Estimating the effect of air quality: spatial versus traditional hedonic price models.

• Author: Neill, Helen R.

1. Introduction

   Traditionally, ordinary least squares (OLS) methods have been used for hedonic modeling. OLS has the advantage that it can be applied to large data sets. It is limited, however, by its inability to account for spatial autocorrelation. For example, geostatistical approaches using Maximum Likelihood Estimation (MLE) can account for spatial autocorrelation, but in turn are limited by current computing techniques to relatively small data sets (i.e., sets of 1000 or fewer, depending on the complexity of the model and number of variables). Consequently, for large data sets modelers must either use OLS and be hampered by inadequately specified estimates of relatively low significance or apply MLE techniques to a sample of the data set.

   These shortcomings suggest two major research questions. The first question, which is central to this paper, relates to the robustness of competing methods of estimation. Here we contribute a method for estimating and predicting hedonic price models while controlling for spatial dependence of proximal homes. The other question, of general interest, deals with the usefulness or validity of hedonic modeling for environmental characteristics; that is, the importance of air quality in explaining sales price of homes. In order to evaluate these two questions we propose and test an approach that uses a block bootstrap method to account for spatial dependence in the data. Our two-step procedure consists of first creating blocks of temporal and spatial information within a larger data set of home sales prices and then implementing the bootstrap by resampling from each block. Next, we apply OLS and geostatistical MLE estimation techniques to generate and store coefficient estimates for each resample. The stored sets of resamples then represent Monte Carlo simulations of the full data set for each of the estimation techniques. These can be compared to evaluate the robustness of the methods based on their respective predictive powers.

   Our findings may be summarized as follows. We find that the geostatistical MLE method, which accounts for spatial effects, outperforms the traditional OLS method in several ways. Theoretically, the geostatistical MLE method better represents the expected spatial autocorrelation between error terms. Empirically, this method provides better results with respect to expected signs, statistical significance, and predictive power. The block bootstrapping technique allows hedonic modelers to effectively address spatial autocorrelation for large data sets. Overall, we find that under these simulated conditions, measured air quality matters for our data set regardless of the method used.

   The remainder of this paper contains five sections. Section 2 reviews the standard econometric methods that incorporate spatial process into estimation of spatially dependent data. Section 3 discusses the econometric and bootstrapping approaches used to evaluate our hedonic price model. Section 4 describes the data and relevant features of the model. Section 5 discusses the empirical results and contrasts predictive accuracy of the OLS and the MLE methods. Section 6 provides a summary and concluding remarks.

2. Background

   A substantial body of research suggests that consumers are willing to pay for environmental goods such as air quality (Smith and Huang 1993, 1995; Kim, Phipps, and Anselin 2003). However, in a recent survey, Boyle and Kiel (2001) report limited empirical evidence for the effect of air quality on sales price of homes. Estimated coefficients of air quality variables are often statistically insignificant and the signs of these estimated coefficients are sensitive to specification. An improper hedonic functional form can cause bias and misleading inferences. (1)

   In order to account for location effects in hedonic price models, empirical studies have employed two basic methods of estimation: the OLS and the MLE. The OLS method is less desirable as it assumes that error terms are not correlated. This ignores information regarding spatial connectedness, which is important because some characteristics, including environmental and structural variables, tend to be strongly related among proximal homes. On the other hand, the MLE method permits modelers to account for spatially autocorrelated error terms. A major challenge for the researcher, however, is to appropriately model the structure of the error terms.

   Two possible approaches have emerged for modeling spatial dependence. The first approach is based on a weight matrix that models the process itself, while the second approach, based on geostatistics, models the variance-covariance matrix of the error terms. Despite numerous empirical studies of the two approaches, the literature has paid little attention to which approach is better. Consequently, it remains ambiguous which of these methods is more practical and which provides greater predictive power.

   The weight matrix approach (Cliff and Ord 1973) emphasizes a distance-decay type of spatial weight to describe spatial features of the data. The approach shares similarities with time-series autoregressive models. MLE is the preferred method for estimating these models (see Can 1992; Pace et al. 2000; Kim, Phipps, and Anselin 2003). The shortcomings of the MLE, however, are that the procedure can be computationally costly when data sizes are large and that the procedure requires restrictive distributional assumptions. To overcome these limitations, Kelejian and Prucha (1999) have developed an alternative--a generalized moments (GM) estimator that is computationally simple and does not require restrictive specifications of the generating process of the error terms. Using micro-level data, Bell and Bockstael (2000) found the GM estimator better at offering low-cost means of obtaining parameter estimates than the MLE method. A further drawback of the MLE procedure is that it is less desirable when dealing with spatially heterogenous data, such as micro-level (cities, regions, etc.) data that are inherently prone to contain non-constant variance and outliers. Borrowing from the Bayesian literature on heteroscedasticity and outliers (Geweke 1994), a number of recent studies (including LeSage 1997; LeSage and Krivelyova 1999) found the Bayesian approach computationally feasible in large samples and powerful in improving forecasting accuracy.

   The second approach, which our study is based on, emphasizes geostatistical relationships directly specifying the variance-covariance structure of the error terms (Matheron 1963). In the spatial hedonic price literature, nearby homes share common characteristics and hence, exhibit high dependence among the error terms. In contrast, distant homes share fewer attributes. A number of functional forms can capture the spatial dependence among the errors. These functions plot the strength of the correlation among the error terms against separation distance, with correlation weakening in strength as separation distance increases. Three often-used functions with desirable decaying properties are the exponential, the Gaussian, and the spherical (Dubin 1988, 1992; Basu and Thibodeau 1998). (2)

   A distinctive feature of the geostatistical approach is kriging, which infers unknown spatial values from known values. Kriging has become a widely used approach to improve the predictive power of spatial hedonic price models. Dubin (2003) undertakes a Monte Carlo experiment with kriging that suggests stronger predictive power of the geostatistical models over the weight-matrix approach.

   The authors are grateful to the editor and two anonymous referees for valuable comments. The authors wish to thank Ms. Debra March, Director of the Lied Real Estate Institute at the University of Nevada, Las Vegas, for supporting this research project with real estate data, and Mr. Russ Merle, Senior Planner from Clark County, for providing air quality data. The views expressed in this paper do not necessarily reflect those of the Lied Real Estate Institute and Clark County government. The authors are responsible for any possible errors.

   Most studies that have attempted to use either the weight-matrix approach or the geostatistical approach to estimate spatial effects have found that the MLE method can be problematic due to the inability of the computer resources to solve these complex models for large data sets (n > 1000). Table 1 summarizes six recent papers that applied spatial approaches to hedonic data sets. The number of observations ranges from 80 to a maximum of 1000, all of which are substantially smaller than data sets commonly available for property value studies. (3) This problem of relatively small data sets used in geostatistical estimation may be overcome by a block bootstrapping Monte Carlo (4) simulation technique that approximates large data sets, which we explore further in the next section.

3. Methods

   In this section we describe the general empirical hedonic model that underlies our research, and the bootstrapping approach we use to evaluate the model for a large data set. First, we discuss traditional linear and non-linear econometric methods. Then, we discuss the specifics of our bootstrapping approach. Finally, we explain how we couple the econometric methods and the bootstrapping approach.

   Econometric Approaches

   Recent contributions to the hedonic price models have focused on the effects of spatial autocorrelation or the proximity of homes through geographic location as determinants of their market values. From this perspective, we consider the following hedonic price model that explains change in the market value of a residential property and its characteristics at a given location,

   Y(s) = X(s)[beta] + u(s), (1)

   where Y(s) indicates home prices at location s, X(s) is the vector of explanatory variables, [beta] is the unknown vector of parameters, and u(s) is the error term associated with site s. To estimate Equation 1, we use OLS...