This paper develops a method for performing inference using spatially dependent data. We consider test statistics formed using nonparametric covariance matrix estimators that account for heteroskedasticity and spatial correlation (spatial HAC). We provide distributions of commonly used test statistics under “fixed-b” asymptotics, in which HAC smoothing parameters are proportional to the sample size. Under this sequence, spatial HAC estimators are not consistent but converge to nondegenerate limiting random variables that depend on the HAC smoothing parameters, the HAC kernel, and the shape of the spatial region in which the data are located. We illustrate the performance of the “fixed-b” approximation in the spatial context through a simulation example.

We develop and apply bootstrap methods for diffusion models when fitted to the long run as characterized by the stationary distribution of the data. To obtain bootstrap refinements to statistical inference, we simulate candidate diffusion processes. We use these bootstrap methods to assess measurements of local mean reversion or “pull” to the center of the distribution for short-term interest rates. We also use them to evaluate the fit of the model to the empirical density.

This article uses evidence from the manuscripts of the 1860 federal census to analyze the wealth of adult males in Boston, New York, Chicago, and Indianapolis. Previous multivariate analyses of wealth from the census have been flawed by reliance on ordinary least squares; we instead use quantile regression. Immigrants fared considerably better in the Midwest than the East: immigrants in the midwestern cities held more wealth than their eastern counterparts, both absolutely and relative to the native-born in their respective cities. We explore the causes of these differences and their consequences for nineteenth-century Americans and their communities.