Testing inequality constraints in linear econometric models

This paper develops three asymptotically equivalent tests for examining the validity of imposing linear inequality restrictions on the parameters of linear econometric models. First we consider the model y = Xβ + ε, where ε is N(0, Σ), and the hypothesis test H: Rβ≥r versus K: β ϵ RK. Later we generalize this testing framework to the linear simultaneous equations model. We show that the joint asymptotic distribution of these test statistics and the test statistics from the hypothesis test H: Rβ = r versus K: Rβ ≥ r is a weighted sum of two sets of independent χ2-distributions. We also derive a useful duality relation between the multivariate inequality constraints test developed here and the multivariate one-sided hypothesis test. In small samples, these three test statistics satisfy inequalities similar to those derived by Berndt and Savin (1977) for the case of equality constraints. The paper also contains an illustrative application of this testing technique.