The Improbable Nature of the Implied Correlation Matrix from Spatial Regression Models

Spatial lag dependence in a regression model is similar to the inclusion of a serially autoregressive term for the dependent variable in time-series context. However, unlike in the time series model, the implied covariance structure matrix from the spatial autoregressive model can have a very counterintuitive and improbable structure. A single value of spatial auto correlation parameter can imply a large band of values of pair-wise correlations among different observations of the dependent variable, when the weight matrix for the spatial model is specified exogenously. We illustrate this using cigarette sales data (1963-92) of 46 US states. We observe that two "close" neighbors can have very low implied correlations compared to distant neighbors when the weighting scheme is the first-order contiguity matrix. However, if the weight matrix can capture the underlying dependence structure of the observations then this unintuitive behavior of implied correlation gets corrected to a large extent. Keeping this in mind, we explore the possibility of constructing the weight matrix (or the overall spatial dependence in the data) that is consistent with the underlying correlation structure of the dependent variable. The results using our suggested procedure are very encouraging.