Regression analysis in a spatial-temporal context: Least squares, generalized least squares, and the use of the bootstrap

In many ecological studies, data are collected at different locations over time, with interest in building a regression model for how one variable depends on its past values and current values of a predictor variable. Our motivation is an example in vestigating the effects of mouse populations on the level of gypsy moth populations within a model accounting for the previous year’s gypsy moth values. With such data, the error term will of ten contain autocorrelationover time as well as contemporaneouscorrelation over series at a fixed time. A common approach in practice is the use of simple least squares or the use of two-stage generalized least squares. The standard errors and tests that are typically used either ignore the correlationin the errorsor, if they do account for the errorstructure, ignore the uncertainty that comes from estimating the correlation structure. There are also con cerns regarding bias in the estimated coefficients and variance/correlation parameters. We in vestigate these issues mainly in the context of our example, which invol ves 8 location and 10 years of data, relying on simulations due to the absence of exact small-sample results. Among other results, we find the potential for gross underestimation of the standard errors of the coefficients with the use of standard generalized least squares methods. A bootstrap method is developed for this context, applied to the example, and evaluated via simulations. The bootstrap estimates of standard errors performed rather well in the settings we examined, supporting the use of the bootstrap in the analysis of our example and suggesting the potential of the bootstrap more generally.