This paper addresses the problem of inference in econometric models where a nuisance parameter is not identified under the null hypothesis. It studies the asymptotic distribution theory for such problems, showing that the asymptotic distributions of test statistics are functionals of chi-square processes. These distributions depend on a large number of unknown parameters, making them difficult to tabulate. A simulation method is proposed to approximate the asymptotic null distribution, improving upon previous bounds. The method is applied to a threshold autoregressive model for GNP growth rates, providing statistical tests that support the presence of a statistically significant threshold effect. The paper discusses various examples of models with unidentified nuisance parameters, including additive non-linearity, Box-Cox transformations, structural change models, threshold models, two-state Markov trend models, common ARMA roots, non-expected utility models, and others. It introduces the concepts of global and pointwise estimates, and discusses the consistency of pointwise estimates under certain conditions. The paper develops a theory for testing structural hypotheses when the nuisance parameter is not identified under the null hypothesis. It examines likelihood ratio, Wald, and Lagrange multiplier (LM) statistics, and shows that only the maximal pointwise Wald and LM statistics have asymptotic distributions robust to heteroskedasticity and serial correlation. The paper also extends the results to t-tests and discusses the distributional theory under uncorrelated errors, homoskedasticity, and autocorrelation. A simulation method is proposed to approximate the asymptotic null distribution, which is shown to be effective in determining critical values for hypothesis testing.This paper addresses the problem of inference in econometric models where a nuisance parameter is not identified under the null hypothesis. It studies the asymptotic distribution theory for such problems, showing that the asymptotic distributions of test statistics are functionals of chi-square processes. These distributions depend on a large number of unknown parameters, making them difficult to tabulate. A simulation method is proposed to approximate the asymptotic null distribution, improving upon previous bounds. The method is applied to a threshold autoregressive model for GNP growth rates, providing statistical tests that support the presence of a statistically significant threshold effect. The paper discusses various examples of models with unidentified nuisance parameters, including additive non-linearity, Box-Cox transformations, structural change models, threshold models, two-state Markov trend models, common ARMA roots, non-expected utility models, and others. It introduces the concepts of global and pointwise estimates, and discusses the consistency of pointwise estimates under certain conditions. The paper develops a theory for testing structural hypotheses when the nuisance parameter is not identified under the null hypothesis. It examines likelihood ratio, Wald, and Lagrange multiplier (LM) statistics, and shows that only the maximal pointwise Wald and LM statistics have asymptotic distributions robust to heteroskedasticity and serial correlation. The paper also extends the results to t-tests and discusses the distributional theory under uncorrelated errors, homoskedasticity, and autocorrelation. A simulation method is proposed to approximate the asymptotic null distribution, which is shown to be effective in determining critical values for hypothesis testing.