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

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 and the MLE methods.

The OLS method requires, among other classical assumptions, independence of the error terms. Since we expect spatial autocorrelation among the error terms, applying OLS to Equation 1 will result in inefficient parameter estimates and overly narrow confidence intervals for the predicted home values. In contrast, the MLE method accounts for spatially correlated errors and yields estimated parameters that are fully efficient and confidence intervals and inferences that are not misleading.

Geostatistical spatial methods have gained increased acceptance in recent hedonic empirical studies, due largely to the kriging method, a widely used spatial-prediction procedure. Kriging assumes that because values in spatially distributed data sets are spatially correlated, unknown values at nearby locations may be accurately predicted by the weighted sum of nearest known points. A spatial autocorrelation function is used in the estimation of the unknown values. Since this function varies with distance, the weights will depend on the known sample distribution. Hence, in our hedonic geostatistical model, data organized by location (e.g., x and y coordinates) included in the estimation of Equation 1 can be used to predict those that were not (Cressie 1991; Basu and Thibodeau 1998; Dubin 2003). These predictions for each x and y coordinate are then compared with actual data that was excluded from the estimation. Kriging selects optimal weights that produce minimum estimation error.

The kriging spatial-prediction procedure can be implemented by using the correlogram or the semivariogram functional form; both approaches model spatial dependence as a separation distance function. The first approach uses the correlogram to model the structure of spatial dependence as function of separation distance. The correlogram function traces out the correlation between the residuals of any two houses as function of separation distance. Nearby houses tend to share common characteristics and hence exhibit a high correlation of the residuals, while distant observations are more likely to display a weak correlation. Examples of this approach can be found in Dubin (1988, 1992, 1998a, 1998b) and Dubin, Pace, and Thibodeau (1999). The other approach consists in employing the semivariogram function to analyze pairwise squared differences among all residuals as function of separation distance. In this framework, clustered observations tend to exhibit low residual variances, whereas the residual variances increase with distance and level off beyond a critical distance when the observations become independent. Examples of this approach can be found in Basu and Thibodeau (1998), Pace, Barry, and Sirmans (1998), and Dubin, Pace, and Thibodeau (1999). In practice, the correlogram and the semivariogram functions are estimated by way of three commonly used spatial functional forms for their desirable stationary properties: the exponential (EXP), the Gaussian (GAU), and the spherical (SPH). (5) Which of these three functional forms best represents the true relationship among error terms has not been established. Nonetheless, we expect that all three functional forms will better capture spatial relationship than the OLS, which cannot account for spatial effect. (6)

Bootstrapping Experiment

Empirical studies show that the MLE method is preferable to the OLS method for data with spatial effects, but existing computing resources restrict the MLE to relatively small data sets (see Dubin 1988, 1992; Can 1992; Basu and Thibodeau 1998; Pace et al. 2000; Kim, Phipps, and Anselin 2003). A bootstrap method can circumvent this limitation. Bootstrapping has proved useful for handling estimation and inference for otherwise intractable computations; for example, when the approximation of the distribution of the test statistics requires complex analytical calculations or when in small sample settings large-sample asymptotic results do not hold (Hall 1992; Brownstone and Valletta 2001).

Bootstrap Sampling Procedure

The bootstrap, developed by Efron (1979), is a Monte Carlo-style procedure for approximating the distribution of an estimator through repeated sampling with replacement. Consider a set of independent, identically distributed data X= ([X.sub.1], [X.sub.2], ...., [X.sub.n]) with unknown probability distribution F. The bootstrap uses the Monte Carlo procedure to estimate the unknown probability distribution F by drawing samples from X with replacement. This gives a resample [X.sup.*] =([X.sup.*.sub.1], [X.sup.*.sub.2], ...., [X.sup.*.sub.n]) with probability distribution F, where P is the estimated or fitted distribution of F. As n approaches infinity, [??] will converge on F. If F is known as a member of a parametric family, parametric bootstrapping yields [??] = F([??]), where [??] is an estimate of the parameter 0 based on X. If, however, the distribution of F is completely unknown, non-parametric bootstrap yields [??], which is the empirical distribution based on X.

The main feature of bootstrapping is that it can estimate the unknown population distribution and provide inference procedures that are asymptotically more accurate than those produced by alternative methods (see Hall 1992). When the original data are comprised of dependent observations, which is the case for time series or spatial data, block bootstrap procedure is the preferred approach (see Carlstein 1986; Kunsch 1989; Lahiri 1999; Horowitz 2001). Data are divided into blocks and the blocks, rather than the individual data, are randomly sampled. Block bootstrapping can capture the dependence structure of proximity effects. The method assumes that correlation between observations is strong within the block but weak between blocks.

There are two block bootstrapping approaches: a non-overlapping approach (Carlstein 1986) and an overlapping approach (Kunsch 1989). For large samples, the difference between the two approaches is shown to be negligible (Lahiri 2003). Suppose we partition the dependent data X=([X.sub.1], [X.sub.2], ...., [X.sub.n]) into n numbers of blocks B=([B.sub.1], [B.sub.2], ...., [B.sub.n]). Following Carlstein's (1986) non-overlapping rule, [B.sup.*] blocks of length l, [B.sup.*] =([B.sub.1.sup.*], [B.sub.1.sup.*], ...., [B.sub.b.sup.*]) are drawn from B by resampling randomly with replacement. This yields the sequence of blocked samples ([X.sub.1.sup.*], [X.sub.2.sup.*]), ...., [X.sub.n.sup.*]) = [X.sub.11.sup.*], ...., [X.sub.1][l.sup.*], [X.sub.2lsup.*], ...., [X.sub.2][l.sup.*], ......, [X.sub.b1].sup.*], ...., [X.sub.b][l.sup.*]). To avoid temporal autocorrelation, we use a non-overlapping bootstrap approach that blocks our data along temporal (month) and spatial (census tract) dimensions.

Block Resampling Strategy

Our resampling strategy is motivated by the two questions central to our study: the relative predictive power of the competing OLS and MLE methods, and the effect of air quality variables in explaining sales price of homes. Accordingly, for the first question, bootstrapping allows us to compare the performance of the regression models, whereas for the second question, bootstrapping provides hedonic pricing inference information.

Our block bootstrapping procedure that accounts for temporal and spatial characteristics of our data follows. First, we sort the whole 15,727 yearly observations by month of sale, which yields 11 blocks of monthly observations. The resulting "time series" blocks are sorted from January to November. December data are omitted because they are incomplete. Second, for each month we draw a sample of 750 observations. The sampling technique is weighted by relative contribution of each of the 110 census tracts represented in the data for that month, which ensures that the spatial distribution of the data set is reflected in the resample. For example, our data set contains 1106 observations for January. Hypothetically, if 5% of the properties (or 55 properties out of 1106) appear in a particular census tract (e.g., census tract 105) for the month of January, then the random sampling procedure maintains that percentage in that census tract by randomly selecting properties from the same census tract and keeping the same percentage for the smaller random sample (or 38 properties out of 750). We repeat this procedure 50 times, each time drawing from the full month data set, generating 50 bootstrap resamples of the 750 observations. We perform steps two and three for each of the first 11 months of the year. The dependence structure of our data set is therefore assured within the block of monthly sales data and not between the 11 blocks of sales data. This yields 550 (11 by 50 = 550) bootstrap resamples, each containing 750 observations. We can now outline steps for addressing the two research questions.

Research Question 1. Do MLE Methods Outperform OLS?

* For each resample, we split the 750 observations into an "estimation" or in-sample group of 675 observations and an "excluded" or comparison group of 75 observations.

* We apply OLS and MLE (EXP, GAU, SPH) methods to the in-sample observations and use the results to predict the "excluded" observations. We then compare OLS prediction errors with kriged MLE (EXP, GAU, SPH) prediction errors derived from the "excluded" observations.

Research Question 2. Does Air Quality Matter?

* Treating the 750 observations in each resample as population data, we compute 550 [beta]coefficient estimates for the key air quality variables using OLS and MLE (EXP, GAU, EXP) methods.

* For each of the methods, we order the 550 estimates from low to high; this represents a distribution of the [beta]-coefficient (see Figures 5 and 6).

[FIGURES 5-6 OMITTED]

4. Data and Model

Data

Our data set contains 15,727 single-family residential properties sold in Las Vegas, Nevada, from January 1999 through November 1999. (7) Table 2 provides a list of variables and Table 3 reports descriptive statistics, mean, and standard deviation for the full data set and the data set organized by month.

Data for our study are from several sources. Property data and measures of congestion came from Metroscan, a multiple listings service that uses data from the Clark County Assessor's office. Location variables came from Clark County, Metroscan, and the Las Vegas Chamber of Commerce (LVCC 1998, 1999). Information pertaining to school zone and school quality came from the Clark County School District. The Clark County Air Quality Division (formerly Office of Air Quality for Clark County) provided monitoring locations and daily air quality levels using a United States Environmental Protection Agency (USEPA) health index for both carbon monoxide (CO) and particulate matter less than ten microns (PM10). Using the daily data, we estimated a monthly average of CO and PM10. We logged the monthly average indices for CO and PM10 and use the terms LMONCO and LPM10 to identify these variables. We combined property data with other characteristics such as environmental and school quality using Geographical Information System (GIS) maps supplied by Clark County Geographical and Information Systems Management to geocode (obtain standard latitude and longitude coordinates using ArcView GIS version 3.0, a GIS program) residential properties. (8)

With respect to environmental factors, the Las Vegas Metropolitan area had only 18 and 16 air pollution monitoring stations in 1999 for CO and PM10, respectively. We obtained addresses for each of the air pollution monitoring stations and geocoded them using the same map used to geocode the residential properties. We matched the closest census tracts to the nearest air quality monitoring station. Next, we merged the average monthly Environmental Protection Agency index values for CO and PM10 linked by census tract by month of sales transaction. Figures 1 and 2 illustrate variation in air quality across the metropolitan area for CO and PM10, respectively. Figures 3 and 4 illustrate variation in air quality by month for CO and PM10, respectively.

[FIGURES 1-4 OMITTED]

Model

We consider a hedonic price model that expresses home price as a function of a number of characteristics. These include the characteristics specific to the home, the market factors, the neighborhood characteristics, and the environmental attributes. Accordingly, the general form of the hedonic price equation is expressed as

P = f(X, Q, W, Z), (2)

where P is the market value of home, X is the set of characteristics associated with home, Q comprises the market factors, W is the neighborhood characteristics, and Z represents the environmental attributes. The partial derivative of price with respect to a given characteristic yields the marginal willingness-to-pay of a household for the particular characteristic (for continuous variables). Below we describe the set of variables that comprise X, Q, W, and Z. (9)

The structural variables (X) reflect the size and quality of the home. These attributes are measured by age of home (AGE), log of square footage of home (LSQFT), size of lot in acreage (ACRES), existence of a pool (POOL), number of fireplaces (FIRE), number of floors (STORIES), and existence of a garage in the home (GARAGE). We hypothesized that LSQFT, ACRES, POOL, FIRE, and GARAGE are positively associated with home price, while AGE and STORIES negatively affect home price.

The market factors (Q) reflect days of transaction (TIME) or month of transfer (JAN, FEB, MAR, APR, MAY, JUN, JUL, AUG, SEP, OCT, and NOV); market transactions periods are likely to have an uncertain effect on home price. Because the Las Vegas economy is growing rapidly, we include the rate of change of both population (POPCH99) and new homes (HOUCH99) in order to avoid the possibility that they would be confounding if omitted. Whereas changes in population size are expected to drive up price, additional new homes are likely to adversely impact home price.

Neighborhood variables (W) capture the character and location of the neighborhood. The impact of these characteristics on home price is ambiguous. Such variables as average fourth-grade test scores where the property is zoned (TEST98) and log of median income (LMEDINC) are expected to improve home price. Other variables have, however, opposite effects. These include the number of children in neighborhood (CHILD99), percentage of new residents (NEWCOM99), percentage of residents living in neighborhood more than 20 years (OVER2099), and percentage of vacant homes in census tract (VACANT98).

Finally, environmental characteristics (Z) include logs of USEPA indices for carbon monoxide (LMONCO) and dust (LPM10). These two variables are hypothesized to negatively affect home price.

5. Empirical Results

Table 4 presents the regression results based on the OLS and the MLE (SPH) methods, respectively. The first two columns of the table report the OLS estimates using the full data set with and without the effects of LMONCO and LPM10, the two environmental variables. The rest of the columns show the OLS and MLE results for the months of February and June. (10) The overall results are consistent with expectations. The coefficients of the structural variables have the expected signs. The size of home (e.g., ACRES and LSQFT), POOL, FIRE, and GARAGE contribute positively to price of home. In contrast, older homes and two-story homes are valued less. Moreover, the signs of the coefficients on the location variables accord with expectations. Change in population (POPCH99) and quality of school (TEST98) contribute positively to value of home, whereas construction of new homes (HO UCH99), addition of new residents (NEWCOM99), percent of children in the zip code (CHILD99), and vacant lots (VACANT98) are associated negatively with residential property values. Finally, the environmental variables LMONCO and LPM10, the variables of interest, have the expected negative coefficients and the majority of these coefficients are statistically significant. The Wald test of the unrestricted model with air quality effect and restricted model without air quality also support the hypothesis that the effects of LMONCO and LPM10 on residential property values are significant (or equivalently that we can reject the null hypothesis that air quality does not matter).

Table 5 presents the characteristics of the parametric and bootstrapped sampling distribution of the OLS and the MLE coefficient estimates ([??]) of LMONCO and LPM10. We report the [??] estimates for the parametric methods and the expected values of the coefficient estimates, E([??]) for the bootstrap approach along with the 5th percentile, mean, median, and 95th percentile values. Table 5 shows the results of the full data set. Appendix Tables A and B group these measures by month.

The coefficients of the parametric OLS and the mean and median values of the coefficients of the bootstrapped OLS and MLE methods share common features. They are consistently negative as expected. In all cases, LMONCO results are statistically significant, with more than 95% of samples resulting in negative coefficients. Findings are similar, although less strong, for LPM10, where 88% to 92% are negative. This latter finding is also in agreement with distributions of both coefficients in Figure 5 for LMONCO and Figure 6 for LPM10. The mean and median of the coefficients do not differ significantly. For all methods, the mean and median coefficient estimates of LMONCO are around -0.02, which corresponds approximately to the mode of the distribution in Figure 5. The corresponding coefficient estimates for LPM10 yield a range of values between -0.02 and -0.04, due to the relative flatness of the LPM10 distribution. (11)

The coefficient estimates of the OLS and the MLE methods present some dissimilarities. In particular, the bootstrapped MLE coefficients tend to be higher in absolute terms than those of the OLS by a margin of 0.1% to 0.2% for LMONCO and 0.4% to 0.9% for the relatively flat LPM10. These discrepancies may trace to the superiority of the MLE over the OLS in adjusting for spatial effects in the hedonic price model. Table 6 presents the merit of the kriged MLE (SPH) over the OLS with respect to predictive performance. Indeed, irrespective of the semivariogram functional form (EXP, GAU, SPH) used, the kriged MLE prediction error are 89% to 93% lower than the OLS prediction error. (12) Similar kriged MLE prediction findings are reported in Appendix C for all the semivariogram functional forms (EXP, GAU, SPH) for the various months of study.

6. Conclusion

This paper was motivated by the need to apply geostatistical analysis to hedonic models with spatially dependent data. The evidence presented here shows that a bootstrap methodology can effectively circumvent computing limitations that currently restrict the MLE method to relatively small data sets. The bootstrap procedure is conceptually straightforward, practical to implement, and generates meaningful outputs. This leads to three significant observations.

First, our analysis confirms that hedonic valuation of residential property sales is most appropriately estimated using techniques that explicitly account for location. MLE provides more accurate results than does OLS, and kriging technique improves this accuracy. Indeed, in our study, the kriged MLE prediction errors are lower than their counterpart OLS prediction errors in 89% or more of the cases.

Second, we expected that environmental characteristics such as measurements of air quality would be important when modeling the sales price of a home using regression analysis. We report in Table 5 that coefficients for carbon monoxide (LMONCO) and particulate matter (LPM10) are generally the expected negative sign and significant for sales price of homes in Las Vegas. This implies that air pollution does matter.

Third, our results show that bootstrapping is an appropriate method for handling spatial effects for large data sets using hedonic price models. For example, our coefficient estimates and the standard errors for the spherical (SPH) semivariogram regarding sill, range, and nuggets are similar to Basu and Thibodeau (1998). But whereas the kriged MLE (SPH) results are more accurate than OLS in 40% to 80% of prediction experiments in Basu and Thibodeau (1998), the three kriged MLE (SPH, GAU, and EXP) outperform OLS in 87% to 93% of our prediction experiments. Moreover, in our simulated approach, we find that between 96% and 97% of the LMONCO coefficient estimates have the correct sign and between 88% and 92% of our LPM10 estimates have the correct sign. This is equivalent to significance at the 5% level for LMONCO and around the 10% level for LPM10. That is, more than 95% of coefficient estimates in the distribution generated through the bootstrapping approach for LMONCO are less than zero (see Figure 5). Likewise, for the Gaussian and spherical MLE, more than 90% of the LPM10 coefficient estimates are less than zero (see Figure 6).

Until recently, the method presented here would have been highly impractical. With increased computing speed, this analysis was feasible, but took substantially longer to run than the OLS alone. Over the next several years, the method described here will become increasingly practical as computer speeds continue to increase exponentially. Until computing methods advance to the point that we can adequately apply spatial methods directly to large data sets, the geostatistical MLE method using Monte Carlo bootstrapping should be favored over the OLS method. Future research should focus both on repeating the bootstrapping method for other data sets and comparing bootstrapping to the weight-matrix approach for accuracy, practicality of application, and consistency of results.

Appendix A

Distributions for LMONCO Regression Coefficients by Month:
OLS versus MLE (SPH)

                        Parametric OLS
                           Estimates

Month         N         [beta]     SE

January     1106        -0.021   0.0102
February    1290        -0.024   0.008
March       1706        -0.038   0.0123
April       1631        -0.044   0.0105
May         1627        -0.04    0.0108
June        1757        -0.022   0.007
July        1554        -0.031   0.01
August      1486        -0.021   0.01
September   1235        -0.02    0.01
October     1174        -0.01    0.009
November    1157        -0.018   0.007

            Bootstrap OLS Estimates by Month Where
                      B = 50 and N = 750

Month       E([beta])     SE       P5       P95

January     -0.0245     0        -0.035   -0.015
February    -0.0236     0        -0.033   -0.014
March       -0.0355     0.002    -0.057   -0.015
April       -0.0402     0.001    -0.058   -0.028
May         -0.0296     0.002    -0.052   -0.01
June        -0.022      0        -0.028   -0.014
July        -0.0247     0.002    -0.043    0.003
August      -0.0216     0.001    -0.034    0
September   -0.0213     0.001    -0.034   -0.01
October     -0.0125     0        -0.021    0
November    -0.0176     0        -0.024   -0.01

               Parametric ML
                 Estimates

Month        [beta]       SE

January     -0.029      0.0168
February    -0.027      0.0105
March       -0.038      0.0196
April       -0.048      0.0133
May         -0.03       0.0178
June        -0.034      0.0102
July        -0.047      0.0162
August      -0.028      0.0119
September   -0.025      0.0129
October     -0.01       0.0138
November    -0.013      0.009

            Bootstrap MLE (SPH) Estimates by Month where
               B = 50 (* = 49 solutions) and N = 750

Month       E([beta])     SE       P5       P95

January     -0.031      0.0014   -0.046   -0.0147
February    -0.025      0.0011   -0.036   -0.01
March       -0.034      0.0021   -0.054   -0.008
April       -0.041      0.0015   -0.062   -0.0264
May         -0.032      0.0026   -0.059   -0.007
June        -0.027      0.001    -0.04    -0.0156
July        -0.0317 *   0.0021   -0.051   -0.008
August      -0.025      0.0012   -0.016   -0.0117
September   -0.021      0.0014   -0.036   -0.005
October     -0.013      0.001    -0.021   -0.005
November    -0.014      0.001    -0.02    -0.007

Appendix B

Distributions for LPM10 Regression Coefficients by Month:
OLS versus MLE (SPH)

                        Parametric OLS
                           Estimates

Month           N       [beta]     SE

January       1106      -0.01    0.0147
February      1290      -0.039   0.016
March         1706      -0.01    0.0128
April         1631      -0.028   0.0141
May           1627      -0.02    0.0147
June          1757      -0.066   0.0165
July          1554       0       0.016
August        1486      -0.055   0.0167
September     1235      -0.046   0.0175
October       1174      -0.241   0.0178
November      1157      -0.053   0.0173

            Bootstrap OLS Estimates by Month Where
                      B = 50 and N = 750

Month       E([beta])     SE       P5        P95

January     -0.008      0.001    -0.021     0.0108
February    -0.0329     0.002    -0.056    -0.01
March       -0.004      0.002    -0.025     0.0248
April       -0.0285     0.002    -0.053     0
May         -0.0225     0.002    -0.051     0.006
June        -0.0727     0.003    -0.1072   -0.046
July        -0.004      0.003    -0.037     0.0323
August      -0.0628     0.002    -0.087    -0.038
September   -0.0458     0.002    -0.068    -0.022
October     -0.023      0.002    -0.04      0
November    -0.0565     0.002    -0.073    -0.04

              Parametric ML
                Estimates

Month        [beta]       SE

January     -0.034       0.03
February    -0.042       0.02
March       -0.014       0.02
April       -0.024       0.02
May         -0.01        0.03
June        -0.054       0.03
July        -0.01        0.03
August      -0.065       0.02
September   -0.041       0.02
October     -0.051       0.03
November    -0.066       0.02

            Bootstrap MLE (SPH) Estimates by month where
                B = 50 (* = 49 solutions) and N = 75

Month       E([beta])     SE       P5         P95

January     -0.0236     0.002    -0.038     0
February    -0.0341     0.002    -0.063     0
March       -0.0101     0.002    -0.042     0.0175
April       -0.0309     0.003    -0.06     -0.01
May         -0.0233     0.003    -0.055     0.0183
June        -0.0664     0.003    -0.1124   -0.031
July        -0.0062     0.003    -0.041     0.0385
August      -0.0641     0.002    -0.092    -0.034
September   -0.0463     0.002    -0.069    -0.016
October     -0.0425     0.002    -0.067    -0.012
November    -0.0671     0.001    -0.088    -0.046

Appendix C

OLS versus Kriged MLE (EXP, GAU, SPH) Prediction Accuracy by Month:
Percentage of the Time Kriged MLE Prediction Error < OLS
Prediction Error

                     January     February   March       April

Kriged MLE (EXP)       90.4%       91.3%      90.9%       87.1%
Kriged MLE (GAU)       91.2        91.5       90.6        86.1
Kriged MLE (SPH)       91          91.2       91          86.1
In-Sample Size        552         644        852         814
Out-of-Sample Size    554         646        854         817
Total-Sample Size    1106        1290       1706        1631

                        May      June       July        August

Kriged MLE (EXP)       92.2%       90.6%      88.3%       89.8%
Kriged MLE (GAU)       93.1%       90.5       88.8        88.7
Kriged MLE (SPH)       92.5%       90.6       88.2        90.2
In-Sample Size        812         877        776         742
Out-of-Sample Size    815         880        778         744
Total-Sample Size    1627        1757       1554        1486

                     September   October    November

Kriged MLE (EXP)       89.7%       88.9%      87.2%
Kriged MLE (GAU)       89.7        88.8       86.6
Kriged MLE (SPH)       89.7        88.9       86.9
In-Sample Size        616         586        577
Out-of-Sample Size    619         588        580
Total-Sample Size    1235        1174       1157

Received May 2005; accepted May 2006.

References

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Bell, Kathleen, and Nancy Bockstael. 2000. Applying the generalized-moments estimation approach to spatial problems involving microlevel data. The Review of Economics and Statistics 82:72-82.

Boyle, Melissa A., and Katherine A. Kiel. 2001. A survey of house price hedonic studies of the impact of environmental externalities. Journal of Real Estate Literature 9:117-44.

Brownstone, David, and Robert Valletta. 2001. The bootstrap and multiple imputations: Harnessing increased computing power for improved statistical tests. Journal of Economic Perspectives 15:129-41.

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Dubin, Robin. 1998a. Predicting house prices using multiple listings data. Journal of Real Estate Finance and Economics 17:35-60.

Dubin, Robin. 1998b. Spatial autocorrelation: A primer. Journal of Housing Economics 7:304-27.

Dubin, Robin. 2003. Robustness of spatial autocorrelation specifications: Some Monte Carlo evidence. Journal of Regional Science 43:221-48.

Dubin, Robin R., Kelly Pace, and Thomas G. Thibodeau. 1999. Spatial autoregression techniques for real estate data. Journal of Real Estate Literature 7:79-95.

Efron, Bradley. 1979. The bootstrap methods: Another look at the jacknife. Annals of Statistics 14:1-26.

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Graves, Phillips, J. Murdoch, and Mark Thayer. 1988. The robustness of hedonic price estimation: Urban air quality. Land Economics 64:221-33.

Hall, Peter. 1992. The bootstrap and Edgeworth expansion. New York: Springer-Verlag.

Graves, Phillips, J. Murdoch, and Mark Thayer. 1988. The robustness of hedonic price estimation: Urban air quality. Land Economics 64:221-33.

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Kammen, Daniel M., and David M. Hassenzahl. 1999. Should we risk it? Exploring environmental health and technological problem solving. Princeton, NJ: Princeton University Press.

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Kim, Chong W., Tim T. Phipps, and Luc Anselin. 2003. Measuring the benefits of air quality improvement: A spatial hedonic approach. Journal of Environmental Economics and Management 45:24-39.

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(1) As further evidence, Boyle and Kiel (2001) cite the results of an earlier study by Zabel and Kiel (2000) where only 29% (23 out of 80) of estimated coefficients of air quality variables are significant, while only 24% (19 out of 80) have the expected negative signs. These unexpected results may trace to econometric theory that provides little guidance for selecting the appropriate form of hedonic price function. Cropper, Derek, and McConnell (1988) and Graves, Murdoch, and Thayer (1988) show that hedonic price model estimation can be sensitive to the selection of regressors.

(2) For general information about these functions, see Millard and Neerchal (2001).

(3) For general information about spatial econometric techniques, see Palmquist (2004).

(4) For general information about Monte Carlo simulation, see Kammen and Hassenzahl (1999).

(5) For details relating semivariogram and correlogram, see Dubin, Pace, and Thibodeau (1999), especially Figures 3 and 4.

(6) We use SAS software for all statistical procedures. The three SAS procedures are PROC MODEL for OLS, PROC MIXED for MLE, and PROC KRIGE2D for KMLE. SAS (1999; pp. 2171-4) reports computational issues on PROC MIXED, a restricted MLE procedure. For MLE, SAS documentation suggests that data sets not exceed 1000 observations. Other packages, including S-Plus and GAUSS, have similar requirements.

(7) We obtained x and y coordinates from a street centerline GIS map of Clark County created on July 10, 1998. We were able to successfully geocode approximately 26,254 out of 31,658 properties (82%). After deleting duplicates, properties with incomplete information, other forms of residential properties that are not separate (condos, townhomes), properties transferred for less than $10,000 (typical in family sales), and homes not considered to be part of the Las Vegas Metropolitan area (e.g., Mesquite and Logandale, Nevada), we used 15,727 observations. The coordinates were originally in standard state plane feet. Coordinates were converted to kilometers using the following process. We subtracted 707,913 from the x values and 26,678,583 from the y values (minimum values of each). Next we divided both x and y coordinates by 3.28 (where 1 meter equals 3.28 feet or 1 foot equals 0.3048 meters) and multiplied it by 0.001 (from meters to kilometers, 1000 meters equals 1 kilometer). The rationale behind this conversion was to follow the same approach as Basu and Thibodeau (1998).

(8) We converted the x and y coordinates in state plane feet to kilometers to make the units consistent with Basu and Thibodeau (1998). We obtained additional GIS maps from Clark County of 22 zip codes, 110 census tracts, and 160 zones for fourth graders in the area (Clark County School District).

(9) The explicit version of the hedonic variable follows: LPRICE = [[beta].sub.0] + [[beta].sub.1](AGE) + [[beta].sub.2](LSQFT) + [[beta].sub.3](ACRES) + [[beta].sub.4](POOL1) + [[beta].sub.5](FIRE) + [[beta].sub.6](STORIES) + [[beta].sub.7](GARAGE1) + [[beta].sub.8](TIME) + [[beta].sub.9](LMONCO) + [[beta].sub.10](LPM10) + [[beta].sub.11](POPCH99) + [[beta].sub.12](HOUCH99) + [[beta].sub.13](TEST98) + [[beta].sub.14](CHILD99) + [[beta].sub.15](LMEDINC) + [[beta].sub.16](NEWCOME99) + [[beta].sub.17](OVER2099) + [[beta].sub.18](VACANT98 ) + u.

(10) The complete results of the regressions for all the months are available from the authors at the following Web site: http://www.unlv.edu/faculty/dmh/Pages%20from%20tables-4a-4b.pdf.

(11) Indeed, compared to LMONCO, LPM10 produces a fat-tail distribution, with the upper tail of the distribution yielding about 6% to 7% of the coefficient estimates with a positive sign.

(12) Tables similar to Table 6 for kriged MLE (EXP) and MLE (GAU) are available upon request. For a comparison of kriged MLE (EXP), MLE (GAU), and MLE (SPH) using monthly data rather than resamples, see Appendix C.

Helen R. Neill, * David M. Hassenzahl, ([dagger]) and Djeto D. Assane ([double dagger])

* Department of Environmental Studies, Greenspun College of Urban Affairs, University of Nevada, Las Vegas, 4505 Maryland Parkway Box 454030, Las Vegas, NV 89154-4030, USA; E-mail [e-mail]; corresponding author.

([dagger]) Department of Environmental Studies, Greenspun College of Urban Affairs, University of Nevada, Las Vegas, 4505 Maryland Parkway Box 454030, Las Vegas, NV 89154-4030, USA; E-mail [e-mail].

([double dagger]) Department of Economics, College of Business, University of Nevada, Las Vegas, 4505 Maryland Parkway Box 456006, Las Vegas, NV 89154-6006, USA; E-mail [e-mail].

Table 1. Summary of Recent Spatial Hedonic Property Value Models

                         Number of
                      Observations in     Total Data
Author                  Estimation            Set

Dubin (1988)                80           Not reported
Dubin (1992)                211              2157
Basu and Thibodeau      218 to 852           5320
  (1998)
Dubin (1998a)              1000              2157
Kim, Phipps, and            609              1121
  Anselin (2003)
Dubin (2003)                200              1000

                           Area            Spatial
Author                    Studied          Approach

Dubin (1988)             Baltimore      Geostatistical
Dubin (1992)             Baltimore      Geostatistical
Basu and Thibodeau        Dallas        Geostatistical
  (1998)
Dubin (1998a)            Baltimore      Geostatistical
Kim, Phipps, and           Seoul         Weight matrix
  Anselin (2003)
Dubin (2003)          Not applicable,   Geostatistical
                      data generated      and weight
                      in Monte Carlo        matrix
                        experiments

Sample sizes are limited in all cases, the largest being 1000
observations-less than half of the available data set.

Table 2. Variable Descriptions and Sources

Variables                       Definition           Units

Dependent Variable
  LPRICE                        Sales Prices         Log(dollars)

Structural Characteristics
  AGE                           Age of home          Years
  LSQFT                         Size of home         Log (Square
                                                       footage)
  ACRES                         Size of lot          Acres
  POOL                          Pool                 0/1 Dummy
  FIRE                          Fireplace            Number
  STORIES                       Accessibility        Number of floors
  GARAGE                        Garage               0/1 Dummy

Temporal Characteristics
  TIME                          Transfer date        Days
  JAN                           Month of transfer    0/1 Dummy
  FEB                           Month of transfer    0/1 Dummy
  MAR                           Month of transfer    0/1 Dummy
  APR                           Month of transfer    0/1 Dummy
  MAY                           Month of transfer    0/1 Dummy
  JUN                           Month of transfer    0/1 Dummy
  JUL                           Month of transfer    0/1 Dummy
  AUG                           Month of transfer    0/1 Dummy
  SEP                           Month of transfer    0/1 Dummy
  OCT                           Month of transfer    0/1 Dummy
  NOV                           Month of transfer    0/1 Dummy

Environmental Characteristics
  LMONCO                        Carbon monoxide      Log (EPA Index)
  LPM10                         Dust particles       Log (EPA Index)
                                  less than 10
                                  microns

Location Variables
  POPCH99                       Rate of change of    Population change
                                  population           by zip code =
                                                       [(pop99-pop98)/
                                                       pop98] x 100
  HOUCH99                       Rate of change of    Housing change by
                                  homes                zip code =
                                                       [(hou99-hou98)/
                                                       hou98] x 100
  TEST98                        Quality of schools   4th grade test
                                                       scores
  CHILD99                       Children             Percentage of
                                                       children in zip
                                                       code
  LMEDINC                       Median income        Log (Median
                                                       Income) in zip
                                                       code
  NEWCOM99                      New residents        Percentage of new
                                                       residents in zip
                                                       code
  OVER2099                      Established          Percentage of
                                  community            residents in zip
                                                       code living in
                                                       neighborhood
                                                       over 20 years
  VACANT98                      Vacant homes         Percentage of
                                                       vacant homes per
                                                       census tract

Spatial Characteristic
  CENSUS *                      Census tract         110 census tracts

Variables                       Source

Dependent Variable
  LPRICE                        Metroscan

Structural Characteristics
  AGE                           Metroscan
  LSQFT                         Metroscan

  ACRES                         Metroscan
  POOL                          Metroscan
  FIRE                          Metroscan
  STORIES                       Metroscan
  GARAGE                        Metroscan

Temporal Characteristics
  TIME                          Metroscan
  JAN                           Metroscan
  FEB                           Metroscan
  MAR                           Metroscan
  APR                           Metroscan
  MAY                           Metroscan
  JUN                           Metroscan
  JUL                           Metroscan
  AUG                           Metroscan
  SEP                           Metroscan
  OCT                           Metroscan
  NOV                           Metroscan

Environmental Characteristics
  LMONCO                        Clark County
  LPM10                         Clark County

Location Variables
  POPCH99                       LVCC
  HOUCH99                       LVCC
  TEST98                        Clark County
                                  School District
  CHILD99                       LVCC
  LMEDINC                       LVCC
  NEWCOM99                      LVCC
  OVER2099                      LVCC
  VACANT98                      Metroscan

Spatial Characteristic
  CENSUS *                      Clark County GIS
                                  map

* While we did not report census tract in our descriptive
statistics or regression results, we used census tract to
organize our data spatially for resampling (see Methods section).

Table 3. Descriptive Statistics, Mean, and Standard Errors in
Parentheses for the Full and Monthly Data Sets in 1999

Variable         Full Data    January    February      March

LPRICE             11.8        11.81       11.8        11.8
                    (0)       (0.01)      (0.01)      (0.01)
ACRES              0.17        0.17        0.17        0.17
                    (0)         (0)         (0)         (0)
AGE                13.4        11.03       11.23       11.12
                  (0.11)       (0.4)      (0.36)       (0.3)
CHILD99            0.34        0.34        0.34        0.34
                    (0)         (0)         (0)         (0)
FIRE               0.76        0.75        0.72        0.75
                    (0)       (0.02)      (0.02)      (0.01)
GARAGE1            0.86        0.89        0.89        0.89
                    (0)       (0.01)      (0.01)      (0.01)
HOUCH99            -0.41       0.53        0.12        0.31
                  (0.13)      (0.48)      (0.46)       (0.4)
LMEDINC            10.77       10.78       10.78       10.8
                    (0)       (0.01)      (0.01)      (0.01)
LMONCO             2.09        2.44        2.25        1.94
                  (0.01)      (0.02)      (0.02)      (0.01)
LPM10              3.16         3.3        3.18        3.21
                    (0)       (0.01)      (0.01)      (0.01)
LSQFT              7.42        7.45        7.42        7.42
                    (0)       (0.01)      (0.01)      (0.01)
NEWCOM99           0.08        0.08        0.08        0.08
                    (0)         (0)         (0)         (0)
OVER2099           0.26        0.26        0.26        0.26
                    (0)         (0)         (0)         (0)
POOL1              0.18        0.15        0.16        0.15
                    (0)       (0.01)      (0.01)      (0.01)
POPCH99            -0.83       0.30        -0.17       0.12
                  (0.13)      (0.51)      (0.48)      (0.42)
STORIES            1.32        1.34        1.33        1.34
                    (0)       (0.01)      (0.01)      (0.01)
TEST98             53.6        53.28       53.46       54.35
                  (0.09)      (0.34)      (0.32)      (0.27)
TIME              162.19       15.89       44.32       74.29
                  (0.72)      (0.25)      (0.24)      (0.23)
VACANT98           0.03        0.03        0.03        0.03
                    (0)         (0)         (0)         (0)
Number of         15,727       1106        1290        1706
  observations

Variable           April        May        June        July

LPRICE             11.8        11.82       11.82       11.81
                  (0.01)      (0.01)      (0.01)      (0.01)
ACRES              0.17        0.17        0.16        0.17
                    (0)         (0)         (0)         (0)
AGE                12.71       12.4        12.91       14.28
                  (0.33)      (0.33)      (0.31)      (0.34)
CHILD99            0.34        0.34        0.34        0.33
                    (0)         (0)         (0)         (0)
FIRE               0.74        0.77        0.77        0.77
                  (0.01)      (0.01)      (0.01)      (0.01)
GARAGE1            0.87        0.88        0.87        0.85
                  (0.01)      (0.01)      (0.01)      (0.01)
HOUCH99            0.09        1.13        0.03        -1.03
                  (0.41)      (0.38)      (0.37)       (0.4)
LMEDINC            10.78       10.78       10.78       10.77
                  (0.01)      (0.01)      (0.01)      (0.01)
LMONCO             1.78        1.63        2.03         1.9
                  (0.01)      (0.01)      (0.02)      (0.01)
LPM10              2.99        3.11          3          3.1
                  (0.01)      (0.01)      (0.01)      (0.01)
LSQFT              7.43        7.44        7.43        7.42
                  (0.01)      (0.01)      (0.01)      (0.01)
NEWCOM99           0.08        0.08        0.08        0.08
                    (0)         (0)         (0)         (0)
OVER2099           0.27        0.26        0.26        0.26
                    (0)         (0)         (0)         (0)
POOL1              0.17        0.18         0.2         0.2
                  (0.01)      (0.01)      (0.01)      (0.01)
POPCH99            -0.11       0.74        -0.36       -1.51
                  (0.44)       (0.4)      (0.39)      (0.42)
STORIES            1.33        1.34        1.32        1.31
                  (0.01)      (0.01)      (0.01)      (0.01)
TEST98             53.4        53.64       54.2        53.61
                  (0.28)      (0.28)      (0.27)      (0.29)
TIME              105.44      135.02       165.2      196.61
                  (0.23)      (0.21)      (0.23)      (0.24)
VACANT98           0.03        0.03        0.03        0.03
                    (0)         (0)         (0)         (0)
Number of          1631        1627        1757        1554
  observations

Variable          August     September    October    November

LPRICE             11.8        11.8        11.79       11.77
                  (0.01)      (0.01)      (0.01)      (0.01)
ACRES              0.17        0.17        0.18        0.17
                    (0)         (0)         (0)         (0)
AGE                14.18       15.64       15.87       17.44
                  (0.34)      (0.39)      (0.39)       (0.4)
CHILD99            0.34        0.34        0.33        0.33
                    (0)         (0)         (0)         (0)
FIRE               0.76        0.78        0.77        0.74
                  (0.01)      (0.02)      (0.02)      (0.02)
GARAGE1            0.86        0.83        0.83         0.8
                  (0.01)      (0.01)      (0.01)      (0.01)
HOUCH99            -0.84       -1.44       -2.11       -2.35
                  (0.42)      (0.48)      (0.47)      (0.44)
LMEDINC            10.76       10.75       10.75       10.72
                  (0.01)      (0.01)      (0.01)      (0.01)
LMONCO             2.16        2.22        2.51        2.53
                  (0.01)      (0.02)      (0.02)      (0.03)
LPM10              3.14        3.03        3.35        3.52
                  (0.01)      (0.01)      (0.01)      (0.01)
LSQFT              7.42        7.41        7.41        7.39
                  (0.01)      (0.01)      (0.01)      (0.01)
NEWCOM99           0.08        0.08        0.08        0.08
                    (0)         (0)         (0)         (0)
OVER2099           0.27        0.27        0.27        0.28
                    (0)         (0)         (0)         (0)
POOL1              0.19        0.19        0.22        0.21
                  (0.01)      (0.01)      (0.01)      (0.01)
POPCH99            -1.38       -2.01       -2.8        -3.13
                  (0.44)      (0.49)      (0.48)      (0.46)
STORIES             1.3         1.3        1.29        1.27
                  (0.01)      (0.01)      (0.01)      (0.01)
TEST98             53.48       53.28       53.32        53
                   (0.3)      (0.32)      (0.34)      (0.34)
TIME              227.52      257.87      287.45      317.41
                  (0.24)      (0.28)      (0.27)      (0.27)
VACANT98           0.03        0.03        0.03        0.03
                    (0)         (0)         (0)         (0)
Number of          1486        1235        1174        1157
  observations

Table 4. Regression Results

                      OLS Year      OLS Year         OLS
Variable             Without Air    With Air      February

Intercept               7.22          7.381         8.052
                     (0.093) ***   (0.098) ***   (0.343) ***
ACRES                   0.405         0.408         0.444
                     (0.01) ***    (0.01) ***    (0.034) ***
AGE                    -0.005        -0.005        -0.005
                       (0) ***       (0) ***     (0.001) ***
CHILD99                -0.367        -0.401        -0.257
                     (0.021) ***   (0.024) ***   (0.074) ***
FIRE                    0.031         0.031         0.024
                     (0.003) ***   (0.003) ***   (0.009) ***
GARAGE1                 0.072         0.071         0.096
                     (0.005) ***   (0.005) ***   (0.017) ***
HOUCH99                -0.002        -0.002        -0.005
                     (0.001) ***   (0.001) ***    (0.002) *
LMEDINC                 0.047         0.043        -0.035
                     (0.008) ***   (0.008) ***     -0.026
LMONCO                               -0.017        -0.024
                                   (0.002) ***   (0.008) ***
LPM10                                -0.022        -0.039
                                   (0.004) ***   (0.016) **
LSQFT                   0.558         0.557         0.578
                     (0.005) ***   (0.005) ***   (0.018) ***
NEWCOM99                -0.33         -0.27        -0.084
                     (0.068) ***   (0.069) ***     (0.214)
OVER2099               -0.334        -0.316        -0.413
                     (0.029) ***   (0.029) ***   (0.099) ***
POOL1                   0.084         0.085         0.076
                     (0.003) ***   (0.003) ***   (0.012) ***
POPCH99                 0.003         0.003         0.005
                     (0.001) ***   (0.001) ***   (0.002) **
STORIES                -0.032        -0.033        -0.041
                     (0.003) ***   (0.003) ***   (0.011) ***
TEST98                  0.002         0.002         0.002
                       (0) ***       (0) ***       (0) ***
TIME                      0             0           0.001
                       (0) ***       (0) ***       (0) ***
VACANT98               -0.417        -0.598        -0.433
                     (0.169) **    (0.169) ***     (0.554)
Adjusted [R.sup.2]      0.761         0.762         0.784
Root MSE                0.155         0.155          0.1
DW                      1.623         1.632         1.686
Variance

SP(SPH)

Residual

Neg2LogLik
Parms
AIC
Number of              15,718        15,718         1289
  observations

                         OLS        MLE (SPH)     MLE (SPH)
Variable                June        February        June

Intercept               7.38          8.26          7.488
                     (0.297) ***   (0.427) ***   (0.429) ***
ACRES                   0.441         0.43          0.439
                     (0.044) ***   (0.037) ***   (0.046) ***
AGE                    -0.005        -0.005        -0.005
                       (0) ***     (0.001) ***   (0.001) ***
CHILD99                -0.321        -0.307        -0.517
                     (0.072) ***   (0.098) ***   (0.11) ***
FIRE                    0.029         0.02          0.025
                     (0.007) ***   (0.008) **    (0.007) ***
GARAGE1                 0.06          0.084         0.061
                     (0.014) ***   (0.018) ***   (0.014) ***
HOUCH99                -0.001        -0.003        -0.001
                       (0.002)       (0.003)       (0.004)
LMEDINC                 0.031        -0.021         0.056
                       (0.023)       (0.034)       (0.035)
LMONCO                 -0.022        -0.027        -0.034
                     (0.006) ***    (0.01) **    (0.01) ***
LPM10                  -0.066        -0.042        -0.053
                     (0.016) ***    (0.02) **    (0.026) **
LSQFT                   0.587         0.533         0.54
                     (0.016) ***   (0.019) ***   (0.017) ***
NEWCOM99               -0.258        -0.037        -0.319
                       (0.213)       (0.309)       (0.330)
OVER2099               -0.374        -0.391         -0.39
                     (0.087) ***   (0.13) ***    (0.132) ***
POOL1                   0.09          0.071         0.09
                     (0.009) ***   (0.011) ***   (0.009) ***
POPCH99                 0.002         0.004         0.001
                       (0.002)       (0.003)       (0.004)
STORIES                 -0.04        -0.025        -0.031
                     (0.009) ***    (0.01) **    (0.009) ***
TEST98                  0.003         0.002         0.003
                       (0) ***     (0.001) ***   (0.001) ***
TIME                      0           0.001           0
                         (0)         (0) **          (0)
VACANT98               -0.335         -0.56        -0.362
                       (0.488)       (0.753)       (0.767)
Adjusted [R.sup.2]      0.787
Root MSE                 0.1
DW                      1.704
Variance                              0.021         0.011
                                    0.001 ***        ***
SP(SPH)                               0.381         1.04
                                    0.033 ***        ***
Residual                              0.004         0.011
                                   0.0004 ***        ***
Neg2LogLik                            -1374         -1860
Parms                                   3             3
AIC                                   -1368         -1854
Number of               1756          1289          1756
  observations

Standard errors in parentheses.

** Significant at the 5% level.

*** Significant at the 1% level.

Table 5. Characteristics of Parametric and Bootstrapped Sampling
Distributions for Air Quality Regression Coefficients

                                   Sample
                                    Size      Method of
Regression Method       B        in Each B    Estimation

OLS Full Data Set            1     15,719     Parametric
OLS                        550        750     Bootstrap
NILE (EXP)                 550        750     Bootstrap
MLE (GAU)                  550        750     Bootstrap
NILE (SPH)                 550        750     Bootstrap
OLS Full Data Set            1     15,719     Parametric
OLS                        550        750     Bootstrap
MLE (EXP)                  550        750     Bootstrap
MLE (GAU)                  550        750     Bootstrap
MLE (SPH)                  550        750     Bootstrap

                       Air
                     Quality      [??] or
Regression Method    Variable     E([??])       Median

OLS Full Data Set     LMONCO      -0.0172        n/a
OLS                   LMONCO      -0.0248       -0.023
NILE (EXP)            LMONCO      -0.0257      -0.0251
MLE (GAU)             LMONCO      -0.0265      -0.0259
NILE (SPH)            LMONCO      -0.0267       -0.026
OLS Full Data Set     LPM10       -0.0221        n/a
OLS                   LPM10       -0.0327      -0.0311
MLE (EXP)             LPM10       -0.0420      -0.0421
MLE (GAU)             LPM10       -0.0365      -0.0358
MLE (SPH)             LPM10       -0.0377      -0.0369

                     Standard
Regression Method     Error          P5          P95

OLS Full Data Set     0.0023      -0.0216      -0.0127
OLS                   0.0005      -0.0475      -0.0084
NILE (EXP)            0.0006      -0.0510      -0.0056
MLE (GAU)             0.0006      -0.0517      -0.0070
NILE (SPH)            0.0006      -0.0507      -0.0077
OLS Full Data Set     0.0041      -0.0300      -0.0141
OLS                   0.0012      -0.0777       0.0085
MLE (EXP)             0.0012      -0.0845       0.0095
MLE (GAU)             0.0012      -0.0866       0.0111
MLE (SPH)             0.0012      -0.0814       0.0096

Table 6. OLS Versus Kriged MLE (SPH) Prediction Accuracy Using
550 Block Bootstrapped Data Sets: Percentage of the Time, Kriged
MLE Prediction Error < OLS Prediction Error

Month        Mean     Median    Standard Error

January     0.9049    0.9070        0.0062
February    0.9305    0.9394        0.0049
March       0.9039    0.9178        0.0064
April       0.9032    0.9167        0.0065
May         0.903     0.9059        0.0066
June        0.9004    0.9056        0.0063
July        0.904     0.9130        0.0063
August      0.9125    0.9143        0.0060
September   0.8892    0.9000        0.0072
October     0.8985    0.9024        0.0069
November    0.8857    0.8889        0.0070

Month       Minimum   Maximum    B Solutions

January     0.8000    0.9744          49
February    0.8286    0.9773          50
March       0.8000    0.9750          48
April       0.8000    0.9750          49
May         0.8000    0.9750          48
June        0.7647    0.9756          50
July        0.7568    0.9730          49
August      0.8000    0.9750          48
September   0.7838    0.9744          49
October     0.7838    0.9706          47
November    0.7857    0.9697          50

Not all maximum likelihood attempts converged on a solution.
Consequently, there is a discrepancy between number of samples
examined (550) and reported (537).