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

在经济学中应用广泛的非线性回归模型中最有趣的形式之一是门限回归模型，其中样本分割值(门限参数)是未知的。也就是说，它根据某个阈值行列式在内部对数据进行分类，将数据分成观察组，每个观察组遵循相同的模型。阈值回归是节俭的，它还允许增加函数形式的灵活性，而不容易受到维度问题的影响。Chan(1993)首先证明了门限估计的渐近分布是复合Poisson过程的泛函，而复合Poisson过程依赖于干扰参数，因此难以推断。Hansen(2000)在门限效应随样本增加而变小的假设下，发展了一个更适用于门限参数估计和回归斜率系数的渐近分布理论，而Caner和Hansen(2004)在外生门限变量框架下允许内生回归变量。Chen等人(2012)在参数自回归模型中将分析从一个独立的外生阈值扩展到两个独立的外生阈值。在这些模型的第二代中，Kourtellos，Stengos和Tan(2016)通过利用从假设误差的联合正态的有限因变量(内生虚拟)文献(例如，Heckman(1979))获得的直觉来允许内生阈值变量。Kourtellos，Stengos和Sun(2018)放宽了正态假设，允许偏差修正项采用半参数结构。我们将扩展Kourtellos，Stengos和Sun(2018)的分析，以允许不止一个内生阈值。我们将首先推导阈值和斜率估计的性质，并通过广泛的蒙特卡罗模拟来分析它们的小样本性质。在应用方面，我们希望重新审视Kourtellos，Stengos和Tan(2013年)，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地探索这一重要方面，特别是考虑到新冠肺炎应承担的新政府债务义务。更直接的是，当前的新冠肺炎大流行是一个重要的案例，拟议的方法可以作为一种新的、新颖的方法来探索一段可能包括不止一个潜在阶段的时期对经济活动的门槛效应。失业率的变化可以模拟为过去失业率变化以及报告的新冠肺炎病例和报告死亡人数变化的函数，这些变化(作为严格封锁等政府遏制行动的函数)可以通过当地(省)或全球国家层面的阈值效应影响更多感染浪潮的发生，并可以在国家(地方)省以及各国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.