Classical threshold models assume that threshold values are constant and stable, which appears overly restrictive and unrealistic. In this article, we extend Hansen's (2000) constant threshold regression model by allowing for a time-varying threshold which is approximated by a Fourier function. Least-square estimation of regression slopes and the time-varying threshold is proposed, and test statistics for the existence of threshold effect and threshold constancy are constructed. We also develop the asymptotic distribution theory for the time-varying threshold estimator. Through Monte Carlo simulations, we show that the proposed estimation and testing procedures work reasonably well in finite samples, and there is little efficiency loss by the allowance for Fourier approximation in the estimation procedure even when there is no time-varying feature in the threshold. On the contrary, the slope estimates are seriously biased when the threshold is time-varying but being treated as a constant. The model is illustrated with an empirical application to a nonlinear Taylor rule for the United States.

中文翻译：

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基于傅里叶近似的时变阈值阈值模型

经典阈值模型假设阈值是恒定和稳定的，这显得过于严格和不切实际。在本文中，我们通过允许由傅立叶函数近似的时变阈值来扩展 Hansen (2000) 的恒定阈值回归模型。提出了回归斜率和时变阈值的最小二乘估计，并构建了阈值效应和阈值恒定性存在的检验统计量。我们还为时变阈值估计器开发了渐近分布理论。通过蒙特卡罗模拟，我们表明所提出的估计和测试程序在有限样本中工作得相当好，即使在阈值中没有时变特征时，估计过程中允许傅里叶近似的效率损失也很小。相反，当阈值随时间变化但被视为常数时，斜率估计值存在严重偏差。该模型通过对美国非线性泰勒规则的经验应用来说明。