On block bootstrapping areal data

Inference for random processes depends on the estimation of the long-run variance of a statistic. Commonly used approaches in time series analysis for estimating the long-run variance include bootstrap approaches and the class of heteroskedasticity and autocorrelation consistent covariance estimators. These approaches have been generalized to deal with point processes and continuous spatial processes, but areal data have received only scant attention as of yet. This paper does not provide any new theoretical results, but provides a heuristic argument why these methods might work with census-type data. This argument is quite trivial, but we can provide no evidence of the application of these methods in the literature. Preliminary results demonstrating the simple application of block bootstrap methods to these data are presented. The two primary ways of estimating the long-run variance of a mean are i) heteroskedasticity and autocorrelation (HAC) variance estimators and ii) bootstrap and subsampling methods that resample the data in blocks that are large enough to preserve much of the dependence in the original dataset. The HAC methodology consists of a weighted sum of the empirical covariance matrix – assigning weights in such a manner that positive definiteness is ensured (Newey and West, 1987). Importantly for the current paper, this method may be alternatively represented as an estimator for the spectral density of the process at the zero frequency. This method has seen only limited application in spatial settings to date. Conley (1999) proposes the method for application to stationary data located on a lattice. Anselin (2002), however, suggests that this method is inappropriate for areal data on a lattice since the stationarity assumption is not tenable in many circumstances. A HAC estimator for areal data has been proposed recently by Kelejian and Prucha (2007) (KP henceforth). That estimator too presupposes that the data are represented on a lattice, but does not place strong stationarity assumptions on the data. Rather than assuming stationarity directly, Kelejian and Prucha instead assume that the sum of the covariance matrix is appropriately bounded. The KP estimator, however, presupposes that the areal sampling units can be situated on a lattice with an appropriate (not necessarily Euclidean) metric characterizing the distance between areal units. Kelejian and Prucha further demonstrate that the estimator remains consistent if the lattice metric contains measurement error. In contrast to the HAC estimators, the bootstrap methods for estimating the long-run variance rely on sampling large, contiguous blocks of data, so that the dependence structure is preserved within each block. Bühlmann and Künsch (1999) show that the block bootstrap variance estimators are asymptotically equivalent to weighted periodogram estimators of the spectral density at the zero frequency. The bootstrap has been studied much more that the HAC variance estimator in the context