non‐parametric hedonic housing prices

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Non‐parametric hedonic housingpricesCarl Mason a & John M. Quigley ba University of California, Berkeley, CA, 94720–7320, USA E-mail:b University of California, Berkeley, USAPublished online: 12 Apr 2007.

To cite this article: Carl Mason & John M. Quigley (1996) Non‐parametric hedonic housing prices,Housing Studies, 11:3, 373-385

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Housing Studies, Vol. 11, No. 3, 1996 373

Non-parametric Hedonic Housing Prices

CARL MASON & JOHN M. QUIGLEY

University of California, Berkeley, USA

ABSTRACT The hedonic price function has long been a standard tool for modeling theprice of complex commodities, such as housing. The theoretical basis of the model issound and appealing, but applications often encounter difficulties. The results of hedonicmodels depend on inclusion of the right independent variables and the correctspecification of the functional form. The functional form assumption is particularlydifficult in the housing context because the hedonic price function summarizes not onlyconsumer preferences and production technology, but also various quantities which arehistorically determined, difficult to measure, and not approachable by theory. In thispaper, the functional form assumption is relaxed by estimating the hedonic pricefunction as a General Additive Model (GAM). The GAM is considerably more generalthan conventional hedonic models and offers significant advantages in comprehensibilityover other non-parametric procedures. The model is used to analyze the substantialdecline in condominium house values in downtown Los Angeles during the 1980s -- aperiod in which apartments lost about 40 per cent of their values.

Introduction

There is increasing academic and business interest in the measurement of realestate prices and the application of standardized methods to produce priceestimates over time and space.

Academic interest arises from the analysis of housing, land and commercialreal estate as economic commodities. For example, economic analysis based onthese price estimates has addressed the responsiveness of supply and demandfor housing to price variation, the efficiency of the markets for housing and land,and the distributional consequences of the market provision of housing.

Business interest arises from the importance of real estate in the investmentportfolios of pension funds and other pools. The efficient allocation of invest-ments by category depends upon the characteristics of returns, that is, uponprice trends for real property.

The interests of both academics and practicing professionals, especially in theUS, have been stimulated by the recent volatility of housing prices and therecognition that individual home owners are the largest bearers of residentialreal estate risk. Home owners are, for the most part, highly leveraged andundiversified. Thus, they have a great deal to gain from the development ofderivatives markets in real estate.

These markets would allow individual consumers to hedge the risks associ-ated with their most important investments, their equity in single-family, owneroccupied housing. The development of a futures market for housing requires an

0267-3037/96/030373-14 © 1996 Journals Oxford Ltd

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374 C. Mason & } . M. Quigley

accurate and replicable method for measuring the value of housing over timeand by region.

The current analytical basis for describing housing price trends is generallyweak. In the US, for example, the most widely reported measure of housingprices is produced by the National Association of Realtors, but this is merely acompilation of the median sale prices of existing single family homes as reportedby member realtors in some 119 US metropolitan areas. The US Bureau of theCensus also publishes an index of new, single family house prices for each offour large census regions in the country. This index is derived from a regressionequation relating the sale prices of houses to a set of independent variables,including a few size and quality measures. (See Musgrave, 1969, for the originalformulation.) Unfortunately, this index is not based upon a representativesample of dwellings bought and sold, or even a random sample of new housescompleted and sold within a given year. The transactions analyzed by the USBureau of the Census include only new houses sold in the 'speculative builder'category in each year, and thus the index ignores about a third of new housescompleted as well as all sales of used housing. (See Peek & Wilcox, 1991, for adiscussion.)

It has been widely recognized that it is appropriate to control statistically forthe various characteristics of properties in inferring price trends. Indeed, theBureau of the Census index, described above, does so in a crude way. During thelast few years, however, there has been increased attention to the statisticalproblems inherent in the estimation of house price indexes. Much of thisattention is addressed to the functional form for the relationship. A variety ofreasons have been advanced to indicate why some particular functional form isconsistent with economic theory.

This paper demonstrates that this attention is misdirected. Indeed, by using asimple example, it is proved that it is a futile exercise to deduce the form of thehedonic relationship from abstract principles of micro-economic theory.

If the form of the hedonic model is purely an empirical matter, it is thenappropriate to consider non-parametric procedures. These methods build on thehedonic price framework but relax the constraints on functional form imposedby the traditional methodology. As an example, an empirical analysis is pre-sented which compares the non-parametric approach with conventional para-metric estimation of housing price indexes. The data for the analysis include anunusual sample of condominium dwellings in the downtown Los Angeles area.The dwellings in the sample are all drawn from a few high-rise buildings locatedwithin a quarter mile of each other. Thus, their locational and public serviceattributes are quite similar. This provides an excellent opportunity for compar-ing the methodology of non-parametric estimation with least squares estimationin a situation where regression analysis works well and where the models arestraightforward.

Hedonic Indexes

Parametric Estimation

Hedonic indices of complex commodities have been developed extensivelyduring the past two decades. The original empirical research using this method-ology dealt with automobiles (Griliches, 1971) and housing (Kain & Quigley,

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Non-parametric Hedonic Housing Prices 375

1970). These techniques were originally designed to distinguish quality changesover time from price changes for complex commodities with many attributes. Inthe intervening period, these methods have been widely diffused, especially inhousing market research.

Hedonic techniques typically result in the estimation of some regressionrelationship between the sale price or monthly rent of individual properties, Vt,their physical and locational characteristics, x, and some specification of time, t

Vt=fe,t) (1)The appropriate interpretation of this relationship depends crucially upon theinclusion of the correct set of the property characteristics, x, and the correctfunctional form, /(.), for the hedonic regression. Conditional upon choice of theappropriate variables and functional form, the hedonic function can be used todisaggregate the intertemporal variation in prices into that attributable tochanges in the qualitative and quantitative characteristics of properties sold, andthat attributable to intertemporal variation in the unit prices of these character-istics. Importantly, the statistical results can be used to produce indices of themarket price for a 'standardized' or quality-adjusted property over time orspace.

These two maintained hypothesis (the 'correct' set of x and the 'correct'functional form/[.]) are not subject to simple tests. The problem of selecting thecorrect set of independent variables in a regression model is familiar in appliedeconomics. However, it is not easily solved, at least not by simple rules.Selecting the correct functional form is also a long-standing problem, but it hasonly recently become a popular area of research.

It has often been asserted that the appropriate functional form for the hedonicregression of housing and real estate prices can be deduced from economictheory. (See Colwell, 1993, for a recent statement.) This is incorrect. The hedonicrelation can, of course, be derived from the underlying micro-economics ofconsumer choice. However, this does not mean that the form or the curvature ofthe relationship can be determined from abstract principles alone.

To demonstrate this, a simple example is presented which is based onrudimentary micro-economic principles. It indicates how the hedonic function isderived from the behavior of consumers. Some simple, but plausible, restrictionsare then placed on consumer preferences, and it is shown that they implypractically nothing at all about the hedonic house price function.

A simple example. Assume that consumers of income y derive utility fromhousing x (a one-dimensional commodity) and other goods z at a price of one(Pz = 1). Housing is sold to the highest bidder. Assume that the stock of housingis fixed and that households vary only in their incomes. Then, if housing is anormal good, competition will force the household with the lowest income tochoose the smallest or lowest quality dwelling in the market. Because housing isa normal good, the function y = G{x), which relates a household's income to itshousing consumption, will be monotonically increasing over the range of hous-ing in the market. The G(x) function can be thought of as the outcome of ahypothetical auction in which each household submits bids for each availablehouse, and the highest bidder for each house wins. Unlike the standard con-sumer maximization problem in which the household may purchase any quan-tity of x at a given price, the market clearing hedonic price of each house arises

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endogeneously from the competition of consumers. The nature of the pricevariation is determined by the interaction of the given housing supply and thedistribution of income across households. The market clearing 'hedonic' pricefunction is determined, not only by preferences of households but also by supplyconditions and by the distribution of income as well.

The hedonic price function V{x) is derived directly from the conditions formaximization of consumer utility and the mapping, y = G(x).

Let U(x,z) represent the consumers' utility functions. The conditions for utilitymaximization ensure that the ratio of the marginal utility of x to that of z equalsthe ratio of their marginal prices. Thus:

dV(x)_dU(x,z)/dxdx BU(x,z)/dz ( '

_dU(x,y-V[x])/dxdU(x,y-V[x])/dz

_8U(x,G[x]-V[x])/dx~dU(x,G[x]-V[x])/dz

The term on the left-hand side is the marginal price of housing, that is, it is thederivative of the hedonic price function. The term on the right-hand side is themarginal rate of substitution of housing for other goods. Substituting G(x) for yin the last line yields a differential equation in x alone. The hedonic pricefunction, V(x), is the solution to this differential equation (with some initialcondition V(x0) = C). Since G(x) is the equilibrium allocation of housing, the V(x)which maintains that allocation and also satisfies equation (2), the first orderconditions of all consumers, is a stable pareto optimal hedonic price function.

Note that a wide variety of hedonic functions with very different propertiescan be specified by equation (2) even for a specific utility function, dependingupon the distribution of income and housing in the local market (that is, theshape of the y = G{x) function).

The function y = G(x) reflects the distribution of income and the distributionof housing attributes in the local market. Ceteris paribus, if the distribution ofincomes is more equal, then G(x) will be more steeply sloped. Small increases inincome will be associated with larger increases in housing consumption. Simi-larly, if there is more variation in the quality of housing, then G(x) will be flatter.Thus, the peculiarities of local housing markets will affect the functional form ofthe hedonic function even if consumers in all markets have the same utilityfunctions and even if the income distribution in all markets is the same.

For a specific example, let G(x) = xd, and let the utility function be Cobb-Douglas, U = x"z^. The differential equation describing the hedonic function is:

^ (3)

With V(l) = 1 as the initial condition, the solution of (3) is:

V(x) = [axs + 5px~x/ll]/[x + 5p] (4)

Even in this very simple case, the shape of the hedonic price function is clearlysensitive to changes in G(x). Figure 1 graphs the hedonic price function for twovalues of <5. As the Figure indicates, the price function is increasing for both

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delta=0.25

Quantity of housing

delta=1.5

15 25 35 " 45 55 65

Quantity of housing

75 85 95

Figure 1. Hedonic price function for different values of delta.

values of 5—dwellings containing more or better housing are more expensive.However, one function is concave and the other is convex. Note that theconsumers' utility functions which underly the two curves are identical. (In bothcases the utility function is Cobb-Douglas with a = /? = 0.5). The only differenceis in the G(x) function. The upper curve is drawn with G(x) = xl/i. The lowercurve is based on G(x) = y?n. Clearly theory alone tells us practically nothingabout the shape of the hedonic price function.

The inability to specify the functional form of the hedonic index has led tosuggestions that the characteristics of properties be standardized with referenceonly to themselves (e.g. Case & Shiller, 1987), and that the effect of intertemporalvariation be analyzed by relying on some very general specification. Standardiz-ing properties with reference to themselves implies an analysis based uponrepeat sales of properties whose characteristics have remained unchanged be-tween sales. However, the benefits of avoiding specifying and measuring hous-ing characteristics using this approach come at considerable cost (Quigley, 1995).

Non-parametric Estimation

Consider, as an alternative, the estimation of (1) by a non-parametric method, forexample the Generalized Additive Model (GAM). The GAM represents acompromise between generality and comprehensibility. At the extreme of com-

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378 C. Mason & f. M. Quigley

prehensibility are conventional (parametric) hedonic regression techniques. Atthe extreme of generality are techniques such as local or kernel regression inwhich smoothed surfaces are generated from neighborhoods of points. Whilethese latter procedures impose very few assumptions, they become difficult tointerpret if the set of independent variables is even of moderate size.

One paper has applied general smoothing procedures to estimate price indicesfor residential housing. Meese & Wallace (1991) used the technique of localregression to estimate housing price indexes for 14 California communitiesduring the period 1970-88. By using non-parametric methods, they were able toavoid assuming the same functional form for each submarket. However, thehigh dimensionality of the data made it difficult to produce understandableresults. Meese & Wallace dealt with this interpretation problem by constructingFisher price indexes. While this is perfectly workable, it adds an additional levelof complexity to the estimation procedure and creates the need for complex bootstrap procedures for the estimation of standard errors.

The alternative method explored in this empirical analysis is a generaladditive model. In place of the completely general techniques of Meese &Wallace, the assumption of additivity of effects is imposed. The cost of thisassumption is the suppression of complex interactions, but the benefits in termsof tractability of the results are significant. (See Hastie & Tibshirani, 1990.)

Consider the relationship:

Vit = y2Sj(xij) + S0(t) + ei, (5)

where Vu is the rent or sale price of dwelling i at time t, xq is housingcharacteristic j measured for observation i and S;(.) is an arbitrary smootherapplied to the ;th variable. Assume that E(e,) = 0 and E(e,2) = a2. Because of theadditive structure, the smooth function S/(jc,y) captures the entire effect attribu-table to Xij. Because Sj(.) is of low dimension, each of the S;(.) can be easilydisplayed and interpreted visually. Because S;(.) is a function of only one (or afew) variables, it is analogous to a coefficient estimated by ordinary least squaresin a linear additive hedonic model.

In a practical situation, the GAM can be represented as:

So = SO (V-*-Si-S2- . . . -Sy) (6)Sl=Si (V-So-*-S2-...-Sj)

s2 = s2 ( V - S o - S i - * - . . . -sj)

S/ = S/ (V-S0-S1-S2-...-*)

where the asterisks are place holders showing the term that is missing in eachrow.

Here lower case s represents a vector of values which are the smooth functionS evaluated at each observation. In each line the argument of the smoothfunction is V minus all of the other smoothers similarly evaluated. Equation (6)is a simultaneous system of / +1 equations. Iterative methods such as theGauss-Seidel algorithm (see Chambers & Hastie, 1992) can solve such a systemefficiently. The algorithm simply recalculates each smoother at each cycle usingthe current values for all of the other smoothers.

In the special case where each of the Sj is the least squares projection, theequation system in (6) reduces to the usual normal equations for ordinary least

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squares. One recent paper applies a special case of (6) to the analysis of realestate markets (Coulson, 1992). In this application one variable (floor area) isrepresented by a general smoother and the others are least squares coefficients.

For a broad class of smoother functions S, estimates can be shown to beconsistent, and exact and approximate standard errors of the estimates can beconstructed (see Hastie & Tibshirani, 1990).

Empirical Analysis

The empirical analysis is based upon a sample of 843 condominium salesrecorded during the 12-year period from January 1980 to December 1991 indowntown Los Angeles. The sample includes essentially every condominiumsale within the downtown area of Los Angeles during this time period. Condo-miniums were located in four different high-rise properties which realtors andreal estate agents consider 'comparables' for the purpose of appraisal. There areno other 'comparables' within several miles of downtown. We gathered infor-mation on the original sellout prices of each of the condominiums in one of thehigh-rise properties completed in 1989 and all subsequent sales of these dwellingunits. We also obtained information on all condominium sales in each of theother three properties beginning in 1980. Property characteristics were obtainedby matching addresses to condominium floor plans. Resale information wasobtained from multiple listing services, from court records and from real estatelenders.

Because the sample consists of properties in only four high-rise buildingslocated within about a quarter mile of each other, the neighborhood and publicservice amenities associated with these properties are identical. The condomini-ums vary in their size and their location within each of the buildings. Werecorded the date of each sale and the real selling price of the property. In noneof the condominiums were the physical characteristics of the sale propertieschanged during the sample period.

The statistical analysis reported below relates the selling prices of theseapartments to the sizes and locations of the properties, the projects in which theyare located and the timing of sales.

Due to the unusual nature of the dataset, the problems associated withchoosing the correct set of independent variables, x, are not particularly serious.Among all of the properties in the sample, there are a limited number of floorplans and locations. Consequently, characteristics such as floor area are notlikely to be seriously distorted by unmeasured features. Further, since theproperties are condominiums, significant changes in physical characteristics arepractically impossible. This makes it possible to concentrate on the effects ofrelaxing the assumptions about functional form, rather than on the difficultiesassociated with selecting the correct set of covariates. The results of estimationusing standard hedonic regression and estimation using the (non-parametric)generalized additive model are reported.

Parametric Hedonic Price Indexes: The Standard Model

Equation (1) is estimated, in the semi-log specification, in three variants usingdifferent representations of the timing of sales. The simplest representationmerely includes the date of the sale, t, as an independent variable:

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log V, = (7)

There are five control variables, x's size (in square feet) and location (storey) anddummy variables for each of three buildings. In this formulation the time of thesale, t, is measured by the number of days since 1 January 1980.

In the second formulation, the same set of control variables are used, but thesimple time trend is replaced with a set of dummy variables measuring year ofsale, beginning 1 January 1980.

In the third formulation, time is represented by a set of dummy variables forquarter year of sale, again beginning with 1980.

Regression estimates of the three variants of the hedonic index are summa-rized in Table 1. Condominium prices vary substantially with the size of thedwellings and their locations—larger dwellings on higher floors command apremium. They also vary significantly with the specific building—projects B andC are more desirable than the other projects. The coefficients are stable across thethree specifications.

The real selling price of otherwise identical buildings clearly varies over time.In the simplest specification, a linear time trend has a t ratio of about 20. Thismodel explains 73 per cent of the variation in selling prices. The most complexspecification, where time is represented quarterly, explains an additional 3 percent of the variation in housing prices.

Table 1. Parametric standard hedonic price indicesfor downtown Los Angeles. Dependent variable:

logarithm of selling price

Size(1000 sq ft)Location(storey)Project A(dummy)Project B(dummy)Project C(dummy)Intercept

Time(1000 days)R2

Model 1linear time

trend

0.867(38.39)

0.096(7.94)

- 0.061(2.01)0.205

(7.17)0.138

(6.91)14.042

(207.18)-0.118(20.32)

0.727

Model 2time inyears

0.862(38.40)

0.095(7.83)

- 0.073(2.23)0.201

(6.80)0.113

(4.63)13.732

(105.33)

0.738

Model 3time in

quarter years

0.876(38.76)

0.096(7.94)

- 0.078(2.30)0.199

(6.53)0.099

(3.96)14.274

(126.27)**

0.759

Notes: Each regression is based on 843 observations on condo-minium sales (t ratios are in parentheses).'Regression also includes 11 variables measuring time in years.

•"Regression also includes 47 variables measuring time in quar-ters.

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Hedonic Model 1

pd

in9

0.05 0.10 0.15 0.20

Sq ft/10,000

0.25 0.30

GA Model 1

Is §

0.05 0.10 0.15 0.20

Sq ft/10,000

0.25

Figure 2. Comparison of the effect of size on downtown Los Angeles condo-minium prices.

Non-parametric Estimation

The non-parametric (or semi-parametric) model which corresponds to equation(7) is:

5

log V, = Si (x) + S2 (x2) + 2 hiXi + S0(t) (8);=3

where Si(.), S2O and So(.) are arbitrary smoothers applied to the variablesmeasuring size, location, and time respectively. The functions are estimatediteratively applying equation (6) using standard computer software (as describedin Chambers & Hastie, 1992). In the second formulation, we replace the timetrend S0(t) with a set of dummy variables measuring year of sale, and in thethird formulation we represent time by dummy variables measuring salesduring each quarter of the period.

For each of the variants, the estimated smooth functions are highly significant,with probability levels of 0.003, 0.060, and 0.007 respectively. Figures 2, 3, and4 illustrate the differences between the hedonic model and the GAM estimator.

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Hedonic Model 1

0.20.1

0.0

-0.1

^ ̂ 7 r f •••,,.,-„•,., Mill INHIH I I I 1

10 15

Poor

20 25 30

GA Mode! 1

Sod

10 15 20 25 30

Figure 3. Comparison of the effect of location on downtown Los Angelescondominium prices.

Each comparison is based upon Model 1. The solid lines represent the estimatedrelationship, while the dotted lines represent the 95 per cent confidence interval.Figure 2 reports the per cent change in price (i.e., the logarithm of sale price) asa function of the size of the dwellings. As compared to the linear relationshipimposed by the hedonic function, the more general GAM relationship reveals aslightly less than proportional relation between the size of dwellings and the percent change in housing prices. Figure 3 presents the same comparison for thevariable indicating the floor (storey) in which the property is located. In thiscomparison, the relationship estimated by the GAM is not quite linear, but it isvery close. Figure 4 presents a comparison of the time trend estimated byimposing a linear relationship and that estimated using the more general GAMtechnique. As the GAM result indicates, the effect of time is not quite linearduring the 1980-91 time period.

In each case, the GAM representation is significantly different, in a statisticalsense, from the simple linear relationship, but none of the differences reportedis very large.

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Hedonic Model 1

1000 2000 3000

Days since January 1,1980

4000

GA Model 1

%pd

1000 2000 3000

Days since January 1,1980

4000

Figure 4. Comparison of time trends in downtown Los Angeles condominiumprices.

Conclusion

As the accurate measurement of housing price movements has become moreimportant, it is natural that increased attention be paid to the theory underlyinghedonic prices and the methods used to estimate them. In this paper it isdemonstrated that the theory of consumer behavior does not provide guidanceabout the choice of the form of the relationship.

Since the form of the relationship is an empirical matter, we suggest thatestimation of the hedonic function by non-parametric methods offers potentialadvantages. The GAM method is quite tractable. It is straightforward to computeand is easily interpreted. Importantly, the GAM estimator imposes no functionalform a priori upon the hedonic relation.

A comparison of models estimated using data on condominium sales indowntown Los Angeles during the decade of the 1980s is suggestive of theadvantages of GAM estimation. The GAM estimates relax the assumptions oflinearity imposed by regression estimation. Graphical comparisons of the GAM

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384 C. Mason & ]. M. Quigley

1980:1 = 100

Model 3 GAM

1980 1985 1990

Year

Figure 5. Quarterly index of condominium prices in downtown Los Angeles.

estimator with the linear regression estimator confirm a non-linear relationshipbetween housing attributes and price. In this application, however, the depar-tures from linearity are rather small, though statistically significant.

Substantively, the analysis of housing prices in downtown Los Angelesdocuments the collapse of the Southern California housing market during thedecade of the 1980s. In particular, Figure 4 shows the decline in housing prices.

As argued in this paper, the most credible estimates of the trend in housingprices during this period are provided by the Generalized Additive Model(GAM). Figure 5 presents the quarterly time trend estimated from Model 3 usingthis method of estimation. As the figure shows, the decline in houseprices during the 1980-91 period was spectacular. The results of Model 3 suggestthat downtown condominiums in Los Angeles lost about 40 per cent of theirvalue during the decade.

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Acknowledgements

Parts of this paper were discussed at the International Conference on Land andUrban Policy, Kyoto, August 1993 and the Seminar on Housing and HousingMarkets, Centre for Housing Research, University of Glasgow, March 1994. Thepaper benefited from the comments of Takahiro Miyao and Christine Whiteheadand from seminar participants.

Correspondence

John M. Quigley, University of California, Berkeley, CA 94720-7320, USA. Email:[email protected]

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New England Economic Review, September/October, pp. 45-56.Chambers, J.M. & Hastie, T.J. (1992) Statistical Models in S (Pacific Grove, CA, Wadsworth and

Brooks).Colwell, P.F. (1993) Comment: semiparametric estimate of the marginal price of floorspace, Journal

of Real Estate Finance and Economics, 7(1), pp. 73-76.Coulson, N.E. (1992) Semiparametric estimates of the marginal price of floorspace, Journal of Real

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non‐parametric hedonic housing prices

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