INFERENCE WHEN A NUISANCE PARAMETER IS NOT IDENTIFIED UNDER THE NULL HYPOTHESIS

Many econometric testing problems involve nuisance parameters which ar e not identified under the null hypotheses. This paper studies the asy mptotic distribution theory for such tests. The asymptotic distributio ns of standard test statistics are described as functionals of chi-squ are processes. In general, the distributions depend upon a large numbe r of unknown parameters. We show that a transformation based upon a co nditional probability measure yields an asymptotic distribution free o f nuisance parameters, and we show that this transformation can be eas ily approximated via simulation. The theory is applied to threshold mo dels, with special attention given to the so-called self-exciting thre shold autoregressive model. Monte Carlo methods are used to assess the finite sample distributions. The tests are applied to U.S. GNP growth rates, and we find that Potter's (1995) threshold effect in this seri es can be possibly explained by sampling variation.