Spatial Heteroskedasticity and Autocorrelation Consistent Estimation of Covariance Matrix

This paper considers spatial heteroskedasticity and autocorrelation consistent (spatial HAC) estimation of covariance matrices of parameter estimators. We generalize the spatial HAC estimators introduced by Kelejian and Prucha (2007) to apply to linear and nonlinear spatial models with moment conditions. We establish its consistency, rate of convergence and asymptotic truncated mean squared error (MSE). Based on the asymptotic truncated MSE criterion, we derive the optimal bandwidth parameter and suggest its data dependent estimation procedure using a parametric plug-in method. The …nite sample performances of the spatial HAC estimator are evaluated via Monte Carlo simulation. Keywords: Asymptotic mean squared error, Heteroskedasticity and autocorrelation, Covariance matrix estimator, Optimal bandwidth choice, Robust standard error, Spatial dependence. JEL Classi…cation Number : C13, C14, C21 Email: [email protected] and [email protected]. Correspondence to: Department of Economics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0508. We thank Graham Elliott Dimitris Politis and Hal White for helpful comments. 1 Introduction This paper studies spatial heteroskedasticity and autocorrelation consistent (HAC) estimation of covariance matrices of parameter estimators. As heteroskedasticity is a well known feature of cross sectional data (e.g. White (1980)), spatial dependence is also a common property due to interactions among economic agents. Therefore, robust inference in presence of heteroskedasticity and spatial dependence is an important problem in spatial data analysis. The …rst discussion of spatial HAC estimation is Conley (1996, 1999). He proposes a spatial HAC estimator based on the assumption that each observation is a realization of a random process, which is stationary and mixing, at a point in a two-dimensional Euclidean space. Conley and Molinari (2007) examine the performance of this estimator using Monte Carlo simulation. Their results show that inference is robust to the measurement error in locations. Robinson (2005) considers nonparametric kernel spectral density estimation for weakly stationary processes on a d-dimensional lattice. Kelejian and Prucha (2007, hereafter KP) also develop a spatial HAC estimator. As in many empirical studies, they model spatial dependence in terms of a spatial weighting matrix. The di¤erence is that the weighting matrix is not assumed to be known and is not parametrized. Typical examples of this type of processes include the spatial autoregressive processes and spatial moving average processes. Local nonstationarity and heteroskedasticity are built-in features of these type of processes. This is in sharp contrast with Conley (1996, 1999) and Robinson (2005) in which the process is assumed to be stationary. KP employ an economic distance to characterize the decaying pattern of the spatial dependence. The covariance of random variables at locations i and j is a function of dij;n, the economic distance between them. As the economic distance increases, the covariance decreases in absolute value and vice versa. The existence of such an economic distance enables us to use the kernel method for the standard error estimation. The estimator is a weighted sum of sample covariances with weights depending on the relative distances, that is, dij;n=dn for some bandwidth parameter dn: We generalize the spatial HAC estimator proposed by KP to be applicable to general linear and nonlinear spatial models and establish its asymptotic properties. We provide the conditions for consistency and the rate of convergence. Let E`n denote the mean of the average number of pseudo-neighbors. By de…nition, two units are pseudo-neighbors if their distance is less than dn: We show that the spatial HAC estimator is consistent if E`n = o(n) and dn ! 1 as n ! 1. This result implies that the rate of convergence of the estimator is E`n=n. Comparing our results with Andrews (1991), we …nd that the properties of the spatial HAC estimator we consider are interestingly parallel to those of the time series HAC estimator, even though they assume di¤erent DGPs and have di¤erent dependence structures. We decompose the di¤erence of the spatial HAC estimator from the true covariance matrix into three parts. The …rst part is due to the estimation error of model parameters and the second and third parts are bias and variance terms even if the model parameters are known. We derive the asymptotic bias and variance and show that the estimation error vanishes faster than the other two terms under some regularity conditions. As a result, the truncated Mean Squared Error (MSE) of the spatial HAC estimator is dominated by the bias and variance terms. This key result provides us the opportunity to select the bandwidth parameter to balance the asymptotic squared bias with variance. We …nd