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Nonlinear Hypotheses, Inequality Restrictions, and Non-Nested Hypotheses: Exact Simultaneous Tests in Linear Regressions

Jean-Marie Dufour

Abstract

In the context of the classical linear model, the problem of comparing two arbitrary hypotheses on the regression coefficients is considered. Problems involving nonlinear hypotheses, inequality restrictions, or non-nested hypotheses are included as special cases. Exact bounds on the null distribution of likelihood ratio statistics are derived. The bounds are based on the central Fisher distribution and are very easy to use. In an important special case, a bounds test similar to the Durbin-Watson test is proposed. Multiple testing problems are also studied: the bounds obtained for a single pair of hypotheses are shown to enjoy a simultaneity property that allows one to combine any number of tests. This result extends to nonlinear hypotheses a well-known result given by Scheffe for linear hypotheses. A method of building bounds induced tests is also suggested.