Inference when a nuisance parameter is not identified under the null hypothesis

Many econometric testing problems involve nuisance parameters which are not identified under the null hypotheses. This paper studies the asymptotic distribution theory for such tests. The asymptotic distributions of standard test statistics are described as functionals of chi-square processes. In general, the distributions depend upon a large number of unknown parameters. We show that a transformation based upon a conditional probability measure yields an asymptotic distribution free of nuisance parameters, and we show that this transformation can be easily approximated via simulation. The theory is applied to threshold models, with special attention given to the so-called self-exciting threshold 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 series can be possibly explained by sampling variation.