Multiple Threshold Semiparametric Regression: Theory and Applications including the Effects of COVID-19

在经济学中有着广泛应用的非线性回归模型中最有趣的形式之一是阈值回归模型，其中样本分裂值（阈值参数）是未知的。也就是说，它在内部根据某个阈值决定因素将数据排序到每个服从相同模型的观测组中。阈值回归是简约的，它也允许增加灵活性的功能形式，而不容易受到维数灾难的问题。Chan（1993）首先证明了阈值估计量的渐近分布是复合泊松过程的泛函，由于它依赖于讨厌的参数，因此对于推论来说太复杂了。汉森（2000）在假设阈值效应随着样本的增加而变小的情况下，为阈值参数估计和回归斜率系数开发了一个更有用的渐近分布理论，而Caner和汉森（2004）在外生阈值变量框架下考虑了内生回归变量。Chen et al（2012）在参数自回归模型中将分析从一个单独的外源性阈值扩展到两个单独的外源性阈值。在这些模型的第二代中，Kourtellos，Stengos和Tan（2016）通过利用从有限因变量（内生虚拟）文献（例如，Heckman（1979））假设误差的联合正态性。Kourtellos，Stengos和Sun（2018）放松了正态性假设，并允许偏倚校正项的半参数结构。我们将扩展Kourtellos，Stengos和Sun（2018）的分析，以允许一个以上的内源性阈值。我们将首先推导出阈值和斜率估计量的属性，并通过广泛的蒙特卡罗模拟来分析它们的小样本属性。在应用方面，我们希望重新审视Kourtellos，Stengos和Tan（2013），并允许同时调查公共债务和外债，以使用我们新开发的方法实证探索这一重要方面，特别是考虑到新冠肺炎导致的新政府债务义务。更直接地说，当前的COVID-19大流行提供了一个重要案例，其中拟议的方法可用作一种新的和新颖的方式，以探索可能包括一个以上潜在阶段的时期内经济活动的阈值效应。失业率的变化可以建模为过去失业率变化以及报告的COVID-19病例变化和报告的死亡人数变化的函数，（作为政府遏制行动的功能，如严格的封锁）可以通过当地的阈值效应影响额外感染波的发生。可以在国家（省）或全球国家级别上进行分析，并且可以在国家（地方）省以及跨国家使用二元阈值回归模型进行分析。

One of the most interesting forms of nonlinear regression models with wide applications in economics is the threshold regression model, where the sample split value (threshold parameter) is unknown. That is, it internally sorts the data, on the basis of some threshold determinant, into groups of observations each of which obeys the same model. Threshold regression is parsimonious and it also allows for increased flexibility in functional form, without being susceptible to curse of dimensionality problems. Chan (1993) was the first to show that the asymptotic distribution of the threshold estimator is a functional of a compound Poisson process which is too complicated for inference as it depends on nuisance parameters. Hansen (2000) developed a more useful asymptotic distribution theory for both the threshold parameter estimate and the regression slope coefficients under the assumption that the threshold effect becomes smaller as the sample increases, while Caner and Hansen (2004) allowed for endogenous regressors, under an exogenous threshold variable framework. Chen et al (2012) extended the analysis from one to two separate exogenous thresholds within a parametric autorogressive model. In the second generation of these models Kourtellos, Stengos and Tan (2016) allow for an endogenous threshold variable by exploiting the intuition obtained from the limited dependent variable (endogenous dummy) literature (e.g., Heckman (1979)) assuming joint normality of the errors. Kourtellos, Stengos and Sun (2018) relax the normality assumption and allow for a semiparametric structure for the bias correction terms. We will extend the analysis of Kourtellos, Stengos and Sun (2018) to allow for more than one endogenous threshold. We will first derive the properties of the threshold and slope estimators and we will analyze their small sample properties by means of extensive Monte Carlo simulations. In terms of applications, we would like to revisit Kourtellos, Stengos and Tan (2013) and allow for a simultaneous investigation of both public and external debt to explore this important aspect empirically using our newly developed approach, especially given the new government debt obligations due to COVID-19. More directly, the current COVID-19 pandemic offers itself as an important case where the proposed methodology can be used as a new and novel way to explore threshold effects on economic activity during a period that may include more than one potential phase. Changes in unemployment can be modelled as functions of past unemployment changes as well as changes in reported COVID-19 cases and changes in reported deaths which (as functions of government containment actions such as strict lockdowns) can affect the occurrence of additional waves of infections through threshold effects at the local (provincial) or global country level and can be analyzed with a bivariate threshold regression model at a national (local) provincial as well as across countries.