This paper addresses the issue of inference in econometric models where nuisance parameters are not identified under the null hypothesis. The asymptotic distributions of test statistics, such as Wald, likelihood ratio (LR), and Lagrange multiplier (LM) statistics, are derived for parametric econometric estimators under general assumptions, allowing for simultaneous equations, stochastic regressors, heterogeneity, and weak dependence. The asymptotic distributions are shown to be represented by the supremum of a chi-square process, which is a quadratic form in a vector Gaussian process indexed by the nuisance parameter. However, these distributions generally depend on a large number of unknown parameters, making tabulation impractical. To address this, a simulation method is proposed to approximate the asymptotic null distribution, which is an improvement over the bounds of Davies (1977, 1987), whose approximation error increases with sample size in many cases of interest. The paper is organized into several sections. Section 2 provides examples of non-identified nuisance parameters from recent literature. Section 3 introduces the concepts of global and pointwise estimates, and conditions for consistent pointwise estimation of structural parameters. Section 4 develops a theory for testing structural hypotheses when the nuisance parameter is not identified under the null hypothesis, examining LR, Wald, LM, and maximal pointwise Wald and LM test statistics. Section 5 presents the asymptotic distribution theory for these test statistics, showing that they are functionals of chi-square processes. Section 6 develops the simulation method for approximating the null asymptotic distribution. Section 7 extends the results to t-statistics. Section 8 applies the theory to threshold models, specifically a threshold autoregressive model of GNP growth rates, and reports formal statistical tests supporting the claim that there is a statistically significant threshold effect in a univariate autoregression for U.S. GNP growth rates.This paper addresses the issue of inference in econometric models where nuisance parameters are not identified under the null hypothesis. The asymptotic distributions of test statistics, such as Wald, likelihood ratio (LR), and Lagrange multiplier (LM) statistics, are derived for parametric econometric estimators under general assumptions, allowing for simultaneous equations, stochastic regressors, heterogeneity, and weak dependence. The asymptotic distributions are shown to be represented by the supremum of a chi-square process, which is a quadratic form in a vector Gaussian process indexed by the nuisance parameter. However, these distributions generally depend on a large number of unknown parameters, making tabulation impractical. To address this, a simulation method is proposed to approximate the asymptotic null distribution, which is an improvement over the bounds of Davies (1977, 1987), whose approximation error increases with sample size in many cases of interest. The paper is organized into several sections. Section 2 provides examples of non-identified nuisance parameters from recent literature. Section 3 introduces the concepts of global and pointwise estimates, and conditions for consistent pointwise estimation of structural parameters. Section 4 develops a theory for testing structural hypotheses when the nuisance parameter is not identified under the null hypothesis, examining LR, Wald, LM, and maximal pointwise Wald and LM test statistics. Section 5 presents the asymptotic distribution theory for these test statistics, showing that they are functionals of chi-square processes. Section 6 develops the simulation method for approximating the null asymptotic distribution. Section 7 extends the results to t-statistics. Section 8 applies the theory to threshold models, specifically a threshold autoregressive model of GNP growth rates, and reports formal statistical tests supporting the claim that there is a statistically significant threshold effect in a univariate autoregression for U.S. GNP growth rates.