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

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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.