This paper develops a statistical theory for threshold estimation in regression models, focusing on both cross-section and time series data. The author addresses the challenge of estimating threshold models, which are widely used in economics for modeling separating and multiple equilibria, empirical sample splitting, and nonparametric function estimation. The paper introduces the threshold autoregressive (TAR) model as an example of a threshold model and discusses its applications in nonlinear time series analysis. The main contribution of the paper is the development of an asymptotic distribution theory for the threshold estimate and the likelihood ratio statistic used for hypothesis testing. The author shows that the distribution of the threshold estimate is nonstandard but can be approximated using a nonstandard Brownian motion process. This allows for the construction of asymptotically valid confidence intervals for the threshold parameter by inverting the likelihood ratio statistic. The paper also provides Monte Carlo simulations to assess the accuracy of the asymptotic approximations and applies the theory to a multiple equilibria growth model. The results demonstrate that the confidence intervals constructed using the proposed methods are asymptotically conservative, even when the threshold effect is relatively large. Overall, the paper provides a comprehensive framework for threshold estimation and testing, filling a gap in the existing statistical literature on threshold models.This paper develops a statistical theory for threshold estimation in regression models, focusing on both cross-section and time series data. The author addresses the challenge of estimating threshold models, which are widely used in economics for modeling separating and multiple equilibria, empirical sample splitting, and nonparametric function estimation. The paper introduces the threshold autoregressive (TAR) model as an example of a threshold model and discusses its applications in nonlinear time series analysis. The main contribution of the paper is the development of an asymptotic distribution theory for the threshold estimate and the likelihood ratio statistic used for hypothesis testing. The author shows that the distribution of the threshold estimate is nonstandard but can be approximated using a nonstandard Brownian motion process. This allows for the construction of asymptotically valid confidence intervals for the threshold parameter by inverting the likelihood ratio statistic. The paper also provides Monte Carlo simulations to assess the accuracy of the asymptotic approximations and applies the theory to a multiple equilibria growth model. The results demonstrate that the confidence intervals constructed using the proposed methods are asymptotically conservative, even when the threshold effect is relatively large. Overall, the paper provides a comprehensive framework for threshold estimation and testing, filling a gap in the existing statistical literature on threshold models.