Selecting the W matrix : parametric vs nonparametric approaches

In spatial econometrics, it is customary to specify a weighting matrix, the so-called W matrix, just choosing one matrix from the different types of matrices a user is considering (Anselin, 2002). In general, this selection is made a priori, depending on the user's judgment. This decision is extremely important because if matrix W is miss-specified in some way, parameter estimates are likely to be biased and they will be inconsistent in models that contain some spatial lag. Also, for models without spatial lags but where the random terms are spatially autocorrelated, the obtaining of robust standard estimates of the errors will be incorrect if W is miss-specified. Goodness-of-fit tests may be used to chose between alternative specifications of W. Although, in practice, most users impose a certain W matrix without testing for the restrictions that the selected spatial operator implies. In this paper, we aim to establish a nonparametric procedure where the chosen by objective criteria. Our proposal is directly related with the Theory of Information. Specifically, the selection criterion that we propose is based on objective information existing in the data, which does not depend on the investigator's subjectivity: it is a measure of conditional entropy. We compare the performance of our criteria against some other alternative like the J test of Davidson and McKinnon or a likelihood ratio obtained in a maximum likelihood framework.