Lagrange-multiplier tests for weak exogeneity: a synthesis

This paper unifies two seemingly separate approaches to test weak exogeneity in dynamic regression models with Lagrange-multiplier statistics. The first class of tests focuses on the orthogonality between innovations and conditioning variables, and thus is related to the Durbin-Wu-Hausman specification tests. The second approach has been developed more recently in the context of cointegration and error correction models, and concentrates on the question whether the conditioning variables display error correcting behaviour. It is shown that the vital difference between the two approaches stems from the choice of the parameters of interest. A new test is derived, which encompasses both its predecessors. The test is applied to an error correction model of the demand for money in Switzerland.