errors in variables and spatial effects in hedonic house price models of ambient air quality

Empirical Economics (2008) 34:5–34DOI 10.1007/s00181-007-0152-3

ORIGINAL PAPER

Errors in variables and spatial effects in hedonic houseprice models of ambient air quality

Luc Anselin · Nancy Lozano-Gracia

Received: 15 January 2007 / Accepted: 16 April 2007 / Published online: 27 July 2007© Springer-Verlag 2007

Abstract In the valuation of the effect of improved air quality through the estimationof hedonic models of house prices, the potential “errors in variables” aspect of theinterpolated air pollution measures is often ignored. In this paper, we assess the extentto which this may affect the resulting empirical estimates for marginal willingness topay (MWTP), using an extensive sample of over 100,000 individual house sales for1999 in the South Coast Air Quality Management District of Southern California. We

This paper is part of a joint research effort with James Murdoch (University of Texas, Dallas) and MarkThayer (San Diego State University). Their valuable input is gratefully acknowledged. The research wassupported in part by NSF Grant BCS-9978058 to the Center for Spatially Integrated Social Science(CSISS), and by NSF/EPA Grant SES-0084213. Earlier versions were presented at the 5th InternationalWorkshop on Spatial Econometrics and Statistics, Rome, Italy, May 2006, the 53th North AmericanMeetings of the Regional Science Association International, Toronto, ON, Nov. 2006, the 2007 Meetingsof the Allied Social Science Assocations, Chicago, IL, Jan 2007, and at departmental seminars at theUniversity of Illinois. Comments by discussants and participants are greatly appreciated. A special thanksto Harry Kelejian for his detailed and patient clarification of the HAC estimator. The usual disclaimerholds.

L. Anselin (B)School of Geographical Sciences, Arizona State University,Tempe, AZ 85287-0104, USAe-mail: [email protected]

N. Lozano-GraciaSpatial Analysis Laboratory (SAL) and Department of Agricultural and Consumer Economics,University of Illinois, Urbana-Champaign, Urbana, IL 61801, USAe-mail: [email protected]

Present Address:N. Lozano-GraciaSchool of Geographical Sciences, Arizona State University, Tempe, AZ 85287-0104, USA

123

6 L. Anselin, N. Lozano-Gracia

take an explicit spatial econometric perspective and account for spatial dependenceand endogeneity using recently developed Spatial 2SLS estimation methods. We alsoaccount for both spatial autocorrelation and heteroskedasticity in the error terms, usingthe Kelejian–Prucha HAC estimator. Our results are consistent across different spatialweights matrices and different kernel functions and suggest that the bias from ignoringthe endogeneity in interpolated values may be substantial.

Keywords Spatial econometrics · Hedonic models · HAC estimation · Endogeneity ·Air quality valuation · Real estate markets

JEL Classification C21 · Q51 · Q53 · R31

1 Introduction

An important aspect of assessing the effectiveness of environmental policies thataddress the improvement of air quality is obtaining a quantitative measure of theeconomic value of the accrued benefits (e.g., Freeman 2003). In the absence of anexplicit market for clean air, several methods have been suggested to estimate thisvalue empirically, such as contingent valuation, conjoint analysis, discrete choicemodels and hedonic specifications. In this paper, we focus on the latter and considersome methodological issues associated with the estimation of an implicit price forclean air by including one or more pollution variables in a hedonic model of houseprices. The rationale behind this approach is that, ceteris paribus, houses in areas withcleaner air will have this benefit capitalized into their value, which should be reflectedin a higher sales price.

The hedonic approach has become an established methodology in environmentaleconomics (e.g., Palmquist 1991). Originating with the classic studies of Ridker andHenning (1967) and Harrison and Rubinfeld (1978), it has generated a voluminousliterature dealing with theoretical, methodological and empirical aspects. Extensivereviews are provided in Smith and Huang (1993, 1995), Boyle and Kiel (2001), andChay and Greenstone (2005), among others.

Recently, empirical econometric work has started to take into account the potentialbias and loss of efficiency that can result when spatial effects such as spatial auto-correlation and spatial heterogeneity are ignored in the estimation process. Spatialeconometric methods (Anselin 1988), which incorporate the spatial dependence incross-sectional data into model specification, estimation and testing have becomefairly commonplace in empirical studies of housing and real estate, leading to so-calledspatial hedonic models. Reviews of the basic specifications and estimation methodsare provided in Anselin (1998), Basu and Thibodeau (1998), Pace et al. (1998),Dubin et al. (1999), Gillen et al. (2001), and Pace and LeSage (2004), among others.In the context of the valuation of environmental amenities, a spatial hedonic approachhas been less common, although some recent applications include Kim et al. (2003),Beron et al. (2004), Brasington and Hite (2005), and Anselin and Le Gallo (2006). Atheoretical perspective is offered in Small and Steimetz (2006).

123

Errors in variables and spatial effects in hedonic house price models of ambient air quality 7

In Chay and Greenstone (2005) (CG), several methodological issues are addressedpertaining to the identification and consistent estimation of the implicit price of air qua-lity, using total suspended particulates as an environmental indicator. Specifically, CGfocus on the potential endogeneity of the pollution variable and suggest an instrumentalvariable approach to estimate it consistently. They also consider potential endogeneitydue to sorting by house purchasers when there is heterogeneity in their preference func-tions with different pollution levels. While considerable care is taken in addressingthese specification problems, the model itself is estimated at a fairly aggregate spa-tial scale of US counties. Bayer et al. (2006) follow Chay and Greenstone (2005)by suggesting the possibility of local air pollution being correlated with unobservedlocal characteristics. They address this form of endogeneity by using the contributionof distant sources to local air pollution as an instrument for air pollution at the countylevel.

In this paper, we focus on a separate source of endogeneity of the air quality variablesin the hedonic specification. We elaborate on an idea outlined in Anselin (2001c), whereit was argued that the use of spatially interpolated values for air quality (or, pollution)results in a prediction error which may be correlated with the overall model disturbanceterm. This would lead to simultaneity bias in an ordinary least squares regression. Wethus consider the treatment of endogeneity in the pollution variable from the particularperspective of an “errors in variables” problem. We use polynomials in the coordinatesof the house locations as instruments to correct for this endogeneity.

In contrast to the aggregate approach of CG, our empirical work is based onobservations for individual house transactions.1 Consequently, we face the mismatchbetween the spatial support of the explanatory variable, a pollution measure collectedat a finite set of monitoring stations, and the dependent variable, the price observed atthe location of the house sales transaction. As outlined in Anselin and Le Gallo (2006),this requires a spatial interpolation operation. Several alternatives are possible, eachwith implications for the precision of the resulting variable.

We take an explicit spatial econometric approach and include a spatiallylagged dependent variable (spatial lag) in the hedonic specification. The combina-tion of the endogeneity of the spatial lag and the air quality variables requires theapplication of spatial two stage least squares estimation (Anselin 1988; Kelejian andRobinson 1993; Kelejian and Prucha 1998; Lee 2003, 2006) and specialized teststatistics (Anselin and Kelejian 1997). In addition, we allow for remaining spatialautocorrelation and heteroskedasticity of an unspecified nature (HAC) and obtainrobust standard error estimates using the method of Kelejian and Prucha (2006a). Webelieve ours is the first true empirical application of spatial hedonic models in whichboth types of endogeneity (spatial and non-spatial) are considered jointly and that usesthe HAC standard errors.

1 CG also employ a panel data set with observations at two points in time, whereas our sample is a purecross-section. CG do not consider spatial effects. In our work, we do not explicitly consider endogeneitydue to sorting. However, from an empirical point of view, the source of the endogeneity is irrelevant onceit is properly accounted for.

123


We assess the extent to which the selection of a particular method affects theparameter estimates in the hedonic function and the derived economic valuation ofwillingness to pay (MWTP) for improved air quality. Specifically, we compare non-spatial to spatial hedonic specifications and estimation with and without instrumentsfor the endogeneity of the air quality variable. We further assess the robustness of ourfindings by carrying out estimation for different spatial weights and different kernelfunctions.

We pursue this empirical assessment by means of an investigation of a sample of115,732 house sales in the South Coast Air Quality Management District of Sou-thern California, for which we have detailed characteristics, as well as neighborhoodmeasures and observations on ozone and particulate matter.2

In the remainder of the paper, we first provide a brief discussion of data sources andvariables included in the model. We next give some methodological background onthe spatial econometric estimators and test statistics used. This is followed by a reviewof the estimation results, with a special focus on the estimates of the parameters ofthe air quality variables. In a brief discussion of policy implications, we compare theestimates for marginal willingness to pay. We close with some concluding remarks.

2 Data and variables

The basic data used in this paper come from three main sources: Experian Company(formerly TRW) for the individual house sales price and characteristics, the 2000 USCensus of Population and Housing for the neighborhood characteristics (at the censustract and block group level), and the South Coast Air Quality Management District forthe measures of ozone (OZ) and particulate matter (TSP) concentration. The houseprice and characteristics are from 115,729 sales transactions of owner-occupied singlefamily homes that occurred during 1999 in the region, which covers four counties: LosAngeles (LA), Riverside (RI), San Bernardino (SB) and Orange (OR). The data weregeocoded, which allows for the assignment of each house to any spatially aggregateadministrative district (such as a census tract, block group or a school district) andfor the computation of accessibility measures and interpolated pollution values forthe location of each individual house in the sample. House price and characteristicsare matched with neighborhood and locational characteristics at the census tract, and,where possible, at the block group level from the 2000 U.S. Census of Population andHousing.3

The variables used in the hedonic specification are essentially the same as those inearlier work by Beron et al. (2004) and Anselin and Le Gallo (2006). This base setis extended with newly computed measures on crime rates, school quality, distance

2 Other studies of the relation between house prices and air quality in this region can be found in Graveset al. (1988), Beron et al. (1999, 2001, 2004), and Anselin and Le Gallo (2006), although only the latter twotake an explicit spatial econometric approach. Also of interest is a general equilibrium analysis of ozoneabatement in the same region, using a hierarchical locational equilibrium model, outlined in Smith et al.(2004).3 We assume that the values obtained for the 2000 Census are representative of the spatial distribution in1999.

123


Table 1 Variable names and description

Variable name Description

Elevation Relative elevation of the house

Livarea Interior living space (10,000 sq.m.)

Landarea Lot size (1,000 sq.m.)

Baths Number of bathrooms

Fireplace Number of fireplaces

Pool Indicator variable for swimming pool

Age Age of the house (10 years)

AC Indicator variable for central air conditioning

Heat Indicator variable for central heating

Beach Indicator variable for location less than 5 miles from beach

Avdistp Average distance to parks in meters

Highway1 Indicator variable for location within a 0.25 km from a highway

Highway2 Indicator variable for location within 0.25–1 km from a highway

Traveltime Average time to work in census tract (CT)

Poverty % of population with income below the poverty level in CT

White % of the population that is white in the census block group (BG)

Over65 % of the population older than 65 years in the census BG

College % of population with college in the CT

Income Median household income in BG (10,000 US$)

Vcrime Violent crime rate for the city (or non urban county rate)

API Average academic performance index for the school district

Riverside Indicator variable for Riverside county

San Bern. Indicator variable for San Bernardino county

Orange Indicator variable for Orange county

OZ Ozone measured in ppb

TSP Total Suspended Particles in µ/m3

to parks, and access to the highway system. All the variables used in the analysisare listed in Tables 1 and 2. We grouped the variables in the Table into five cate-gories: house-specific characteristics from the Experian data set; location-specificcharacteristics, such as accessibility measures, computed from the house coordinates;neighborhood characteristics, obtained from the Census, supplemented with variablescalculated from the FBI Uniform Crime Reports and the State of California Depart-ment of Education school performance scores; county dummies; and interpolated airpollution values.

Five new variables are included in the current analysis that were not used in Anselinand Le Gallo (2006): Vcrime, API, Avdistp, Highway1 and Highway2. They werecomputed from different sources. Crime rates for violent crimes taking place during1998 were obtained from the FBI Uniform Crime database. This measure is reportedat the city as well as the county level. Where possible, we assigned the city level crime

123


Table 2 Basic descriptive statistics for all variables

Variable name Mean Std. Deviation Min Max

House price 243,346 210,000 20,000 5,345,455

Ln(house price) 12.213 0.571 9.900 15.490

Elevation 0.995 0.145 -4.000 6.588

Livarea 0.160 0.073 0.050 3.182

Landarea 8.900 19.072 0.8 2818.332

Baths 1.924 0.799 0.500 9.500

Fireplace 0.643 0.560 0 7

Pool 0.150 0.357 0 1

Age 4.287 2.023 0.1 10

AC 0.407 0.491 0 1

Heat 0.277 0.447 0 1

Beach 0.012 0.111 0 1

Pavdist 5.637 0.991 4.447 8.992

Highway1 0.091 0.288 0 1

Highway2 0.342 0.475 0 1

Traveltime 2.936 0.412 1.014 4.717

Poverty 0.120 0.091 0 0.670

White 0.570 0.221 0 1

Over65 0.105 0.059 0 0.868

College 0.259 0.176 0 0.800

Income 5.946 2.588 0 20.000

Vcrime 0.142 0.057 0.037 0.348

API 5.948 0.920 4.271 8.918

Riverside 0.056 0.230 0 1

San Bern. 0.118 0.323 0 1

Orange 0.172 0.378 0 1

OZ 8.111 1.838 4.717 13.467

TSP 82.101 14.435 54.729 121.240

rate to each house in the city. Where crime rates were not available at the city scale,we used the non-urban crime rate for the county in which the house is located.

A measure of the average school quality is computed from the Academic Perfor-mance Index (API), published by the California Department of Education.4 This isthe primary indicator used by the state to evaluate school performance. The API is anindex calculated using both base and growth values of student rankings in the StateStandardized tests. It is based on a scale from 200 to 1,000 with the target being 800.

4 http://www.cde.ca.gov/ta/ac/ap/.

123


The average 1999 API value for all schools in a school district is calculated and thenassigned to all the houses in the district.5

We supplement the beach access variable with three other indicators of accessibilityto amenities. First, we obtained the locations for each park in the four counties from theGeographic Names Information System website.6 For each house location, we thencomputed the average distance to parks as a summary measure. We also supplementedthe Census travel time measure with two other indicators of access to the highwaysystem. These are intended to capture both the negative externalities (such as noise)experienced from being very close to the highways, as well as positive externalitiesdue to shorter travel distances. We used ArcGIS and detailed highway maps7 to definebuffers of 0.25 km around the highways and to create two indicator variables. The firsttakes the value of one if the house is within 0.25 km of a highway, the second takesthe value of one if the house is between 0.25 and 1 km from a highway.

Air quality is measured as ambient air pollution. In the literature, hedonic specifica-tions typically include either ozone (OZ) or total suspended particulate matter (TSP) aspollutants, since these are most visible in the form of “smog.” In addition, local newsoutlets report daily measures of these pollutants and broadcast alerts when dangerouslevels are reached. Consequently, it is reasonable to assume that these pollutants enterinto the utility function of potential buyers, although the question remains to whatextent a continuous measure of air quality is the appropriate metric.8 We include bothpollutants in the specification, in order to minimize omitted variable problems.9

We use the average of the daily maxima during the worst quarter of 1998 from thehourly observations recorded at monitoring stations for ozone and suspended particles.It should be noted that the number and locations of stations in the South Coast AirQuality Management District (SCAQMD) is not the same for each pollutant. In 1998,there were measurements for OZ for 28 monitoring stations, while TSP only had 12.The location of the monitoring stations relative to the houses in the sample is illus-trated in Fig. 1. This yields a reasonable coverage of the spatial distribution of houselocations for OZ, but much less so for TSP, which has fewer than half the number ofstations.

We interpolate the values at the monitoring stations to the location of every housein the sample using ordinary kriging. Anselin and Le Gallo (2006), find ordinary kri-ging to be the most reliable among several interpolation methods, including Thiessen

5 It would have been preferable to use a measure of school quality from the year previous to the year inwhich the house sale takes place, as we do for the air quality measures. However, information for the APIin California school districts is only available starting in 1999.6 http://geonames.usgs.gov/pls/gnispublic/.7 ESRI Data & Maps CD-ROM (2002). Redlands, CA, USA: Environmental Systems Research Institute.8 In Anselin and Le Gallo (2006) discrete categories were also considered. In the current paper, our focusis on endogeneity and we leave the issue of the proper metric for a separate analysis.9 We also ran the analysis for specifications with only one pollutant in the equations and the results andconclusions were qualitatively similar to what we found here. Detailed results are not reported, but availablefrom the authors.

123


Fig. 1 Spatial distribution of houses and location of monitoring stations

polygons, inverse distance weighting and splines. Figures 2 and 3 show the resultinginterpolated values of ozone and particles, with darker color representing higher levelsof the pollutant.10 The spatial pattern is very different for the two measures of air pol-lution. For ozone, lower levels are observed closer to the ocean and air quality seemsto worsen as one moves North-East with a suggestion of separate air quality “bands.”For TSP, generally lower pollution is observed in the North-West corner of the Basin,with increasing levels as one moves towards the South-East.

The precision of the interpolated value varies across the sample, becoming worsefor locations further removed from monitoring sites. To correct for a possible biasingeffect of such “high-error” interpolated values, the house locations within the upper5% of the prediction error distribution for either pollutant were dropped from thesample. This resulted in a final set of 103,867 house locations, of which 67,864 arein LA county, 17,914 in OR county, 12,266 in SB and 5,823 in Riverside county.The observed sales price ranges from $20,000 to $5,345,455, with an overall mean of$243,346. There is considerable variability across counties. For example, the averagehouse price for observations in LA county is $ 261,946, while it is $269,081 in OR,$148,948 in SB and $146,249 in RI. Figure 4 illustrates the spatial distribution of houseprices, with higher prices represented through darker colors. Some concentration ofhigh prices per squared meter can be seen in the coast of LA and OR, although overall,

10 Kriging interpolations were carried out using the ESRI ArcGIS Geostatistical Analyst extension. Aspherical model allowing for directional effects was used for both pollutants. For OZ the model chosenincluded 8 lags with a lag size of 9 km, and the estimated parameters were 303.4 and 9 for the direction(angle), 4.16 for the partial sill, 68,604 and 68,236 for the major ranges and 59,381 and 68,236 for the minorranges. The model chosen for TSP included 9 lags with a lag size of 6km, and the estimated parameterswere 352.8 and 9 for the direction, 546.84 for the partial sill, 50,969 and 50,959 for the major ranges and11,303 and 50,959 for the minor ranges.

123


Fig. 2 Kriging interpolation: OZ

Fig. 3 Kriging interpolation: TSP

there is considerable complexity in the spatial distribution of prices. Basic descriptivestatistics for all the variables included in the analysis are given in Table 2.

3 Spatial econometric issues

We estimate a hedonic function in log-linear form and take an explicit spatial econo-metric approach. This includes testing for the presence of spatial autocorrelation and

123


Fig. 4 Spatial distribution of house prices (Price/sq.m.)

estimating specifications that incorporate spatial dependence.11 We follow Anselin(1988) and distinguish between spatial dependence in the form of a spatially laggeddependent variable, and a model with a spatially correlated error term. We refer tothese as spatial lag and spatial error models, respectively.

Formally, a spatial lag model is expressed as:

y = ρW y + Xβ + u, (1)

where y is a n × 1 vector of observations on the dependent variable, X is a n × kmatrix of observations on explanatory variables, W is a n × n spatial weights matrix,u a n × 1 vector of i.i.d. error terms, ρ the spatial autoregressive coefficient, and β ak × 1 vector of regression coefficients.

The theoretical motivation for a spatial lag specification is based on the literatureon interacting agents and social interaction. For example, a spatial lag follows as theequilibrium solution of a spatial reaction function (Brueckner 2003) that includes thedecision variable of other agents in the determination of the decision variable of anagent (see also Manski 2000). In the current setting, which is purely cross-sectional, itis difficult to maintain such a theoretical motivation, since it would imply that buyersand sellers simultaneously take into account prices obtained in other transactions. Analternative interpretation is provided by focusing on the reduced form of the spatiallag model:

y = (I − ρW )−1 Xβ + (I − ρW )−1u, (2)

11 For a general overview of methodological issues involved in the specification, estimation and diagnostictesting of spatial econometric models, we refer to Anselin (1988, 2001b, 2006) and Anselin and Bera (1998),among others.

123


where, under standard regularity conditions, the inverse (I −ρW )−1 can be expressedas a power expansion

(I − ρW )−1 = I + ρW + ρ2W 2 + · · · . (3)

The reduced form thus expresses the house price as a function of the own characte-ristics (X ), but also of the characteristics of neighboring properties (W X , W 2 X ),albeit subject to a distance decay operator (the combined effect of powering thespatial autoregressive parameter and the spatial weights matrix). In addition, omit-ted variables, both property-specific as well as related to neighboring properties areencompassed in the error term. In essence, this reflects a scale mismatch between theproperty location and the spatial scale of the attributes that enter into the determinationof the equilibrium price. From a purely empirical perspective, one can also argue thatthe spatial lag specification allows for a filtering of a strong spatial trend (similar todetrending in the time domain), i.e., to ensure the proper inference for the β coeffi-cients when there is insufficient variability across space. Formally, the spatial filterinterpretation stresses the estimation of β in:

y − ρW y = Xβ + u. (4)

In contrast, spatial error autocorrelation results when omitted variables follow aspatial structure such that the error variance-covariance matrix is no longer diagonal:

Var[uu′] = E[uu′] = �, (5)

where� �= I, with I as the identity matrix. Arguably, such spatially structured omittedvariables may be addressed by means of spatial fixed effects, e.g., by including adummy variable for each census tract or block group. This rests on the assumptionthat the spatial range of the unobserved heterogeneity/dependence is specific to eachspatially delineated unit. In practice, there may be spatial units (such as school districts)where such a spatial fixed effects approach is sufficient to correct for the problem.However, the nature of omitted neighborhood variables tends to be complex, as is thedefinition of the correct “neighborhood.” Instead of including spatial fixed effects, weassume a process for the error terms that allows the externalities to spill over throughoutthe system. More specifically, in contrast to most earlier work, we do not impose aspecific functional form, but take a non-parametric perspective, implementing therecent results of Kelejian and Prucha (2006a).

By means of the spatial weights matrix W , a neighbor set is specified for eachlocation. The positive elements wi j of W are non-zero when observations i and j areneighbors, and zero otherwise. By convention, self-neighbors are excluded, such thatthe diagonal elements of W are zero. In addition, in practice, the weights matrix istypically row-standardized, such that

∑j wi j = 1. Many different definitions of the

neighbor relation are possible, and there is little formal guidance in the choice of the“correct” spatial weights.12 The term W y in Eq. (1) is referred to as a spatially lagged

12 For a more extensive discussion, see Anselin (2002, pp. 256–260), and Anselin (2006, pp. 909–910).

123


dependent variable, or spatial lag. For a row-standardized weights matrix, it consistsof a weighted average of the values of y in neighboring locations, with weights wi j .

In our application, we consider three spatial weights to assess the sensitivity of theresults to this important aspect of the model specification. One weight is derived fromthe contiguity relationship for Thiessen polygons constructed from the house loca-tions. This effectively turns the spatial representation of the sample from points intopolygons. The resulting weights matrix is symmetric and extremely sparse (0.006%non-zero weights). On average it contains 6 neighbors for each location (ranging froma minimum of 3 neighbors to a maximum of 35 neighbors, with 6 as the median).We supplement this with two weights based on a nearest neighbor relation among thelocations, for respectively 6 and 12 neighbors. The corresponding weights matrix isasymmetric, but equally sparse (respectively 0.006 and 0.012% non-zero weights).The three weights matrices are used in row-standardized form.

We first obtain ordinary least squares (OLS) estimates for the hedonic model andassess the presence of spatial autocorrelation using the Lagrange Multiplier test sta-tistics for error and lag dependence (Anselin 1988), as well as their robust forms(Anselin et al. 1996).13 The results consistently show very strong evidence of positiveresidual spatial autocorrelation, with an edge in favor of the spatial lag alternative (seeSect. 4). This matches earlier results obtained in Anselin and Le Gallo (2006). Wetherefore focus on the estimation of the spatial lag model but allow remaining spatialerror autocorrelation of unspecified form, as well as heteroskedasticity of unspecifiedform.

Our paper takes two distinctive approaches towards estimation and inference of thespatial hedonic model that warrant further elaboration. First, we use a spatial two-stage least squares estimator (S2SLS) that allows for a spatial lag as well as otherendogenous variables. Consider the spatial lag model (1) with an additional term:

y = ρW y + Yν + Xβ + u, (6)

where Y is a n × p matrix of endogenous variables, with associated coefficient vectorν. In our model, the endogenous variables are the air quality variables, say y2 and y3.Since the actual pollution is not observed at the locations i of the house transactions, it isreplaced by a spatially interpolated value, such as the result of a kriging prediction. Thisinterpolated value measures the true pollution with error, for example, at location i :

y2i = y∗2i + ψi , (7)

where y∗2i is the true air quality that enters into the agent’s utility function, y2i is the

“observed” value (the interpolated value), and ψi an error term. Note that this error isrelated to the interpolation error to the extent that the predicted item is also what entersinto the utility function. An additional source of error would be a discrepancy betweenwhat is predicted as air quality and what is included into the agent’s utility function as

13 See Anselin (2001a), for an extensive review of statistical issues.

123


air quality.14 From a practical perspective, due to the nature of the kriging predictor,the prediction error will be highly spatially structured. We suggest that it therefore islikely to mimic the spatially structured equation disturbance u. In addition, the failureto predict air quality correctly at a location may be due to similar omitted variablesas those that affect the error of the hedonic specification (e.g., the omitted presenceof noxious facilities). As a result, it is likely that E[ψi ui ] �= 0, causing simultaneousequation bias due to errors in variables.

Using traditional notation, Eq. (6) can be rewritten concisely as:

y = Zγ + u, (8)

with Z = [W y,Y, X ] and γ = [ρ, ν′, β ′]′.The spatial two stage least squares estimator is an extension of the standard two

stage least squares estimator that includes specific instruments for the spatially laggeddependent variable (see Anselin 1980, 1988; Kelejian and Robinson 1993; Kelejianand Prucha 1998; Kelejian et al. 2004; Lee 2003, 2006). Specifically, consider theq × n matrix of instruments Q, with q ≥ k + p + 1:

Q = [X,W X, H ], (9)

where W X is a matrix consisting of the spatially lagged explanatory variables (exo-genous variables only, and excluding the intercept), and H is a matrix of instrumentsfor the other endogenous variables (the air quality variables).

The use of W X as instruments for the spatial lag is based on the reduced formof the model. The selection of instruments for the errors in variables problem is lessstraightforward. Proper instruments should be correlated with the unobserved truepollution value y∗ and uncorrelated with the regression error u. The effects on theestimates of using weak instruments have been widely discussed in the literature (seee.g., Staiger and Stock 1997) and the question of how to specify the right instrumentsremains unresolved for many economic problems. We chose instruments that are ableto proxy the overall spatial pattern of the pollution as a global spatial trend. Theytherefore are unlikely to be correlated with the hedonic error terms, which reflect localspatial patterns of omitted variables. Specifically, we use the latitude, longitude andtheir product as the instruments. Note that these instruments may also aid in correctingendogeneity due to other factors, such as sorting. As long as they are uncorrelated withthe error term, they will yield consistent estimates. However, if the instruments do notaccurately capture the causal mechanism underlying the other sources of endogeneity,the resulting estimates will not be most efficient. This needs to be considered toge-ther with other sources of inefficiency, such as unobserved heterogeneity and spatialautocorrelation in the error term. In order for the asymptotic properties of the HACestimator to hold, we only need consistency of the estimates in the first stage, which

14 An early application of instrumental variables in this context within the economic literature is Friedman(1957), where a measurement problem appears when using annual income as a proxy for permanent incomein estimating a consumption function.

123


will be satisfied by our instruments (as long as they are uncorrelated with the errorterm).

With the instrument matrix in hand, we obtain the S2SLS estimates as:

γS2SL S = [Z ′Q(Q′Q)−1 Q′Z ]−1 Z ′Q(Q′Q)−1 Q′y. (10)

Inference is based on the asymptotic variance matrix:

AsyV ar [γS2SL S] = σ 2[Z ′Q(Q′Q)−1 Q′Z ]−1, (11)

with σ 2 = (y − Z γS2SL S)′(y − Z γS2SL S)/n.

We relax the assumption of homoskedasticity used in (11) and allow for heteroske-dasticity of unspecified form. A direct application of the approach outlined in White(1980) yields an alternative estimate for the asymptotic variance matrix as:

AsyV ar [γS2SL S−W ] = [Z ′Q (Q′�Q)−1

Q′Z ]−1, (12)

with (Q′�Q)−1 = (Q′SQ)−1, where S is a diagonal matrix containing the squared

S2SLS residuals.15

We also continue to test for remaining spatial error autocorrelation, using thegeneralized LM tests for 2SLS residuals (Anselin and Kelejian 1997).

The second distinctive methodological aspect of our approach is that we allow forremaining spatial error autocorrelation of unspecified form. Since the specificationtests indicate the presence of such autocorrelation (see Sect. 4), we apply the recentlydeveloped heteroskedastic and autocorrelation robust (HAC) approach of Kelejianand Prucha (2006a). This builds upon the framework outlined in Conley (1999) as anextension to the spatial domain of the well-known Newey and West (1987) result fromtime series analysis (see also Andrews 1991).

The core of the HAC technique is a non-parametric estimator for the spatialcovariance, using weighted averages of cross-products of residuals, the range of whichis determined by a kernel function.16 Formally, we need to obtain an estimate of thematrix� = Q′�Q, where� is a non-diagonal spatial variance–covariance matrix forthe error terms. As Kelejian and Prucha (2006a) show, the estimator for the individualr, s elements of the matrix � is given by:

ψr,s = (1/n)∑

i

∑

j

qir q js ui u j K (di j/d), (13)

15 For a recent discussion of technical aspects associated with heteroskedastic robust estimation in spatialmodels, see Kelejian and Prucha (2006b) and Lin and Lee (2005).16 The origins of this approach can be found in Hall and Patil (1994).

123


where the subscripts refer to the individual elements of the matrix Q and residualvector u, and K is a kernel function.17 In the case of OLS, Q is replaced by X , thematrix of observations on the explanatory variables.

The kernel function K ( ) determines which pairs i, j are included in the crossproducts in (13). The kernel function is a real, continuous and symmetric functionthat is bounded and integrates to one, similar to a probability density function.18 Inthe current context, the kernel is formulated as K (di j/d), where di j is the distancebetween i and j , and d is the bandwidth, such that K (di j/d) = 0 for di j ≥ d. In ourapplication, we use three different kernel functions: the triangular or Bartlett kernel,with K (z) = 1 − z (with z = di j/d), the Epanechnikov kernel, with K (z) = 1 − z2,and the bisquare kernel, with K (z) = (1 − z2)2. Note that for each of these K = 1for di j = 0. We implement this using a variable bandwidth, based on the distances tothe 40 nearest neighbors.

Using the estimates for � from (13), the HAC variance for the S2SLS estimates isobtained as:

AsyV ar [γS2SL S−H AC ] = (Z ′q Zq)

−1 Z ′Q(Q′Q)−1�(Q′Q)−1 Q′Z(Z ′q Zq)

−1,

(14)with Z ′

q Zq = Z ′Q(Q′Q)−1 Q′Z .One final methodological note pertains to the assessment of model fit. In spatial

models, the use of the standard R2 measure is not appropriate (see Anselin 1988,Chap. 14). In order to provide for an informal comparison of the fit of the variousspecifications, we report a pseudo-R2 measure, computed as the ratio of the varianceof the predicted value to the variance of the observed values. In the classical linearregression model, this is equivalent to the R2, but in the spatial models this measureshould be interpreted with caution.

In the spatial lag model, the spatially lagged dependent variable W y is endoge-nous. We therefore obtain the predicted value from the expression for the conditionalexpectation of the reduced form:

y = E[y|X ] = (I − ρW )−1 X β (15)

This operation requires the inverse of a matrix of dimension n ×n, which we approxi-mate by means of a power method, accurate up to 6 decimals of precision.

4 Estimation results

We begin the review of our empirical results by focusing on the coefficients obtai-ned using the four estimation methods under consideration: OLS, IV (standard non-spatial 2SLS with the pollutants treated as endogenous), LAG (spatial 2SLS with

17 In practice, the term (1/n) cancels out in the final expression for the variance matrix in (14). We includeit here to be consistent with the notation in Kelejian and Prucha (2006a).18 See, among others, Härdle (1990, Chap. 3), Andrews (1991, pp. 822–823), Simonoff (1996, Chap. 3),and Cameron and Trivedi (2005, pp. 299–300).

123


Table 3 Coefficient estimates: traditional hedonic variables— queen weights

Variable name OLS IV LAG LAG-end

Constant 12.12 12.5281 8.1169 8.4700

Landarea 0.0011 0.0012 0.0009 0.0009

Livarea 2.6057 2.5864 2.2326 2.2237

Elevation −0.0004∗ −0.0029∗ 0.0027∗ 0.0007∗Baths 0.0471 0.04724 0.0415 0.0416

Fireplace 0.0457 0.0441 0.0363 0.0352

Pool 0.0505 0.0508 0.0438 0.0440

Age −0.0166 −0.0197 −0.0130 −0.0153

Age2 0.0190 0.0153 0.0140 0.0113

AC −0.0249 −0.0229 −0.0159 −0.0150

Heat 0.0386 0.0363 0.0245 0.0229

Beach 0.2405 0.2661 0.1719 0.1934

Distance Parks −0.0287 −0.0395 −0.0213 −0.0298

Highway1 −0.0199 −0.0234 −0.0130 −0.0155

Highway2 0.0028∗ 0.0027∗ 0.0043∗∗ 0.0043∗∗Travel time −0.0649 −0.0579 −0.0541 −0.0494

Poverty 0.0142∗ −0.0210∗ 0.0201∗ −0.0454

White 0.3230 0.3179 0.2282 0.2253

Over65 0.1125 0.0376∗∗ 0.0327∗∗ −0.0213∗College 1.0155 0.9192 0.5988 0.5342

Income 0.0212 0.0224 0.0085 0.0096

Vcrime −0.3938 −0.2450 −0.2446 −0.1261

API 0.0007∗ 0.0011∗ 0.0007∗ 0.0017∗Riverside −0.1405 −0.0025∗ −0.0977 0.0006∗San Bern. −0.1411 −0.0652 −0.0938 −0.0413

Orange −0.0077∗∗ 0.0579∗∗ −0.0126 0.0370

R2(var ratio) 0.7761 0.7947 0.7814 0.8017∗ Not significant∗∗ Significant at 5%

a spatially lagged dependent variable), and LAG-end (spatial 2SLS with a spatiallylagged dependent variable and the pollutants treated as endogenous). We separate theresults into those for the traditional hedonic variables, reported in Table 3, and those forthe pollutant coefficients, reported in Table 4 together with some model diagnostics.The tables only contain results for the queen spatial weights (to create the spatiallylagged dependent variable). The complete set of estimates for all three spatial weightsis given in the Appendix.

First, consider the OLS results. Overall, the coefficients of the house characteristicsare significant and of the expected sign, in accordance with earlier findings in the lite-rature. The only exception is relative elevation, which was not found to be significant.House prices increase as both land and living area increase. Similarly, houses with

123


Table 4 Pollutant coefficients by estimator—queen weights

Variable Name OLS IV LAG LAG-end

OZ −0.0253 −0.0137 −0.0195 −0.0099

TSP −0.0047 −0.0102 −0.0032 −0.0073

ρ − − 0.3314 0.3266

RLM-LAG 2357.271 − − −RLM-ERR 1339.671 − - −DWH 2,540 − − −A-K 18323.48 60.24 137.46

more bathrooms, fireplaces, as well as with AC and heating systems are higher valued.As the literature suggests (see among others Bourassa et al. 1999; Beron et al. 2004)there appears to be a quadratic relationship between age and price: prices are higherfor more recently built houses. There is also a vintage effect of age on prices that isreflected in the positive sign of the quadratic term.

In terms of access variables, there is a significant premium for houses that arelocated closer to the beach and closer to parks, but the effect of the immediate vicinityto the highway is that of a nuisance. Location in a zone 0.25–1 km from the highwayis not significant (for OLS; it is positive and becomes significant at p < 0.05 in thespatial models).

The results for the neighborhood variables are also in accordance with conventionalwisdom: travel time and crime are negatively valued, whereas % white, the proportionof college graduates and median income have a positive effect. Poverty and the schoolquality score were not found to be significant. The percentage elderly is positive, butthis finding is not stable across estimators (see below).

Los Angeles county was used as the base case, which resulted in a negative value forthe dummy variables for Riverside and San Bernardino, but no significant differencefor Orange county.

The overall fit is very satisfactory, with an R2 of 0.78. However, as the modeldiagnostics indicate (Table 4), OLS suffers from a number of problems. First, theDurbin–Wu–Haussman test statistic for endogeneity strongly rejects the null hypo-thesis that the interpolated pollutants are exogenous. In addition, there is evidence ofvery high residual spatial autocorrelation, with the robust LM test statistic suggestingthe lag specification as the proper alternative.

We next consider the effect on the estimates for the traditional hedonic variablesof treating the pollutants as endogenous (column IV in Table 3), including a spa-tially lagged dependent variable (column LAG), and combining both spatial lag andendogeneity of the pollutants (column LAG-end). Note that the A–K test for residualspatial autocorrelation also rejected the null for all three non-OLS cases, even after aspatially lagged dependent variable was included. The latter is highly significant, withestimates for the spatial autoregressive coefficient around 0.3. The A–K test pointsto the need to account for remaining spatial error autocorrelation through the HACapproach. The most appropriate specification is therefore the LAG-end with HAC

123


variance estimates. The other results are provided to assess the effect of addressingendogeneity and spatial effects in isolation versus in combination.

For the individual house characteristics and accessibility variables, the estimatedcoefficients remain fairly stable across methods, with only marginal changes. Theestimates obtained with LAG-end are slightly smaller in absolute value, but all thesignificance remain the same. This is not the case for the estimates of the neighborhoodcharacteristics. These vary considerably across methods, both in magnitude as well asin significance. For example, Poverty, which is not significant for OLS, IV and LAG,becomes significant and negative in the LAG-end model. In the reverse direction, the% elderly, which is significant in OLS, gradually loses significance (significant onlyat p < 0.05 for IV and LAG) to become insignificant in LAG-end. The absolutevalue of the coefficients for Income, College and Vcrime in LAG-end is less than halfthe magnitude for OLS. These variables are measured at an aggregate scale (censustract or block group, or city for the crime variable) and therefore the disturbancesfrom the model may be correlated within the aggregation groups (Moulton 1990). It islikely that houses in the same census tract share unobservable characteristics leadingto correlation in the error terms. We surmise that the inclusion of a spatially laggeddependent variable filters out some of this error and yields more accurate estimates.

The pollution variables are similarly affected by the estimation method. Bothcoefficients of Ozone and TSP are negative and highly significant throughout. Ho-wever, their absolute value varies considerably across methods. Taken individually,the effect of controlling for endogeneity seems to be strongest, resulting in a changebetween OLS and IV of −0.025 to −0.014 for Ozone, and of −0.005 to −0.010 forTSP. Between OLS and LAG, the change is much smaller. In LAG-end, accountingfor both the spatial effects and the endogeneity yields a coefficient of −0.0099 forOzone and −0.0073 for TSP. This suggests that a reduction of 1 ppb in OZ levelswould raise house prices by 0.99% and a decrease of 1 µ/m3 in TSP values wouldincrease house values by 0.73%.

Since the joint consideration of spatial effects and endogeneity is new in the currentpaper, there are no results available in the literature to compare our findings to directly.However, our OLS estimates are in line with previous published results. For example,in a meta-analysis of 37 studies, Smith and Huang (1995) suggest that a decrease of1µ/m3 in the TSP values will result in an increase of house values ranging between0.05 and 0.10%. Using an IV estimator Chay and Greenstone (2005) estimate that achange in 1µ/m3 will produce a 0.2–0.4% change in house prices in the oppositedirection. These estimates are considerably lower than those obtained in the currentstudy, but it is important to keep in mind that their results are obtained for countyaggregates. The OLS results in Beron et al. (2001) suggest that a decrease in one ppbof OZ would produce an increase in house prices ranging from 2.3 to 7.1%, whichis consistent with our OLS estimates. Relative to OLS, when accounting for bothendogeneity and spatial autocorrelation in the LAG-end model, the effect of ozone onhouse prices appears to be significantly smaller in absolute terms, while the effect ofTSP is larger in absolute value.

As shown in Table 4, the A–K test in the LAG-end model still shows significantremaining spatial error autocorrelation. We assess the effect of this on the precision ofthe estimates for both pollutants by computing three sets of standard errors: classical,

123


Table 5 Standard errors: OZ

Coeff. Standard errors

OZ Classical White HAC-Ep HAC-Tr HAC-Bi

OLS −0.0253 0.0008 0.0008 0.0018 0.0016 0.0016

IV −0.0137 0.0010 0.0011 0.0026 0.0023 0.0024

LAG Queen −0.0195 0.0007 0.0008 0.0012 0.0011 0.0011

LAG-end −0.0099 0.00099 0.0010 0.0017 0.0016 0.0016

LAG Knn6 −0.01822 0.00078 0.00087 0.00125 0.00115 0.00115

LAG-end −0.00895 0.00099 0.00108 0.00175 0.00157 0.00158

LAG Knn12 −0.01802 0.00078 0.00086 0.00123 0.00113 0.00114

LAG-end −0.00853 0.00098 0.00107 0.00170 0.00154 0.00155

Table 6 Standard errors: TSP

Coeff. Standard errors

TSP Classical White HAC-Ep HAC-Tr HAC-Bi

OLS −0.0047 0.00010 0.00010 0.00021 0.00019 0.00019

IV −0.0102 0.00019 0.00019 0.00046 0.00041 0.00041

LAG Queen −0.0032 0.00010 0.00010 0.00010 0.00014 0.00014

LAG−end −0.0073 0.00018 0.00020 0.00032 0.00029 0.00030

LAG Knn6 −0.0032 0.00009 0.00010 0.00015 0.00014 0.00014

LAG−end −0.0073 0.00018 0.00020 0.00031 0.00028 0.00028

LAG Knn12 −0.0032 0.00009 0.00010 0.00015 0.00013 0.00014

LAG−end −0.0073 0.00018 0.00020 0.00031 0.00028 0.00029

White (heteroskedastic consistent), and HAC. The results are reported in Tables 5and 6, for the three spatial weights matrices and three kernel functions. The estimatesfor the pollution variables are essentially the same across the three spatial weights,with only a slight difference for ozone. However, accounting for remaining heteros-kedasticity and spatial error correlation has a dramatic effect on the precision of theestimates. The standard errors are up to twice as large for the HAC as the classical andWhite results with consistently the largest value for the Epanechnikov kernel. By andlarge, the numerical values are essentially the same across kernels and spatial weights,which provides some evidence of the robustness of our findings. The more realisticmeasure of the standard errors of the estimates will be important in assessing theprecision of the derived welfare measures, such as the MWTP, to which we turn next.

5 Policy analysis

We conclude this empirical exercise by comparing the valuation of air quality com-puted from the parameter estimates obtained by the alternative methods. In a hedonic

123


model, the implicit price of any characteristic may be obtained as the derivative ofthe hedonic price equilibrium equation with respect to the characteristic of interest.In a non-spatial log-linear model, the MWTP equals the estimated coefficient for thepollution variable times the price (P), or:19

MW T Pg = ∂P

∂g= βg P, (16)

where g is either OZ or TSP.As shown in Kim et al. (2003), a spatial multiplier effect needs to be accounted for

to accurately compute the MWTP in a spatial lag model. For a uniform change in theamenity across all observations the MWTP then follows as:

MW T P = βg P(1

1 − ρ), (17)

with ρ as the estimate of the spatial autoregressive coefficient.The distinction between (16) and (17) is important in light of the recent discussion

by Small and Steimetz (2006). They considered the different interpretation of welfareeffects between the direct valuation in (16) and the multiplier effect included in (17).In their view, the multiplier effect should only be considered as part of the welfarecalculation in the case of a technological externality associated with a change inamenities. In the case of a purely pecuniary externality, the direct effect is the onlycorrect measure of welfare change. A strong argument in favor of using a spatiallag specification (where warranted by the data) is that it allows the two effects to beconsidered explicitly.

In Tables 7 and 8 we report the calculated MWTP for OZ and TSP for the fourestimation methods. For the lag models, we include both the direct effect as well asthe total effect. In addition to point estimates, we list a confidence band which consistsof ± two standard errors around the point estimate. In the non-spatial models and forthe direct effect computation, the standard errors are those reported for the regressioncoefficients. In the spatial multiplier, the standard error of both β and ρ needs to beaccounted for jointly, which we implement by means of the delta method (see e.g., forfurther details Greene 2003). We report the results for the three spatial weights and withstandard errors based on the classic form, the White and the three HAC formulations.The MWTP are estimated for a change of 0.1 ppb for ozone and 1 µ/m3 for particleswhich correspond to changes of 1.1% on average.

For both pollutants, we note a striking difference between the OLS estimate andthe result from the LAG-end model, but not in the same direction. For ozone, the OLSresult would suggest a point estimate of $616 compared to $330–$358 as the rangeacross spatial weights for the total effect for LAG-end, with $208–$241 as the rangefor the direct effect. For TSP, the direction of change is opposite, with an OLS result

19 In all cases we use the mean house price in the sample to calculate the MWTP.

123


Table 7 MWTP for reductions in OZ levels

OZ OLS IV LAG LAG LAG-end LAG-endDirect With multiplier Direct With multiplier

Queen

Estimate 616 335 475 710 241 358

Classic 575–657 281–388 433–513 652–768 193–289 286–430

White 573–659 277–392 433–517 649–770 189–293 281–435

HAC-Ep 526–705 204–466 412–538 618–802 154–329 229–489

HAC-Tr 536–695 220–450 417–532 626–794 163–320 242–475

HAC-Bi 535–696 218–452 417–532 625–794 162–320 241–475

KNN-6

Estimates – – 444 684 218 334

Classic – – 405–4,818 625–743 170–266 260–408

White – – 401–486 623–770 165–271 257–435

HAC-Ep – – 382–505 592–776 133–303 204–465

HAC-Tr – – 387–500 600–768 141–295 218–451

HAC-Bi – – 387–500 600–767 141–295 217–452

KNN-12

Estimate – – 439 706 208 330

Classic – – 400–4768 644–768 160–255 254–406

White – – 397–481 649–770 155–260 281–435

HAC-Ep – – 379–499 592–776 125–291 204–465

HAC-Tr – – 383–494 620–793 132–283 211–450

HAC-Bi – – 383–494 619–793 132–283 210–450

of $1,148 contrasted with a range of $2,640–$2,713 for the total effect using LAG-end, and $1,705–$1,778 as the range of the direct effect. Taking into account thestandard errors, including the much wider ones suggested by the HAC estimates, wecan characterize these differences as significant.

Even though it is sometimes suggested that OLS results may be appropriate asestimates of the total effect, our findings do not support this.20 It is also interes-ting to note that the direction of the difference is opposite between the two pollu-tants, something earlier studies that included only a single pollutant were not ableto ascertain. The main conclusion is therefore that OLS estimates are likely to bemisleading, but not that they over- or underestimate the total effect in a specificdirection.

A closer consideration of the results seems to suggest that the primary differenceis due to accounting for endogeneity, rather than the inclusion of the spatial lag.

20 This is separate from the issue that on technical grounds OLS will yield inconsistent and impreciseestimates, due to the presence of both spatial autocorrelation and endogeneity.

123


Table 8 MWTP for reductions in TSP levels

TSP OLS IV LAG LAG LAG-end LAG-endDirect With multiplier Direct With multiplier

Queen

Estimate 1148 2489 783 1,170 1,778 2,640

Classic 1,097–1,198 2,392–2,585 733–832 1,096–1,245 1,687–1,868 2,511–2,769

White 1,099–1,197 2,394–2,585 732–833 1,099–1,242 1,678–1,877 2,511–2,769

HAC-Ep 1,042–1,253 2,261–2,716 733–832 1,105–1,236 1,620–1,935 2,419–2,861

HAC-Tr 1,054–1,241 2,289–2,689 712–853 1,071–1,270 1,634–1,922 2,439–2,841

HAC-Bi 1,053–1,243 2,285–2,693 712–853 1,070–1,271 1,632–1,924 2,437–2,844

KNN-6

Estimate – – 775 1,195 1,740 2,671

Classic – – 726–823 1,119–1,245 1,651–1,830 2,538–2,773

White – – 724–825 1,124–1,241 1,642–1,838 2,542–2,769

HAC-Ep – – 700–849 1,086–1,279 1,587–1,893 2,441–2,870

HAC-Tr – – 706–843 1,096–1,269 1,601–1,880 2,430–2,850

HAC-Bi – – 706–843 1,095–1,270 1,599–1,881 2,459–2,852

KNN-12

Estimate – – 743 1,196 1,705 2,713

Classic – – 694–791 1,117–1,249 1,616–1,794 2,576–2,777

White – – 693–793 1,125–1,241 1,607–1,803 2,586–2,767

HAC-Ep – – 670–816 1,084–1,282 1,553–1,857 2,486–2,867

HAC-Tr – – 675–811 1,093–1,273 1,564–1,845 2,505–2,848

HAC-Bi – – 675–811 1,093–1,274 1,563–1,847 2,503–2,850

The main contribution of including the latter is that it becomes possible to distin-guish the direct effect from the total effect. However, taking into account the stan-dard errors (especially from the HAC effects), there does not seem to be a signifi-cant difference between the total effect in LAG-end and the estimate obtained forIV. The latter is significantly different from the direct effect estimate under LAG-end, so that it would not be appropriate to use the IV results as a welfare measurewhen pecuniary externalities are underlying the spatial multiplier. A similar com-parison holds between the results of OLS and those of the LAG model withoutendogeneity.

It should be noted that the inclusion of the spatial lag has important consequences forthe other parameters in the model (such as the neighborhood characteristics) and thatwe are not suggesting that it should be ignored. However, from a policy perspective, ifthe sole concern is with an estimate of MWTP irrespective of its composition betweendirect effects and spatial multiplier effects, the results from a model that accounts forendogeneity (but ignores the spatial lag) may be acceptable. However, ignoring spatialeffects leads to unrealistic indications of precision (narrow confidence intervals) whichmay be misleading in a decision support setting.

123


6 Conclusion

In this paper, we contribute to the empirical literature on the valuation of ambientair quality in spatial hedonic models by considering three novel aspects. First, weconsidered endogeneity in the form of errors in variables for the interpolated mea-sures of air pollution. This led to a the use of spatial two stage least squares estimationwith instruments for the spatially lagged dependent variable as well as the inclu-sion of the coordinates of house locations and their interaction as instruments for theinterpolated pollution values. Second, we implemented the recently developed hete-roskedastic and spatially autocorrelation consistent (HAC) estimates for the standarderrors to obtain more robust results for the precision of the computed MWTP. Third,we extended the scope of the analysis by including two pollutants in the specifica-tion, rather than the traditional focus on a single pollutant. Additionally, we carrythis out for one of the largest samples used in the empirical study of spatial hedonicmodels.

Our results underscore the importance of correcting for the errors in variables natureof the interpolated pollution values. The effect is both significant with respect to thecoefficient estimates in the hedonic model, as well as for the calculation of the MWTP.For the coefficient estimates, the main changes are seen for the pollution variables andthe neighborhood measures. The coefficients for the individual house characteristicswere found to be only marginally affected by the estimation method. In all cases,strong evidence was found of spatial error autocorrelation, which persisted even aftera spatially lagged dependent variable was included in the model. This provides asolid argument in favor of using HAC estimates of the standard errors. In practice,classical and even White standard errors seriously underestimate the imprecision of theestimates in the presence of remaining spatial correlation and spatial heterogeneity.Interestingly, there is no consistent direction of the bias of the OLS estimates forthe pollution variables. Further insight into the precise mechanisms underlying thisphenomenon requires additional investigation.

The computation of the MWTP is similarly affected by the choice of the estimationmethod. The need to account for endogeneity is clear and OLS-based calculations arelikely to be misleading. Moreover, a spatial lag specification allows for a distinctionbetween direct effects and the role of a spatial multiplier, which are combined in theestimates of the non-spatial models.

A number of aspects of estimation were not taken into consideration and remainthe subject of future work. Foremost among these is the role of spatial heteroge-neity. The strong evidence of remaining heterogeneity and spatial correlation wouldsuggest that perhaps a different scale of analysis might be more appropriate. Forexample, this might include an explicit accounting for submarkets or for possiblesorting of households by preference regarding environmental quality. Finally, the evi-dence presented here only applies to a single case study, and additional empiricalwork is needed to start establishing the foundations for general results. It is hopedthat accounting for errors in variables of the interpolated pollution measures willbecome a routine aspect of applied work in spatial hedonic models of ambient airquality.

123


Appendix

Table 9 Estimates from alternative models and standard errors

Variable OLS IV Queen weights knn6 weights knn12 weights

LAG LAG-end LAG LAG-end LAG LAG-end

Constant 12.12 12.5281 8.1169 8.4700 7.8676 8.1911 7.5316 7.900

Classical (0.0188) (0.0219) (0.0730) (0.07645) (0.0715) (0.0746) (0.0727) (0.0761)

White (0.02118) (0.0246) (0.1299) (0.1336) (0.1388) (0.1414) (0.1344) (0.1379 )

HAC-Tr (0.0373) (0.0457) (0.1471) (0.1525) (0.1607) (0.1646) (0.1593) (0.1649)

HAC-Ep (0.0418) (0.0513) (0.1521) (0.0764) (0.1668) (0.1711) (0.1653) (0.1715)

HAC-Bi (0.0380 ) (0.0465) (0.1480) (0.1535) (0.1622) (0.1662) (0.1608) (0.1666)

Landarea 0.0011 0.0012 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009

Classical (0.00004) (0.00004) (0.00004) (0.00004) (0.00004) (0.00004) (0.00004) (0.00004)

White (0.00028) (0.0003) (0.0002) (0.0002) (0.0002) (0.0002) (0.0002) (0.0002)

HAC-Tr (0.00028) (0.0003) (0.0002) (0.0002) (0.0002) (0.0002) (0.0002) (0.0002)

HAC-Ep ( 0.0002 ) (0.0003) (0.0002) (0.00004) (0.0002) (0.0002) (0.0002) (0.0002)

HAC-Bi (0.0002) (0.0003) (0.0002) (0.0002) (0.0002) (0.0002) (0.0002) (0.0002)

Livarea 2.6057 2.5864 2.2326 2.2237 2.1689 2.1587 2.2004 2.1942

Classical (0.0207) (0.02108) (0.0204) (0.0206) (0.0204) (0.0206) (0.0201) (0.0203)

White (0.1235) (0.1228) (0.1104) (0.1098) (0.1089) (0.1082) (0.1084) (0.1080)

HAC-Tr (0.1259) (0.1252) (0.11141) (0.1108) (0.1103) (0.1096) (0.1094) (0.1090)

HAC-Ep (0.1269) (0.1262) (0.1120) (0.0206) (0.1110) (0.1103) (0.1100) (0.1096)

HAC-Bi ( 0.1259) (0.1253) (0.1112) (0.1107) (0.1102) (0.1093) (0.1093) (0.1089)

Elevation −0.0004 −0.0029 0.0027 0.0007 0.00007 −0.0018 0.0002 −0.0017

Classical (0.0057) (0.0058) (0.0053) (0.0054) (0.0053) (0.0053) (0.0053) (0.0053)

White (0.0066) (0.0067) (0.0062) (0.0062) (0.0061) (0.0062) (0.0061) (0.0062)

HAC-Tr (0.0074) (0.0077) (0.0068) (0.0071) (0.0066) (0.0068) (0.0068) (0.0071)

HAC-Ep (0.0075) (0.0079) (0.0069) (0.0073) (0.0066) (0.0069) (0.0069) (0.0072)

HAC-Bi (0.0074) (0.0078) (0.0069) (0.0071) (0.0066) (0.0068) (0.0069) (0.0071)

Baths 0.0471 0.04724 0.0415 0.0416 0.0416 0.0417 0.0409 0.0411

Classical (0.0019) (0.0019) (0.0017) (0.0017) (0.0017) (0.0017) (0.0017) (0.0017)

White (0.0075) (0.0074) (0.0061) (0.0059) (0.0058) (0.0059) (0.0061) (0.0059)

HAC-Tr (0.00769) (0.0076) (0.0062) (0.0062) (0.0060) (0.0060) (0.0061) (0.0061)

HAC-Ep (0.0077) (0.0472) (0.0063) (0.0063) (0.0060 ) (0.0060) (0.0061) (0.0061)

HAC-Bi (0.0077) (0.0076) (0.0062) (0.0062) (0.0060) (0.0060) (0.0061) (0.0061)

Fireplace 0.0457 0.0441 0.0363 0.0352 0.0348 0.0337 0.0352 0.0342

Classical (0.0017) (0.0017) (0.0016) (0.0016) (0.0016) (0.0016) (0.0016) (0.0016)

White (0.0023) (0.0023) (0.0020) (0.0020) (0.0019) (0.0019) (0.0019) (0.0019)

HAC-Tr (0.00277) (0.0028) (0.0023) (0.0023) (0.0022) (0.0022) (0.0022) (0.0023)

HAC-Ep (0.0028) (0.0029) (0.0023) (0.0024) (0.0023) (0.0023) (0.0023) (0.0023)

HAC-Bi (0.0028) (0.0028) (0.0023) (0.0023) (0.0022) (0.0022) (0.0023) (0.0023)

Pool 0.0505 0.0508 0.0438 0.0440 0.0435 0.0437 0.0434 0.0436

123


Table 9 continued



Classical (0.0025) (0.0025) (0.0023) (0.0023) (0.0023) (0.0023) (0.0023) (0.0023)

White (0.0031) (0.0031) (0.0027) (0.0027) (0.0026) (0.0026) (0.0026) (0.0026)

HAC-Tr (0.0034) (0.0034) (0.0028) (0.0028) (0.0028) (0.0028) (0.0028) (0.0028)

HAC-Ep (0.0035) (0.0035) (0.0029) (0.0029) (0.0028) (0.0028) (0.0029) (0.0029)

HAC-Bi (0.0034) (0.0034) (0.0028) (0.0028) (0.0028) (0.0028) (0.0028) (0.0028)

Age −0.0166 −0.0197 −0.0130 −0.0153 −0.0102 −0.0125 −0.0104 −0.0127

Classical (0.0017) (0.0018) (0.0016) (0.0016) (0.0016) (0.0016) (0.0016) (0.0016)

White (0.0026) (0.0026) (0.0023) (0.0023) (0.0022) (0.0022) (0.0022) (0.0022)

HAC-Tr (0.0037) (0.0038) (0.0028) (0.0029) (0.0027) (0.0028) (0.0027) (0.0028)

HAC-Ep (0.0039) (0.0041) (0.0030) (0.0031) (0.0028) (0.0029) (0.0028) (0.0029)

HAC-Bi (0.0037) (0.0038) (0.0029) (0.0029) (0.0027) (0.0028) (0.0027) (0.0028)

Age Sqrd. 0.0190 0.0153 0.0140 0.0113 0.0113 0.0087 0.0110 0.0084

Classical (0.0017) (0.0018) (0.0016) (0.0016) (0.0016) (0.0016) (0.0016) (0.0016)

White (0.0027) (0.0027) (0.0024) (0.0024) (0.0023) (0.0023) (0.0023) (0.0023)

HAC-Tr (0.0038) (0.0039) (0.0029) (0.0030) (0.0028) (0.0028) (0.0028) (0.0029)

HAC-Ep (0.0041) (0.0042) (0.0031) (0.0032) (0.0029) (0.0030) (0.0029) (0.0030)

HAC-Bi (0.0039) (0.0040) (0.0029) (0.0030) (0.0028) (0.0029) (0.0028) (0.0028)

Beach 0.2405 0.2661 0.1719 0.1934 0.1721 0.1925 0.01688 0.1903

Classical (0.0079) (0.0081) (0.0075) (0.0076) (0.0074) (0.0076) (0.0074) (0.0076)

White (0.0119) (0.0119) (0.0114) (0.0115) (0.0112) (0.0113) (0.0112) (0.0113)

HAC-Tr (0.0247) (0.0247) (0.0182) (0.0184) (0.0174) (0.0175) (0.0169) (0.0171)

HAC-Ep (0.0275) (0.0275) (0.0199) (0.0200) (0.0190) (0.0191) (0.0183) (0.0185)

HAC-Bi (0.0254) (0.0254) (0.0186) (0.0188) (0.0177) (0.0179) (0.0172) (0.0175)

AC −0.0249 −0.0229 −0.0159 −0.0150 −0.0148 −0.0138 −0.0144 −0.0136

Classical (0.0021) (0.0022) (0.0020) (0.0020) (0.0020) (0.0020) (0.0020) (0.0020)

White (0.0021) (0.0021) (0.0020) (0.0020) (0.0019) (0.0019) (0.0019) (0.0019)

HAC-Tr (0.0029) (0.0029) (0.0024) (0.0024) (0.0023) (0.0023) (0.0023) (0.0023)

HAC-Ep (0.0032) (0.0031) (0.0025) (0.0025) (0.0024) (0.0024) (0.0024) (0.0025)

HAC-Bi (0.0030) (0.0030) (0.0024) (0.0024) (0.0023) (0.0024) (0.0024) (0.0024)

Heat 0.0386 0.0363 0.0245 0.0229 0.0235 0.0219 0.0234 0.0219

Classical (0.0024) (0.0024) (0.0022) (0.0022) (0.0022) (0.0022) (0.0022) (0.0022)

White (0.0025) (0.0025) (0.0024) (0.0024) (0.0024) (0.0024) (0.0024) (0.0024)

HAC-Tr (0.0033) (0.0033) (0.0028) (0.0028) (0.0028) (0.0028) (0.0028) (0.0028)

HAC-Ep (0.0035) (0.0035) (0.0029) (0.0030) (0.0029) (0.0029) (0.0029) (0.0029)

HAC-Bi (0.0386) (0.0030) (0.0028) (0.0028) (0.0028) (0.0028) (0.0028) (0.0028)

Travel time −0.0649 −0.0579 −0.0541 −0.0494 −0.0519 −0.0472 −0.0516 −0.0472

Classical (0.00243) (0.0024) (0.0022) (0.0023) (0.0022) (0.0022) (0.0022) (0.0022)

White (0.0025) (0.0025) (0.0023) (0.0023) (0.0023) (0.0023) (0.0023) (0.0023)

HAC-Tr (0.0049) (0.0049) (0.0035) (0.0035) (0.0033) (0.0034) (0.0033) (0.0034)

HAC-Ep (0.0056) (0.0056) (0.0038) (0.0039) (0.0037) (0.0037) (0.0036) (0.0037)

123


Table 9 continued



HAC-Bi (0.0050) (0.0050) (0.0035) (0.0036) (0.0034) (0.0034) (0.0033) (0.0034)

Poverty 0.0142 −0.0210 0.0201 −0.0454 −0.0137 −0.0386 −0.0137 −0.0379

Classical (0.0152) (0.0154) (0.0142) (0.0143) (0.0141) (0.0142) (0.0140) (0.0142)

White (0.0178) (0.0181) (0.0164) (0.0166) (0.0163) (0.0165) (0.0163) (0.0165)

HAC-Tr (0.0302) (0.0318) (0.0216) (0.0230) (0.0206) (0.0220) (0.0205) (0.0219)

HAC-Ep (0.0336) (0.0356) (0.0233) (0.0251) (0.0222) (0.0239) (0.0218) (0.0236)

HAC-Bi (0.0307) (0.0324) (0.0218) (0.0232) (0.0207) (0.0221) (0.0206) (0.0220)

White 0.3230 0.3179 0.2282 0.2253 0.2241 0.2208 0.2173 0.2151

Classical (0.0059) (0.0060) (0.0057) (0.0058) (0.0057) (0.0057) (0.0057) (0.0057)

White (0.0059) (0.0060) (0.0063) (0.0063) (0.0063) (0.0063) (0.0062) (0.0062)

HAC-Tr (0.0117) (0.0120) (0.0088) (0.0089) (0.0086) (0.0087) (0.0084) (0.0086)

HAC-Ep (0.0133) (0.0136) (0.0096) (0.0098) (0.0094) (0.0096) (0.0092) (0.0094)

HAC-Bi (0.0119) (0.0122) (0.0088) (0.0090) (0.0086) (0.0088) (0.0085) (0.0087)

Over65 0.1125 0.0376 0.0327 −0.0213 0.0231 −0.0297 0.0239 −0.0276

Classical (0.0173) (0.0177) (0.0162) (0.0164) (0.0161) (0.0163) (0.0057) (0.0163)

White (0.0190) (0.0189) (0.0165) (0.0168) (0.0162) (0.0164) (0.0163) (0.0165)

HAC-Tr (0.0340) (0.0337) (0.0240) (0.0246) (0.0226) (0.0235) (0.0224) (0.0235)

HAC-Ep (0.0378) (0.0374) (0.0260) (0.0267) (0.0244) (0.0256) (0.0241) (0.0254)

HAC-Bi (0.0349) (0.0346) (0.0245) (0.0250) (0.0229) (0.0238) (0.0228) (0.0238)

College 1.0155 0.9192 0.5988 0.5342 0.5807 0.5156 0.5341 0.4751

Classical (0.0093) (0.0098) (0.0113) (0.0114) (0.0111) (0.0111) (0.0113) (0.0113)

White (0.0110) (0.0110) (0.0147) (0.0141) (0.0150) (0.0144) (0.0149) (0.0144)

HAC-Tr (0.0216) (0.0216) (0.0204) (0.0196) (0.0204) (0.0196) (0.0207) (0.0199)

HAC-Ep (0.0242) (0.0244) (0.0220) (0.0212) (0.0219) (0.0211) (0.0221) (0.0213)

HAC-Bi (0.0221) (0.0221) (0.0207) (0.0199) (0.0207) (0.0199) (0.0210) (0.0201)

Income 0.0212 0.0224 0.0085 0.0096 0.0083 0.0093 0.0073 0.0085

Classical (0.0005) (0.0005) (0.0005) (0.0005) (0.0005) (0.0005) (0.0005) (0.0005)

White (0.0009) (0.0009) (0.0007) (0.0007) (0.0007) (0.0007) (0.0007) (0.0007)

HAC-Tr (0.0014) (0.0014) (0.0009) (0.0009) (0.0009) (0.0009) (0.0009) (0.0009)

HAC-Ep (0.0015) (0.0015) (0.0010) (0.0010) (0.0010) (0.0010) (0.0010) (0.0010)

HAC-Bi (0.0014) (0.0014) (0.0010) (0.0009) (0.0009) (0.0009) (0.0009) (0.0009)

Vcrime −0.3938 −0.2450 −0.2446 −0.1261 −0.2283 −0.1130 −0.2098 −0.0940

Classical (0.0236) (0.0251) (0.0222) (0.0233) (0.0220) (0.0231) (0.0231) (0.0231)

White (0.0239) (0.0255) (0.0234) (0.0242) (0.0234) (0.0243) (0.0234) (0.0242)

HAC-Tr (0.0451) (0.0496) (0.0325) (0.0356) (0.0311) (0.0342) (0.0308) (0.0340)

HAC-Ep (0.0510) (0.0562) (0.0355) (0.0394) (0.0340) (0.0378) (0.0332) (0.0372)

HAC-Bi (0.0459) (0.0504) (0.0327) (0.0359) (0.0312) (0.0344) (0.0309) (0.0341)

API 0.0007 0.0011 0.0007 0.0017 0.0011 0.0013 0.0016 0.0018

Classical (0.0013) (0.0013) (0.0012) (0.0012) (0.0012) (0.0012) (0.0012) (0.0012)

White (0.0014) (0.0014) (0.0012) (0.0013) (0.0063) (0.0012) (0.0012) (0.0012)

123


Table 9 continued



HAC-Tr (0.0028) (0.0029) (0.0020) (0.0020) (0.0019) (0.0019) (0.0018) (0.0019)

HAC-Ep (0.0032) (0.0033) (0.0022) (0.0019) (0.0021) (0.0022) (0.0020) (0.0021)

HAC-Bi (0.0029) (0.0029) (0.0020) (0.0021) (0.0019) (0.0020) (0.0019) (0.0019)

Distance Parks −0.0287 −0.0395 −0.0213 −0.0298 −0.0210 −0.0292 −0.0205 −0.0289

classical (0.0011) (0.0012) (0.0010) (0.0011) (0.0010) (0.0011) (0.0012) (0.0011)

White (0.0011) (0.0012) (0.0010) (0.0011) (0.0010) (0.0011) (0.0010) (0.0011)

HAC-Tr (0.0021) (0.0025) (0.0015) (0.0017) (0.0014) (0.0017) (0.0014) (0.0017)

HAC-Ep (0.0024) (0.0028) (0.0016) (0.0019) (0.0015) (0.0018) (0.0015) (0.0018)

HAC-Bi (0.0021) (0.0025) (0.0015) (0.0018) (0.0014) (0.0017) (0.0014) (0.0017)

SB −0.1411 0.0652 −0.0938 −0.0413 −0.0897 −0.0384 0.0870 −0.0376

Classical (0.0040) (0.0044) (0.0038) (0.0041) (0.0038) (0.0041) (0.0038) (0.0041)

White (0.0037) (0.0040) (0.0036) (0.0037) (0.0063) (0.0036) (0.0036) (0.0036)

HAC-Tr (0.0068) (0.0084) (0.0050) (0.0060) (0.0049) (0.0058) (0.0048) (0.0058)

HAC-Ep (0.0077) (0.0096) (0.0055) (0.0068) (0.0053) (0.0065) (0.0052) (0.0065)

HAC-Bi (0.0069) (0.0085) (0.0051) (0.0061) (0.0049) (0.0058) (0.0049) (0.0058)

RI −0.1405 −0.0025 −0.0977 0.0006 −0.0938 0.0021 −0.0915 0.0021

Classical (0.0054) (0.0065) (0.0050) (0.0059) (0.0050) (0.0059) (0.0050) (0.0058)

White (0.0054) (0.0062) (0.0052) (0.0057) (0.0052) (0.0057) (0.0052) (0.0057)

HAC-Tr (0.0102) (0.0125) (0.0073) (0.0085) (0.0070) (0.0082) (0.0069) (0.0081)

HAC-Ep (0.0114) (0.0142) (0.0079) (0.0094) (0.0076) (0.0091) (0.0074) (0.0089)

HAC-Bi (0.0103) (0.0127) (0.0073) (0.0086) (0.0070) (0.0083) (0.0069) (0.0081)

OR −0.0077 0.0579 −0.0126 0.0370 −0.0084 0.0397 −0.0097 0.0384

Classical (0.0032) (0.0040) (0.0030) (0.0037) (0.0030) (0.0030) (0.0036) (0.0036)

white (0.0032) (0.0039) (0.0030) (0.0037) (0.0029) (0.0036) (0.0029) (0.0036)

HAC-Tr (0.0066) (0.0085) (0.0047) (0.0060) (0.0044) (0.0057) (0.0044) (0.0057)

HAC-Ep (0.0075) (0.0098) (0.0052) (0.0067) (0.0049) (0.0064) (0.0048) (0.0063)

HAC-Bi (0.0067) (0.0087) (0.0047) (0.0061) (0.0045) (0.0058) (0.0044) (0.057)

Highway1 −0.0199 −0.0234 −0.0130 −0.0155 −0.0116 −0.0121 −0.0129 −0.0154

Classical (0.0030) (0.0030) (0.0028) (0.00283) (0.0028) (0.0028) (0.0028) (0.0028)

White (0.0030) (0.0032) (0.0029) (0.0030) (0.0029) (0.0030) (0.0029) (0.0030)

HAC-Tr (0.0051) (0.0054) (0.0038) (0.0040) (0.0035) (0.0038) (0.0036) (0.0039)

HAC-Ep (0.0056) (0.0060) (0.0040) (0.0043) (0.0037) (0.0041) (0.0038) (0.0041)

HAC-Bi (0.0052) (0.0055) (0.0038) (0.0041) (0.0036) (0.0038) (0.0036) (0.0039)

Highway2 0.0028 0.0027 0.0043 0.0043 0.0043 0.0040 0.0046 0.0047

Classical (0.0018) (0.0018) (0.0017) (0.0017) (0.0017) (0.0017) (0.0017) (0.0017)

White (0.0018) (0.0018) (0.0017) (0.0017) (0.0017) (0.0017) (0.0017) (0.0017)

HAC-Tr (0.0034) (0.0036) (0.0024) (0.0025) (0.0023) (0.0024) (0.0022) (0.0024)

HAC-Ep (0.0038) (0.0040) (0.0026) (0.0028) (0.0025) (0.0027) (0.0024) (0.0026)

HAC-Bi (0.0034) (0.0036) (0.0024) (0.0026) (0.0036) (0.0024) (0.0023) (0.0024)

OZ −0.0253 −0.0137 −0.0195 −0.0099 −0.0182 −0.0089 −0.0180 −0.0085

123


Table 9 continued



Classical (0.0008) (0.0010) (0.0007) (0.0009) (0.0007) (0.0009) (0.0007) (0.0009)

White (0.0008) (0.0011) (0.0008) (0.0010) (0.0008) (0.0010) (0.00008) (0.0010)

HAC-Tr (0.0016) (0.0023) (0.0011) (0.0016) (0.0011) (0.0015) (0.0011) (0.0015)

HAC-Ep (0.0018) (0.0026) (0.0012) (0.0017) (0.0012) (0.0017) (0.0012) (0.0017)

HAC-Bi (0.0016) (0.0024) (0.0011) (0.0016) (0.0011) (0.0015) (0.0011) (0.0015)

TSP −0.0047 −0.0102 −0.0032 −0.0073 −0.0031 −0.0071 −0.0030 −0.0070

Classical (0.0001) (0.0001) (0.0001) (0.0001) (0.00009) (0.0001) (0.00009) (0.0001)

White (0.0001) (0.0001) (0.0001) (0.0002) (0.0001) (0.0002) (0.0001) (0.0002)

HAC-Tr (0.0001) (0.0004) (0.0001) (0.0002) (0.0001) (0.0002) (0.0001) (0.0002)

HAC-Ep (0.0002) (0.0004) (0.0001) (0.0003) (0.0001) (0.0003) (0.0001) (0.0003)

HAC-Bi (0.0001) (0.0004) (0.0001) (0.0003) (0.0001) (0.0002) (0.0001) (0.0002)

ρ − − 0.3314 0.3266 0.3514 0.3484 0.3787 0.3716

Classical − − (0.0058) (0.0059) (0.0057) (0.0057) (0.0058) (0.0058)

White − − (0.0105) (0.0104) (0.0112) (0.0111) (0.0109) (0.0108)

HAC-Tr − − (0.0117) (0.0117) (0.0129) (0.0128) (0.0128) (0.0127)

HAC-Ep − − (0.0121) (0.0121) (0.0134) (0.0132) (0.0132) (0.0131)

HAC-Bi − − (0.0118) (0.0117) (0.0130) (0.0129) (0.0129) (0.0128)

R2(var ratio) 0.7761 0.7947 0.7814 0.8017 0.7833 0.8038 0.7849 0.8055

AK test 18323.48 60.24 137.46 146.9 242.05 564.72 646.04

p-value 0 0 0 0 0 0 0

References

Andrews DW (1991) Heteroscedasticity and autocorrelation consistent covariance matrix estimation. Eco-nometrica 59:817–858

Anselin L (1980) Estimation methods for spatial autoregressive structures. Regional Science Dissertationand Monograph Series, Cornell University, Ithaca

Anselin L (1988) Spatial econometrics: methods and models. Kluwer, DordrechtAnselin L (1998) GIS research infrastructure for spatial analysis of real estate markets. J Housing Res

9(1):113–133Anselin L (2001a) Rao’s score test in spatial econometrics. J Stat Plan Inf 97:113–139Anselin L (2001b) Spatial econometrics. In: Baltagi B (ed) A companion to theoretical econometrics.

Blackwell, Oxford, pp. 310–330Anselin L (2001c) Spatial effects in econometric practice in environmental and resource economics. Am J

Agric Econ 83(3):705–710Anselin L (2002) Under the hood. Issues in the specification and interpretation of spatial regression models.

Agric Econ 27(3):247–267Anselin L (2006) Spatial econometrics. In: Mills T, Patterson K (eds) Palgrave handbook of econometrics,

vol 1, Econometric Theory. Palgrave Macmillan, Basingstoke, pp 901–969Anselin L, Bera A (1998) Spatial dependence in linear regression models with an introduction to spatial

econometrics. In: Ullah A, Giles DE (ed) Handbook of applied economic statistics. Marcel Dekker,New York, pp 237–289

Anselin L, Kelejian HH (1997) Testing for spatial error autocorrelation in the presence of endogenousregressors. Int Reg Sci Rev 20:153–182

123


Anselin L, Le Gallo J (2006) Interpolation of air quality measures in hedonic house price models: spatialaspects. Spat Econ Anal 1:31–52

Anselin L, Bera A, Florax R, Yoon M (1996) Simple diagnostic tests for spatial dependence. Reg Sci UrbanEcon 26:77–104

Basu S, Thibodeau TG (1998) Analysis of spatial autocorrelation in house prices. J Real Estate FinanceEconomics 170(1):61–85

Bayer P, Keohane N, Timmins C (2006) Migration and hedonic valuation: the case of air quality. NBERWorking Papers Series, Paper No. 12106

Beron KJ, Murdoch JC, Thayer MA (1999) Hierarchical linear models with application to air pollution inthe South Coast Air Basin. Am J Agric Econ 81:1123–1127

Beron K, Murdoch J, Thayer M (2001) The benefits of visibility improvement: New evidence from LosAngeles metropolitan area. J Real Estate Finance Econ 22(2–3):319–337

Beron KJ, Hanson Y, Murdoch JC, Thayer MA (2004) Hedonic price functions and spatial depen-dence: implications for the demand for urban air quality. In: Anselin L, Florax RJ, Rey SJ (eds)Advances in spatial econometrics: methodology, tools and applications. Springer-Verlag, Berlin,pp 267–281

Bourassa S, Hamelink F, Hoesli M, MacGregor B (1999) Defining residential submarkets. J Housing Econ8:160–183

Boyle MA, Kiel KA (2001) A survey of house price hedonic studies of the impact of environmentalexternalities. J Real Estate Lit 9:117–144

Brasington DM, Hite D (2005) Demand for environmental quality: a spatial hedonic analysis. Reg SciUrban Econ 35:57–82

Brueckner JK (2003) Strategic interaction among governments: An overview of empirical studies. Int RegSci Rev 26(2):175–188

Cameron AC, Trivedi PK (2005) Microeconometrics: methods and applications. Cambridge UniversityPress, Cambridge

Chay KY, Greenstone M (2005) Does air quality matter? evidence form the housing market. J Polit Econ113(2):376–424

Conley TG (1999) GMM estimation with cross-sectional dependence. J Econom 92:1–45Dubin R, Pace RK, Thibodeau TG (1999) Spatial autoregression techniques for real estate data. J Real

Estate Lit 7:79–95Freeman AM III (2003) The measurement of environmental and resource values, theory and methods, 2nd

edn. Resources for the Future Press, WashingtonFriedman M (1957) A theory of the consumption function. Princeton University Press, PrincetonGillen K, Thibodeau TG, Wachter S (2001) Anisotropic autocorrelation in house prices. J Real Estate

Finance Econ 23(1):5–30Graves P, Murdoch JC, Thayer MA, Waldman D (1988) The robustness of hedonic price estimation: urban

air quality. Land Econ 64:220–233Greene WH (2003) Econometric analysis, 5th edn. Prentice Hall, Upper Saddle RiverHall P, Patil P (1994) Properties of nonparametric estimators of autocovariance for stationary random fields.

Prob Theory Related Fields 99:399–424Härdle W (1990) Applied nonparametric regression. Cambridge University Press, CambridgeHarrison D, Rubinfeld DL (1978) Hedonic housing prices and the demand for clean air. J Environ Econ

Manage 5:81–102Kelejian HH, Prucha IR (1998) A generalized spatial two-stage least squares procedure for estimating

a spatial autoregressive model with autoregressive disturbances. J Real Estate Finance Econ 17(1):99–121

Kelejian HH, Prucha IR (2006a) HAC estimation in a spatial framework. J Econom (forthcoming)Kelejian HH, Prucha IR (2006b) Specification and estimation of spatial autoregressive models with auto-

regressive and heteroskedastic disturbances. Working paper, Department of Economics, University ofMaryland, College Park

Kelejian HH, Robinson DP (1993) A suggested method of estimation for spatial interdependent modelswith autocorrelated errors, and an application to a county expenditure model. Papers Reg Sci 72:297–312

Kelejian HH, Prucha IR, Yuzefovich Y (2004) Instrumental variable estimation of a spatial autoregressivemodel with autoregressive disturbances: large and small sample results. In: LeSage JP, Pace RK (eds)Advances in econometrics: spatial and spatiotemporal econometrics. Elsevier Science Ltd., Oxford,pp 163–198

123


Kim CW, Phipps T, Anselin L (2003) Measuring the benefits of air quality improvement: a spatial hedonicapproach. J Environ Econ Manage 45:24–39

Lee L-F (2003) Best spatial two-stage least squares estimators for a spatial autoregressive model withautoregressive disturbances. Econom Rev 22:307–335

Lee L-F (2006) GMM and 2SLS estimation of mixed regressive, spatial autoregressive models. J Econom(forthcoming)

Lin X, Lee L-F (2005) GMM estimation of spatial autoregressive models with unknown heteroskedasticity.Working paper, The Ohio State University, Columbus

Manski CF (2000) Economic analysis of social interactions. J Econ Perspect 14(3):115–136Moulton BR (1990) An illustration of a pitfall in estimating the effects of aggregate variables on micro

units. Rev Econ Stat 72:334–338Newey WK, West KD (1987) A simple, positive semi-definite, heteroskedasticity and autocorrelation

consistent covariance matrix. Econometrica 55:703–708Pace RK, LeSage JP (2004) Spatial statistics and real estate. J Real Estate Finance Econ 29:147–148Pace KR, Barry R, Clapp JM, Rodriguez M (1998) Spatial autocorrelation and neighborhood quality. J Real

Estate Finance Econ 17(1):15–33Palmquist RB (1991) Hedonic methods. In: Braden JB, Kolstad CD (eds) Measuring the demand for

evironmental quality. North Holland, Amsterdam, pp 77–120Ridker R, Henning J (1967) The determinants of residential property values with special reference to air

pollution. Rev Econ Stat 49:246–257Simonoff JS (1996) Smoothing methods in statistics. Springer, HeidelbergSmall KA, Steimetz S (2006) Spatial hedonics and the willingness to pay for residential amenities. Econo-

mics working paper no. 05-06-31, University of California, IrvineSmith VK, Huang J-C (1993) Hedonic models and air pollution: 25 years and counting. Environ Resource

Econ 3:381–394Smith VK, Huang J-C (1995) Can markets value air quality? a meta-analysis of hedonic property value

models. J Polit Econ 103:209–227Smith VK, Sieg H, Banzhaf HS, Walsh RP (2004) General equilibrium benefits for environmental impro-

vements: projected ozone reductions under EPA’s Prospective Analysis for the Los Angeles air basin.J Environ Econ Manage 47:559–584

Staiger D, Stock JH (1997) Instrumental variables regression with weak instruments. Econometrica65(3):557–586

White H (1980) A heteroskedastic-consistent covariance matrix estimator and a direct test for heteroske-dasticity. Econometrica 48:817–838

123

errors in variables and spatial effects in hedonic house price models of ambient air quality

Documents