Myung Hwan Seo

Testing for Threshold Effects in Regression Models

In this article, we develop a general method for testing threshold effects in regression models, using sup-likelihood-ratio (LR)-type statistics. Although the sup-LR-type test statistic has been considered in the literature, our method for establishing the asymptotic null distribution is new and nonstandard. The standard approach in the literature for obtaining the asymptotic null distribution requires that there exist a certain quadratic approximation to the objective function. The article provides an alternative, novel method that can be used to establish the asymptotic null distribution, even when the usual quadratic approximation is intractable. We illustrate the usefulness of our approach in the examples of the maximum score estimation, maximum likelihood estimation, quantile regression, and maximum rank correlation estimation. We establish consistency and local power properties of the test. We provide some simulation results and also an empirical application to tipping in racial segregation. This article has supplementary materials online.

The lasso for high-dimensional regression with a possible change-point

Summary We consider a high dimensional regression model with a possible change point due to a covariate threshold and develop the lasso estimator of regression coefficients as well as the threshold parameter. Our lasso estimator not only selects covariates but also selects a model between linear and threshold regression models. Under a sparsity assumption, we derive non‐asymptotic oracle inequalities for both the prediction risk and the l1‐estimation loss for regression coefficients. Since the lasso estimator selects variables simultaneously, we show that oracle inequalities can be established without pretesting the existence of the threshold effect. Furthermore, we establish conditions under which the estimation error of the unknown threshold parameter can be bounded by a factor that is nearly n−1 even when the number of regressors can be much larger than the sample size n. We illustrate the usefulness of our proposed estimation method via Monte Carlo simulations and an application to real data.

Testing for structural stability in the whole sample

The paper examines a Lagrange Multiplier type test for the constancy of the parameter in general models with dependent data without imposing any artificial choice of the possible location of the break. In order to prove the asymptotic behaviour of the test, we extend a strong approximation result for partial sums of a sequence of random variables. We also present a Monte-Carlo experiment to examine the finite sample performance of the test and how it compares with tests which assume some knowledge of the possible location of the break.

Testing for a Debt-Threshold Effect on Output Growth

Abstract Using the Reinhart–Rogoff dataset, we find a debt threshold not around 90 per cent but around 30 per cent, above which the median real gross domestic product (GDP) growth falls abruptly. Our work is the first to formally test for threshold effects in the relationship between public debt and median real GDP growth. The null hypothesis of no threshold effect is rejected at the 5 per cent significance level for most cases. While we find no evidence of a threshold around 90 per cent, our findings from the post‐war sample suggest that the debt threshold for economic growth may exist around a relatively small debt‐to‐GDP ratio of 30 per cent. Furthermore, countries with debt‐to‐GDP ratios above 30 per cent have GDP growth that is 1 percentage point lower at the median.

SPECIFICATION TESTS FOR LATTICE PROCESSES

We consider an omnibus test for the correct specification of the dynamics of a sequence S0266466614000310_inline1 in a lattice. As it happens with causal models and d = 1, its asymptotic distribution is not pivotal and depends on the estimator of the unknown parameters of the model under the null hypothesis. One first main goal of the paper is to provide a transformation to obtain an asymptotic distribution that is free of nuisance parameters. Secondly, we propose a bootstrap analog of the transformation and show its validity. Thirdly, we discuss the results when S0266466614000310_inline2 are the errors of a parametric regression model. As a by product, we also discuss the asymptotic normality of the least squares estimator of the parameters of the regression model under very mild conditions. Finally, we present a small Monte Carlo experiment to shed some light on the finite sample behavior of our test.

Oracle Estimation of a Change Point in High Dimensional Quantile Regression

ABSTRACT In this article, we consider a high-dimensional quantile regression model where the sparsity structure may differ between two sub-populations. We develop ℓ1-penalized estimators of both regression coefficients and the threshold parameter. Our penalized estimators not only select covariates but also discriminate between a model with homogenous sparsity and a model with a change point. As a result, it is not necessary to know or pretest whether the change point is present, or where it occurs. Our estimator of the change point achieves an oracle property in the sense that its asymptotic distribution is the same as if the unknown active sets of regression coefficients were known. Importantly, we establish this oracle property without a perfect covariate selection, thereby avoiding the need for the minimum level condition on the signals of active covariates. Dealing with high-dimensional quantile regression with an unknown change point calls for a new proof technique since the quantile loss function is nonsmooth and furthermore the corresponding objective function is nonconvex with respect to the change point. The technique developed in this article is applicable to a general M-estimation framework with a change point, which may be of independent interest. The proposed methods are then illustrated via Monte Carlo experiments and an application to tipping in the dynamics of racial segregation. Supplementary materials for this article are available online.

Local M-estimation with discontinuous criterion for dependent and limited observations

This paper examines asymptotic properties of local M-estimators under three sets of high-level conditions. These conditions are sufficiently general to cover the minimum volume predictive region, conditional maximum score estimator for a panel data discrete choice model, and many other widely used estimators in statistics and econometrics. Specifically, they allow for discontinuous criterion functions of weakly dependent observations, which may be localized by kernel smoothing and contain nuisance parameters whose dimension may grow to infinity. Furthermore, the localization can occur around parameter values rather than around a fixed point and the observation may take limited values, which leads to set estimators. Our theory produces three different nonparametric cube root rates and enables valid inference for the local M-estimators, building on novel maximal inequalities for weakly dependent data. Our results include the standard cube root asymptotics as a special case. To illustrate the usefulness of our results, we verify our conditions for various examples such as the Hough transform estimator with diminishing bandwidth, maximum score-type set estimator, and many others.

Is There a Jump in the Transition

This article develops a statistical test for the presence of a jump in an otherwise smooth transition process. In this testing, the null model is a threshold regression and the alternative model is a smooth transition model. We propose a quasi-Gaussian likelihood ratio statistic and provide its asymptotic distribution, which is defined as the maximum of a two parameter Gaussian process with a nonzero bias term. Asymptotic critical values can be tabulated and depend on the transition function employed. A simulation method to compute empirical critical values is also developed. Finite-sample performance of the test is assessed via Monte Carlo simulations. The test is applied to investigate the dynamics of racial segregation within cities across the United States.

High-Dimensional Predictive Regression in the Presence of Cointegration

While a great number of predictive variables for stock returns have been suggested, their prediction power is unstable. We propose a Least Absolute Shrinkage and Selection Operator (LASSO) estimator of a predictive regression in which stock returns are conditioned on a large set of lagged covariates, some of which are highly persistent and potentially cointegrated. We establish the asymptotic properties of the proposed LASSO estimator and validate our theoretical findings using simulation studies. The application of this proposed LASSO approach to forecasting stock returns suggests that cointegrating relationships among the persistent predictors leads to a significant improvement in the prediction of stock returns over various competing models in the mean squared error sense.