Quantile Regression for Spatial Data

69 Pages·2013·2.246 MB·English

by  Daniel P. McMillen (auth.)

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
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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.