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

Abstract

Spatial lag dependence in a regression model is similar to the inclusion of a serially autoregressive term for the dependent variable in a 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 autocorrelation 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. This is illustrated using cigarette sales data (1963-1992) of 46 US states. It can be seen that that two close neighbours can have very low implied correlations compared to distant neighbours 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 behaviour of implied correlation is corrected to a large extent. From this, 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 is explored. The suggested procedures produced very positive results indicating further research.