Kung-Sik Chan of the University of Iowa, Bruce E. Hansen of the University of Wisconsin-Madison, and Allan Timmermann of the University of California, San Diego

This special issue of the Journal of Business & Economic Statistics (April) on regime switching and threshold models is motivated by the mounting empirical evidence of important nonlinearities in regression models commonly used to model the dynamics in macroeconomic and financial time series. Commonly cited examples include the very different behavior of second moments for many macroeconomic time series before and after the Great Moderation in the early eighties, the different behavior of U.S. interest rates during the Federal Reserve’s Monetarist Experiment from 1979–1982, and the behavior of a variety of risk indicators during the more recent global financial crisis. These are episodes that can be difficult to model in the context of standard linear regression models.

The key difference between Markov switching models and threshold models is that the former assume that the underlying state process that gives rise to the nonlinear dynamics (regime switching) is latent, whereas threshold models commonly allow the nonlinear effect to be driven by observable variables but assume the number of thresholds and the threshold values to be unknown. However, it is often overlooked that the general formulation of the threshold model includes the Markov switching model. Thus, these two classes of models share many common features. From an econometric perspective, both classes of models are affected by the presence of unidentified parameters under the null, which poses challenges to inference, including the number of thresholds (or regimes) and their location. Empirically, both types of models can, by design, allow for discrete, nonlinear effects.

The papers brought together in this special issue highlight both similarities and differences for threshold- and regime-switching models, offering many novel insights along methodological, computational, and empirical lines.

Luc Bauwens, Jean-Francois Carpantier, and Arnaud Dufays, in “Autoregressive Moving Average Infinite Hidden Markov Switching Models,” study a class of Markov switching models in which regime switches only affect some parameters, while other parameters remain the same across regimes. Limiting regime switches to a subset of the parameters can lead to simpler models with fewer unknown parameters and better out-of-sample forecasting performance. In particular, the authors propose to decouple the regime switching dynamics for the mean and variance parameters.

The methodology—developed by Bauwens, Carpantier, and Dufays—allows the number of regimes to be determined as part of the estimation process and so has no need to use extraneous criteria for selecting the number of regimes. Detailed empirical applications to quarterly U.S. GDP growth and monthly U.S. inflation show that the new class of “sticky infinite hidden Markov switching autoregressive moving average” models can lead to better forecasts than more conventional models. These findings are corroborated on a set of 18 additional macroeconomic variables.

In their paper “Forecasting Macroeconomic Variables Under Model Instability,” Davide Pettenuzzo and Allan Timmermann compare a range of methods in common use in macroeconomic forecasting for handling parameter instability. Specifically, the paper focuses on comparing and contrasting approaches that assume small but frequent changes to the model parameters (time-varying parameter models) versus models that assume rare, but large (discrete) breaks to the model parameters. The paper considers breaks in the parameters of both the first and second moments of the modeled process and studies their impact using predictive accuracy measures that focus on either the conditional mean or the entire probability distribution of the outcome.

In an empirical out-of-sample forecasting exercise for U.S. GDP growth and inflation, the authors find that models that allow for parameter instability generate more accurate density forecasts than constant-parameter models. Conversely, such models fail to produce better point forecasts. Overall, a model specification that allows for both time-varying parameters and stochastic volatility is found to perform best. Model combination methods also deliver gains in the performance of density forecasts, but fail to improve on the predictive accuracy of the time-varying parameter model with stochastic volatility. These results suggest that accounting for model instability can deliver better probability forecasts for key macroeconomic variables, whereas gains in predictive accuracy for traditional point forecasts are harder to come by.

Jesus Gonzalo and Jean-Yves Pitarakis, in “Inferring the Predictability Induced by a Persistent Regressor in a Predictive Threshold Model,” introduce a predictive regression model with threshold effects and use it to construct tests that have power to detect episodic predictability arising from a persistent predictor. The null hypothesis being tested in the paper is one of no predictability versus the alternative of predictability triggered by threshold effects associated with a particular predictor variable. The tests developed are easy to implement and robust to possible threshold effects for auxiliary predictors not of interest to the forecaster. Moreover, the proposed test statistic is robust to the presence of certain unidentified nuisance parameters. An empirical application to predictability of stock market returns by means of valuation ratios finds evidence that the predictive power of these variables is stronger around recessions and reveals state-dependence in return predictability.

Kung-Sik Chan, in “Testing for Threshold Diffusion,” studies continuous time diffusion processes that assume piece-wise linear drift and diffusion terms and develops a test for threshold nonlinearities in the drift of the process. In particular, Chan develops a quasi-likelihood test under the assumption that the diffusion term is constant, thus side-stepping the problem that, in general, the functional form of the diffusion term is unknown. A test for a single threshold is shown to have an asymptotic null distribution, which is a distribution of a functional of centered Gaussian processes. Chan develops ways to efficiently compute the p-value of the test statistic by bootstrapping its asymptotic null distribution. He also shows that the test statistic is consistent, derives its local power function, and extends the test to allow for multiple thresholds. Finally, simulations and empirical analysis of the term structure of U.S. interest rates are used to demonstrate the performance and usage of the test statistic.

Bruce Hansen, in “Regression Kink With an Unknown Threshold,” develops methods for estimation and inference in regression kink models that can have an unknown threshold. The class of regression kink models explored are threshold regressions required to be everywhere continuous, but can have a kink at an unknown threshold. Hansen develops a toolkit for inference and estimation, including ways to test for the presence of the threshold and for estimating model parameters and conducting inference on the regression parameters and, more broadly, on the regression function. Inference on the regression function is shown to be non-standard due to the nondifferentiability of the regression function with respect to the model parameters. Empirically, the paper applies its methods to the study of the possibly nonlinear (threshold) relationship between growth and debt using a long-span time-series for the United States.

Young-Joo Kim and Myung Huan Seo develop a procedure for testing for jumps in smooth transition processes in the paper, “Is There a Jump in the Transition?” The null model under the proposed test methodology is a threshold regression, while the alternative model is a smooth transition model. To conduct the test, the authors develop the asymptotic distribution of a quasi-Gaussian likelihood ratio statistic. The asymptotic distribution is defined as the maximum of a two-parameter Gaussian process that has a non-zero bias term. Kim and Seo show that asymptotic critical values can be tabulated and these depend on the transition function employed. Empirical critical values can be computed through simulations. The authors evaluate the finite sample performance of the test by means of Monte Carlo simulations and provide an empirical illustration through a model for the dynamics of racial segregation within cities across the United States.

The paper, “Sharp Threshold Detection Based on Sup-Norm Error Rates in High-Dimensional Models,” by Laurent Callot, Mehmet Caner, Anders Bredahl Koch, and Juan Anders Riquelme develops a high-dimensional threshold regression model. The authors propose a new threshold-scaled Lasso estimator suitable for this class of models. The authors’ main theoretical contribution is a new sup-norm bound on the estimation error. This bound can be used to provide sharper insights into variable selection properties. The authors also provide an empirical investigation into the impact of debt on GDP growth using a multi-country data set.

The paper, “Status Traps,” by Steven Durlauf, Andros Kourtellos, and Chih Ming Tan is an empirical application of threshold regression methods to study intergenerational mobility. The authors explore nonlinearities in children’s outcomes based on parental education and skills. The uncovered threshold processes imply persistent dynastic effects, which are interpreted as “status traps.” The empirical investigation is conducted with three distinct U.S. data sets (the PSID, NLSY, and administrative data), and the consistent findings across these data sets indicate the threshold effects are quite robust.

Biqing Cai, Jiti Gao, and Dag Tjostheim, in “A New Class of Bivariate Threshold Cointegration Models,” study nonlinear cointegration effects in the context of cointegrated vector autoregressive processes with threshold effects. The paper only imposes cointegration between the processes outside a compact region and shows cointegrating parameters converge at the conventional rate (T). In addition, the authors establish a faster convergence rate for the estimators of the remaining in the cointegrated region (T^1/2) than in the non-cointegrated region (T^1/4). The authors study the finite sample properties of the estimators in a Monte Carlo simulation. In an empirical application to a two-state bivariate model consisting of the federal funds rate and the three-month treasury bill rate, the authors show the cointegrating coefficients are identical across regimes, while the coefficients determining the short-run dynamics differ across regimes.

In “On Mixture Double Autoregressive Time Series Models,” Quodong Li, Qianqin Zhu, Zhao Liu, and Wai Keung Li study a class of mixture double autoregressive models whose mixing probabilities are allowed to vary over time. Double autoregressive processes allow autoregressive dynamics to affect both the conditional mean and the conditional variance as the heteroscedasticity of such processes are driven by past squared values of the process. Such processes resemble autoregressive models with ARCH dynamics. Li, Zhu, Liu, and Li establish stochastic properties for this class of processes, including conditions guaranteeing the existence of their moments. Further, they discuss methods for maximum likelihood estimation and inference with particular attention to the logistic mixture double autoregressive model. A simulation study and empirical example are used to illustrate the properties of the proposed model.

In the paper, “Inference for Heavy-Tailed and Multiple-Threshold Double Autoregressive Models,” Yaxing Yang and Shiqing Ling develop methods for conducting inference on a class of multi-threshold double autoregressive models that can have heavy tails. Specifically, the authors establish consistence and convergence properties of the estimators of the thresholds. Other (nonthreshold) parameter estimators are shown to be asymptotically normal. Methods for determining the number of thresholds and diagnostic tools are developed in the paper. Finally, Yang and Ling illustrate their approach in an empirical application for daily crude oil prices.

In “Threshold Estimation via Group Orthogonal Greedy Algorithm,” Ngai Hang Chan, Ching-Kang Ing, Yuanbo Li, and Chun Yip Yau develop a computationally efficient algorithm for estimating a self-exciting threshold autoregressive model with known autoregressive order and delay, but unknown number of thresholds. It is a three-step procedure that first uses a group orthogonal greedy algorithm (GOGA) to compute a solution path for screening potential thresholds, then applies a new high-dimensional information criterion and trimming procedure to eliminate spurious thresholds. The proposed method is shown to achieve consistent estimation of the thresholds at the rate of convergence, where is the sample size. Simulation experiments reported in the paper indicate the GOGA outperforms the group LASSO for estimating the thresholds. The authors illustrate their approach by revisiting the analysis of the U.S. real GNP data.