Uniform Critical Values for Likelihood Ratio Tests in Boundary Problems

Abstract

Limit distributions of likelihood ratio statistics are well-known to be discontinuous in the presence of nuisance parameters at the boundary of the parameter space, which lead to size distortions when standard critical values are used for testing. In this paper, we propose a new and simple way of constructing critical values that yields uniformly correct asymptotic size, regardless of whether nuisance parameters are at, near or far from the boundary of the parameter space. Importantly, the proposed critical values are trivial to compute and at the same time provide powerful tests in most settings. In comparison to existing size-correction methods, the new approach exploits the monotonicity of the two components of the limiting distribution of the likelihood ratio statistic, in conjunction with rectangular confidence sets for the nuisance parameters, to gain computational tractability. Uniform validity is established for likelihood ratio tests based on the new critical values, and we provide illustrations of their construction in two key examples: (i) testing a coefficient of interest in the classical linear regression model with non-negativity constraints on control coefficients, and, (ii) testing for the presence of exogenous variables in autoregressive conditional heteroskedastic models (ARCH) with exogenous regressors. Simulations confirm that the tests have desirable size and power properties. A brief empirical illustration demonstrates the usefulness of our proposed test in relation to testing for spill-overs and ARCH effects.