Testing jointly for structural changes in the error variance and coefficients of a linear regression model

Abstract

We provide a comprehensive treatment for the problem of testing jointly for structural changes in both the regression coefficients and the variance of the errors in a single equation system involving stationary regressors. Our framework is quite general in that we allow for general mixing‐type regressors and the assumptions on the errors are quite mild. Their distribution can be nonnormal and conditional heteroskedasticity is permitted. Extensions to the case with serially correlated errors are also treated. We provide the required tools to address the following testing problems, among others: (a) testing for given numbers of changes in regression coefficients and variance of the errors; (b) testing for some unknown number of changes within some prespecified maximum; (c) testing for changes in variance (regression coefficients) allowing for a given number of changes in the regression coefficients (variance); (d) a sequential procedure to estimate the number of changes present. These testing problems are important for practical applications as witnessed by interests in macroeconomics and finance where documenting structural changes in the variability of shocks to simple autoregressions or vector autoregressive models have been a concern.

Pitfalls of Two-Step Testing for Changes in the Error Variance and Coefficients of a Linear Regression Model

“…Our results illustrate a need for a joint approach to test for structural changes in both the coefficients and the variance of the errors. We provide some evidence that the procedures suggested by Perron et al (2019) provide tests with good size and power.…”

“…For TP-3, the log-likelihood function under 0 is An online supplement shows that the results remain qualitatively the same for the following extended cases: (a) Models with lagged dependent variables (Supplement I); (b) models with multiple structural changes (Supplement II); and (c) CUSQ tests for a change in variance that correct for potential correlation in the error variance; for example, conditional heteroskedasticity (Supplement III). Perron et al (2019) provided a comprehensive treatment for the problem of testing jointly for structural changes in the regression coefficients and the variance of the errors. Here, we consider two versions of their tests to illustrate how they solve the size and power problems.…”

“…All methods are shown to suffer from important size distortions and/or power losses. We then present evidence that the joint approach of Perron et al (2019) provides tests with good size and power. Our work is related to that of Pitarakis (2004) and can be viewed as complementary although the tests considered are different.…”

“…In a companion paper, Perron et al (2019) provided a comprehensive treatment of the problem of testing jointly for structural changes in both the regression coefficients and the variance of the errors in a single equation regression model involving stationary regressors, allowing the break dates for the two components to be different or overlap. Their framework is quite general with assumptions no stronger than those in Qu and Perron (2007).…”

“…Hence, what is needed is a joint approach when changes are suspected in both the coefficients and variance. This was covered in Perron et al (2019) and we present simulation results showing that their procedures yield tests with good size and power.…”