Linear Wald Methods for Inference on Covariances and Weak Exogeneity Tests in Structural Equations

Inference about the vector of covariances between the stochastic explanatory variables and the disturbance term of a structural equation is an important problem in econometrics. For example, one may wish to test the independence between stochastic explanatory variables and the disturbance term. Tests for the hypothesis of independence between the full vector of stochastic explanatory variables and the disturbance have been proposed by several authors. When more than one stochastic explanatory variable is involved, it can be of interest to determine whether all of them are independent of the disturbance and, if not, which ones are. We develop simple large-sample methods which allow us to construct confidence regions and test hypotheses concerning any vector of linear transformations of the covariances between the stochastic explanatory variables and the disturbance of a structural equation. The main method described is a generalized Wald procedure which simply requires two linear regressions. No nonlinear estimation is needed. Consistent tests for weak exogeneity hypotheses are derived as special cases.