Sample splitting and threshold estimation

Threshold models have a wide variety of applications in economics. Direct a pplications include models of separating and multiple equilibria. Other app lications include empirical sample splitting when the sample split is based on a continuously-distributed variable such as firm size. In addition, thr eshold models may be used as a parsimonious strategy for nonparametric func tion estimation. For example, the threshold autoregressive model (TAR) is p opular in the nonlinear time series literature. Threshold models also emerge as special cases of more complex statistical f rameworks, such as mixture models, switching models, Markov switching model s, and smooth transition threshold models. It may be important to understan d the statistical properties of threshold models as a preliminary step in t he development of statistical tools to handle these more complicated struct ures. Despite the large number of potential applications, the statistical theory of threshold estimation is undeveloped. It is known that threshold estimate s are super-consistent, but a distribution theory useful. for testing and i nference has yet to be provided. This paper develops a statistical theory for threshold estimation in the re gression context. We allow for either cross-section or time series observat ions. Least squares estimation of the regression parameters is considered. An asymptotic distribution theory for the regression estimates (the thresho ld and the regression slopes) is developed. It is found that the distributi on of the threshold estimate is nonstandard. A method to construct asymptot ic confidence intervals is developed by inverting the likelihood ratio stat istic. It is shown that this yields asymptotically conservative confidence regions. Monte Carlo simulations are presented to assess the accuracy of th e asymptotic approximations. The empirical relevance of the theory is illus trated through an application to the multiple equilibria growth model of Du rlauf and Johnson (1995).