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SpringerBriefs in Regional Science For furthervolumes: http://www.springer.com/series/10096 Daniel P. McMillen Quantile Regression for Spatial Data 123 Daniel P. McMillen Department of Economics Instituteof Government andPublic Affairs Universityof Illinois Urbana,IL USA ISSN 2192-0427 ISSN 2192-0435 (electronic) ISBN 978-3-642-31814-6 ISBN 978-3-642-31815-3 (eBook) DOI 10.1007/978-3-642-31815-3 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2012942925 (cid:2)TheAuthor(s)2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserof thework.Duplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsofthe CopyrightLawofthePublisher’slocation,initscurrentversion,andpermissionforusemustalwaysbe obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright ClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Acknowledgments I am grateful to Roger Koenker, Paul Carrillo, Catia Nicodemo, and Mark Partridge for their helpful comments and suggestions. v Contents 1 Quantile Regression: An Overview. . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 A Monte Carlo Study of Gentrification . . . . . . . . . . . . . . . . . . 3 1.2 Quantile Regression Estimates . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Implied Distribution of Sales Prices. . . . . . . . . . . . . . . . . . . . . 5 1.4 Nonlinear Quantile Regression . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Linear and Nonparametric Quantile Regression . . . . . . . . . . . . . . 13 2.1 Linear Quantile Regression: Simulated Data. . . . . . . . . . . . . . . 13 2.2 Simulating the Distribution of the Dependent Variable. . . . . . . . 17 2.3 The Effect of a Discrete Change in an Explanatory Variable . . . 18 2.4 Nonparametric Quantile Regression. . . . . . . . . . . . . . . . . . . . . 22 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 A Quantile Regression Analysis of Assessment Regressivity. . . . . . 29 3.1 A Monte Carlo Analysis of Assessment Ratios. . . . . . . . . . . . . 30 3.2 Assessment Ratios in DuPage County, Illinois . . . . . . . . . . . . . 32 3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 Quantile Version of the Spatial AR Model . . . . . . . . . . . . . . . . . . 37 4.1 Quantile Regression with an Endogenous Explanatory Variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2 An Application to Hedonic House Price Functions . . . . . . . . . . 41 4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 vii viii Contents 5 Conditionally Parametric Quantile Regression . . . . . . . . . . . . . . . 49 5.1 CPAR Quantile Regression for Spatial Data. . . . . . . . . . . . . . . 50 5.2 An Empirical Example: House Prices in Tacoma, WA. . . . . . . . 51 5.3 Assessment Ratios in Cook County, IL . . . . . . . . . . . . . . . . . . 57 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6 Guide to Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Abstract Quantile regression analysis differs from more conventional regression models in its emphasis on distributions. Whereas standard regression procedures show how the expected value of the dependent variable responds to a change in an explan- atory variable, quantile regressions imply predicted changes for the entire distri- butionofthedependentvariable.Despiteitsadvantages,quantileregressionisstill notcommonlyusedintheanalysisofspatialdata.Theobjectiveofthisbookisto makequantileregressionproceduresmoreaccessibleforresearchersworkingwith spatialdatasets.Theemphasisisoninterpretationofquantileregressionresults.A series of examples using both simulated and actual data sets shows how readily seeminglycomplexquantileregressionresultscanbeinterpretedwithsetsofwell- constructedgraphs.Bothparametricandnonparametricversionsofspatialmodels are considered in detail. Keywords Quantileregression(cid:2)Spatialeconometrics(cid:2)Nonparametric(cid:2)Locally weighted regression ix Chapter 1 Quantile Regression: An Overview Linear regression is the standard tool for empirical studies in most of the social sciences. When the relationship between a dependent variable, y, and a set of explanatory variables, X, can be written as y¼Xbþu, a simple ordinary least squares (OLS) regression of y on Xcan potentiallyprovide unbiased estimatesof theparameters,b,andapredictedvalue,by ¼Xbbisthebestguessofthevalueofy given values for X. A glance at any journal in the social sciences quickly reveals the dominance of regression analysis as the tool for empirical analysis This heavy reliance on linear regression models has been carried over to the analysis of spatial data. The most commonly used spatial model adds a weighted average of nearby values for the dependent variable to the list of explanatory variables: y¼qWY þXbþu. In this model, W is a ‘‘spatial weight’’ matrix that specifies the relationships between observations. For example, a municipality’s choice of tax rate may be influenced by the tax rates of its neighbors. If a municipalityhasfourneighborsandnoneighborhasmoreinfluenceonthetaxrate choice than another, then WY simply defines the average value of the tax rate across the four neighbors. As another example, the sale price of a home may be influencedbythesalespricesofnearbyhomes.Ineitherexample,theobjectiveof a regression analysis is to estimate the coefficients, q and b, and to obtain pre- dictions of y at given values of X. Regression analysis is not well suited to explaining the distribution of a vari- able. When the predicted values from a regression are y ¼Xb^, then the distri- b butionofthepredictedvaluessimplymimicsthedistributionofthevariablesinX. The implied effect of a change in one of the explanatory variables is to cause a parallelshiftofybyanamountdeterminedbythevariable’sestimatedcoefficient. b Though a parallel shift may be reasonable in some cases, it is limitation that a researcher may not want to impose beforehand. Agoodexampleofthe restrictionsimposedbyregressionanalysisistheeffect of gentrification on house prices. A regression analysis of gentrification would seek to determine how much prices rise in neighborhoods that become gentrified. D.P.McMillen,QuantileRegressionforSpatialData, 1 SpringerBriefsinRegionalScience,DOI:10.1007/978-3-642-31815-3_1, (cid:2)TheAuthor(s)2013 2 1 QuantileRegression:AnOverview Afterdeterminingwhichneighborhoodsinacityaregentrified,theresearcheradds the gentrification dummy variable to a set of controls for characteristics of the housing.Asignificantlypositivecoefficientforthegentrificationvariableindicates thathousepricesarehigheringentrified neighborhoods,otherthingsbeingequal. This finding may be a cause for concern to housing advocates because it implies thatlow-incomeresidentsmaybeforcedtomovefromgentrifyingneighborhoods as the prices of all homes rise. The conclusion that all prices rise in gentrifying neighborhoods may be a vast oversimplification. Suppose that the wealthy people who move into a neighbor- hood bid up the price of the upper 10 % of the homes in the neighborhoods, and that the previous homeowners simply move elsewhere. The top 10 % of home prices increases, while the lower 90 % are left unchanged. Yet a regression of home prices on a set of variables that includes an indicator of gentrification will indicate higher expected sales prices in gentrifying neighborhoods. A researcher could easily be led to the erroneous conclusion that gentrification harms low- incomehouseholdsbyraisinghousingcostswhen,infact,priceshavenotchanged at all for lower-priced homes. Quantile regression is much better suited to analyzing questions involving changes in the distribution of a dependent variable. Roughly speaking—why this caveatisnecessarywillbecomeevidentintheexamplesconsideredinthebook— quantile regressions allow for separate effects of an explanatory variable on dif- ferentpointsofthedependentvariabledistribution.Inthegentrificationexample,a quantile regression for the upper part of the sale price distribution would reveal a significant effect of gentrification on sale prices, while a regression for lower quantiles would reveal little or no effect. Although quantile regression has become common, the full implications of the estimation procedure are not always realized. Researchers typically report regressions for various quantiles—the 10, 25, 50, 75, and 90 % quantiles are a commonchoice.Thecoefficientestimatesarethenfrequentlyinterpretedasbeing analogous tostandard linear regression estimates, albeit for different points in the distributionofthedependentvariable.Itislesscommonlyrecognizedthatquantile regression can produce estimates of changes in the full distribution of the dependentvariablewhenthevaluesoftheexplanatoryvariableschange.Thesetof coefficients produced for the gentrification variable imply a change in the full distributionofsalespriceswhenaneighborhoodbecomesgentrified.Graphsofthe results can be used to show that the sale price distribution shifted to the right at high prices while leaving the bulk of the distribution unchanged. Special issues do not necessarily arise when estimating quantile regressions using spatial data. Several researchers have proposed variants of the spatial autoregressive (AR) model, y¼qWY þXbþu; for quantile analysis. These procedures treat WY as just another endogenous explanatory variable. The spatial ARmodelmaynotnecessarilybethebestchoiceforspatialmodeling,particularly forlargedatasetscomprisingindividualgeographicpointsratherthanlargezones or tracts. In situations where the distribution of the dependent variable changes smoothly over space, a nonparametric procedure may be a much better approach.

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