t r i n s i c value then the assessor w i l l never be i n a position to determine objectively 'actual value 1. 3 The cost approach implies that the value of a property can be broken into separately measurable components. R a t c l i f f argues that the value of a property is the result of the interaction of the components and that the only way to break value into 2A components i s by subjective procedures. He also argues that i t i s not possible to measure a decline i n value measured by depreciation u n t i l the property actually s e l l s . Thus any measure of determining value of improvements by calculating replacement costs new less depreciation i s based on subjectivity. In the light of these inadequacies of the cost approach to value, i t can be seen why the real property tax has been so widely c r i t i c i z e d by economists, politicians and the public at large. But i n spite of i t s inadequacies, the real property tax has been maintained as the major source of revenue for municipal governments i n Canada. There have been five provincial commissions of enquiry i n Canada 25 since 1963 that have considered the problems of the property tax. White has summarized the findings of the commissions and states: Dislike for the tax centered on i t s regressivenessj poor administration, especially assessment practices which permitted a glaring lack of uniformity of assessments; and cariations i n f i s c a l capacity of municipalities brought about by the unequal distribution of real property values within a province. 2" But, i n spite of i t s many drawbacks, the commissions recommended that the institution of the real property tax not be abolished. ...the tax was recognized as being the only major one which i s capable of being administered s a t i s f a c t o r i l y i n a municipality of any size and thus i t offers l o c a l authorities an opportunity to exercise responsibility i n a meaningful sense.2'' Deciding that the tax was the best of the worst, the commissions' objective became to reform the tax; that i s , to construct a more eff i c i e n t system of real property taxation. An important area of reform centered White states that: The c r i t e r i a i n the selection of a satisfactory-basis of assessment are that i t should be: (1) simple to understand; (2) certain; (3) capable of being estimated i n the majority of cases by reference to data which w i l l permit the quality of assessments to be tested by s t a t i s t i c a l methods and (4) amenable to estimate with the resources available. It i s important to analyze the basis of assessment i n Burnaby with respect to each of these c r i t e r i a . F i r s t l y , the basis of assessment must be simple to understand. The basis of assessment in Burnaby i s simple to understand as i t i s based on actual value. However, this simplicity i s lost when different methods for calculating taxable assessed values are introduced. It i s therefore recommended that the taxable assessed value of both land and improvements be based dire c t l y on actual value. Certainty i n the basis of assessment requires that the taxing statute define the basis of assessment i n a clear and unambiguous manner. The vagueness of the definition of the basis of assessment in the statutes has already been mentioned. The law must provide a clear definition of the basis of assessment. The law should also recognize the appraisal methodology that enables the basis of assessment to ensure equality of treatment. Certainty w i l l be best introduced by defining actual value to be most probable s e l l i n g price where most probable s e l l i n g price i s determined by the market approach to value. The requirement that the basis of assessment be measured by s t a t i s t i c a l methods to measure the quality of the assessment i s necessary in order to permit the implementation of equality of treatment called for White states that: The data must be objective, and the only basis of assessment that w i l l meet this requirement i s current market value. Market transactions w i l l supply a continuous stream of evidence of value for most kinds of property, and i t can be used at w i l l . No such evidence i s available for any other basis of assessment. The data used for valuation purposes in Burnaby is primarily data for the cost approach to value. This data i s not as objective as the data used in the market approach to value. It i s recommended, therefore, that market data be used for the basis of assessment. The fourth c r i t e r i a i s that the basis of assessment be amenable to estimate with the resources available. The amount of such resources available for assessment purposes i s a p o l i t i c a l decision and i s largely influenced by considerations of cost. The prohibitive cost of employing an appraisal staff to assess each property at market value has been a large factor i n fostering the acceptance of the cost approach to value. New aids i . to the assessor i n the form of computers and s t a t i s t i c a l tools now make i t possible for the assessment of municipal properties at market value. By adopting these new techniques for valuing large numbers of properties, the assessor may be i n a position for the f i r s t time to adopt objective market comparisons for valuation at a re l a t i v e l y reasonable cost. The costs of implementing a valuation system based on multiple regression analysis i s i l l u s t r a t e d i n Appendix B. This chapter has illustrated that the assessor has been forced to rely on an i l l o g i c a l cost approach to generate 'actual values'. It has also shown how 'judicious fudging' of these cost estimates generates a value that i s lagging behind current market values. Assessors have been aware of the contradiction between what the law demanded of them and what current assessment techniques were able to provide. However, the cost approach to value was the only method that would enable them to generate values for land and improvements on the scale required. Many assessors now recognize that adoption of current market value as the basis of assessment i s the best approach to valuation for ensuring the existence of equality of treatment. White states: The reasons for general agreement in favour of current market value as the basis of assessment are expressed in the New Brunswick Report: "... uniformity of assessment, especially between individual taxpayers, can only be secured where there is an objective measure of value, and the only objective yardstick i s the one based on competition i n the market."-** There i s now a s t a t i s t i c a l technique of valuation that permits assessors to value thousands of single family residential properties by the market approach to value. This s t a t i s t i c a l technique i s called multiple regression analysis. Together with the tremendous processing capabilities of the modern electronic computer, i t i s enabling assessors to use the concept of market value as the basis of assessment. A valuation system u t i l i z i n g multiple regression analysis meets the four c r i t e r i a that White indicates a good assessment must have. F i r s t , i t w i l l enable legislators to be more explicit i n their definition of actual value and how to achieve i t . Value under this system w i l l be a market determined 'most probable se l l i n g price'. Second, i t w i l l create a basis of assessment that w i l l be easier to understand than the complicated cost approach to valuation that now exists. Third, i t w i l l permit the quality of assessments to be measured in an objective manner, since the whole foundation of multiple regression analysis is based on the objectivity of s t a t i s t i c a l concepts. Fourth, i t w i l l provide a re l a t i v e l y inexpensive method of valuing a l l single family residential properties. The end product of such a valuation system w i l l be the generation of 'actual values' where 'actual value' i s current market value or most probable s e l l i n g price. If the provisions of the taxing statutes were changed so that the assessments should be provided for the t o t a l property rather than separate values for land and improvements, this modern valuation system would create the ultimate in objectivity for the assessor. However as long as land and improvements must be valued separately, some of the objectivity w i l l be lost when the values of the process are a r b i t r a r i l y separated into value for land and improvements. But the technique is undeniably useful i n providing an objective starting point for the assessor. To the extent that the market approach to valuation i s more objective than other assessment procedures, this new method of valuation w i l l f a c i l i t a t e the equitable distribution of the burden of the tax within municipalities and among municipalities. Before discussing the technical details of multiple regression analysis i n a valuation context, trends i n appraisal theory and econometrics which have argued for, and provided the means for the intro-duction of a s t a t i s t i c a l approach to value w i l l be introduced and investigated. Appraisal theory has been able to provide an answer to the assessor's dilemma for some time. Assessors must apply this new valuation theory and s t a t i s t i c a l technique to their own valuation problem now. The application of these new concepts can solve the assessor's dilemma and ensure that the institution of the real property tax w i l l produce equality of treatment among real property owners. Finnis, Property Assessment In Canada, p . l Kenneth Grant Crawford, Canadian Municipal Government, (Toronto: University of Toronto Press, 1954), p. 261. Finnis, Property Assessment in Canada, p . l . K.E. Boulding, Principles of Economic Policy, (Englewood C l i f f s , N.J.: Prentice-Hall, Inc., 1958), p.30. Equality of treatment is the treatment of each real property owner in accordance with the intentions of the taxing statute. Rating and  Income Tax, (London: Solicitors Law Stationery Soc.Ltd.) Vol.XLII (1949), 488 and Vol.XLV (1952), 675. White, Land Taxation i n Canada, p.12 Ibid. J.F. Due, Government Finance, (Homewood, 111.: Richard D.Irwin, Inc., 1963), Chapter v i . J.C. Bonbright, The Valuation of Property (Charlotteville,Va.: The Mitchie Company, 196^ 5) I, p.500 M.A. Cameron, Property Taxation and School Finance in Canada, (Toronto: Canadian Education Association, 1945), pp.20-21. J.R. Hicks, V.K.Hicks and C.E.V.Lester, The Problem of Valuation for  Rating, National Institute of Economic and Social Research. Occasional Paper No.7 (London: Cambridge University Press, 1944), pp. 6-11. Finnis, Property Assessment i n Canada, p.83. White, Land Taxation i n Canada, p. 14* Real Estate Institute of B r i t i s h Columbia, Local Assessment and  Taxation, lesson 4» p. 2. 15 M**t P»3 1 6 Ibid. 1 7 Ibid. 18 Finnis, Property Assessment in Canada, p.84. 19 R.U. R a t c l i f f , Valuation Theory, Unpublished course material (Vancouver: Faculty of Commerce and Business Administration, The University of Br i t i s h Columbia, 1970), Chapter V, p.12. O f ) * w Finnis, Property Assessment ln Canada, p. 70. 2 1 Ibid., p.71. 2 2 Ibid., p.70. 23 R.H. Craig, "Property Assessment at Market Value," Appraisal Institute  Magazine. Vol.XXXIX, No.2, (April, 1971), p.3. 2 ^ Ra t c l i f f , Appraisal Theory. Chapter V. 25 Report of the Royal Commission on Finance and Municipal Taxation i n  New Brunswick, Queen's Printer, Fredericton,N.B., 1963. Report of the Manitoba Royal Commission on Local Government Organization  and Finance, Queen's Printer, Winnipeg, Manitoba, 1964. Report of the Roval Commission on Taxation r Province of S w B l c a t f t h « w H T i T Queen's Printer, Regina, Saskatchewan, 1965. Report of the Royal Commission on Taxation (Government of Quebec). Queen's Printer, Quebec, 1965. The Ontario Committee on Taxation, Report, 1967, Queen's Printer, Toronto, 1967. 26 White, 27 Ibid. 28 Ibid.. 29 Ibid.. 30 Ibid. 31 Ibid.. VALUATION THEORY AND MULTIPLE REGRESSION ANALYSIS INTRODUCTION This chapter w i l l i l l u s t r a t e that there i s a growing movement within the appraisal community, both at the academic and at the practising levels, away from the cost approach to valuation. The new movement i s c a l l i n g for the abolition of the cost approach to valuation and for the adoption of a more modern market approach to valuation. The assessor must also move in the direction of this new valuation theory as i t provides the only answer to three very serious problems: 1 The assessor's statutory obligation to assess at market value 2 The increasing number of properties that have to be valued with limited assessment staffs and budget 3 The use of the i l l o g i c a l cost approach to value. TRENDS IN VALUATION THEORY The cost approach to value has long been condemned as unsound. Even back in 1932, the o f f i c i a l standards of Appraisal Practice of the National Association of Real Estate Boards (of the United States) commented on the inadequacies of the summation process (cost approach): Such summation appraisals are condemned as unsound, inaccurate and misleading because this method bases the opinion of value on the addition of values that may not simultaneously obtain, and ignores the effect of over-, under-, and misplace improvement, and disregards the interrelation between land value and value of the improvement...it i s unethical for an appraiser to issue a report on a property i n which the t o t a l reported value i s derived by adding to-gether the fractional parts of the property. In particular, i n appraising the security for a loan, i t i s unethical for an appraiser to issue an appraisal report on a property in which the t o t a l reported value i s derived by adding together the market value of the land as i f unimproved, or the value of the land as i f improved to i t s highest and best use, and the reproduction cost of the improve-ments less accrued structural depreciation.^ Current appraisal literature increasingly condemns the cost approach to valuation. Babcock states: ...the cost approach i s suspect as a market valuation process. 2 Wendt states: ...the replacement-cost method has only limited and special applications i n real estate valuation. ...contrary to accepted appraisal theory, the division of depreciation between 'physical', 'functional' and 'economic' is a r t i f i c i a l and cannot be accurately measured.' Ra t c l i f f states: For nearly 40 years commentaries have appeared on the cost of reproduction less accrued depreciation approach which have t o t a l l y destroyed i t on lo g i c a l grounds as a device for estimating market value.^ R a t c l i f f argues that the most useful measure of market value i s 5 'most probable s e l l i n g price'. Most probable s e l l i n g price is an estimate or forecast of what the property i s l i k e l y to s e l l for in the real estate market. Most probable se l l i n g price i s thus a function of consumer or investor behavior and not the a r t i f i c i a l value t r a d i t i o n a l l y accepted in appraisal theory based on the doctrine of w i l l i n g buyers and willi n g s e l l e r s . Being a forecast of l i k e l y s e l l i n g price, most probable s e l l i n g price i s not a certainty, and is qualified by R a t c l i f f by expressing i t as a range of estimates with attached probabilities expressing the likelihood of most probable s e l l i n g price being in a particular portion of the range.' Academics have been quick to give R a t c l i f f support for this notion and together they a l l have firmly entrenched this new concept of value into appraisal theory. Wendt states: Market value should be viewed as the central concept i n real estate valuation.'' Smith and Racster indicate that there should be: ...provision of probability estimates with value or s e l l i n g estimates.** Kinnard comments: Perhaps the most stimulating and challenging new development in appraisal theory i s Dr. R a t c l i f f s contention that the objective of appraisal analysis should be the estimation or prediction of most probable s e l l i n g price, rather than market value. Dr. R a t c l i f f argues that this i s more nearly descriptive of what the market does. This concept i s more orientated to economic and market factors rather than being based on l e g a l i s t i c notions of an abstract 'economic man' environment.9 ...when modern methods of s t a t i s t i c a l probability analysis are applied to appraising which i s aimed at estimating most probable s e l l i n g price, both the r e l i a b i l i t y of the single value estimate and the odds on the range around i t can be estimated with a high (and predictable and measurable) degree of confidence.1° Ratcliff also argues that there are only two approaches to estimating most probable selling price; simulation and statistical inference. Simulation is used when there is l i t t l e market sales activity on which to base the appraisal. This ...approach to price prediction is simulation of the price establishment process in the market. Here the appraiser 'takes into account1 a l l of the factors relationships and processes which could affect the selling price i f the subject property were exposed to the market and arrives at a judgmental conclusion. 1 1 {On the other hand) given some records of past market price behavior in the market, the appraiser can process these market facts and derive a predic-tion by statistical inference. Thus as a product of statistical analysis, the record of relevant past market behavior in respect of properties like the subject property becomes the basis for expectations concerning the probable selling price of the subject property.*2 This process of valuation by statistical inference has been able to progress from the realm of academic theory in the realm of practicability for appraisers due to recent developments in econometrics and computer technology. In the field of econometrics, a method of making inferences about the nature of a population by multiple regression analysis enables valuation to take place in the context of Rat c l i f f s statistical inference. Also, new developments in statistical analysis enable simple measures to be made of the quality of valuation made from this process of statistical inference. Kinnard states: Through the development of Bayesian statistics and the application of finite mathematics, analysis of the probabilities of errors and the range of error inherent in small samples (which most appraisal studies almost necessarily are) offers for the f i r s t time the prospect of an evaluation of the conclusions reached i n appraisal analysis.13 While this multiple regression technique i s appealing, i t suffers from the problem of requiring a large number of very laborious calculations that tend to make i t an inoperable proposition for any fee appraiser or assessor. New computer technology has solved this problem however. Kinnard The most important single prospect of the application of electronic data processing to the appraisal f i e l d at the moment i s that i t permits rapid and systematic analysis of large volumes of data. Once a bank of reliable market data (sales, rents, or whatever) i s available, then i t i s possible to talk with increasing degrees of accuracy about what difference a deficiency or excess actually does make on the market. The c r i t i c a l factor, then, i s that electronic data processing equipment permits large volumes of data to be manipulated quickly and consistently so as to produce appropriate r e s u l t s . ^ Case states: In combination with newly developed mathematical tools, computers can provide invaluable assistance i n a number of c r i t i c a l decisions such as: 1 simple and accurate methods for processing data for large numbers of comparable properties; 2 substantiation of the accuracy of value estimates Ratcliff's method of s t a t i s t i c a l inference for valuation purposes then, i3 an operable concept. This has great implications for assessors. states: NEW APPRAISAL THEORY AND THE ASSESSOR S t a t i s t i c a l inference w i l l enable the assessor to u t i l i z e the objectiveness of the market approach to value, and computer technology w i l l enable large data banks to be assembled to store the vast quantities of information that would be subject to this inferential process. The computers w i l l also enable the multitude of calculations required i n multiple regression analysis to be handled with speed and with a minimal element of human error. The assessor should t r y to adopt those valuation techniques that the entire appraisal community adopts. Hinshaw states: A modern assessor's office must apply the same principles and procedures to a l l property as appraisers apply to individual appraisal assignments. 1" The most objective measure of market value that both appraisers and assessors can use is R a t c l i f f s concept of most probable s e l l i n g price. This i s the most objective measure of market value since i t i s the result of actual buyer and s e l l e r behavior. It should be the value concept adopted for the basis of assessment as i t alone can guarantee objectivity and equality of treatment i n mass appraisal. By adopting these new concepts into his valuation system, the assessor w i l l at last be in a position to meet his statutory obligations. If he adopts most probable s e l l i n g price as the basis of his assessment, he w i l l be assured of creating the equality of treatment that the statutes require he produce. If he adopts multiple regression analysis as a computational aid, he w i l l have at his disposal a means of producing most probable selling prices for the thousands of single family residences that need to be valued in his municipality each year. RECENT TRENDS IN ASSESSMENT TECHNIQUE To date, very few assessment offices have shed the cost approach to valuation and implemented a market approach to valuation by statistical inference. There have been movements towards approximating the market approach by 'judiciously fudging' values generated by the cost approach, and there has been a great deal of application of the computer to facilitate the simple, repetitive calculations of the cost approach. A first and natural application of the computer to the assessor has been to eliminate the tedium of the cost approach. Given an adequate computer fa c i l i t y that can store basic variables required in the cost approach for a l l properties in a municipality and given an adequate computer program to do the required arithmetic calculations, the assessor simply has to describe the subject property and the machine will perform a l l the repetitive calculations required. The basic advantages of this type of system are that i t saves many man hours of work from the time required to produce an assessment r o l l ; i t cuts down on the staff requirements for clerical positions, and i t permits the assessor's staff to spend more and more time in valuation problems that require the application of appraisal expertise. An example of the computer and 'judicious fudging' of the cost approach is provided in the study area, Burnaby. Here a property is first valued by the cost approach to value on the basis of a detailed cost manual. Thereafter, a detailed analysis of real property assessment sales ratios in the munciipality indicates annual rates of real property price changes. This provides the basis for the construction of the index, or factor, vhich i s applied to the original cost approach value to bring that value up to the current level of prices as indicated by the sales analysis. This factor i s then used by the computer to adjust a l l the old values that are stored within the computer system. While this type of procedure may very accurately reflect price changes, i t s t i l l does not eliminate the original i l l o g i c a l method of the cost approach to value. The danger of inequality of treatment can therefore, s t i l l exist. These aids to the assessor, although they save time and dollars, do not allow the assessor to l i v e up to his f u l l statutory obligation of valuing at actual value. To do this he must adopt the market basis of assessment. The cost approach to value cannot be guaranteed to produce a current market value. A system of valuation that makes use of multiple regression analysis w i l l generate market values (most probable s e l l i n g prices). Such a system w i l l introduce a new era of objectivity into the assessment process and w i l l guarantee equality of treatment. ECONOMETRIC MODELS FOR THE ASSESSOR To understand the role of multiple regression analysis for the assessor i t i s essential to realize that the multiple regression equation i s an econometric model that can describe or predict relevant market structure. Econometrics i s the application of s t a t i s t i c s to estimate and 17 predict l o g i c a l relationships suggested by economic theory. By using multiple regression analysis the assessor i s actually building a s t a t i s t i c a l model that relates market factors to s e l l i n g prices. Brigham and McAllister suggest that there are three steps of model 18 formulation with which an assessor must contend. F i r s t l y , a model must be formulated of the relevant market structure. For example, i n the assessment of single family residential homes, intimate knowledge of the factors that affect this market are essential. R a t c l i f f considers this as becoming aware of the multitude of factors that affect the real 19 property's productivity. Knowledge must be gained of the physical characteristics of the land, on site and off site improvements, structural improvements and the importance of location to the properties. In addition insti t u t i o n a l factors, national, regional and l o c a l economic conditions that affect the market are also relevant i n any attempt to construct a model which brings together a l l of the factors that influence market value. Basically then, this step involves ascertaining those factors that have some logi c a l effect on market value and attempting s t a t i s t i c a l l y to relate these factors to the s e l l i n g price of the properties. The s t a t i s t i c a l model found most relevant for discovering relationships among different factors i n real property valuation i s called multiple regression analysis. Basically, multiple regression analysis i n a valuation context i s a method of f i t t i n g a formula to estimate the market value of a sample of sales through s t a t i s t i c a l observations based on minimizing the sum of squares of the difference between the actual and estimated values. Ordinarily the multiple regression formula takes the form; Y = a + b 1(x 1) • b 2(x2) • + b n(x n) where Y • moat probable s e l l i n g price of the real property. This variable i s the dependent variable as i t depends or is formulated by the other parts of the equation. a s a constant sum s t a t i s t i c a l l y determined by the multiple regression technique. b = a multiplier or coefficient s t a t i s t i c a l l y determined by the multiple regression process. x = a factor that affects r e a l estate value as discussed above. n = a subscript denoting any given number of items. The power of multiple regression analysis is f u l l y u t i l i z e d when i t i s used to estimate the market value of properties outside the sample used to construct the equation (that i s , to predict the most probable s e l l i n g prices of the population of single family residential properties). However, due to the fact that a sample i s very seldom a perfect representation of the population from which i t i s drawn, there are bound to be imperfect predictions of population s e l l i n g prices. Hence i t is useful to determine how closely the multiple regression formula derived from the sample describes the actual relationship between the dependent and independent variables i n the population. This i s a matter of determining the quality of the multiple regression formula. Measures such as the standard error of the estimate, the coefficient of determination, the regression coefficients and their standard errors and t tests for each variable have been devised to measure the s t a t i s t i c a l quality of the multiple regression formula. These s t a t i s t i c a l measures allow the assessor to express his confidence that value estimates w i l l be contained within o assessor to express in probabilistic terms the degree of estimating error inherent in the value predictions. No attempt other than the preceding breif description w i l l be made to explain the mechanics of multiple regression analysis as there are many texts available that explain i t in 20 considerable depth. The second step of Brigham and McAllister's model formulation i s to actually take sold properties, quantify many of their property characteristics, including s e l l i n g price, and submit them to multiple regression analysis. The output w i l l be the formula i l l u s t r a t e d e a r l i e r with the various other s t a t i s t i c s that measure the quality of the regression formula. The third step in the process is to use the multiple regression formula to predict the prices of unsold properties. Multiple regression analysis then, i s a way of predicting the prices of unsold properties through observation and analysis of past market transactions. Smith states: ...multiple regression analysis is another way of analyzing market data. The technique of multiple regression analysis i s just a labor saving short cut for applying the market data approach where one would not normally have the necessary time or resources to apply i t properly. 2^ One study says of multiple regression analysis and the market approach to value: S t a t i s t i c a l valuation [is] an attempt to apply the traditional market approach to mass appraisal conditions. In this context the potential of multivariate analysis is that i t seems to provide a systematic framework for making the subjective decisions associated with the adjustments of comparables on a scale suited to mass valuations. Yet just as the market approach cannot be used when few good comparables for a particular property are available, statistical valuation . is limited by the extent of recent sales data. Our work so far suggests that large numbers of urban dwellings in a typical urban center are amenable to some form of statistical valuation. Obviously when so many comparables must be utilized in the analysis the traditional concept of very exact comparables must be altered. Ratcliff argues that: The primary criterion for selection of the comparables is substitutability in the eyes of the buyer, for the subject property is a medium for attaining housing goals.23 Thus non-identical homes are substitutable in the eyes of buyers i f they have differing combinations of factors that produce the same ut i l i t y or satisfaction to the buyer. Also, comparables may vary in neighborhood,quality of improvements, lot size, etc. MULTIPLE REGRESSION ANALYSIS AND EQUALITY OF TREATMENT A primary purpose of Chapter Two was to show how, in the absence of principles of taxation on which to base the fairness of the property tax, that i t is necessary to ensure equality of treatment by basing the tax on standards provided by the statutes, that is on a market value basis of assessment. White was shown to argue that for assessment purposes, the data used in the valuation process must be objective, and that the only basis of assessment which met this requirement was current market value. It i s a fundamental theme of this thesis that market value or most probable s e l l i n g price i s the onlylogical basis on which to ensure equality of treatment. The implications of multiple regression analysis are at once obvious. On a mass scale i t processes objective market transactions by a very objective s t a t i s t i c a l approach to produce market values. A l l single family residential properties within a municipality can be valued this way. The subjectivity of valuation and the danger of unequal treatment caused among taxpayers by inconsistent implementation of the cost figures and depreciation estimates by assessors w i l l be eliminated. If this valuation process i s also adopted by other munciipalities, i t would be possible to ensure equalization of assessments as well as equality of treatment within each municipality. Multiple regression analysis seems to be the best aid available to help the assessor meet his statutory obligations. The case for an econometric approach to valuation is neatly summed up by Renshaw: Perhaps the most important area of application for the econometric method of appraisal i s in the f i e l d of assessment. It is common knowledge that property tax assessment is often carried out i n a manner that does not insure a f a i r valuation of a l l property. Some studies of property assessment relative to sales value have even indicated that assessment could be improved from the standpoint of increasing equality of treatment, by assigning to a l l property within the taxing d i s t r i c t the average sales value of property sold. Since the l a t t e r system i s tantamount to making no appraisal of the individual properties concerned, i t i s immediately apparent that a model of the real estate market which explained only a small fraction of the variance i n sales would be an improvement over the actual assessment procedure followed in some d i s t r i c t s . The econometric approach to real estate assessment, almost by definition, assures equal treatment. He who pays taxes based on the approach can easily ascertain that he i s being treated impartially, even though he may f e e l that he i s being treated unfairly by a model not always accurate in i t s predictions of sale value. Further advantages can be cited for the econometric approach when the issue i s i n t e r d i s t r i c t equalization. The popular assessment-sales ratio method of determin-ing equalization factors i s fraught with the d i f f i c u l t y that the sample of properties sold maybe unrepresenta-tive of average assessment practice. The problem of sampling error is particularly acute when certain classes of property are sold infrequently. These d i f f i c u l t i e s are largely avoided by the econometric approach, since sales observations used to validate the assessment model can be taken from a l l d i s t r i c t s which are s t a t i s t i c a l l y comparable. Further, where functional relationships are known to exist, the sales observations actually used may not have to be repres-entative of either average property value or average assessment practice to obtain a good f i t . 2 ^ In Chapter Two i t was i l l u s t r a t e d how d i f f i c u l t i t was for the assessor to be able to observe the nature of real property price trends. Changes of value that appeared in one year could not be reflected in revised assessments u n t i l the assessor could be sure that the change was r e a l l y an established trend. This often takes two years to establish and i s a major reason why assessments are notorious for their lagging behind actual market conditions. Finnis stated: As sales and rental analyses should cover at least a two year period i f reliable trends are to be established, i t would be...unlikely that assessments would be made at so called current market values.25 The application of multiple regression analysis offers a solution to this problem. As w i l l be shown in more detail in Chapter Four, multiple regression analysis permits the assessor to ascertain annually the nature of price changes. Thus, each year the nature of current price trends w i l l be able to be reflected on the assessment r o l l . This w i l l shorten considerably the lag between assessed values and market value and i n many cases w i l l permit assessed values to be a t the same level as actual market values. CURRENT USE OF MULTIPLE REGRESSION ANALYSIS IN ASSESSMENT A f i r s t application of multiple regression analysis i n an 26 assessment context has taken place in Orange County, California. Here assessment records for 275,000 single family residences were transformed to computer storage disks. For each property was recorded the appraised land value, construction type, area, age, number and type of room, construction cost, descriptions and cost of extra plumbing, a i r -conditioning, - a l l the essential appraisal cost factors and market data. In a l l 200 million bits of information were programmed, coded and keypunched into the computer. In addition to this appraisal record, every sale of property in the county and i t s property characteristics were stored to be used as the sample on which the multiple regression formula was to be calculated. The v a l i d i t y and v e r s a t i l i t y of multiple regression analysis was tested under differing conditions, using five geographical areas: Area I regular, f l a t lots; sales prices are between $15,000 and $32,000; homes from one to twelve years old; with 1,000 to 2,100 square feet of l i v i n g area. Area II much li k e Area I, except most homes are two to. three years old (most recorded sales.in this area were original sales, not resales). Area III an older part of a city in transition; sales prices are between $8,000 and $30,000; homes are from six to 45 years old; have from 600 to 3000 square feet of l i v i n g area, in three to eight rooms. Area IV r o l l i n g h i l l lots from 6,000 to over 100,000 square feet; sale prices are between $15,000 and $150,000; homes are from one to twenty-six years old; have from 900 to 4000 square feet of l i v i n g area, in four to ten rooms. Area V near the ocean; sale prices are between $12,000 and $65,000; homes are from one to 40 years old; have from 600 to 3000 square feet of l i v i n g area in four to 11 rooms. For each area approximately 200 sales comprised the basis for multiple regression analysis. For each property there were 166 property characteristics that were to be considered i n determining the functional relationship between sales price and property characteristics. The literature on this particular application of multiple regression analysis does not provide the detailed descriptions of the equation that this thesis w i l l provide in describing the quality of the multiple regression formula. The equation that resulted from this study i s reported as follows: Estimated Selli n g Price = $14,500. • 14.32 - 637. - 479. •f 876. • 343. - 918. • 193. - 1,160. - 809. - 1,318. - 391. + .77 1.46 3.29 .48 • 2.42 • 222. • 440. .93 1.18 .52 + 1.082 50.57 - 412. i f lot i s irregular i f lot i s on a cul-de-sac i f neighborhood value trend i s above standard i f available financing i s above standard i f condition i s below standard i f condition i s above standard per bathroom i f no central heating i f no central cooling per quality class per square foot of l i v i n g area per dollar of cost of uncovered patio per dollar of a i r conditioning cost per dollar of extra kitchen cost per dollar of extra plumbing cost i f detached garage i f attached garage per dollar of f l a t work cost per dollar of fence cost per dollar of miscellaneous cost per dollar of replacement cost new per month for age of sale i f view is below standard The quality of the equation i s judged by comparing estimated s e l l i n g price with actual se l l i n g price. The results are shown in Table IV. TABLE IV DISTRIBUTION OF DIFFERENCES BETWEEN ESTIMATED SELLING PRICE AND ACTUAL SELLING PRICE Number of Cases % Deviation Area I Area II Area III Area 17 Area V Total 1 48 75 31 35 28 217 2 40 40 19 29 15 143 3 32 46 28 25 27 158 4 32 18 25 27 17 119 5 9 23 16 15 15 78 6 7 21 17 19 12 76 7 9 8 8 13 12 50 8 1 5 10 9 16 41 9 1 5 8 7 6 27 10 - 1 5 10 7 23 10 to 15 1 4 11 13 18 47 15 & over wm 2 8 1 3 14 Source: Andrew J. Hinshaw, "The Assessor and Computerization of Data", The Appraisal Journal, Vol. XXXVII, No.2 (April, 1969) Table I, p. 287. in 72$ of the cases (715 out of 993) estimated s e l l i n g price differed from actual s e l l i n g price by less than 5%. In 94$ of a l l cases (932 out of 993) the difference was less than 10$. Judged by the standards most researchers in this f i e l d have formulated, these are impressive results. One study stated: As a rough and ready guide we regard predictions with 10$ of "observed value" with a r e l i a b i l i t y level of 95$ as suitable for research and devel-opment.2' Another measure of the quality of the procedure involves computing the average amount by which the estimated s e l l i n g price differs from the 21 actual s e l l i n g price both i n dollars and as a co-efficient of dispersion. These measures of quality are presented i n Table V. TABLE V AVERAGE DEVIATIONS CF ESTIMATED SELLING PRICE FROM ACTUAL SELLING PRICE AND COEFFICIENTS OF DISPERSION Mean se l l i n g Average Coefficient Area price Deviation of dispersion I $ 20,496 448 • 2.3% II 24,500 705 1 2.9 III 19,429 901 * 4.6 IV 35,590 1,354 * 3.8 V 22,668 1,082 * 4.8 Source: Andrew J. Hinshaw, "The Assessor and Computerization of Data", The Appraisal Journal, Vol. XXXVII, No.2 (April, 1969) Table II, p. 287. By both measures this application of multiple regression analysis was very successful. This clearly i l l u s t r a t e s that this econometric approach to valuation i s of a very high calibre. By using multiple regression analysis as an aid i n the valuation process, the assessor i s released from the computational and other routine tasks which now absorb such a great deal of hi3 time. The assessor i s then l e f t to concentrate on evaluating estimated market values and on spotting those cases where the computor fai l e d to estimate value. But i t must be understood that values predicted by multiple regression analysis do not substitute for appraisal opinion. Value predictions produced by multiple regression analysis must be reviewed by qualified personnel in the assessment o f f i c e . Only a qualified assessor can discover those cases where prediction i s outside of the range of quality accepted. Multiple regression analysis i s only a computational aid. As such, the quality of information produced i s a function of the quality of information put into the system. It i s up to the assessor to ensure that reliable unbiased simple information i s put into the process and i t i s also up to the assessor to observe those predictions of value that are unacceptable and which must then be done by the assessor himself. system that u t i l i z e s real estate sales i n the valuation process. The functional elements of a t o t a l assessment system aret A TOTAL ASSESSMENT SYSTEM Multiple regression is actually only a sub-routine of a larger Function Description 1 preparation of appraisal f i l e accumulate information on property character-i s t i c s for each residential property i n the municipality 2 preparation of sales f i l e accumulate information on a l l recent s a l e s — property characteristics and sales price 3 cost updating perform a l l required cost calculations on the appraisal and sales f i l e U profile analysis examine sample and population to ascertain i f sample i s unbiased calculation of estimating equation 6 calculate market value for a l l properties i n appraisal f i l e 7 ensure quality of estimated prices 8 transfer values to tax r o l l . apply equation to a l l properties i n the municipality examine output and watch for exceptional cases. A. detailed example of a t o t a l system of assessment and i t s cost i s presented in Appendix A. It can be seen then, that multiple regression analysis i s just part of a t o t a l valuation system. It does not substitute for appraisal expertise A t o t a l assessment system requires subjective decisions i n each and every functional area including multiple regression analysis. Whereas before the assessor spent the majority of hi3 time in laborious calculation, he now can be freed to use his time in those areas of the valuation process that require the application of appraisal expertise. LIMITATIONS IN USING MULTIPLE REGRESSION ANALYSIS The application of multiple regression analysis to the assessment process certainly seems like an attractive answer to the assessor's problems However, there are very real problems of technique, costs of implementing such a system, and h o s t i l i t y of assessors that must be overcome before the process w i l l be able to be successfully implemented. Dasso and Swaden state that multiple regression analysis: 1 Can only be applied when many properties have similar characteristics that vary i n some manner relative to s e l l i n g price; that i s , an adequate number of sales of a given property type must have occurred i n order to give r e l i a b i l i t y to value estimates. Therefore multiple regression analysis cannot be applied to special prupose properties such as churches, department stores, and manufact-uring plants. 2 The system cannot be used in small communities where rela t i v e l y few property sales occur each year. 3 The elements of attributes contributing to explanation of value i n one community cannot be u t i l i z e d to explain value i n another community. 9 Another study states: Throughout our research we have had d i f f i c u l t y with both data collection and computer f a c i l i t i e s . Our data collection problems have resulted from the need to convert information designed for other purposes into a form suited to multivariate analysis. These problems seem unavoidable in the context of the administrative arrangements which prevail i n most assessment offices. Information on replacement cost for example, i s not ideal for market analysis using s t a t i s t i c a l methods. Similarly, i n many cases, sales data i s not integrated with information on property characteristics, and few offices have any information suited to immediate data processing for analytic purposes.30 As Appendix B i l l u s t r a t e s , the cost of data collection for assessment under this system i s very large. In the case of most municipalities, especially those which have no computer aids i n their present assessment systems, these costs could be prohibitive. Multiple regression analysis i n an appraisal context has been c r i t i c i z e d by Lessinger, who argues that i t is an improper method of real estate valuation because of the inter-relationship among the dependent and independent variables;(this problem i s called multicollinearity). He states the process produces large average errors and intolerable maximum errors. Current research however, has certainly lowered these errors, and i n many cases the valuation by multiple regression analysis i s as good as or better than this quality of appraisals done without this technique. Lessinger's c r i t i c i s m concerning multicollinearity i s that multiple regression analysis does not show the true contribution of a property characteristic to s e l l i n g price, as i t is distorted by multicollinearity. But for practical purposes, the assessor is concerned with predictability more than with causality. 1 Quoted in Frederick M. Babcock, The Valuation of Real Estate, (New York: McGraw H i l l Book Company, Inc., 1932), pp. 178-179. 2 Frederick M. Babcock, "The Three Approaches" The Real Estate Appraiser, Vol. 36, No. 5, (July-August, 1970), p.35. 3 Paul F. Wendt, "Recent Developments in Appraisal Theory", Appraisal Institute Magazine, Volume K, Book 1 (Spring, 1970), p. 35. ^ Richard U. Ratcliff, Current Practices in Income Property Appraisal -A Critique, Research Report 30 (Berkeley: The Centre for Real Estate and Urban Economics, Institute of Urban and Regional Development, University of California, 1967), p. 11. ^ Richard U. Ratcliff, Modem Real Estate Valuation (Madison, Wisconsin: Democrat Press, 1965), p. 36. ^ Ratcliff, Valuation Theory, Chapter iv, pp. 10 - 16. ? Wendt, Recent Developments in Appraisal Theory, p. 35. ® H.C. Smith and R.L. Rocster, "Should the Traditional Appraisal Process Be Restructured," The Real Estate Appraiser, Vol. 36, No. 7, (November - December, 1970), p. 11. ^ William N. Kinnard, Jr., "New Thinking in Appraisal Theory", The Real  Estate Appraiser, Vol. 32, No.5 (August, 1966), p. 12. 1 0 Ibid., p. 13. 1 1 Ratcliff, Valuation Theory, chapter v i i i , p. 2. 12 Ibid. ^ Kinnard,"New Thinking in Appraisal Theory", p. 6. U Ibid. 15 Fred E. Case "New Decision Tools for the Appraiser", The Appraisal Journal, Vol.XXXV, No.l (January, 1967)', p. 21. Andrew J. Hinshaw, "The Assessor and Computerization of Data", The Appraisal Journal, Vol. XXXVII, No.2 (April, 1969), p. 283. 17 Eugene F. Brigham and Donald M. McAllister, "Applying Econometric Models", The Appraisal Journal. Vol.XXXVI, No. 4, (October,1968), p. 541. 1 8 Ibid* 1 9 R a t c l i f f , Valuation Theory, chapter v i , pp. 27-32. 20 The two main references for this thesis are: More^cai Ezekiel and K.A. Fox, Methods of Correlation and Regression  Analysis, (New York: John Wiley and Sons, 1959), 548 pages. Taro Yamane, Statistics - An Introductory Analysis, (2nd ed., New York: Harper and Row Publishers, 1964), 825 pages. 2^ Theodore R.Smith, "Multiple Regression and the Appraisal of Single Residential Properties", The Appraisal Journal, Vol.XXXIX, No. 2, (April, 1971), p. 284. 22 Methodology Section, Assessment Standards Branch, Ontario Department of Municipal Affairs, Multivariate Analysis and Residential Property  Valuation i n Ontario, (Toronto: Ontario Department of Municipal Affairs, October, 1970), p. 8. 2^ R a t c l i f f , Valuation Theory, chapter v i , p. 53. ^ E.F. Renshaw, "S c i e n t i f i c Appraisal", National Tax Journal, December, 1958, p. 320. 25 Finnis, Property Assessment in Canada, p. 70. Hinshaw, The Assessor and Computerization of Data, p. 283. 2? Ontario Department of Municipal Affairs, Multivariate Analysis in  Ontario, p. 4« Coefficient of dispersion i s a measure of the percent that the average deviation bears to the mean value. Average deviation is the sum of a l l the deviations, divided by the number of items. Jerome Dasso and Paul Swaden, "Data Processing Implications for Property Taxation", The Appraisal Journal, Vol.XXXVIII, No. 1 (January, 1970), p. 54. Ontario Department of Municipal Affairs, Multivariate Analysis ln  Ontario, p. 34. Jack Lessinger, "Econometrics and -Appraisal", The Appraisal Journal, Vol.XXXVII, No.4 (October, 1969), p. 508. TECHNICAL ASPECTS OF MULTIPLE REGRESSION ANALYSIS FOR THE ASSESSOR INTRODUCTION Lessinger has been strong in his arguments against the use of multiple regression analysis in real property valuation. He argues: Users of multiple regression w i l l recaite coefficients of correlation and coefficients of standard errors. These coefficients purport to give precise information about the sample under study; but they are only vali d under very stringent assumptions. The assumptions apply infrequently i n the typical case of real estate appraisal-cases where co l l i n e a r i t i e s and interactions are present to a high degree. Furthermore, any three variables are liable to exhibit both problems simultaneously. For example, bathrooms and bedrooms are not only collinear but they are also interactive on price. Multiple regression holds forth an intriguing promise. However, the facts underlying the valuation of real estate are much too complex for the simple additive theory on which i t i s based. Certainly, one gets answers-but i t must be kept in mind that such answers are true only in some fantastic sense which mutilates r e a l i t y . Lessinger i s pointing out that interaction among variables contained in the multiple regression equation destroy any qualities of causality. On the other hand, he states the multiple regression technique does result in predictions. The problem of multicollinearity or interaction among variables should not be allowed to prevent the use of a tool which has good predictive a b i l i t y . The assessor needs a technique that predicts with a high degree of accuracy. He is not overly concerned with causality. The problem of multicollinearity in the assessor's valuation system w i l l be explored in this chapter. Also, those parts of multiple regression analysis that are of importance in understanding the application of multiple regression analysis in a valuation context w i l l be introduced and analyzed with respect to the problems of the assessor. USES OF THE MULTIPLE REGRESSION MODEL There are two broad uses to which a multiple regression model can be put to use. Pendleton states: Two uses for this type of s t a t i s t i c a l model of the real estate market should be distinguished. The f i r s t i s analysis of particular economic phenomena which are expressed through the functioning of the market. The second use, in contrast, employs the s t a t i s t i c a l model as a cheap and accurate means of estimating market value of properties for which recent or reliable sales data are available. 2 Ideally, i t would be best i f a model could perform well in each of these two areas. Davis relates these two uses: The key to prediction i s not only to examine market price as such, but to understand the determinants of market price. R a t c l i f f has stated that the ' c i r -cumstances of the transactions and the real estate market conditions and prospects at the time' are factors to consider in predicting market price from recent sales of similar properties. To find meaning in market prices for prediction purposes requires a good understanding of the market. This in i t s e l f i s a d i f f i c u l t job. Rat c l i f f has stated: ' i t i s indeed a formidable task to convert the history of market transactions into a form which the appraiser can use with confidence for predictive purposes. The raw materials for processing are separate transactions, each one of which was conditioned by a complex and unique set of circumstances. A l l parcels of real estate are dissimilarj transactions involving many types of real estate are infrequent; and the discovery of the essential facts surrounding the transaction i s often very d i f f i c u l t . The next step i s to relate cause and effect by the use of a s t a t i s t i c a l system for translating market data into price predictions. This step has formed a veritable barrier to appraisal progress. R a t c l i f f states 'to give meaning to the transaction prices of the past, i t i s necessary to measure or judge the impact on price of each of the many factors which entered into i t s determination. This i s a problem i n multiple correlation so complicated and involving so many incommensurables that no known methods are f u l l y adequate. In the present state of the art, we are able only to identify the more important of the independent variables and to estimate their relative importances on the basis of general knowledge of real estate market phenomena.1 While causality i s a desired feature of this valuation system, the assessor cannot afford to lose any degree of predictive quality at the expense of making the multiple regression formula more causal. Gustafson states: The acid test i s not how the value was derived but rather how well the equation measures the f u l l cash value.^ Actually, the problem of causality i s a function of factors that are considered in estimating market value. Market prices of single family homes can be a function of many variables, including physical property characteristics, l o c a l , regional and national economic conditions, available mortgage financing, neighborhood trends, etc. There have been two approaches to choosing variables to predict real property values. The f i r s t approach u t i l i z e s variables that are of a socio-economic type. A model formulated by Wendt i s presented here as being representative of the type of causal model that urban land economists have formulated to 5 explain cause and effect relationships of real property prices. The Wendt model i s as follows: V = f x(P,Y,S,P u, PI) - I ( T + 0 C+I l m * D L F F I) R, C g) where V B value of urban land fx" expectations P = population Y - average income S = supply of competitive land Pu = competitive p u l l of area PI = public investment T = l o c a l taxes 0 C S operative costs Iim = interest on improvements DJJJJ* depreciation on improvements i = interest rates R = investment risk Cg s capital gain p o s s i b i l i t y This particular model i s aimed more at identifying the causes of secular and c y c l i c a l changes in aggregate land values, but i t does i l l u s t r a t e the nature of the causal relationships with which urban land economists are concerned. The other approach to valuation u t i l i z e s variables more closely associated with the actual physical real property. Gustafson, for example, analyzes factors such as neighborhood features, topography, land attributes, improvement attributes and certain cost data. (A detailed l i s t of these property features i s contained on the residential fact sheets in Appendix D). This is the approach that has been followed in most research in the assessment context to date. Its wide acceptability i s probably due to the fact that i t u t i l i z e s data that i s re l a t i v e l y objective and accessible to any assessment off i c e . One study states: We assume that the socio-economic characteristics which distinguish between values in different neighborhoods are reflected by broad location variables. As our analysis proceeds we might find that this type of assumption is unwarranted. In general, however, we hope to confine our models to characteristics directly associated with real property.? In this thesis a similar line of reasoning w i l l be followed. In studying such a small area of Burnaby i t i s reasonable to assume that these socio-economic variables w i l l be r e l a t i v e l y constant. As a result, socio-economic variables w i l l be excluded from the test since i t is desirable to analyze only those factors that vary among single family residences. The inclusion of socio-economic variables w i l l be considered only i f the predictive quality of the equation based on analysis of the physical and locational characteristics leaves much to be desired. Lessinger's problem of multicollinearity i s a problem that w i l l be encountered in both approaches to variable selection. Multicollin-earity effects w i l l be reduced in this thesis by choosing variables that appear to have minimal interaction. There has developed a type of multiple regression analysis called 'stepwise multiple regression' that minimizes the problem of g multicollinearity. The stepwise regression procedure w i l l be used in this thesis i n an attempt to reduce the effects of multicollinearity and increase the causality of the model. As long as multiple regression analysis i s able to estimate market price within acceptable limits, this problem of multicollinearity should not be considered a serious one. For practical purposes the assessor i s concerned only with predictability. In fact, some researchers who are concerned solely with predictability attempt to introduce multicollinearity 9 in order to increase the predictive quality of the estimating equation. If there i s a strong presence of multicollinearity, i t w i l l not be possible to use the equation as an estimate of the relative significance of individual variables. It i s worth noting, nevertheless, that violation of the non multicollinearity assumption does not necessarily destroy the overall predictive a b i l i t y of the equation. In fact, i t may be possible purposely to introduce terms that w i l l result in multicollinearity in an attempt to increase the correlation between the estimate and the actual values. A particular technique Is the use of some form of polynomial progression such as: L a c k i n g i n ITuniber Only t Y = V o * b l X l * b2 X2 b3 X3 + b4 X4 + b5 X5 where ASPECTS OF MULTIPLE REGRESSION ANALYSIS There are certain problems that w i l l be encountered by an assessor in the application of the multiple regression technique to mass valuation. Some of these problems and their solution w i l l be il l u s t r a t e d i n this section. The f i r s t problem i s the great number of variables that the assessor has to consider for inclusion into the equation. Normal multiple regression procedure requires that the researcher f i r s t select several variables and submit them to the multiple regression procedure. A t test i s performed for each variable at a specified level of significance. Thus the researcher can t e l l which variables in that particular model are significant for purposes of inclusion into a valuation model. The t t e 3 t i s simply a test of the hypothesis that the variable's coefficient i s significantly different than zero. Given a standard t distribution, the researcher simply uses the number of degrees of freedom and the level of significance required to find that t s t a t i s t i c that sets the c r i t e r i a of acceptance or rejection of the hypothesis. If the t s t a t i s t i c calculated for the variable from the test i s larger than the t s t a t i s t i c significantly different from zero. If the t s t a t i s t i c reveals that the variable's coefficient i s significantly different than zero, then the variable i s deemed to be significant enough to be included i n the estimating model.. The researcher would have to go through this process many times in an attempt to find those variables that are significant. Variables selected from each test as significant could then be run together i n a f i n a l attempt to determine the f i n a l significant variables. This i s a laborious process and fraught with d i f f i c u l t i e s for the inexperienced researcher. Fortunately, there has developed a variation of multiple regression analysis called stepwise regression which can automatically determine which few of a hundred or more variables to be considered make the most contribution to the equation. This i s the s t a t i s t i c a l technique that w i l l be used in this thesis. The technical aspects of this stepwise regression technique w i l l not be discussed here. There are many excellent 10 sources that describe i t i n great d e t a i l . Several s t a t i s t i c a l measures are produced by this technique: 1 the coefficients of the regression equation 2 the standard error of each regression coefficient 3 the t s t a t i s t i c for each variable U the standard error of the estimate 5 the coefficient of determination 6 the F probability for the entire equation 7 printer plots of the residuals. These seven s t a t i s t i c a l measures combined permit the researcher to determine the quality of the regression equation. Two important outputs which judge the quality of the equation i n this thesis are the coefficient of determination and the standard error of the estimate. Shenkel discusses these two measures. With respect to the coefficient of determination he comments; Predictability judged on this value i s indeterminate because of the interaction between variables and the close association of property characteristics. Experience indicates, though, other measures of predictability w i l l f a l l below acceptable limits i f the coefficient of determination i s not over 90 per cent. He says of the standard error of the estimates This value measures the probability that the predicted value w i l l f a l l within certain value li m i t s . Generally, predictability for assessing purposes w i l l be acceptable i f the standard error of the estimate is ten per cent or less of the average sample sales price The standard error of the estimate divided by the average sample se l l i n g price creates a measure known as the relative standard error of 12 the estimate. Given that other measures, especially the coefficient of determination, are within acceptable limits, attention w i l l be addressed primarily to this rati o . While Shenkel strives of obtain a relative 13 standard error of the estimate of under ten percent, other studies have tried to achieve a ratio of under five percent as a minimum standard of acceptance. L i t t l e research has been done to ascertain the quality of predictions attained i n appraisals. However, Gustafson states: One old wives' tale has i t that most appraisals are within plus or minus ten per cent of the correct valuer^ Current research i s more specific in the degree of predictiveness required of a valuation system: As a rough and ready guide we regard predictions within plus or minus ten percent of 'observed value' (actual value) with a r e l i a b i l i t y level of 95% as suitable for research and development. Estimated values w i l l be within plus or minus one relative standard error of the estimate from actual or observed value for 68% of the cases in the sample and within plus or minus two relative standard errors of the estimate for 95% of the cases}^ Hence, when the standard error of the estimate i s 5% or less of the mean sample s e l l i n g price, one can be assured that estimated values on the average w i l l be within plus or minus 1056 of the actual values for 95% of the cases i n the sample. Prior research has yielded relative standard errors of the estimate which were considered as indicating a high degree of predictiveness for the estimating equation. Gustafson reports a relative standard error of the estimate of 3,5% and an Ontario study reports a relative standard error of the estimate of 3.8%. In this thesis a meaningful relative standard error of the estimate w i l l be considered to be five per cent or less. An F probability i s calculated for the entire equation and permits an examination of the overall significance of the equation. This probability i s the probability of obtaining a value of the coefficient of determination greater or equal to the one calculated, given that there is no association between the dependent and independent variable, that i s , given that the true regression coefficients Bn. B_ are a l l zero. If this probability i s less than 0.05, i t i s usually concluded that the coefficient of determination i s significantly different than zeroV 3 The F ratio and associated probability i s calculated for each of the regression coefficients. An F test i s performed to test the significance of each regression coefficient. The probability of obtaining a value of F^ greater than or equal to the one calculated, given that "B^" (that i s , the population's coefficient) equals zero i s also printed. If this probability is less than 0.05 i t i s usually concluded that is sig n i f i c a n t l y different than zero.^9 The t s t a t i s t i c i s easily calculated from the F ratio as the t and the F distributions are related by Davis gives a very rough guideline to the value of the t s t a t i s t i c that i s acceptable at various levels of significances One author gives a theoretical example of a test of a hypothesis in which the sign of the regression coefficients i s known from non-statistical considera-tions. The t value for a one tailed test and a level of significance of 5% was given as 2.02 for 5 degrees of freedom and 1.65 for 1000 degrees of freedom. It was concluded that as a 'rough' rule of thumb i f the calculated t value i s greater than 2 one may conclude that i s significantly different than zero, no matter what the degrees of freedom. Another problem to be considered i s the l i n e a r i t y of the data. I n i t i a l l y , the study w i l l assume that the data does display linear tendencies. As a result, the regression technique w i l l be multiple linear regression analysis. An analysis of the printer plots of the residuals w i l l indicate non li n e a r i t y . If the plot of the residuals indicates a condition of non-linearity exists, then the technique of multiple non-linear regression analysis w i l l be used. The use of multiple regression analysis does enable assessments to be made that are more i n line with current market values. Trends i n market values since a previous r o l l can be ascertained by examining sales that have taken place during this period. However, the inclusion of the independent variable 'time of sale' (in this thesis i t i s month of sale) into the multiple regression analysis complicates the valuation process for the assessor. When month of sale i s found to be a significant variable in the estimating equation i t w i l l not be possible to extrapolate beyond the range of this variable without the loss of considerable predictive quality. Hence, the most current valuation date for the assessor would be the last month of sale considered in the sample. Where month of sale was highly correlated with se l l i n g price, indicating a market in rapid transition, i t is obvious that there would be a difference in prices between the last month of sale considered i n the sample and any later assessment date. While multiple regression analysis would capture the nature of recent trends i t would s t i l l produce values that lagged behind current market prices by several months. Appendix C il l u s t r a t e s the typical sequence of events followed i n a proposed valuation system using multiple regression analysis and ill u s t r a t e s how the problem of a lag in values may result. It shows that the closer the sample of sales i s to the assessment date, the shorter w i l l t e the lag between assessed values and current market values. A very low correlation between month of sale and se l l i n g price usually indicates a relatively stable market. In this case, there i s l i t t l e likelihood that values five or six months before the assessment date w i l l be different than current market values. Hence values produced by multiple regression analysis with a sample of sales where the last month of sale was six months before the assessment date would be able to produce market values that were indeed very current. These values would t r u l y reflect most probable selling prices i n the market at the time of the assessment. An important test that the assessor must conduct then, i s to examine the multiple regression equation and the correlation matrix to ascertain the nature of recent price trends. Where month of sale i s significant and indicates rapid price changes, the values produced by the estimating equation w i l l be lagging behind actual values at the assessment date. However, instead of lagging by the traditional one or two years they w i l l now lag by only a few months. In fact, they w i l l lag by the difference between the assessment date and the date at the end of the sales sample. Thus, the closer the sample can be collected to the assessment date, the more current w i l l be assessed values. Where month of sale i s not considered to be significant, the assessor can be sure that the values produced w i l l be almost, i f not exactly, current market values as of the assessment date. One caution that must be taken in the application of multiple regression analysis i s that one cannot extrapolate beyond the range of any of the independent variables considered i n the sales data. To do so distorts the quality of the predictions. It must be emphasized that a great deal of experimentation must take place i n producing equations that accurately predict prices. The stepwise regression procedure eliminates much of the experimentation concerned with choosing significant variables. The real experimentation, however, comes from refining the sales data. This i s largely a matter of stratifying the data in such a way that more significant results are produced. Methods of stratifying data w i l l be discussed in the following chapter. ^ Lessinger, Econometrics and Appraisal, p. 508. 2 William C. Pendleton, " S t a t i s t i c a l Inference i n Appraisal and Assessment Procedures," The Appraisal Journal, Vol.XXXIII, No.l, (January, 1965), p. 79. 3 Irving F. Davis, Jr., A S t a t i s t i c a l Approach to Real Estate Value with Application to Farm Appraisal, Study No.12, (Fresno, California: State of California, Division of Real Estate, 1965), pp. 30-31. ^ Robert H. Gustafson, Data Banks and Computerized Annual Updating of Assessment Rolls, Paper prepared for delivery at the Annual Conference of the International Association of Assessing Offices, September 7 - 1 0 , 1969, Denver, Colorado, p. 20. c J Paul F. Wendt, "Theory of Urban Land Values", Journal of Land Economics, Vol.XXXIII, (August, 1957), pp. 228-240. ^ Gustafson, Data Banks and Assessment Rolls, p.4. ^ Ontario Department of Municipal Affairs, Multivariate Analysis in  Ontario, p. 9. 8 Ibid., p. 11. 9 Smith, Appraisal of Residential Properties, p. 280 10 An excellent source i s N.R. Lraper and H. Smith, Applied Regression  Analysis,(New York: John Wiley & Sons, Inc., 1966), 407 pages. H William M. Shenkel, "Valuation Studies", International Property Assessment Administration, Proceedings of the 35th Annual International Conference on Assessment Administration (Chicago: International Association of Assessing Offices, 1969), p. 106. Ontario Department of Municipal Affairs, Multivariate Analysis i n  Ontario, p. 31. REFERENCES CITED - CHAPTER 17 Shenkel, Valuation Studies, p. 106. U Ibi^., p. 105. *5 Ontario Department of Municipal Affairs, Multivariate Analysis  i n Ontario, p.A. ^ Gustafson, Data Banks and Assessment Rolls, p.15. 1 7 Ibid., p. 19. 1 8 J.H. Bjerring and R.H. Hall, Triangular Regression Package, (Vancouver: University of Bri t i s h Columbia, Computing Centre, 1968), p. 69. 1 9 Ibid. Draper and Smith, Applied Regression Analysis, p. 25. 21 Davis, A S t a t i s t i c a l Approach to Real Estate Values, p. 63. TESTING MULTIPLE REGRESSION ANALYSIS IN BURNABY INTRODUCTION In addition to arguing for the adoption of multiple regression analysis in a real property valuation system, a further purpose of this thesis i s to test the ap p l i c a b i l i t y of the procedure for the valuation of single family residences in Burnaby. The test w i l l be designed to provide answers to the following problems? 1 The adaptability of current assessment records to multiple regression analysis 2 The sampling procedures to be followed i n implement-ing the multiple regression technique 3 Whether the frequency of sales is enough to enable the technique to be applied to a wide enough range of properties U The sampling procedures that w i l l have to be followed in implementing such a technique to valuation 5 The problem of causality versus predictability 6 How to determine the optimum balance between the number of regression equations and the degree of homogeneity desired. Shenkel l i s t s five main steps to be followed i n testing for the ap p l i c a b i l i t y of multiple regression analysis. They are: 1 sample selection 2 selection of value indicating variables 3 coding of variables U s t a t i s t i c a l processing 5 data refinement 1 The application of multiple regression analysis to the assessment of single family residential properties i n Burnaby w i l l be analyzed in this chapter under these major headings. The establishment of a multiple regression equation i s the heart of a t o t a l valuation system based on market value. A test such as the one being undertaken in this thesis i s a key step to be taken in testing the f e a s i b i l i t y of adopting this valuation technique. Once i t can be proven that the multiple regression procedure has the a b i l i t y to estimate the se l l i n g prices accurately, the way should be cleared for the transformation of a l l assessment records into computer processing form, and for the preparation and installation of the computer programs that w i l l perform various sundry functions such as editing sales, cost calculations, s t a t i s t i c a l analysis of the sample population and the calculation of estimated s e l l i n g price. SAMPLE SELECTION Multiple regression i s a method of s t a t i s t i c a l inference that permits assessors to make inferences about a population on the basis of sample characteristics. Inferences of this sort are only possible when units in the sample adequately represent the units in the population. This study w i l l make inferences about single family residential properties that have not recently sold by inferring their price from the prices of similar sold properties. Market value i s deemed to be most probable sales price and i s regarded as a function of consumer and investor behaviour. This function i s reflected in the relations between sale price and various physical and locational characteristics of recently sold properties. By > establishing these relations through s t a t i s t i c a l analysis the value of similar properties that have not sold can be predicted. Of crucial importance to the inferential process i s the use of proper sampling procedures. Normally one knows the kind of population he wants to make inferences about and a process of random sampling i s init i a t e d to generate an unbiased sample, representative of the population. But this process cannot be rigorously applied in s t a t i s t i c a l approaches to real property valuation. For example, i n the case of single family dwellings, an "ideal" population includes a l l single family dwellings in the municipality. But information can only be collected about sale prices for those properties that have sold. Consequantly, a sample cannot be chosen so that a l l single family dwellings in the municipality are adequately represented. The best sample that can be devised consists of a l l single family dwellings that have sold in the recent past. This i s forced upon the assessor by real estate markets. Therefore, i t i s possible to make inferences only about those kinds of properties that are adequately represented by the recent sales contained in this sample. Two problems are thus created. F i r s t l y , the existence of scanty sales data means that there w i l l be some types of single family dwellings that cannot be valued by using the techniques associated with multiple regression analysis. Secondly, the fact that the characteristics of the sample properties determines the population upon which inferences can be made means that the population cannot be specified u n t i l a detailed analysis of recent sales has been completed. This second point is very important and should be c l a r i f i e d . For each sample property there w i l l be associated a number of specific property characteristics, (for example, livable square footage, lot depth, number of rooms, sales value). There is a range of measurement associated with each property characteristic. This range sets the limit or bounds for the characteristics of the population. For instance, i f analysis of square footage of livable area showed a sample range of 800 to 2,000 square feet, i t would be dangerous to t r y to estimate the value of a house that had 4,000 square feet. Making inferences beyond the range of the associated 2 property characteristic is called extrapolation. Generally speaking, one can only make inferences for values contained i n the sample. To do otherwise greatly reduces the r e l i a b i l i t y of the inference. In addition, to the range of each property characteristic, i t i s also helpful to have a mean value and standard deviation for each property characteristic. These w i l l be provided for each property characteristic considered i n this thesis One problem worth consideration i s the wide range of types of homes The sample can refer to different populations that are to be valued; for example the population of $10,000 to $20,000 homes and the population of $35,000 to &50,000 homes. It i s very unlikely that sales of the lower priced homes would be able to be useful in predicting the values of the higher priced homes. Thus, i t would seem necessary to use different multiple regression models for different groups of rela t i v e l y similar single family dwellings. There are many different ways of looking at population groups to be derived from the t o t a l population of a l l single family dwellings. It is important then, to s t r a t i f y the sample of sales into component markets, groups or market aggregations i n a way that w i l l permit the development of models which w i l l accurately predict the value of individual properties. In fact, this process of s t r a t i f i c a t i o n w i l l be one of the key experimental issues i n this study. Prior research has shown that poor predictive qualities of multiple regression analysis i s often due to improper s t r a t i f i c a t i o n . Further experiments i n s t r a t i f i c a t i o n often 3 can enhance the predictive quality of the valuation model. Shenkel states: The sales sample must represent a f a i r l y uniform class of property. Best results are obtained when sales cover highly uniform property types A Gustafson's research has developed different multiple regression models for various areas in a c i t y where value influences and property 5 types created a sales sample of f a i r l y homogeneous sales. This study w i l l proceed along the same lines and w i l l identify one neighborhood or area of Burnaby, where i t is f e l t the value influences are the same and the property types are of a f a i r l y homogeneous type. It i s important to note that greater accuracy i n prediction created from a more homogeneous type of sales sample i s generated at the added cost of more models. That i 3 , the greater the number of homogeneous areas subject to valuation, the higher the cost of the valuation process to the assessor. Thus, i t i s of key importance to the assessor to get as diverse a sample as possible (subject to the limits imposed by the degree of predictability required). This sample can be found only by a great deal of testing especially through the use of sales s t r a t i f i c a t i o n . A danger i s created by this s t r a t i f i c a t i o n process. The more fin e l y the assessor divides the whole sales sample, the fewer sales there are for each part of the s t r a t i f i e d sales sample and hence the danger of poor predictability i s created. This study w i l l not attempt to devise a number of multiple regression equations that together w i l l be able to predict the estimated sale prices of a l l single family residential dwellings i n Burnaby. Rather i t w i l l seek an area of Burnaby where dwellings are rather homogeneous with respect to a l l property characteristics and where value influences on these properties are very similar. Similar value influences simply means that a l l properties i n the study area are similarly influenced by major industrial areas, t r a f f i c congestion, pollution, travel time to major employment centers, etc. The study area chose in Burnaby comprises Di s t r i c t Lots 97, 98, 99 H9, 150, 156, 157, 158, 159, 175, Group 1, New Westminster Land Di s t r i c t (see Appendix E). It was chosen s p e c i f i c a l l y because of the lack of special or exceptional value influences that could affect real property values. In order to i l l u s t r a t e the type of single family homes examined in the study, i t w i l l be necessary to examine the c l a s s i f i c a t i o n of single family residential dwellings developed by the Appraisal Manual of the Province of Br i t i s h Columbia, This manual i s the municipal assessor's guide to the valuation of single family properties. Houses are f i r s t l y grouped into a one storey c l a s s i f i c a t i o n which i t s e l f contains ten sub-groups; a one and a half storey (split level) c l a s s i f i c a t i o n (ten sub-groups also); and a two storey cl a s s i f i c a t i o n (ten sub-groups also). The one storey cl a s s i f i c a t i o n w i l l have ten sub-groups, 1-1 through 1-10. This sub-grouping f a c i l i t a t e s a quality ranking based on age, architectural and construction quality. Within this one storey c l a s s i f i c a t i o n , 1-1 properties are the poorest quality and the 1-10 properties the best quality. This same procedure works for the one and a half and two storey dwellings. In order to get a clear picture of the sales sample chosen from this area i t w i l l be helpful to i l l u s t r a t e the property types obtained. In 1970, the mean assessed Values of these property types were as found in Table VI. It i s apparent from Table VI that the 1-9 property type has the highest frequency of sales i n the sample, and hence w i l l l i k e l y provide a reliable sales base for value prediction of the population of the 1-9 property type. In fact, i t i s estimated that in the study area, approximately 60 to 75$ of the homes were of this type. I f this property type provides a useful s t r a t i f i c a t i o n and results in significant results then there i s a chance that one estimating equation could predict the values of 60 to 75$ of the population of the study area. DISTRIBUTION OF SAMPLE BY PROPERTY TYPE AND MEAN ASSESSED VALUE Property Mean Assessed Type Valtte Frequency 1-5 $11,900 31 1-6 18,600 16 1-8 15,400 65 1-9 20,400 161 M 1-9 26,700 24 1-10 35,1X30 8 1-15 17,800 IS 1-2& 21,200 8 1-3S 15,600 6 1-19 20,400 12 1-2D 27.200 6 1-21 39,500 4 1-27 19,600 7 1-26 25,600 4 1-31 30,700 4 1-32 42,500 2 It i s also apparent in. eEamaa&ng Tails VI that t2»3« la IsasaafflsfeBt. sales ofefca f u r most jirj>p&rti> -types t© -vaibBe separately tibelr respsstifc** popplat3ion» T J H B , a &ey experiment T d l l toe ^whether a. single eqaatieB on tiie sal«s cf a l l property types toill bs afcls ^ bo ffeneasofee -the 3pre«Hletjt»e qnality ttesirad <or -wfaBTfcber stratif %-ca53xm sby property -types -will i» nstassary- Ftir ExampJlaB^  i f aane «qna±4ian -the «ntl3» sample tew* the -predictive -qiral^lry -desiraii, a strstlfiBarfdaan -of -fibs 1—8"s4 1*9"«^ or l-S'Vs and l-9'"-:s izBfgsthBr •nrjuia ibe triatL These a-fctsmpts vdJH be d&seu&s i^l in sn *nsOT±ng ssctiEnu In a valuation system based on statistical inference, i t is important to ensure that the sales included in the sample are representa tive market transactions and not of an inter-family or inter-company nature. Value in this study is regarded as a function of consumer or investor behavior in the open real estate market. To permit anything but normal market transactions into the sample would destroy the objectlveness of this valuation procedure and hinder the process from predicting the most probable selling price of unsold dwellings. Each sale in the sample has been examined by the assessment Commissioner as having been the result of a normal market transaction. The sample selection of property types is just the first step in determining the population which will be subject to the inferential process. It is not enough to know the population is likely to be two or three or maybe more property types. It is essential to have measures of central tendency and dispersion for property characteristics of the different property types in order to gain a fuller understanding of the true nature of the population. PROPERTY CHARACTERISTICS AND THEIR CODING Multiple regression analysis creates a functional relationship between the dependent variable sales price and the independent variables the property characteristics. This section will describe the property characteristics that have been gathered from the records of the Burnaby Assessment Department, how these property characteristics determine the population of the inferential process and finally how the coding of these property characteristics can affect the quality of the predictions in the inferential process. Renshaw l i s t s six desirable qualities that variables of the regression program (both dependent and independent variables) should haves 1 A l l variables should be objective: the data pertaining to them should be obtainable from reliable sources. 2 Both the form of the equation and the measurement of the variables included should be determined independ-ently of individuals whose property is being assessed. 3 Each indpeendent variable should represent dire c t l y or indirectly an important value dimension} a sufficient number of variables should be included in order to represent a l l of the principal dimensions of property value. U The dependent variable should correspond closely to actual sales value. 5 The equation should be as sensitive to low valued property as to high. 6 The equation should yield a high correlation and be theoretically plausible, without undue complexity that would make the formula administratively unfeasible." This study uses these six qualities as a standard for variable selection. Many subjective characteristics such as view can only be obtained from physical inspection. Even then physical inspection could not assure the application of standards to guaging variations of this variable. In order to avoid this type of problem the data f o r each property was obtained from improvement assessment records, land assessment records and maps in the Burnaby assessment off i c e . An aim of the thesis i s to u t i l i z e only those property characteristics that can be objectively measured. Variable Computer Identification :Coding Type.of Variable 1 Sales price of sold properties 2 Month of sale 3 1971 improvement assessment 4 Lot frontage 5 Lot depth 6 Area of lot 7 Is the lot of irregular shape? 8 Is the l o t a corner lot? 9 Is the lot a cul-de-sac lot? 10 Is the lot a ravine lot? 11 Is there a sidewalk? 12 Does property have rear access(lane)? 13 What i s the t r a f f i c volume? 14 On a truck route? 15 Within 2 blocks of bus route? 16 Distance to elementary schools 17 Distance to shopping 18 Area of dwelling 19 Effective age of dwelling 20 Number of storeys 21 Number of plumbing pieces 22 Type of heating 23 Is there a fireplace? 24 Basement area 25 Finished basement area 26 Garage? 27 Attached garage? 28 Is there a carport? 29 Is there a patio? SALEPR MONSAL RCNLDA FRF00T DEPTHL AERALT IRRSHP C0RL0T CLDESC RAVLOT SDWALK RACCSS TRAVOL TRKRUT BUSRUT SKLDIS SHPDIS DWELAR AGE IMP STORYS PLMPCS HEATNG FIREPL BASEAR FBASAR GARAGE GARATT CARPRT PATTIO ;/ quantitative 6 l: • » ' . • • ll * !• -•" ; b qualitative -:• i >? ti • •••J' r. ' •' iv ii> f !'.; quantitative*: 6.*/ ( 1 , b -. W-'-\ . V 'I !vrn/<-. qJuaii;ta"6iy©, . vj* :3a • it V', ~\' ^^quantitative* (? tt 'qualitative" ti >.."• -• tt a quantitative variables are continuous variables ^ qualitative variables are dummy variables II • .Characteristics MEASUREMENT OF PROPERTY CHARACTERISTICS Approximate Range Standard • \ Means of Coding Mean Deviation Low High S) . . . j ' " . « * actual value ! 123,990. 4,146.00 15,200 33,450 1 to 21 • | 10.32 6.007 1 27 : j actual assessment j 11,580. 2,591. 6,280 16,880 f' ;•' ac'tual. frontage ! 57.62 12.09 32.0 105.0 r actual, depth 132.1 29.01 67.0 202.0 actual area (square feet) 7,605.0 2,397.0 2,805.0 12,405.0 ; yes » 1 ; no o 0 0.0372 0.1900 0 1 . yes = 1 no r 0 0.1491 0.3573 0 1 j yes = 1 no s 0 0.0124 0.1111 0 1 xyea a 1 . no = 0 0.0186 0.1356 0 1 '. yes - 1 no • 0 0.1491 0.3573 0 1 . yes',- 1 no s 0 0.8696 0.3378 0 1 under 10000 r l j over 10000 - 0 0.9565 0.2046 0 1 yes s 1 . no s 0 0.0745 0.2635 0 1 v yes m l no a 0 0.6522 0.6448 0 1 in miles and tenths 0.3211 0.3971 .1 1.6 it 0.8292 0.3633 .1 1.7 square feet 1,072.0 159.6 675.0 1,402. actual years of age 16.96 4.99 1.0 56.0 actual number of storeys 1.0 0.0 1.0 2.0 4 pieces r 1; 2 pieces • 1.189 0.3748 1.0 2.5 hot a i r ,'s 1; other r 0 0.9689 0.1740 0 1 yes s 1 no s 0 0.9752 0.1561 0 1 square foot 923.6 341.8 200.0 1,605.0 » square foot 302.7 295.4 150.0 725.0 yes r 1 no r 0 , 0.3106 0.4642 0 1 ;yes r 1 no r 0 I 'yes = 1 no = 0 •{ iv yes = 1 no' r 0 0.0062 0.0788 0 1 0.4348 .4973 0 1 0.1118 .3161 0 1 Renshaw'g third quality relates to the problem of model selection raised i n Chapter Four. Socio-economic variables used i n model by urban land economists in their research have not been obtained because of the d i f f i c u l t y of this type of data collection and the ease of collection of data of a physical and locational nature from the Burnaby assessment records. Appendix D includes a "residential fact sheet" used i n one study to l i s t and record the kinds of property characteristics that could be used 7 for multiple regression analysis. These many characteristics can be grouped into the following broad classifications: neighborhood data, land attributes, topography, and improvement data. These four c l a s s i f i c a -tions of data form the broad types of data that were sought in this study. Table VII enumerates the twenty-nine property characteristics collected for each property in the sample. The computer "identification" for each variable i s a computer code that identifies each of the property characteristics. The "means of coding" indicates whether the character-i s t i c i s a qualitative or quantitative variable and how i t was measured. Also included i s the mean, standard deviation and range of the property characteristic. An explanation must be made of the section entitled "measurement of property characteristics". As the t o t a l sample i s s t r a t i f i e d into smaller samples, the range, standard deviation and mean of each sub-group w i l l naturally vary. Thus, there i s created a problem of whether to use the entire sales sample or one of the several sub-groups to generate these measurements. Table VII w i l l provide the measurement of these property characteristics from the sample of sales that has achieved the highest level of predictive quality. It is the property characteristics of the sample that provides the best s t a t i s t i c a l results that w i l l determine what the population of the inferential process w i l l be. Therefore, i t w i l l be necessary to jump ahead and reveal the nature of the property characteristics from the only sub-group that had significant s t a t i s t i c a l results. Together the mean, standard deviation and range of these property characteristics describe the property characteristics of the population. The range describes i n broad terms the variation of the property characteristics while the standard deviation and mean i l l u s t r a t e more sophisticated measurements for the dispersion and central tendency of the property characteristic. Certain variables need some explanation. With respect to month of sale, the sales in the sample range over a twenty-one month period from October 1968, to June 1970. Each sale i s given a number from one to twenty-one, according to the month in which i t sold. The 1971 improvement assessment is based on actual value. Improvement assessment (RCNLDA) was included since other research had found i t s inclusion was necessary in order to achieve the desired level of p r e d i c t a b i l i t y . 8 T r a f f i c volume measures the 1970 volume per day on the street where the property i s situated. The shopping centers that related to sales properties are of the type strung out along major arteries in most neighborhoods of North America. Distance to major lower mainland shopping areas or downtown Vancouver, or other major shopping and employment centers,was not obtained. The number of plumbing pieces refers to the number of bathroom fixtures. A standard bathroom contains four plumbing pieces: a t o i l e t , sink, tub and shower. This is measured for coding as 1. A powder room that contains only a sink and t o i l e t i s measured by 1/2. The property characteristics or variables are the most objective variables that could be obtained for such a large sample of properties. The variables are of two types; quantitative and qualitative. The quantitative variables are a l l of a continuous nature and therefore easily quantified for computer analysis by simply recording the actual measurement. The qualitative variables however, must be quantified by recording them as "dummy" quantitative variables. The quantification of these qualitative variables i s done by coding each variable as 1 or 0. For a variable that requires a yes or no, the yes is 1 and the no i s 0. Similarly with t r a f f i c volume, under 10,000 i s 1 and over 10,000 i s 0. This study w i l l proceed on the basis of simple dummy variables measured by 1 and 0, for the measurement of qualitative data. No way could be found to devise a wider or more continuous type of quantification for these variables. Poor predictive qualities of the multiple regression equation might later suggest this c l a s s i f i c a t i o n i s too crude and poses the problem of having to quantify these variables to make them more sensitive to s t a t i s t i c a l analysis. In implementing a multiple regression technique, i t i s essential to ascertain whether the data is of a linear nature. If the data i s not linear, then the s t a t i s t i c a l technique for the inferential process must be multiple curvilinear rather than linear regression analysis. It i s essential to ascertain the degree of l i n e a r i t y of the data at the outset of the study. There are two quick means to establish the degree of l i n e a r i t y . F i r s t , one can examine the nature of the relationship between each of the independent variables and the dependent variable s e l l i n g price by observing printer plots of these relationships. This was done for each of the quantitative (continuous) variables in this study. In each case there was a high degree of l i n e a r i t y . A second method i s to observe the plot of residuals. If the residuals appear in anything but a random scattering then there is reason to suspect the data is non-linear and that perhaps coding the variables in logarithmic form or other forms to f a c i l i t a t e multiple curvilinear regression analysis would be advisable. The plot of residuals was observed and in each test the residuals were randomly scattered indicating therefore a high degree of l i n e a r i t y in the data. Another problem that is generated in data collection i s the degree of interaction among variables, that i s , multicollinearity. The presence of multicollinearity i s indicated by a simple test. If the correlation matrix constructed between a l l the independent variables exhibits high positive or negative correlations, then the presence of multicollinearity is suspect. The correlation among the independent variables and also between the dependent variables and the independent variables in this test i s low, suggesting the absence of a substantial amount of multicollinearity. The objective of s t a t i s t i c a l testing in this study i s to define a sample of sales that w i l l permit multiple regression analysis to estimate i t s s e l l i n g prices with a high degree of accuracy. The testing procedure w i l l proceed from an analysis of the largest sample possible and proceed to narrow down the sample size by s t r a t i f i c a t i o n when the results are not within acceptable limits. The s t a t i s t i c a l testing w i l l consist essentially of experimentation i n s t r a t i f i c a t i o n , that i s , trying to find that sample or samples of sales where sample s e l l i n g prices can be relia b l y predicted. Only when this sample i s defined can the population of the inferential process be defined. As heterogeneous a sample as possible i s desired, since stratifying the sample to increase homogeneity w i l l increase costs and make the valuation process more complicated as more and more equations become necessary in the valuation process. To achieve the required degree of predictability usually involves considerable experimentation. The nature of the sales sample guarantees t h i s . It cannot be known what the population i s because of the forced nature of the sales sample and i t also cannot be known i f there are enough sales of different property types to permit heterogeneous testing. Only continued experimentation can solve these problems. When the results of a particular s t a t i s t i c a l test do not meet the required standards i t w i l l be necessary to ascertain why and to attempt to improve the quality of the predictability. Shenkel l i s t s three main reasons why the s t a t i s t i c a l testing w i l l not produce the desired results: Poor quality sales data—untypical sales, that i s , sales that are not the result of normal market transactions may have been improperly included in the sales transactions prevail becomes necessary. Non-homogeneous sample of sales—samples that measure non-homogeneous property produce regression formulas that loose predictability i f value i s not explained by a common set of variables. In this instance a re-designed sample w i l l usually improve predictability. Improper coding of variables—conversion of variables to show a curvilinear association with price may indicate that values do not move in arithmetic proportions but in some other way. Processing of data to measure the correct relationship w i l l improve predictability." In this thesis the problem of poor quality sales data has been eliminated since the sales analyzed have been examined by the assessment Commissioner as having been the result of normal market transactions in linear form as plotting of sales price against each variable and examination of residuals in the s t a t i s t i c a l testing did not reveal any marked curvilinear relationships. The results of each test w i l l be presented in the following manner. F i r s t l y , s t a t i s t i c a l quality w i l l be examined by observing the o coefficient of determination (R ), the "relative" standard errors of the estimate (Syx/y) and the t s t a t i s t i c s of the significant variables. The following guidelines w i l l be followed: 2 1 R : experience indicated that other measures of predictability w i l l f a l l below acceptable limits of R 2 i not over 90%.10 2 Syx/y: five per cent or less 3 t s t a t i s t i c : "as a rough rule of thumb, i f the calculated t value ie greater than 2, one may conclude that bj (coefficient of the variable) i s significantly different than zero no matter what the degrees of freedom". The higher the t s t a t i s t i c the more one can be sure that the regression co-efficient is significantly different than zero. 1 1 Appendix F w i l l contain the detailed results of each s t a t i s t i c a l test. For each test w i l l be presented the number of observations, degrees of freedom, and level of significance, R , F Probability of the entire equation, the standard error of the estimate, Syx/y, and a l i s t of the variables chosen as significant at the specified level of significance. With each variable w i l l be presented i t s coefficient, standard error, F ratio, F Probability, and t s t a t i s t i c . Finally, R 2 w i l l be examined at each step of the stepwise regression procedure to give a clearer picture of the v a r i a b i l i t y in s e l l i n g price explained by the inclusion of each of the variables deemed to be of a significant nature. Test number one consisted of regressing s e l l i n g price against a l l of the independent variables except replacement cost new less depreciation (RCNLDA). It was f e l t that inclusion of RCNLDA would decrease the causality of the model. The model was being formed to show how price was the result of market factors and RCNLDA was deemed to have been the result of the i l l o g i c a l cost approach and hence not market determined. Every sale in the sample was used in the f i r s t test (376 observations, 375 degrees of freedom.) The results are as follows: a. R » 0.7693 b. Syx/y . 13.2% c. The variables closen as significant and their t s t a t i s t i c s : Variable t S t a t i s t i c DWELAR 13.1 AGEIMP 10.5 FBASAR 4.64 BASEAR 3.60 MONSAL 4.93 PLMPCS 3.20 CARPRT 2.45 HEATNG 2.84 TRAVOL 3.38 FRFOOT 3.26 These variables together do not provide the predictive quality desired. R i s much too low and Syx/y i s too high. Test number two involves trying to improve the s t a t i s t i c a l predictability of test number one by stratifying the sample of sales into a more homogeneous group of sales as represented by the 1-9 property types This narrows considerably the degree of heterogeneity of the sample while maintaining a large enough sales sample to ensure for the regression procedure to be able to produce reliable results. Test number two consist of 185 sales of the 1-9 property type (184 degrees of freedom). The results are as follows: a. R2 r 0.7424 b. Syx/y = 8.9% c. The variables chosen as significant and their t s t a t i s t i c s ! Variable t S t a t i s t i c DWELAR 8.24 BASEAR 4.79 MONSAL 5.34 FBASAR 4.79 AGEIMP 4.88 CARPRT 3.90 TRAVOL 2.50 FRFOOT 4-09 GARAGE 2.63 PATTIO 2.47 AERALT 2.36 While the relative standard error of the estimate has been lowered, i t i s 2 s t i l l not within acceptable bounds and R has f a l l e n . A primary purpose of this study has been to strive for causality i well as predictability. RCNLDA was not considered as an independent variable since i t was not considered to be a causal factor. Taken alone, RCNLDA i s an estimate of improvement value based on the cost approach. However, taken together with other independent variables, i t can be considered as a proxy to several independent variables. RCNLDA can be considered as a proxy to variables such as age, quality of structural improvements and many of the physical characteristics associated with the improvement. Thus in an effort to improve predictability RCNLDA w i l l be considered as an independent variable. While causality may suffer due to the effects of multicollinearity i t i s hoped that inclusion of RCNLDA w i l l increase the quality of prediction as i t has done in other similar research. Test number three involves regressing s e l l i n g price against the tota l sample (376 observations, 375 degrees of freedom) with RCNLDA included as an independent variable. The results are as follows: a. R 2 = 0.9082 b. Syx/y - 10.7% c. The variables chose as significant and their t s t a t i s t i c s : Variable t S t a t i s t i c RCNLDA 33.91 BASEAR 5.21 MONSHL 7.04 FRF00T 4.64 TRAVOL 3.34 FBASAR 2.87 IRRSHP 2.24 R 2 has now come within acceptable limits and Syx/y ha3 improved significantly over test number one which also involved the entire sample. RCNLDA has proved to be an extremely significant variable (t o 33.91) and accounts for 87% of the variance of s e l l i n g price. A variable of concern i n the above three models i s month of sale (MONSAL). It3 inclusion a 3 a significant variable means that t he estimating equation cannot be extrapolated with reliable results beyond the time period oovered by the sample of sales. The inclusion of this variable into the valuation process would mean the assessor would have to value property as at the end of the time period covered by the sales sample. Valuation would have to be at a recent instant of time, then, rather than valuation at the present. For practical assessment purposes i t i s advisable to avoid the inclusion of this variable into the valuation equation. Its exclusion would enable the assessor to value with consider-ably more f l e x i b i l i t y . Test number four w i l l repeat test number three i n order to ascertain i f exclusion of MONSAL results in a lower quality of prediction. The results are as follows: a. R 2 z 0.8960 b. Syx/J = 11.0$ c. The variables chosen t s t a t i s t i c s : as significant and their Variable t S t a t i s t i c RCNLDA 27.29 BASEAR 5.17 FRFOOT 4-U TRAVOL 2.88 PLMPCS 2.15 IRRSHP 2.12 There i s a very weak correlation between month of sale (MONSAL) and selling price (SALEPR). The correlation i s O.0785 which seems to indicate a relative degree of s t a b i l i t y between prices and time. The exclusion of MONSAL from the regression equation marginally lowers the predictive qualities of the equation. The coefficient of determination drops by 0.0122 and the relative standard error of the estimate rises by 0.3%. Further evidence of the small contribution of MONSAL i s evidenced by the small increase in R2 attributed to i t s inclusion in each of the nine tests (See Appendix F). In a practical application of multiple regression analysis this test would normally recommend to the assessor that MONSAL be excluded from the equation because of i t s marginal contribution. For the purposes of this thesis however, i t w i l l be retained since i t i s desired to use every possible means of improving the predictive quality of the equation. four by producing a more homogeneous sample of sales by stratifying the sample into the 1-9 property types. There are 185 observations and 184 degrees of freedom. The results are as follows: Test number five tries to improve the results of tests three and a. R 2 = 0.8649 b. Syx/y 8.7% c. The variables chosen t s t a t i s t i c s : as significant and their Variable t S t a t i s t i c RCNLDA 19.16 MONSAL 6.91 AGE IMP 3.22 CARPRT 2.42 BASEAR 2.32 This s t r a t i f i c a t i o n has produced better results. Syx/y i s the s t a t i s t i c of primary concern to the assessor a 3 i t measures directly how closely the independent variables are able to predict the sel l i n g prices. This s t r a t i f i c a t i o n has confirmed the belief that a more homogeneous group of properties would be needed to produce results with good predictive qualities. Further analysis of the 1-9 property types showed there were two broad categories of 1-9 property types. One was the actual 109's with sales ranging from approximately $17,000 to $33,000 and the other was a higher priced group of 109's called M109's whose values ranged from approximately $33,000 to $55,000. Further analysis revealed that there were twenty four sales of the Ml-9 type that were included in the broad 1-9 cl a s s i f i c a t i o n . Te3t number six involves excluding the Ml-9's in an attempt to define an even more homogeneous sample. There are 161 sales in this revised sample with 160 degrees of freedom. The results are as follows: a. R 2 a 0.8887 b. Syx/y . 8.35$ The variables chosen as significant and their t s t a t i s t i c s : Variable t S t a t i s t i c RCNLDA 20.09 6.12 4.65 2.12 MONSAL AGEIMP CARPRT It i s significant to note that Syx/y has fallen and R i s within tolerable l i m i t s . However, Syx/y i s s t i l l too high for practical assessment purposes. Test number seven involves testing the refined sample of 1-9's for seasonability. It i s generally believed that the market for residential property i s most active in the months of April, May, June and July, and not so active during the rest of the year. The sales of 1-9's were further s t r a t i f i e d into these two sample groups and a test was made to ascertain i f seasonality existed. The results of the test for seasonality are as follows: Apr i l , May, June, July A l l other months R 2 . 0.7304 R 2 = 0.7891 Syx/y = 8.6% Syx/y - 8.2% Variable t S t a t i s t i c Variable t S t a t i s t i c RENLDA 11.02 MONSAL 5.11 FRFOOT 2.87 RCNLDA 7.01 MONSAL 2.27 TRKRUT 2.66 DWELAR 2.45 AGEIMP 2.05 BASEAR 2.19 CARPRT 3.16 While Syx/y has improved marginally, R has been lowered considerably throwing into question the r e l i a b i l i t y of each Syx/y. Hence i t is concluded that s t r a t i f i c a t i o n of 1-9's for seasonality does not improve predictive results. It would appear that there i s l i t t l e element of seasonality in the sample sales prices. Test number eight involves taking the 1-8 property types and subjecting them to multiple regression analysis. There are 65 sales and 64 degrees of freedom. The results are as follows: a. R 2 - 0.6689 b. Syx/y - 9.6% c. The variables chosen as significant and their t s t a t i s t i c s : Variable t St a t i s t i c BASEAR A.17 RCNLDA 5.UU TRKRUT 5.33 MONSAL 4.73 AERALT 3.42 Str a t i f i c a t i o n into the 1-8 property type clearly does not result in a high degree of predictability. Test number nine involves combining the 1-8 and 1-9 property types in an attempt to achieve a more heterogeneous sample for the inferential process. There are 226 sales and 225 degrees of freedom. The results are as follows: a. R 2 - 0.8334 b. Syx/y = 8.8$ c. The variables chosen as significant and their t s t a t i s t i c s : Variable t S t a t i s t i c RCNLDA 17.59 MONSAL 7.84 AGE IMP 2.91 BASEAR 3.a FRFOOT 4.47 TRKRUT 4.81 Clearly, any improvement i n the predictive quality of the 1-8 property type i s due to the inclusion of the 1-9 property type into the sample. Tests were not performed on the other property types for two reasons: 1 there were not enough sales of each property type to produce s t a t i s t i c a l l y significant results; 2 i t would lead to the creation of too many models to be used in any practical application of the technique. ANALYSIS OF THE TESTS The nine tests undertaken on the Burnaby sales data have not produced an equation "that could be used for practical assessment purposes. In each case the predictive quality which was generated was not of suf f i c i e n t l y high calibre to result in accurate predictions for the assessor. The coefficient of determination and the relative standard error of the estimate for each run are presented i n Table VIII. Test 2 number six has the greatest predictive capabilities with an R of 0.8887 and a relative standard error of the estimate of 8.35$. This i s equivalent to saying that for the 1-9 property types the estimated market values w i l l be within plus or minus 16.7$ of actual value for 95$ of the cases considered i n the sample. Though the results are not useful for practical purposes of valuation, they do indicate how s t a t i s t i c a l testing i s capable of improving the predictability of the estimating equation. The test actually has quite meaningful results, considering that this i s only a f i r s t attempt in one small area of Burnaby, with rel a t i v e l y few property characteristics, in relation to other studies which have tried to implement the technique. In fact, the nine tests have produced results which suggest the technique of valuation by multiple regression analysis might be improved by further research involving other areas of Burnaby, or more sophisticated analysis of the current study. Analysis of the study in light of current research methodology leads to suggestions for improving the predictive quality of future research. One possible source of improvement might be to improve the number and quality of property characteristics from each property that i s being considered for analysis. Other studies have u t i l i z e d many more variables than this study. Appendix D presents a useful guide to the kinds and COEFFICIENT OF DETERMINATION AND "RELATIVE" STANDARD ERROR OF ESTIMATE FOR EACH TEST IN THE STUDY COEFFICIENT OF RELATIVE STANDARD TEST NUMBER DETERMINATION ERROR OF THE ESTIMATE 1 0.7693 13.2% 2 0.7424 8.9% 3 0.9082 10.7% 4 0.8960 11.0% 5 0.8649 8.7% 6 0.8887 8.35% 7 a 0.7304 8.6% 7 b 0.7891 8.2% 8 0.6689 9.6% 9 0.8334 8.8% x Relative standard error of the estimate - standard error of the estimate i mean sales price of test sample. number of property characteristics that have been used i n successful applications of multiple regression analysis i n an assessment context. A useful means of improving predictability may be to devise a means of making the qualitative property characteristics more sensitive to analysis by constructing a three or four way cla s s i f i c a t i o n to measure their variance. Perhaps this could be done by devising two qualitative variables out of each one presently being considered and using a dummy variable for each of the two new qualitative variables. For example, the dummy variable now considering whether a lot i s of irregular shape could be s p l i t up into two dummy variables in order to record more precisely the nature of the irregular shape. One of the dummy variables could be used to determine i f the lot was long and narrow while the other could be used to measure some other feature of shape. One half of the variables considered in this study are of a qualitative nature. If the preceding suggestions can improve the sensitivity of these variables and the predictive qualities of the regression equation, then this area should be explored i n depth. The tests have definitely revealed how st r a t i f i c a t i o n of the sample i s often necessary in order to improve the predictive qualities of the equation. The cla s s i f i c a t i o n of single family homes by property types has permitted the use of a convenient means of stratifying the sales data. By confining the study to a small geographic area, such as the one considered in this study, there i s created the danger of too few sales of each property type. This could lower considerably the predictive qualities of the equation. There are two possible means of overcoming this problem. F i r s t , the assessor can group property types in the study into sub-samples according to ranges of selling price. For example, the property types known to be in the value range of $20,000 to $30,000 might be analyzed as a group. For those property types that have sold infrequently, the predictive qualities w i l l probably remain poor, in spite of any grouping attempted to bring more sales together for analysis. The second approach suggested by this study is to test a property type or group of similar property types within a much larger geographic area. For example, the 1 -9 property type or the 1-8 and 1-9 property types together could be tested for a l l of Burnaby. This would provide a much greater number of comparable properties for analysis and would increase the a b i l i t y of multiple regression analysis to predict the sample s e l l i n g prices. For practical assessment purposes the estimating equations must be of a more heterogeneous nature. To have several equations in such a small geographic area w i l l probably prove to be too cumbersome a procedure to implement. The greater the number of single family residences whose value can be estimated from a single equation the more useful i s the technique. At present, only the 1-9 property type can produce results that are significant. Hopefully, the two approaches suggested above might provide an easy means of avoiding the problem of equations that are too numerous and too homogeneous i n nature. The examination of residuals i n each of the nine tests supported the view that there was a relatively good linear relationship between the dependent variable and each of the independent variables. In each case, the residuals were randomly distributed. The multiple regression model found to be most useful in this study (from t est number six) found that 85% of the variance i n se l l i n g price was explained by RCNLDA. While the inclusion of RCNLDA is a very significant variable for predictive purposes, i t does not appear to increase the causality of the model. Taken alone, RCNLDA is simply a figure representing the cost approach to value for the improvements. But together with property characteristics, i t can be considered useful as being a proxy to other features of the property that are basic considerations of buyers and sellers in the market place. For example, RCNLDA can be a proxy for age, architectural quality and quality of construction. To derive a highly causal model, the use of this variable should be avoided. Perhaps the inclusion of more locational variables, physical property characteristics and socio-economic variables into the regression analysis would eliminate the need to use RCNLDA. It is interesting to note that only current study of multiple regression analysis in an assessment context in Canada has avoided the use of RCNLDA and has achieved very significant results. IMPLEMENTATION OF THE TECHNIQUE A test such as the one conducted i n this study reveals the immense task ahead of any assessor who decides to adopt a valuation system based on multiple regression analysis. The costs and time required to adopt such a system are large mainly because of the need to adopt current assessment records into a form suitable for the computer application of multiple regression analysis. Burnaby is in a unique position in that i t is in the process of putting many of the property characteristics needed in their current cost approach into data banks for manipulation by computer. From this i t i s a small matter (technically) to devise a form such as illu s t r a t e d in Appendix D to collect the data required for the multiple regression technique. Such forms can probably be best completed by the assessor's staff at the time of physical inspection of the single family residences in the munipality. In an area the size of Burnaby this would probably involve three to four years to complete. Such a time period would permit the multiple regression technique to be introduced gradually thus giving the assessor and his staff the time to learn the complexities of the approach. But while the approach has high fixed costs to i n i t i a t e , once the basic appraisal f i l e containing the property characteristics is completed there is very l i t t l e cost involved i n maintaining the appraisal f i l e and constructing the multiple regression equations (see Appendix B). Once these i n i t i a l tasks are completed, the assessor is in an excellent position to experiment with the data. With property characteristics of a l l properties on record he can very simply request the computer to construct estimating equations from any number of different stratifications of the sales sample he should desire. The testing in this study involved large amounts of time in finding the property characteristics of each sold property, coding these characteristics for input into a computer system and preparing the physical input for the computer. Once a t o t a l valuation system i s implemented, experimentation w i l l not be held back by so much preparation. Experimentation at that time w i l l be a highly automated procedure which w i l l enable the assessor to quickly determine highly predictive estimating equations to value the real property in his assessment d i s t r i c t . REFERENCES CITED - CHAPTER V x William M. Shenkel, "Valuation Studies", International Property  Assessment Administration, Proceedings of the 35th Annual International Conference on Assessment Administration (Chicago: International Association of Assessing Officers, 1969), p. 103. 2 Richard C. Clelland, et.al., Basic Statistics With Business Application,(New York: John Wiley & Sons,Inc., 1966), p. 463. 3 Ontario Department of Municipal Affairs, Multivariate Analysis in Ontario, p. 10. ^ William M. Stenkel, "Computer Valuation by Multiple Regression Analysis", International Property Assessment Administration, Proceedings of the 34th Annual International Conference on Assessment Administration, (Chicago: International Association of Assessing Officers, 1968),p.32. 5 Robert H. Gustafson, E.S.P. and the Appraiser, paper prepared for delivery at the 35th Annual Conference of the National Association of Tax Administrators, San Francisco, June 14, 1967, p. 7. ^ Renshaw, Sci e n t i f i c Appraisal, p. 319. 7 Gustafson, Data Banks and Assessment Roll3, Appendix. 8 Ibid., p. 16. 9 Shenkel, Valuation Studies, pp. 106-107. 1 0 Ibid., p. 106. 11 Davis, A S t a t i s t i c a l Approach to Farm Appraisal, p. 63. SUMMARY AND CONCLUSIONS This f i n a l chapter includes a restatement of the study, a summary of the findings, conclusions of the study, and suggestions for further research to be directed at improving the application of the multiple regression technique to real property valuation. RESTATEMENT The intent of the investigation undertaken in this thesis has been twofold. Both aspects have centered on the questionability of logic and technique employed by assessors in the traditional approach taken to the valuation of single family residences. The main intent was to reveal that traditional assessment techniques were grounded in poor appraisal methodology. The second intent was to suggest an alternate method of valuation; that i s , one based on sound appraisal theory. A further intent was to test the applicability of this method with actual assessment data. It was shown that i t i s of paramount importance for the assessor to apply techniques of valuation that ensure equality of treatment among real property owners. A Fair distribution of the burden of the tax can be attained only when equality of treatment is an inherent part of the valuation system. Because the real property tax i s not based on •principles of taxation 1, i t became necessary for legislators to define an objective standard for attaining equality of treatment. This standard was identified as valuation at 'actual value'. It was further revealed why assessors are unable to value objectively and f a i r l y using the cost approach to value. U n t i l recently, the cost approach to value was the only valuation technique that would enable the assessor to value separately land and improvements for so many thousands of properties. But a great deal of subjectivity i s inherent i n the cost approach because of the use of cost and depreciation estimates. It was shown that replacement cost new less depreciation very seldom i s equal to current market values. Market value was shown to be the only approach to value that could ensure equality of treatment. Most probable s e l l i n g price was suggested to be the most objective concept to value. It was then shown that most probable s e l l i n g price, value i n exchange, current market value and •actual value' were synonomoU3. Multiple regression analysis was suggested as an aid to the assessor as i t can replicate the forces i n the market that determine the most probable s e l l i n g price of real property. The technique not only permits a most probable s e l l i n g price estimate, but also generates measures that enable the assessor to gauge the degree of predictability inherent in the estimating equation. The a p p l i c a b i l i t y of the technique was tested on sales data in Burnaby in an area where i t was believed there would be available a SUMMARY OF FINDINGS The purpose of the tests on the Burnaby sales date was to discover how heterogeneous a sample could be valued with one multiple regression equation. It was found that a single equation could not produce the predictive quality desired. The main experimentation in the study then became to s t r a t i f y the data into groups of a more homogeneous nature i n an e f f o r t to increase the quality of results. The sales data, contained in the 1-9 property type, for example, was able to generate an equation of much higher predictive quality than the original sample containing a l l the property types together. While the results of these tests were not of sufficient quality for Implementation in an actual assessment situation, they were signif-icant enough to suggest that further research and experimentation with more detailed and sophisticated sales data could probably raise the predictive quality of the valuation procedure. CONCLUSIONS t The application of the market approach to value as the basis of assessment i s the only way that assessors can be sure of building equality of treatment into their valuation systems. The use of multiple regression analysis as a computational aid in the market approach w i l l enable the assessor for the f i r s t time to be able to apply a market approach to value for the thousands of single family residences he is required to value. The use of multiple regression analysis w i l l make i t possible to revalue annually each property within the municipality. It w i l l considerably shorten the lag in values that generally exists between current values and values on the r o l l . The assessor w i l l be in a position to ascertain quickly the nature of recent value trends and w i l l therefore, be able to reflect annual changes in values on the r o l l rather than having to wait two or three years to ascertain the nature of the trends. The adoption of multiple regression analysis into the f i e l d of real property valuation w i l l herald a new era into assessing. .The* assessor's dilemma as i t now exists w i l l vanish. When i t i s adopted real property owners w i l l be assured that they are being treated impartially and objectively. The application of multiple regression analysis w i l l ensure that each real property owner bears a ' f a i r ' burden of the tax. If such a valuation system i s adopted on a provincial scale, i t w i l l also ensure equality of treatment between municipalities where t o t a l assessed values are used as the basis for apportioning provincial grants among the municipalities. SUGGESTIONS FOR FURTHER STUDY It has been shown in this thesis how s t a t i s t i c a l testing can improve the predictive quality of the multiple regression equation. But these results must be improved substantially before such a system of valuation can become of practical importance. One suggestion for further study concerns the independent variables of the tests. More variables might be introduced into the model, in an attempt to ensure that a l l the major factors affecting market value are subject to consideration by the multiple regression procedure. The inclusion of other variables found to be of a significant nature could improve the overall predictive quality of the equation by mak-ing i t unnecessary to use replacement cost new less depreciation (RCNLDA) as an independent variable. Much current research i s being directed at preliminary analysis of the independent variables in order to ascertain the degree of l i n e a r i t y of the data and the degree of multicollinearity inherent i n *the data. This research uses simple tabulations and histograms, 'Automatic Interaction Detection' (a s t a t i s t i c a l device that identifies interaction effects between variables and f a c i l i t a t e s the s t r a t i f i c a t i o n of data into homogeneous groups for use i n subsequent regression sub-models) and factor analysis i n a preliminary exploration to evaluate the nature of the data, determine i t s scope of application, and identify areas of multicollinearity and interaction. On the basis of the findings from such analyses, multiple regression models can be structured to maximize predictive relationships, by minimizing both the inclusion of spurious terms and the exclusion of significant factors, and by combining factors so as to represent the interactions identified i n the preliminary analysis. As more sophisticated s t a t i s t i c a l techniques are developed and implemented into the real estate valuation problem, there w i l l undoubtedly develop valuation models that contain both a high degree of predictability and causality. To reap the benefits of such a valuation system, assessors must act now to learn the problems and solutions posed by these technique in their own municipalities. Assessors must not let the seeming complexity of this valuation system outweigh the benefits that can be gained from i t s adoption. Assessors must actively advocate for the implementation of these techniques into their own valuation systems, in order to reap the obvious benefits. These techniques offer the only means of basing mass valuation upon sound appraisal methodology. A. Books Babcock, Frederick M. The Valuation of Real Estate. New York: McGraw H i l l Book Company, Inc., 1932. Bjerring, James H. and Seagraves, Paul. Triangular Regression Package. Vancouver, B.C.: The University of British Columbia Computing Center, 1970. Bonbright, J.C. The Valuation of Property. Charlotteville, Va.: The Mitchie Company, 1965. Boulding, K.E. Principles of Economic Policy. Englewood Cl i f f s , N.J.: Prentice-Hall Inc., 1958. v v -Cameron, M.A. Property Taxation and School Finance in Canada. Toronto: Canadian Education Association, 1945. Clelland, R.C.; deCani, John S.j Brown, Francis E.; Bursk, J.Parker, and Murray, Donald S. Basic Statistics with Business Application. New York: John Wiley and Sons, 1966. Crawford, Kenneth Grant. Canadian Municipal Government. Toronto: University of Toronto Press, 1954. Draper, N.R. and Smith, H. Applied Regression Analysis. New York: John Wiley & Sons, Inc., 1966. Due, J.F. Government Finance. Homewood, 111.: Richard D.Irwin, Inc. 1963. Ezekiel, Mordecai and Fox, K.A. Methods of Correlation and Regression  Analysis. New York: John Wiley and Sons, 1959. International Association of Assessing Officers. Proceedings of the 34th  Annual International Conference on Assessment Administration. Chicago: International Association of Assessing Officers. Proceedings of the 35th  Annual International Conference on Assessment Administration. Chicago: Klein, Lawrence R. A Textbook of Econometrics. Evanston: Row, Peterson and Company, 1956. McCracken, Daniel D. A Guide to Fortran Programming. New York: John Wiley and Sons, 1965. Netzer, Dick. Economics of the Property Tax. Washington, D.C. The Brookings Institution, 1966. Ratcliff,, Richard U. Modern Real Estate Valuation. Madison, Wisconsin: Democrat Press, 1965 Yamane, Taro. Statistics - An Introductory Analysis. 2nd ed. New York: Harper and Row, Publishers, 1964. B. Monographs Davis, Irving F., J r . A S t a t i s t i c a l Approach to Real Estate Value with  Applications to Farm Appraisal. Study No.12. Fresno, California: State of California, Division of Real Estate, 1965. Finnis, F.H. Property Assessment in Canada. Canadian Tax Paper No.50. Toronto: Canadian Tax Foundation, 1970. t**S Hicks, J.R.; Hicks, U.K. and Lester, C.E.V. The Problem of Valuation  for Rating. National Institute of Economic and Social Research. Occasional Paper No.7. London: Cambridge University Press, 1944. Methodology Section, Assessment Standards Branch, Ontario Department of Municipal Affairs. Multivariate Analysis and Residential Property  Valuation in Ontario. Toronto: Ontario Department of Municipal Af f a i r s . 1970. R a t c l i f f , Richard U. Current Practices in Income Property Appraisal -A Critique. Research Report No.30. Berkeley: The Center for Real Estate and Urban Economics, Institute of Urban and Regional Development, University of California, 1967. C. Periodicals Babcock, Frederick M. "The Three Approaches." The Real Estate Appraiser, Vol. 36, No. 5 (July- August, 1970), pp.5 - 9. Blettner, Robert A. "Mass Appraisals via Multiple Regression Analysis." The Appraisal Journal, Vol.XXXVII, No.4 (October, 1969), pp.513-521. Brigham, Eugene F. and Donald M. McAllister. "Applying Econometric Models." The Appraisal Journal, Vol. XXXVI, No.4 (October, 1968), pp.541-548. Case, Fred. E. "Computer Applications in Real Estate Appraisal" Colloquium on Computer Applications in Real Estate Investment.  Analysis. Edited by Richard U. R a t c l i f f . Vancouver: Faculty of Commerce and Business Administration, University of Br i t i s h Columbia, 1968, pp. 65-76. Case, Fred, E. "Electronic Data Processing and the Appraisal Function." The Real Estate Appraiser, Vol.32, No.9 (September, 1966), pp. 2-9. Case, Fred, E. "New Decision Tools for the Appraiser." The Appraisal  Journal, Vol.XXXVI, No.4 (October, 1968), pp.21-27. Craig, Robert H. "Property Assessment at Market Value." The Appraisal  Institute Magazine, Vol.14, Bk.4 (Winter 1970-71), pp.2-16. Dasso, Jerome and Swaden, Paul. "Data Processing Implications for Property Taxation." The Appraisal Journal, Vol.XXXVIII, No.l (January, 1970), pp.52-56. Eisenlauer, Jack F. "Mass Versus Individual Appraisals." -The Appraisal  Journal, Vol. XXXVI, No.4 (October, 1968), pp. 532-540. Hihshaw, Andrew J. "The Assessor and Computerization of Data." The  Appraisal Journal, Vol.XXXVII, No.2 (April, 1969), pp. 283-288. Kinnard, William N.Jr. "New Thinking in Appraisal Theory." The Real  Estate Appraiser. Vol.32, No.8 (August, 1966), pp.2-H. Lessinger, Jack. "Econometrics and Appraisal." The Appraisal Journal, Vol.XXXVII, No.4 (October, 1969), pp.501-512. Lessinger, Jack. "Toward a New Method and Theory of Appraisal." The Real  Estate Appraiser. Vol.34, No.2 (March, 1968), pp. 42-47. Pendleton, William C. " S t a t i s t i c a l Inference in Appraisal and Assessment Procedures." The Appraisal Journal, Vol.XXXIII, No.l (January, 1965), pp.73-82. Rating and Income Tax. London: Solicitors Law Stationery Society Ltd. Passim. Renshaw, E.F. " S c i e n t i f i c Appraisal." National Tax Journal, December,1958, pp.314-322. Shenkel, William M. "Computer Valuation by Multiple Regression Analysis." International Property Assessment Administration. Proceedings of the 34th Annual International Conference on Assessment Administration. Chicago: International Association of Assessing Officers, 1968, pp.26-41. Shenkel, William M. "Valuation Studies." International Property Assessment Administration. Proceedings of the 35th Annual International Conference on Assessment Administration. Chicago: International Association of Assessing Officers, 1969, pp.98-115. Smith, H.C., and Racster, R.L. "Should the Traditional Appraisal Process Be Restructured." The Real Estate Appraiser, Vol.36, No.7 (November -December, 1970), pp.6-11. Smith, Theordore R. "Multiple Regression and the Appraisal of Single Family Residential Properties." The Appraisal Journal, Vol.XXXIX, No. 2 (April, 1971), pp.277-284. Wendt, Paul F. "Recent Developments in Appraisal Theory." The Appraisal  Institute Magazine, Vol.14, Bk.l (Spring, 1970), pp.35-45. Wendt, Paul F. "Theory of Urban Land Values." Journal of Land Economics, Vol.XXXIII (August, 1957), pp.228-240. D. Unpublished Material Clettenberg, Karl J. "Computer Applications to Real Estate Assessment." Paper presented at the "Symposium on the Application of Econometric Methods to the Appraisal Process" Las Vegas, Nevada, October 27, 1970. Gustafson, Robert H. "Computers and Statistics as an Aid to the Appriaser of Single Family Residences." Paper prepared for delivery at the Symposium on the Application of Econometric Methods to the Appraisal Process, Las Vegas, Nevada, October 27, 1970. Gustafson, Robert H. "Computer Application in Real Property Tax Assess-ments." Paper delivered at Colloquium on Computer Applications in Real Estate Investment Analysis, The University of B r i t i s h Columbia, February 2, 1968. Gustafson, Robert H. "Data Banks and Computerized Annual Updating of Assessment Rolls." Paper delivered at the 35th Annual Conference on Assessment Administration, Denver, Colorado, September 7-10, 1969. Gustafson, Robert H. "E.S.P. and the Appraiser." Address delivered at the 35th Annual Conference of the National Association of Tax Administrators, San Francisco, June 14, 1967. Gustafson, Robert H. "Estimated Sales Price Programs." Paper delivered at the 18th Annual Conference of the Western States Association of Tax Admin,jistrators, Las Vegas, Nevada, October 20-24, 1969. Hinshaw, Andrew J. "Analysis of the Assessment Function." Paper presented at the International Association of Assessing Officers, 1970 Annual Conference, Las Vegas, Nevada, October 26, 1970. "Local Assessment and Taxation" Real Estate and Appraisal Diploma Course. Unpublished course material, The University of British Columbia, Vancouver, B.C. 1966. Ratcliff, Richard U. Valuation Theory. Unpublished course material. Vancouver: Faculty of Commerce and Business Administration, The University of British Columbia, 1970. Shenkel, William M. "Valuation by Multiple Regression Analysis - Selected Case Studies." Paper presented before the 36th Annual International Conference on Assessment Administration, Las Vegas, Nevada, October,1970. White, Philip H. Land Taxation in Canada. Unpublished paper. Vancouver: Faculty of Commerce and Business Administration, The University of British Columbia, 1969. APPENDICES COMPONENTS OF A TOTAL ASSESSMENT SYSTEM It has been emphasized that the multiple regression procedure is just part of a total valuation system. But i t is a very large part of such a total system and is, in fact, the heart of such a system. A total system of valuation can be broken into three main sub-components: 1 that part concerned with the collection, coding and transformation of data from present assessment records into a form suitable for use by the computer. The greatest deal of time and effort in developing a valuation system will be expended here. This sub-component13 main function is to supply the inputs to the multiple regression procedure. 2 that part concerned with the testing and preparation of the actual multiple regression equation. 3 that part which utilizes the multiple regression equation to calculate the most probable selling prices which will be used on the real property assessment r o l l . The accompanying figure identifies seventeen functional areas of such a total valuation system: i n Appendix C are used to collect the basic data to be used in the valuation system. The property characteristics on the form describe the building, the land, the t o t a l property and the neighborhood. The number of character-i s t i c s is primarily a function of the homogeneity of the area under study and the precision desired in the end product. The more diversified the properties comprising a municipality, the more characteristics are needed to properly reflect the t o t a l municipality. FUNCTION 2 The coded characteristics are entered into the computer's data bank through a series of programs that update the f i l e and edit the data. The editing routine checks both for factual error conditions, such as omission of data or impossible coding. The resulting edit l i s t contains those f a t a l errors which are obviously wrong and also 'informational errors' such as the inclusion of a six-room house containing four bedrooms and five baths. FUNCTION 3 Characteristics on each single family residential unit are maintained within the computer system and constitute the assessor's appraisal f i l e . The characteristics multiple regression analysis finds relevant in the regression equation can then be pulled from this f i l e for each property to be appraised. FUNCTION 4 Information on sales that have occurred in the municipal-i t y are sent to the local assessor from the lo c a l Land Registration Office. These sales have been edited by the Assessment Commissioner to ensure that they are the result of normal market transactions. FUNCTION 5 The sales information i s transferred into the computer system and forms the sales f i l e . This sales f i l e plays a key role in the system, as the primary criterion to value is the market approach. The sales data are f i r s t added to the property characteristics i n the basic appraisal f i l e where the last three sales on each property w i l l be retained with the appraisal f i l e . At the same time the sales are moved into the appraisal f i l e , the property characteristics for that property at that point in time are moved to the sales f i l e . These property characteristics contained in the sales f i l e are not to be updated as they represent the status of the property at the time of the sale. It w i l l be this f i l e that i s used for regression and therefore the market approach. FUNCTION 6 Thi3 sales f i l e contains the basic data for the detailed sales ratios studies that are used by the assessor. FUNCTION 7 AND 8 Both the basic appraisal f i l e and the sales f i l e are advanced once each year through the cost update phase. This cost update consists of a series of table look-ups for new cost factors and depreciation. In this system calculation of replacement cost new and replacement cost new less depreciation i s retained both for preparation of these factors as variables to be included in the multiple regression analysis and to maintain the cost approach to value within the computer system u n t i l the market approach to valuation is t o t a l l y accepted. Until the problems of multiple regression analysis have been solved, i t would be advisable to keep the two systems operable. FUNCTION 9 Both f i l e s are then passed through a series of programs that produce a set of s t a t i s t i c a l profiles. Such profiles can show for each property characteristic: mean value, standard deviation, frequency of occurrence and minimum and maximum value data. Computer generated histograms can also be devised to more precisely depict the distribu-tion of the characteristics that are measured on a continuous 3cale. These profiles w i l l make i t possible to answer the question "Are the sold properties unlike the sold properties?" When i t i s found that the sold properties do not cover the f u l l span of characteristics, i t i s possible to enter older sales to complete the picture or exclude certain properties that are not representative of the population. FUNCTION 10 After careful review of the characteristics of both the sold and unsold properties, certain items may be eliminated. FUNCTION 11 There w i l l be those characteristics that do not occur within the area under study. Others w i l l occur but w i l l not vary throughout the area, and hence w i l l be . liminated. S t i l l others w i l l occur but with such low frequency that they w i l l not be used. Property characteristics associated with the sales are processed through a program that performs multiple regression analysis. Various tests w i l l be performed on the data to ensure that an equation of substantial predictive quality results. FUNCTION 12 The regression coefficients are stored for future use with the appraisal f i l e . FUNCTION 13 The regression coefficients are applied to the value of the variables found to be significant for each property in the appraisal f i l e . The estimated s e l l i n g price of each property in the appraisal f i l e and in the sales f i l e are calculated for further analysis. FUNCTION H. A value sheet i s generated for each property in the current appraisal year. A value sheet contains most of the property characteristics contained in the data bank as well as the indicated values by multiple regression analysis and the cost approach. FUNCTION 15 Analysis reports can be generated to show trends in value by property type or for geographic areas in the munic-i p a l i t y . FUNCTION 16 Exception l i s t s are generated to indicate those properties where the estimated value f a i l s to meet certain minimum c r i t e r i a of acceptance. These exceptions w i l l have to be carefully reviewed by the assessor and i n many cases manual appraisals of these exceptions w i l l have to be performed. FUNCTION 17 The last component of the system is a sales r e c a l l program. Since the primary emphasis on the value sheet i s the estimated sales price, i t i s essential to have the a b i l i t y to r e c a l l at random a l i s t i n g of sales that are comparable to the subject property. This would be particularly useful in helping the assessor to value those properties that the computer has identified on the exception l i s t . THE COSTS OF CONVERTING TO A NEW VALUATION SYSTEM Gustafson has described the costs of implementing a t o t a l valuation system based on multiple regression analysis in several counties in California. A f i r s t major expense involves collecting the data on a form such as illustrated i n Appendix D. Costs varied between counties and ran from 01.00 to $1.50 per property. The variation in cost is a function of the detail already contained in the appraisal f i l e . Obviously, the better the f i l e s , the less costly the conversion. The cost to maintain the data banks with annual updates and annual printouts varied between $0.10 and $0.15 per property. The cost of maintaining the data in the memory units, from the f i e l d point of view i s dependent upon the size of the f i e l d staff assigned to the function. Some neighborhoods w i l l require l i t t l e work beyond analyizing new building permits, while others, especially those neighborhoods in transition, w i l l require more f i e l d investigation time. The costs of acquiring the computer programs for this valuation system are re l a t i v e l y inexpensive. The programs required to produce the multiple regression equation, calculate estimated se l l i n g price and printout value sheets were acquired at a cost of about $30,000. The programs required to build, edit and maintain the appraisal f i l e was obtained for about $15,000. A more detailed examination of the costs of implementing and maintaining such a t o t a l valuation system in Kern County, California, is presented here for analysis. The costs in Kern County have been used along with similar costs from several other counties to derive the above costs. COSTS OF IMPLEMENTING A TOTAL VALUATION SYSTEM IN KERN COUNTY, CALIFORNIA Area for Analysis: 1 10,000 single family residences 2 240 million f u l l cash value 3/1/69 3 25 percent assessment level 3/1/69 4 60 million assessed value 3/1/69 5 Last reappriasal as of 3/1/69 Cycl i c a l program a. Four year reappraisal cycle b. Reappriasal cost: 10,000 + 8/day - 1,250 f 230 days/year = 5.4 man years @ $12,000 a $64,800 c. Annual costs to pick up new work and permits between reappraisals: .5 man years @ $12,000 s $6,000 d. Costs increasing at 5 percent per year. 2 Annual updating using regression a. Conversion crew (1) 5 appraisers f 4 clerks • 2 key punch operators = 5 x $12,000 • 6 x $6,000 = $96,000. (2) can do 57,500 per year or 250 per day. (3) 10,000 - 57,500 = .17 crew years @ $96,000 = $16,300. b. Annual costs to pick up new work and permits: (1) .5 man years @ $12,000 • $6,000. (2) on year of conversion this cost i s only $3,000. c. Physical inspection of each property every fourth year: 10,000 f 500/day s .08 crew years @ $96,000 s $8,000. d. Annual review of E.S.P. and cost for enrolling (1) 10,000 f 100/day a 100 days or .43 man years @ $12,000 s $5,160. (2) $1,000 for computer support Cost increasing at 5 percent per year 3 Costs over an eight year period (two reappraisal cycles) are as follows: Year 1969 1970 1971 1972 1973 1974 1975 1976 1977 Cyclical Program COST 6000 6000 6000 64800 6000 6000 6000 64800 1.00 1.05 1.1025 1.1576 1.2155 1.2763 1.3401 1.4071 = ft Total 1970-1977 6,000 6,300 6.615 75,012 7,293 7,658 8,041 91.180 $208,099 16300 (6000 (6000 (6000 (6000 (6000 (6000 (6000 Annual Program 3000 - 6160 - $25,460 6160)x 1.05 - 12,768 6160)x 1.1025 - 13,406 6160)x 1.1576 - 14,076 6160 x 8000)x 1.2155 - 24,504 6160)x 1.2763 - 15,120 6160)x 1.3401 - 16,296 6160)x 1.4071 - 17,110 $138,740 x Computer assisted Values: 1 Growth i n f u l l cash value i s 6 percent per year. 2 On c y c l i c a l program where only new work is done between reappriasals assessed value w i l l grow by .5 percent rather than 6 percent since there w i l l be no revaluation. 3 Values over an eight year period are as follows: VALUE F.C.V. Year Value % Chg. (Million)  1969 240 — 1970 254.4 6 1971 270.0 6 1972 285.8 6 1973 303.0 6 1974 321.2 6 1975 340.4 6 1976 360.9 6 1977 382.5 6 Total 1970-77 -Cyclical Program Assd.Val. %Chg. Ratio (Million) 60 - 25.0% 60.3 .5 23.7 60.6 .5 22.4 60.9 .5 21.3 75.7 24.3 25.0 76.1 .5 23.7 76.5 .5 22.5 76.8 .5 21.3 95.6 25.0 642.5 Annual Program* Assd.Val %Chg. Ratio (Million)  60 - 25.0% 63.6 6 25.0 67.4 6 25.0 71.5 6 25.0 75.7 6 25.0 80.3 6 25.0 85.1 6 25.0 90.2 6 25.0 95.6 6 25.0 689.4 TYPICAL SEQUENCE OF EVENTS IN THE ASSESSMENT CYCLE The sequence of events that would occur in a valuation system based on multiple regression would be as follows, given a hypothetical March 1970 assessment date and a July 1970 r o l l date: 1 During June of 1969 a l l costs would be updated. 2 A l l sales occurring during the twelve months prior to July, 1969, would be selected and run through the profile programs. 3 Having selected the sales and reviewed and selected the variables, regression would be run during July, 1969, for and area (s) or property type(s) under valuation. t\ By the f i r s t of August, 1969, the f i r s t printouts of the value sheets would be available, together with exception l i s t s and analysis reports. 5 Between August, 1969, and May, 1970, the value sheets would be reviewed either in the office or in the f i e l d , depending upon the degree of review required. 6 The values would be ready for the July, 1970, r o l l . This system has about an eight month time lag built into i t , which can be reduced by using a mid-fall cut-off rather than a July 1 date. Throughout the year the data would be updated i n the data bank by means of new building permits and review of a l l properties in the f i e l d at least once in four years. The maintenance of accurate current data in the data bank is essential to the efficiency of this system. A R E A N O . r , . I .RESIDEN" FACY SHEET F A D C t L N O . * * • I : : T R A C T C O T • L O C K SITUS R E C O R O O A T * L A N O A T T R I B U T E ! • U l L O I N G O A T A C O S T O A T A M U L T I P L E O A T A t a i *«>'i . Date I : 1 ! 1 '• : '• 101 w . a i h F I . 401 Con*. C | Moo. 1 M ; A a t i . A 301 C a n Le-rtl B a t . Vr . <0I Un.tt F « r n . | • 102 Crnp>arM N a . I 102 Oeo ln F t . 407 C o n n . Vr . : ; ; 302 C l a n l i t F i r . i i 602 U n t i l \Jt*t+tn. { • 103 U M C O S * 1; J O ) V Q . F I . (actual) ! • : ! A O 3 C i t . Vr . ! : SO) C lat i 2nd F i r . i '• SO) 1 r > » j r o o m | : 104 T o u r Prop. V e » . | » • ! i • 304 S Q . F l . (uviaDie) : • : : 404 D « p . Table : 304 A r ja (or M o d . |* 1 i ! 634 2 B r t f o s r n | 10S L V t V d . | j * • 1 • • • i . 30} Acret | \ . 401 L i te : 303 «M 3 bedroom f : 10} Imp. V » l . - I : : ; : ! l o n e Actual r - • 40* JOS l t t F i r . A r e a ; 1 : : 354 Stvo<o I : 107 Per Prop. V4l . * t * * • • 1 * 102 Zone Confor m. N V 407. 307 2nd F i r . A r e * : 1 ! • 337 R>c. R o o m i j 103 & I U U I V i i . I ; ; 1 30 S Typical N V 403 Dining R o o m i 303 2nd F i r . Factor , 1 l i = 373 1 Sa in | t 101 C J M L O I Feet.* 1 •' • •' 3 0 , Irregular N V 404 F a m . - Oen i s 309 3rd F i r . Area 303 2 t a i n t | no 110 Cu l -Oe See N V 410 N o . o l Bedroom! ] \ 310 3rd F i r . Factor 1 i I i 310 1 r-i'm 1 * T E M P O R A R Y V A L U E 311 ejon.Tnru-St. N V 411 No . of Balnf 1 S l l At t ic Area : : i : til 111 Part Compl . A 312 Non-St. -Frontage N V 412 Utfl . Rooms a 1 312 Att ic Factor SI2 P t n l n i Saace/unlt T. • Board Ac l lon a 313 Corner N V 413 3 1 1 fitmt. Area | ; ' J ; ; 313 Parkln>-Fr>e F : P. '3 p Otner c 114 Al lay N r 414 314 Btmt. Factor ? 614 112 311 U M . U V G . N V 415 Funct . Plan a A p 313 A d d n . Area : i : : 613 11) lit C u r b l L G u t t e d N V 416 Condit ion G A p 316 A d d n . Factor 616 114 • 317 S » o e « » e l l - s 1 •4 V 417 Workmanship G A p 317 Gar. Area 1 * 617 ns lit St. L lgnt i N V 411 S I , Space O A p 3 1 * Gar. C.ai t : 1 611 N E I O M S O » M O O O 319 Parkwayi H V 419 319 Pen. Area | I N C O M E O A T A 201 A « 320 Parhitr la i 3 * * ' f i " ' ' V 420 320 Pen. Factor i I i 701 G r o i l A n n . Inc.*" 102 M M . 0««TlJntf * G A p 321 C o m m o n Green M V 421 Heating (Oucfed) N V 321 Mi i c . C o n ! : ! 702 A c t . Rent : . '. . : 20) A P 372 C o m m o n A t e . M v 422 Cooling (Oucted) e R 322 70) E x ^ n e e l : ; 204 T r t - « d a S o 32) H . 4 a. U U N Y 42) 323 Fireplace Co l t » : » # : : 1 ' 704 R . S . L . I » r l . ) : 209 A M 324 424. Gar A O 324 A i r C o n d . Cott i S i 701 V a c . CM 1 i 20* 321 V iaw Qual . O A P 421 Carport H V 323 70* 207 0 « « M « Oce. N V 326 Landicaoe O A P 476 Pa i l* H V 326 P i t lo Area | '. ; • i s 707 2-0 S * • » . A r U N V 37 7 Traff ic F low L A H 427 Pool N V 327 Patio Factor 701 203 N V »a O u t . to pys. N A t 4 2 ! Fence N V 32S Pool Area \ ; j 1 709 210 12* 479 StO'aoe N V 329 Pool Eetrat f 1—;— 710 *>> pVch Attf. G A p T O P O G R A P H Y 4)0 Cuett Mouie N V 3 ) 0 211 212 r Q 3)0 L o w L • f£»en C M i « « H 4 ) 1 3)1 Ftatworki Cott •: i ' 1 S A L - C S O A T A 21) i S 1)1 Leeet L 4 32 Fireplace N V 3)2 Eat . » ' S , C o i l ~." r~T T : •' i ' • 01 Rac 'd . Dale ) ! 1 ; ! : 214 N V Mllly M 4 ] ] Bullt - lni N V 3 ) 3 Built- in Co i t ! • 1 : • 32 Satf i Price 21) NuiuACt Iff! N » S t o p . s 4)< 3 ) 4 • 03 2 1 « •ami a 4)* 333 Mite. Imp. Cott • • * • • 04 2 1 » O t n M 0 4JC 336 Ml i c . Imo. E f f . V r . • 03 an 132 41) 117 M I I C . Imp. L l l e | ' to* r*> n X a 1 <-3 ;» CO CO o w X O 3» rs r-< M (Ji 01 M H ca M o ca rO a LO „ L E T APPENDIX E HE GEOGRAPHIC AREA UNDER STUDY Sales Area District Lots 97, 98 , 99, H9, 150, 156, 157, 158, 159, 175: Al l of Group 1 New Westminster Land District, B U R N A B Y DISTRICT MUNICIPALITY BRITISH COLUMBIA DETAILED RESULTS OF EACH STATISTICAL TEST Test 1 Sales price i s run against a l l independent variables except RCNLDA, 376 observat ions 375 degrees of freedom 95% level of significance Coefficient of determination (R ) : 0.7693 FProb = 0 . 0 Standard Error of the Estimate (Syx) =2814.4095 Syx/mean sales price (Syx/y) 1 13.2% b. Variable Coefficient Standard Error of Coefficient FRatio FProb t S t a t i s t i c CONSTANT MONSAL FRFOOT TRAVOL DWELAR AGE IMP HEATNG BASEAR FBASAR CARPRT PLMPCS 3628.7173 134.7424 39.4611 2439.8560 9.2676 147.6389 1544.8901 1.7663 3.5645 871.0176 1759.1777 1348.2293 27.0210 12.0482 722.1424 0.7053 14.0195 544.6133 0.4902 0.7665 355.8317 546.2736 24.8661 10.7273 11.4152 172.6379 110.9013 8.0467 12.9810 21.6270 5.9919 10.3705 0.0000 0.0013 0.0010 0.0 0.0 0.0049 0.0005 0.0 0.0143 0.0016 4.93 3.26 3.38 13.1 10.5 2.84 3.60 4.64 2.45 3.20 c. Explained variance at each step of stepwise regression procedure, Variable Entering Increase in Step 1 DWELAR 0.4734 2 AGSIMP 0.6127 .1393 3 FBASAR 0.6970 .0843 4 BASEAR 0.7169 .0199 5 MONSAL 0.7346 .0177 6 PLMPCS 0.7434 .0088 7 CARPRT 0.7501 .0067 8 HEATNG 0.7569 .0068 9 TRAVOL 0.7615 .0046 10 FRFOOT 0.7693 .0078 Te3t 2 Separation of the 109 property type from the main sample. a. 185 observations 184. degrees of freedom 95% level of significance R 2 FProb « Syx = Syx/y a 0.7424 0.0 2138.9204 8.9% b. Standard Error of Variable Coefficient Coefficient FRatio FProb t Statis CONSTANT 3465.5161 1956.2397 MONSAL 161.4618 30.2491 28.4914 0.0 5.34 FRFOOT 88.0083 21.4967 16.7612 0.0001 4-09 AERALT -0.2569 0.1087 5.5851 0.0185 2.36 TRAVOL 2280.3938 913.2343 6.2353 0.0131 2.50 DWELAR 11.3421 1.3764 67.9004 0.0 8.24 AGEIMP -200.4191 40.9707 23.9294 0.0 4.88 BASEAR 2.8363 0.5916 22.9817 0.0 4-79 FBASAR 3.2705 0.6609 24.4859 0.0 4.99 GARAGE 1124.6123 427.3723 6.9246 0.0098 2.63 CARPRT 1530.7921 393.6852 15.1194 0.0002 3.90 PATTIO 1403.2437 564.1180 6.1877 0.0134 2.47 c. Explained Variance at each step of stepwise regression procedure. Variable 2 Entering _R~ Increase Step 1 DWELAR 0.4470 2 BASEAR 0.5348 .0878 3 MONSAL 0.5793 .0445 4 FBASAR 0.6196 .0403 5 AGEIMP 0.6644 .0448 6 CARPRT 0.6871 .0227 .7 TRAVOL 0.6973 .0102 8 FRFOOT 0.7139 .0166 9 GARAGE 0.7240 .0101 10 PATTIO 0.7326 .0086 11 AERALT 0.7424 .0098 Test 3 Sales price against a l l variables including RCNLDA. a. 376 observations 375 degrees of freedom 95% level of significance 0.9082 0.0 2489.6361 10.7% R FProb Syx Syx/y b. Standard Error or Variable Coefficient Coefficient FRatio FProb t St a t i s t i c CONSTANT 2683.4390 921.7630 MONSAL 154-6481 21.9704 49.5467 0.0 7.04 RCNLDA 1.1322 0.0333 1155.4117 0.0 33.91 FRFOOT 44.4502 9.5612 21.6131 0.0 4.64 IRRSHP -1391.0068 622.6727 4.9904 0.0247 2.24 TRAVOL 2022.6670 604.6226 11.1919 0.0011 3.34 BASEAR 1.9322 0.3716 27.0346 0.0 5.21 FBASAR 1.3634 0.4745 8.2548 0.0044 2.87 c. Explained variance at each step of the stepwise regression procedure. Variable Entering Step 1 RCNLDA 2 3 4 5 6 7 BASEAR MONSAL FRFOOT TRAVOL FBASAR IRRSHP 0.8766 0.8874 0.8988 0.9023 0.9050 0.9070 0.9082 Increase in R .0108 .0114 .0035 .0027 .0020 .0012 Test 4 Same as Test 3, except month of sale is excluded a. 376 observations 375 degrees of freedom 95% level of significance R - 0.8960 FProb = 0.0 Syx = 2646.4195 Syx/y = 11.0% b. Variable CONSTANT RCNLDA FRFOOT IRRSHP TRAVOL BASEAR PLMPCS Coefficient 4014.2339 1.1050 45.3136 -1409.1536 1849.6301 2.0005 807.3157 Standard Error of Coefficient 948.1767 0.0401 10.1847 663.3355 642.7478 0.3864 375.2062 FRatio 761.1594 19.7954 4.5128 8.2811 26.8007 4.6296 FProb t S t a t i s t i c 0.0 0.0 0.0324 0.0043 0.0 0.0303 27.29 4.44 2.12 2.88 5.17 2.15 c. Explained variance at each step of the stepwise regression procedure. Variable Entering Step 1 RCNLDA 2 3 4 5 6 BASEAR FRFOOT TRAVOL PLMPCS IRRSHP 0.8766 0.8874 0.8913 0.8936 0.8947 0.8960 Increase in R .0108 .0039 .0023 .0011 .0013 a. 185 observations 184 degrees of freedom 95% level of significance R " 0.8649 FProb = 0.0 Syx = 2132.7388 Syx/y = 8.7% b. Variable Coefficient Standard Error of Coefficient FRatio FProb t Sta t i s t i c CONSTANT MONSAL RCNLDA AGEIMP BASEAR CARPRT 2400.3831 186.1436 1.3772 128.0049 1.2821 813.2051 1345.8720 26.9450 0.0699 39.7063 0.5527 331.8654 47.7245 387.7357 10.3929 5.3812 6.0045 8.0 0.0 0.0017 0.0204 0.0146 6.91 19.61 3.22 2.32 2.45 c. Not available for this test. Test 6 Testing 109 property types after editing sample. a. 161 observations 160 degrees of freedom 95% level of significance FProb Syx Syx/y 0.8887 0.0 2087.7897 8.35% b. Variable Coefficient Standard Error of Coefficient FRatio FProb t S t a t i s t i c CONSTANT MONSAL RCNLDA AGE IMP CARPRT 525.2463 187.2319 1.6190 207.1495 727.4358 1724.9618 28.2934 0.0891 45.1180 342.5732 43.7913 329.8302 21.0798 4.5090 0.0 0.0 0.0 0.0334 6.12 20.09 4.65 2.12 c. Explained var iance at each step of the stepwise regression procedure. Variable Entering Step 1 RCNLDA 2 MONSAL 3 AGEIMP 4 CARPRT J3~ 0.8567 0.8694 0.8745 0.8887 Increase in R .0127 .0051 .0142 Teat 7 a Testing for seasonality — months: April, May, June and July. a. 56 observations 55 degrees of freedom 95% level of significance R 2 • 0.7304 FProb s 0.0 Syx s 2122.0392 Syx/y = 8.6% b. Variable Coefficient Standard Error of Coefficient FRatio FProb t S t a t i s t i c CONSTANT MONSAL RCNLDA FRFOOT 5955.1211 113.3480 1.2278 51.5676 1811.5112 49.9199 0.1114 17.9576 5.1556 121.5749 8.2462 0.0259 0.0 0.0059 2.27 11.02 2.87 c. Explained variance at each step of the stepwise regression procedure. Variable Entering R 2 Increase ln R^  Step 1 RCNLDA 0.6533 2 FRFOOT 0.7036 .0503 3 MONSAL 0.7304 .0268 Test 7 b Testing for seasonality — a l l other months. a. 105 observations 104 degrees of freedom 95% level of significance R 2 = 0.7891 FProb • 0.0 Syx * 1932.0645 Syx/y » 8.2% b. Variable Coefficient CONSTANT MONSAL RCNLDA TRKRUT DWELAR AGEIMP BASEAR CARPRT -818.6973 188.2757 1.2051 -1931.6555 4.8871 108.5582 1.3877 1277.2817 Standard Error of Coefficient 1915.2069 36.8524 0.1717 726.4381 1.9922 52.9479 0.6555 404.2491 FRAtio 26.1010 49.2697 7.0707 6.0176 4.2037 4.4819 9.9833 FProb t S t a t i s t i c 0.0 0.0 0.0090 0.0153 0.0407 0.0348 0.0023 5.11 7.01 2.66 2.45 2.05 2.19 3.16 c. Explained variance at each step of the stepwise regression procedure. Variable Entering R 2 Increase in R 2 Step 1 RCNLDA 2 MONSAL 3 AGE IMP 4 CARPORT 5 TRKRUT 6 DWELAR 7 BASEAR J 2 : 0.6526 0.7037 .0511 0.7395 .0358 0.7545 .0150 0.7700 .0155 0.7793 .0093 0.7891 .0098 Te3t 8 Testing the 108 property types. a. 65 observations 64 degrees of freedom 95% level of significance R2 = 0.6689 FProb = 0.0 Syx = 1733.0099 Syx/y = 9.6% b. Variable Coefficient CONSTANT MONSAL RCNLDA AERALT TRKRUT BASEAR 6887.2695 173.8240 0.2824 0.2824 -3474.1182 2.4411 Standard Error of Coefficient 1343.7158 36.8056 0.1605 0.0827 651.9912 0.5859 FRatio 22.3045 29.5256 11.6594 28.3926 17.3608 FProb t S t a t i s t i c 0.0 0.0 0.0012 0.0 0.0001 4.73 5.44 3.42 5.33 4.17 c. Explained variance at each step of the stepwise regression procedure. Variable Entering R^ Increase in R Step 1 BASEAR 0.3546 2 RCNLDA 0.4601 .1055 3 TRKRUT 0.5419 .0818 4 MONSAL 0.6160 .0741 5 AERALT 0.6689 .0529 a. 226 observations 225 degrees of freedom 95% level of significance R = 0.8334 FProb = 0.0 Syx = 1939.2208 Syx/y = 8.8% b. Variable Coefficient Coefficient FRatio FProb t S t a t i s t i c CONSTANT MONSAL RCNLDA FRFOOT TRKRUT AGEIMP BASEAR 894.4810 172.1344 1.3596 49.9213 -2232.0068 96.0231 1.3679 1228.6825 21.9800 0.0768 11.1725 464.6546 33.0729 61.3312 313.1585 19.9651 23.0744 8.4296 0.0 0.0 0.0 0.0 0.0041 0.4014 11.6144 0.0009 7.84 17.59 4.47 4.81 2.91 3.41 Explained variance at each step of the stepwise regression procedure, Variable Entering Step 1 RCNLDA 2 MONSAL 3 AGEIMP 4 BASEAR 5 TRAVOL 6 FRFOOT 7 TRKRUT 8 TRAVOL (leaving) 0.7353 0.7796 0.8030 0.8113 0.8192 0.8308 0.8350 0.8334 Increase in R .0443 .0234 .0083 .0079 .0116 .0042 (.0016) (decrease) 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0101845/manifest

Comment

Related Items