Sample Splitting and Threshold Estimation
样本划分与门槛估计AI 解读 ↓
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样本划分与门槛估计
- 研究问题
- 当样本按连续变量划分为不同回归区间时，如何刻画门槛及斜率估计的统计性质，并构造可用于推断的置信区间？
- 方法与思路
- 作者在横截面或时间序列回归中采用最小二乘估计，推导门槛与斜率的渐近分布，并通过反演似然比统计量构造置信区间。
- 材料陈述的贡献
- 摘要称门槛估计的极限分布非标准，所构造置信区域渐近保守，并以蒙特卡洛模拟和多重均衡增长模型应用考察该理论。
- 阅读关注点
- 阅读全文时应核对门槛变量与误差项的条件、非标准极限的适用范围，以及似然比反演方法和模拟中的置信区间覆盖表现。
- 阅读边界
- 摘要未说明渐近理论的具体条件、模拟覆盖率或应用估计结果，因而不能判断有限样本表现或增长模型的经济结论。
原文摘要
Threshold models have a wide variety of applications in economics. Direct applications include models of separating and multiple equilibria. Other applications include empirical sample splitting when the sample split is based on a continuously-distributed variable such as firm size. In addition, threshold models may be used as a parsimonious strategy for nonparametric function estimation. For example, the threshold autoregressive model (TAR) is popular in the nonlinear time series literature. Threshold models also emerge as special cases of more complex statistical frameworks, such as mixture models, switching models, Markov switching models, and smooth transition threshold models. It may be important to understand the statistical properties of threshold models as a preliminary step in the development of statistical tools to handle these more complicated structures. Despite the large number of potential applications, the statistical theory of threshold estimation is undeveloped. It is known that threshold estimates are super-consistent, but a distribution theory useful for testing and inference has yet to be provided. This paper develops a statistical theory for threshold estimation in the regression context. We allow for either cross-section or time series observations. Least squares estimation of the regression parameters is considered. An asymptotic distribution theory for the regression estimates (the threshold and the regression slopes) is developed. It is found that the distribution of the threshold estimate is nonstandard. A method to construct asymptotic confidence intervals is developed by inverting the likelihood ratio statistic. It is shown that this yields asymptotically conservative confidence regions. Monte Carlo simulations are presented to assess the accuracy of the asymptotic approximations. The empirical relevance of the theory is illustrated through an application to the multiple equilibria growth model of Durlauf and Johnson (1995).