Quantile Regression Estimates of Hong Kong Real Estate Prices Stephen Mak, Lennon Choy and Winky Ho [Paper first received, October 2008; in final form, July 2009] Abstract Linear regression is a statistical tool used to model the relation between a set of housing characteristics and real estate prices. It estimates the mean value of the response vari- able, given levels of the predictor variables. The quantile regression approach comple- ments the least squares by identifying how differently real estate prices respond to a change in one unit of housing characteristic at different quantiles, rather than estimat- ing the constant regression coefficient representing the change in the response variable produced by a one-unit change in the predictor variable associated with that coefficient. It estimates the implicit price for each characteristic across the distribution of prices and allows buyers of higher-priced properties to behave differently from buyers of lower- priced properties, even if they are within one single housing estate. Thus, it provides a better explanation of the real-world phenomenon and offers a more comprehensive picture of the relationship between housing characteristics and prices. 1. Introduction Residential property is a multidimensional commodity which can be considered as a bundle of utility-bearing attributes that consumers value. These attributes are char- acterised by their physical inflexibility, durability and spatial fixity such that dif- ferent combinations of them can produce a heterogeneous good. In the real estate literature, housing price is defined as a func- tion of a bundle of inherent attributes (i.e. flat size, age, floor and balcony), neighbour- hood characteristics (i.e. view), accessibility (i.e. transport, the presence of recreation facilities, community services and school) and environmental quality (waterfront or natural beauty) that yield utility or satisfac- tion to homebuyers. Specifically, a hedonic price model involves first the specification of a housing price function which relates the observed housing expenditure to the selected physical, neighbourhood and acces- sibility characteristics that are considered to influence prices (Bailey et al., 1963; Ridker and Henning, 1967; Kain and Quigley, 1970; Wilkinson, 1973; Freeman, 1979; Pollakowski, 1982; Epple, 1987; Can, 1992; Cheshire and Sheppard, 1995; Can and Megbolugbe, 1997). Based on the estimated coefficients of housing attributes, the second stage is to construct price indexes. Winky Ho (corresponding author), Stephen Mak and Lennon Choy are in the Department of Building and Real Estate, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China. E-mail: [email protected]; [email protected]; and [email protected]. 1–12, 2010 0042-0980 Print/1360-063X Online Ó 2010 Urban Studies Journal Limited DOI: 10.1177/0042098009359032 Urban Stud OnlineFirst, published on May 18, 2010 as doi:10.1177/0042098009359032
Transcript
Quantile Regression Estimates ofHong Kong Real Estate Prices
Stephen Mak, Lennon Choy and Winky Ho
[Paper first received, October 2008; in final form, July 2009]
Abstract
Linear regression is a statistical tool used to model the relation between a set of housingcharacteristics and real estate prices. It estimates the mean value of the response vari-able, given levels of the predictor variables. The quantile regression approach comple-ments the least squares by identifying how differently real estate prices respond toa change in one unit of housing characteristic at different quantiles, rather than estimat-ing the constant regression coefficient representing the change in the response variableproduced by a one-unit change in the predictor variable associated with that coefficient.It estimates the implicit price for each characteristic across the distribution of prices andallows buyers of higher-priced properties to behave differently from buyers of lower-priced properties, even if they are within one single housing estate. Thus, it providesa better explanation of the real-world phenomenon and offers a more comprehensivepicture of the relationship between housing characteristics and prices.
1. Introduction
Residential property is a multidimensionalcommodity which can be considered asa bundle of utility-bearing attributes thatconsumers value. These attributes are char-acterised by their physical inflexibility,durability and spatial fixity such that dif-ferent combinations of them can producea heterogeneous good. In the real estateliterature, housing price is defined as a func-tion of a bundle of inherent attributes (i.e.flat size, age, floor and balcony), neighbour-hood characteristics (i.e. view), accessibility(i.e. transport, the presence of recreationfacilities, community services and school)
and environmental quality (waterfront ornatural beauty) that yield utility or satisfac-tion to homebuyers. Specifically, a hedonicprice model involves first the specificationof a housing price function which relatesthe observed housing expenditure to theselected physical, neighbourhood and acces-sibility characteristics that are considered toinfluence prices (Bailey et al., 1963; Ridkerand Henning, 1967; Kain and Quigley,1970; Wilkinson, 1973; Freeman, 1979;Pollakowski, 1982; Epple, 1987; Can, 1992;Cheshire and Sheppard, 1995; Can andMegbolugbe, 1997). Based on the estimatedcoefficients of housing attributes, the secondstage is to construct price indexes.
Winky Ho (corresponding author), Stephen Mak and Lennon Choy are in the Department of Buildingand Real Estate, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China. E-mail:[email protected]; [email protected]; and [email protected].
Urban Stud OnlineFirst, published on May 18, 2010 as doi:10.1177/0042098009359032
Linear regression is a statistical tool usedto model the relation between a set of pre-dictor variables and a response variable. Itestimates the mean value of the responsevariable for given levels of the predictor var-iables. Suppose we are interested in inves-tigating the relationship between housingprices and a set of predictors, such as apart-ment size, age, floor level, view, directionand car park. The dataset used for this exam-ple contains a total of 5947 cross-sectionalintertemporal transaction data from CityOne Sha Tin, a private housing estate locatedin Sha Tin, the New Territories, for theJanuary 1997–October 2004 period. The lin-ear regression model for this example is asfollows1
This model estimates how, on average, theseproperties’ characteristics impact on realestate prices. The car park predictor variable,CP; compares the effect of having a car parkon property prices with not having a car park.While this model can address the question ofwhether or not a car park matters inthe price determination, it cannot answeranother important question: ‘‘Does a carpark influence property prices differentlyfor low-priced properties than for median-priced properties?’’. One can obtain a morecomprehensive picture of the effect of thepredictors on the response variable by usingquantile regression, which models the rela-tion between a set of predictor variablesand the specific percentiles (or quantiles) of
the response variable. It specifies changes inthe quantiles of the response. For example,a median regression (the 50th percentile) ofproperty prices on properties’ characteristicsspecifies the changes in the median propertyprices as a function of the predictors. Theeffect of car park on median property pricescan be compared with its effect on otherquantiles of property prices.
In linear regression, the regression coeffi-cient represents the change in the responsevariable produced by a one-unit change inthe predictor variable associated with thatcoefficient. The quantile regression parame-ter estimates the change in a specified quan-tile of the response variable produced bya one-unit change in the predictor variable.
This allows for a comparison of how specificpercentiles of property prices may be moreaffected by certain properties’ characteris-tics than other percentiles. This is reflectedin the change in the size of the regressioncoefficient.
The objective of this paper is to estimateempirically how specific quantiles of prop-erty prices respond differently to a one-unitchange in the properties’ characteristics. Asan alternative to OLS regression, this studyadopts quantile regression to identify theimplicit prices of housing characteristics forthe different percentiles of the distributionof housing prices. This explicitly allowshigher-priced apartments to have differentimplicit prices for a property’s characteristic
P ¼� 819906:8þ 6980:7GFA� 2:8GFA2 � 62374:2AGE þ 2328:3AGE2
than lower-priced apartments. Heckman(1979) suggests that the issues associatedwith truncation could possibly be avoidedsince quantile regression makes use of theentire sample rather than the mean value ofthe response variable. This will eliminatethe problem of biased estimates that is cre-ated when OLS is applied to housing pricesub-samples (Newsome and Zietz, 1992).
This paper is organised as follows. Section2 briefly presents a literature review of thequantile regression. Section 3 discusses themodel specification adopted in this paper,while the data quality and sources will bepresented in section 4. Section 5 presentsand discusses the empirical results, utilisinghousing transaction data from one very largehousing estate, the City One Sha Tin, locatedin Shatin, the New Territories for the periodbetween January 1997 and October 2004. Thelast section summarises the major findings.
2. Literature Review
Quantile regression is based on the mini-misation of weighted absolute deviationsfor estimating conditional quantile (percen-tile) functions (Koenker and Bassett, 1978;Koenker and Hallock, 2001). For the median(quantile 5 0.5), symmetrical weights areused, while asymmetrical weights are emp-loyed for all other quantiles (0.1, 0.2, ..., 0.9).While the classical OLS regression estimatesconditional mean functions, quantile reg-ression can be employed to explain thedeterminants of the dependent variable atany point of the distribution of the depen-dent variable. For hedonic price functions,quantile regression makes it possible toexamine statistically the extent to whichhousing characteristics are valued differentlyacross the distribution of housing prices.Although one may argue that the same goalmay be accomplished by utilising the priceseries sub-samples according to the uncondi-tional distribution and then applying OLS to
the sub-samples, Heckman (1979) arguesthat the truncation of the dependent variablemay create biased parameter estimates andshould be avoided if possible. Since quantileregression employs the full dataset, a sampleselection problem does not arise in the firstplace.
Koenker and Hallock (2001) suggest thatthere is a rapidly expanding empiricalquantile regression literature in economicsthat, when taken as a whole, makes a persua-sive case for the value of going beyondmodels for the conditional mean in empir-ical economics. This methodology has beenintensively applied to the issues in laboureconomics, such as union wage effects,returns to education and labour marketdiscrimination. Chamberlain (1994) findsthat, for manufacturing workers, the unionwage premium at the first decile is 28 percent and declines monotonically to 0.3 percent at the upper decile. The least squaresestimate of the mean union premium of15.8 per cent is thus captured mainly bythe lower tail of the conditional distribu-tion. Other studies exploring these issuesin the labour market include the influentialwork of Buchinsky (1994, 1997), Schultzand Mwabu (1998) and Kahn (1998). Thework of Machado and Mata (2005) is partic-ularly notable, since it introduces a usefulway to extend the counterfactual wagedecomposition approach of Oaxaca (1973)to quantile regression and provides a generalstrategy for simulating marginal distribu-tions from the quantile regression process.Arias et al. (2001), employing data on iden-tical twins, interpret observed heterogeneityin the estimated returns to education overquantiles as an indicator of an interactionbetween observed educational attainmentand unobserved ability.
In demand analysis, Deaton (1997) offersan introduction to quantile regression.Employing food expenditure data fromPakistan, his study finds that, although the
STEPHEN MAK ET AL. 3
median Engel elasticity of 0.906 is similar tothe ordinary least squares estimate of 0.909,the coefficient at the tenth quantile is 0.879and the estimate at the 90th percentile is0.946, indicating a pattern of heterosce-dasticity. In another demand application,Manning et al. (1995) investigate the demandfor alcohol using survey data from theNational Health Interview Study and suggestthe presence of considerable heterogeneity inthe price and income elasticities over the fullrange of the conditional distribution.
Utilising the American Housing Surveydata, Gyourko and Tracy (1999) adopt thequantile regression approach to investigatechanges in housing affordability between1974 and 1997. Without controlling forchanges over time in housing characteristics,real house prices in 1997 had risen by 35 percent at the 0.9 quantile and had fallen by 28per cent at the 0.1 quantile, while medianprices did not change. Controlling for changesin housing characteristics over time, thereal house price with 1974 characteristicsincreased by only 1 per cent at the 0.9 quantile,while real prices increased by 33 per cent at the0.1 quantile. The quantile estimates indicatethat real house prices with 1974 characteristicsat the 0.9 quantile increased by about 31 percent over the time-period, which is muchcloser to the price increase for those situationswhen changes over time in housing character-istics are not controlled. Real housing priceswith 1974 characteristics increased by about20 per cent at the 0.1 quantile, less than thatindicated from the mean-based estimates,but much more than for those situationswhen changes over time in housing character-istics are not controlled. These results suggestthat quantile effects are important, while aver-age quality has worsened at the bottom of thehouse price distribution.
Employing housing transaction data fromChicago in 1993 through 2005, McMillen andThornes (2006) suggest that quantile regres-sion has advantages over the conventional
mean-based approaches to estimate thehousing price index. A median-based quan-tile estimator which reduces the outlier effectsuffers less bias from unobserved renovationsthan a standard mean-based estimator. Theproblem of outliers is particularly importantfor the repeat-sales estimator, which is vul-nerable to an upward bias when the sampleincludes renovated houses and there is noway to identify which homes have beenupgraded. In this situation, a more realisticview of the housing market may be gainedby constructing indexes using lower quantilesas the target point. Zietz et al. (2008) utilisequantile regression, with and without acco-unting for spatial autocorrelation, to identifythe coefficients of a large set of diverse varia-bles across different quantiles. Their resultssuggest that, while buyers of higher-pricedhomes value square footage and the numberof bathrooms differently from buyers oflower-priced homes, other variables, such asage, also vary across the distribution of hous-ing prices.
To the best of our knowledge, our paperis the first of its kind to use the quantileregression technique, based on Hong Konghousing transaction data, to investigate theimplicit prices of housing characteristics indifferent quantiles of prices. Hong Kong, formany reasons, presents an interesting case.It is a densely populated territory, with themajority of its citizens residing in housingestates instead of stand-alone residentialbuildings or houses. Frequent transactionsof residential properties within even onesingle housing estate (typically with 20–30blocks of buildings) over time provide re-searchers with adequate observations (froma sample of similar location-specific charac-teristics) to employ a quantile regressiontechnique to identify how differently realestate prices respond to a change in oneunit of housing characteristic at differentquantiles, without the need to account forspatial autocorrelation.
4 HONG KONG REAL ESTATE PRICES
3. Model Specification
For the purpose of this study, the hedonicpricing model of residential real estate takesthe following forms:
Pi ¼ fðHi;Ni;a;bÞ ð2Þ
where, Pi is the residential sales price of prop-erty i; Hi is a vector of physical housingattributes associated with an apartment; Ni
is a vector of neighbourhood/locational var-iables; and a and b are the estimated param-eters associated with the exogenous variables.
A variety of econometric issues arisesfrom estimating hedonic models, includingthe model specification, function form, theproblems associated with heteroscedasti-city and spatial correlations. Ideally, modelspecification and function form should bedetermined by a theoretical framework.Unfortunately, there is little theoretical guid-ance regarding model specification and res-trictions imposed on function form, withthe exception of the guidance in respect ofthe expected signs of certain coefficients asso-ciated with the variables. On the one hand,model specification is largely determinedby data availability and a priori beliefs aboutthe type of location and structural amenitiesthat are relevant to each household. On theother hand, the choice of functional form islargely evaluated by empirical evidence. A typ-ical approach is to compare the goodness offit, Akaike information criterion (AIC) orBayesian information criterion (BIC) fromalternative functional forms and then pickup the best-fitting model.
In the ordinary least square (OLS) estima-tion, one of the classical assumptions is thatthe endogenous and residual variables arehomoscedastic, which requires the variancesof error terms to be constant across observa-tions. However, heteroscedasticity is oftenfound to exist in cross-sectional or paneldata due to the properties of the data. For
example, larger or older dwelling units tendto have a larger error term than those ofsmaller or relatively new units. If this classicalassumption is not true, inaccurate standarderrors and inefficient estimators are expectedfrom the results. To test for the assumptionof homoscedasticity, White’s (1980) testcan be performed, which involves an auxil-iary regression of the squared residuals onthe original regressors and their squares totest for the null hypothesis of no heterosce-dasticity against heteroscedasticity of someunknown general form. The test statistic iscomputed by an auxiliary regression, wherethe squared residuals are regressed on all pos-sible (non-redundant) cross-products of theregressors.
Following Koenker and Hallock’s (2001)methodology, an alternative methodology isthe use of a quantile regression which gener-alises the concept of an unconditional quan-tile to a quantile that is conditioned on one ormore covariates. The quantile can be definedthrough a simple alternative expedient as anoptimisation problem. For example, thesample mean could be defined as the solutionto the problem of minimising a sum of squareresiduals and the median could be defined asthe solution to the problem of minimisinga sum of absolute residuals. The symmetryof the piecewise linear absolute value func-tion implies that the minimisation of thesum of absolute residuals must equate withthe number of positive and negative resid-uals. Hence, it ensures that there are the samenumbers of positive and negative observa-tions above and below the median. As thesymmetry of the absolute value yields themedian, minimising a sum of asymmetricallyweighted absolute residuals (i.e. simply givingdiffering weights to positive and negativeresiduals) would yield the quantiles. Solvingequation (3)
minj2<
Xrt yi � jð Þ ð3Þ
STEPHEN MAK ET AL. 5
where, the function rt �ð Þ is the tilted abso-lute value function that yields the tth samplequantile as its solution. Least squares regres-sion offers a model for how to define condi-tional quantiles in an analogous fashion. Ifthere is a random sample y1; y2; :::; ynf g,we can solve it
minm2<
Xn
i¼1
yi � mð Þ2 ð4Þ
Then the sample mean and an estimate of theunconditional population mean, EY ; can beobtained. If we replace the scalar m by a para-metric function m x;bð Þ and solve
minb2<r
Xn
i¼1
yi � m xi;bð Þð Þ2
we can then obtain an estimate of the condi-tional expectation function E Y jxð Þ:
For quantile regression, we can simply gofurther to obtain an estimate of the condi-tional median function by replacing the sca-lar j in equation (3) by the parametricfunction j xi;bð Þ and setting t to 1=2: Toobtain estimates of the other conditionalquantile functions, we can replace the abso-lute values by rt �ð Þ and solve
minb2<r
Xrt yi � j xi;bð Þð Þ
When j x;bð Þ is formulated as a linear func-tion of parameters, the resulting minimisa-tion problem can then be solved veryefficiently by linear programming methods.
The standard errors and confidence limitsfor the coefficient estimates can be obtainedwith asymptotic and bootstrapping meth-ods.2 Both methods provide robust results(Koenker and Hallock, 2001), with the boot-strap method considered more practical(Buchinsky, 1982; Efron, 1982; Hao andNaiman, 2007). Gould (1992, 1997) also sug-gests that the standard errors of coefficient
estimates using the bootstrap method are sig-nificantly less sensitive to heteroscedasticitythan the standard error estimates based onthe method suggested by Rogers (1993).
4. Data Sources
To minimise the spatial effects upon resi-dential property prices, a feasible approachis to select a sample with similar location-specific characteristics, relatively homoge-neous household tastes and least variationsin building design and quality such that thenet effects of inherent attributes and loca-tion-specific factors tend to be similar. Forthe purpose of this study, we choose CityOne, Sha Tin, for our case study because itcomprises 10 642 small to medium-sizedunits in 52 residential blocks of differentsizes and layouts, but with a relatively homo-geneous design (see Figure 1). It is a standardmass housing estate located in the NewTerritories with a high trading volume atall times. Since the current study casts a focuson only one housing estate, the accessibilitycharacteristics (such as accessibility to trans-port, amenities and schools, etc.) and theexternal environment are more or less iden-tical for all dwelling units of the estate.
Data on housing prices, physical and loca-tion-specific characteristics are generatedfrom the government official property trans-action records compiled by a major realestate valuation firm. Observations withmissing data for any of the variables are drop-ped from the analysis. This process yieldsa sample of 5947 housing transactions. Realestate prices, P; represent the transactionprice (total consideration) of a residentialproperty, which is recorded in HK dollars,inflation adjusted. GFA represents the totalgross floor area of a residential property,which is measured in square feet. AGE repre-sents the age of a residential property in years,which can be measured by the differencebetween the date of issue of the occupation
6 HONG KONG REAL ESTATE PRICES
permit and the date of transaction. FL repre-sents the floor level of a property in a residen-tial building block. Apartment size, age andfloor level are included as quadratic effectsfor the test of their non-linear effect on pri-ces. (See Table 1 for descriptive statistics.)
View is divided into three categories:building view, obstructive view and openview. It represents the type of view a propertyis facing and equals 1 if a property is facinga particular view, 0 otherwise. The omittedcategory is open view so that coefficientsmay be interpreted relative to this category.The direction a property is facing is dividedinto four categories: NE; SE; SW and NW :It represents the direction a property is facingand equals 1 if a property is facing a particulardirection, 0 otherwise. The omitted categoryis NW so that coefficients may be interpreted
relative to this category. The car park, CP;represents whether a residential propertytransaction is associated with the sale of carparking. It is a dummy variable which equals1 if the transaction has such a tie-in sale,0 otherwise.
The financial variable is measured in realterms by using the Monthly Price Indicesfor Selected Popular Private DomesticDevelopments to deflate the series. This pricedeflator series is published by the Rating andValuation Department, with the base year of1999/2000 5 100. The indices are based onan analysis of price paid for apartments inselected housing developments, as recordedin their sale and purchase agreement. Apartfrom the overall price indices for all residen-tial properties within the selected housingestates, the indices are further broken down
Figure 1. City One, Sha Tin. Source: www.maps.google.com.
STEPHEN MAK ET AL. 7
into price indices for the small to median-sized properties and luxury properties, andcan be further sub-divided by their price seriesin the urban areas and the New Territoriesrespectively. Data are obtained from theRating and Valuation Department.
5. Empirical Results
Most analysis using the hedonic pricing modelhas employed conventional least squares re-gression methods. However, it has been recog-nised that the resulting estimates of variouseffects on the conditional mean of real estateprices are not necessarily indicative of thesize and nature of these effects on the lowertail of the price distribution. A more completepicture of covariate effects can be provided byestimating a family of conditional quantilefunctions. At any chosen quantile, one canask how different are the corresponding realestate prices, given a specification of the otherconditioning variables. Table 2 presentsa summary of the empirical results obtainedby the traditional hedonic pricing model andthe quantile regression. The estimated coeffi-cient estimates for the linear regression andthe 5th, 10th, 25th, 50th, 75th, 90th and 95thquantile regression coefficient estimates forproperty prices (along with their t-statistics),goodness of fit measures and diagnostic statis-tics are shown. To correct for the observedheteroscedasticity and correlations amongobservations in cross-sectional data, this study
employs HAC covariance to estimate theimplicit prices of the housing attributes inthe OLS specification. Most variables are sta-tistically significant at conventional levelsand have the expected signs.
The apartment size, age and floor levelenter the model as quadratic effects. Acco-rding to the linear regression model, whileGFA tends to increase real estate prices up tothe size of 1257 square feet, it tends to decreaseprices beyond 1257 square feet. AGE tends todecrease prices up to 13.4 years and increaseprices beyond 13.4 years. FL tends to increaseprices up to 24th floor level and decreaseprices after that threshold level. For quantileregression, the ‘optimal size’ becomes bigger,with the exception of size at t ¼ 0:05 andt ¼ 0:1: At lower quantiles, such as att ¼ 0:1 it is about 1212 square feet. At higherquantiles, it is about 1348 square feet att ¼ 0:9; and 1417 square feet at t ¼ 0:95:For the ‘optimal age,’ it is lower than themean age at all ranges. At lower quantiles oft ¼ 0:05 and t ¼ 0:1; the ‘optimal floor level’is lower than the mean floor level. At higherquantiles, the optimal floor levels are 26, 27and 33 at t ¼ 0.75, 0.9 and 0.95 respectively,all of which are greater than the mean floorlevel.
Homebuyers generally do not favourproperties that have a building or obstructiveview; most prefer properties with an openview, a green view, or a sea view. Empiricalresults demonstrate that homebuyers of
higher-priced properties are more concernedabout the type of view their properties haveand they are not willing to opt for propertieswith a building or obstructive view unlessa bigger discount is offered to them than tothe homebuyers of lower-priced properties.This phenomenon is represented by biggerand negative estimated coefficients of thesetwo variables at higher quantiles than thoseof their mean values and the lower quantiles.
An apartment with a car park space obvi-ously commands a greater price premiumthan an apartment without one, aboutHK$167 000, according to the ordinary leastsquares estimates of the mean effect, but isclear from the quantile regression resultsthat the disparity is much larger in the lowerquantiles of the distribution and consider-ably smaller in the higher tail of the distri-bution. For example, a car parking spacecommands HK$314 000 at the 0.05 quantile,but only costs about HK$52 000 at the 0.95quantile. The least squares estimate of themean car park effect is thus mainly capturedby the lower tail of conditional distribution.The conventional least squares confidenceinterval does a poor job of representing thisrange of disparities.
6. Concluding Remarks
The objective of this paper is to investigatehow differently homebuyers value specifichousing characteristics across different quan-tiles of conditional distribution. Althoughlinear regression estimates the mean value ofthe response variable for given levels of thepredictor variables, the results are quite differ-ent from the specific data points within the
sample, depending on which side of the distri-bution those particular points of interest lie.Specifically, the quantile regression parameterestimates the change in a specified quantile ofthe response variable produced by a one-unitchange in the predictor variable. This allowsfor a comparison of how specific percentilesof real estate prices may be more affected bycertain properties’ characteristics than otherpercentiles. This is reflected in the change inthe size of the regression coefficient.
The distinction between linear and quan-tile regression is best explained by Mostellerand Tukey
What the regression curve does is give a grandsummary for the averages of the distributionscorresponding to the set of x’s. We could gofurther and compute several different regressioncurves corresponding to the various percentagepoints of the distributions and thus get a morecomplete picture of the set. Ordinarily this isnot done, and so regression often gives a ratherincomplete picture. Just as the mean gives anincomplete picture of a single distribution, sothe regression curve gives a correspondingincomplete picture for a set of distributions(Mosteller and Tukey, 1977, p. 266).
Empirical results suggest that homebuyers’tastes and preferences for specific housingattributes vary greatly across different quan-tiles of conditional distribution. This is sim-ply due to the fact that individual tastes andpreferences are unique, so that some home-buyers place a higher valuation on certainhousing characteristics than others. The quan-tile regression approach complements theleast squares by identifying how differentlyreal estate prices respond to a change in
a one-unit of housing characteristic at differ-ent quantiles, rather than estimating the con-stant regression coefficient representing thechange in the response variable produced bya one-unit change in the predictor variableassociated with that coefficient. It allowsbuyers of higher-priced properties to behavedifferently from buyers of lower-priced prop-erties even if they are within one single hous-ing estate, which is a better explanation of thereal-world phenomenon.
Notes
1. For the data definitions and sources, pleaserefer to section 4.
2. See Hall (1992) for detailed discussions on howto employ the bootstrapping method to estimatethe standard errors of the coefficient estimates.
Acknowledgments
The authors have benefited greatly fromcareful review and thoughtful suggestionsby four anonymous referees. They also thankthe Hong Kong Polytechnic University forthe research grant and financial support forthis research project (PolyU Research GrantA-SA46). The authors are, of course, respon-sible for the contents.
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