Likelihood-ratio-based confidence sets for the timing of structural breaks

Abstract

We propose the use of likelihood-ratio-based confidence sets for the timing of structural breaks in parameters from time series regression models. The confidence sets are valid for the broad setting of a system of multivariate linear regression equations under fairly general assumptions about the error and regressors, and allowing for multiple breaks in mean and variance parameters. In our asymptotic analysis, we determine the critical values for a likelihood ratio test of a break date and the expected length of a confidence set constructed by inverting the likelihood ratio test. Notably, the likelihood-ratio-based confidence sets are more precise than other confidence sets considered in the literature. Monte Carlo analysis confirms their greater precision in finite samples, including in terms of maintaining accurate coverage even when the sample size or magnitude of a break is small. An application to postwar U.S. real gross domestic product and consumption leads to a shorter 95% confidence set for the timing of the "Great Moderation" in the mid-1980s than previously found in the literature. Furthermore, when taking co-integration between output and consumption into account, confidence sets for structural break dates become even shorter and suggest a "productivity growth slowdown" in the early 1970s and an additional large, abrupt decline in long-run growth in the mid-1990s.Keywords. Inverted likelihood ratio, multiple breaks, system of equations, Great Moderation, productivity growth slowdown.JEL classification. C22, C32, E20.Yunjong Eo: yunjong.eo@sydney.edu.au James Morley: james.morley@unsw.edu.auWe thank Frank Schorfheide, Elie Tamer, and three anonymous referees for helpful editorial suggestions. Graham Elliott, Junsoo Lee, Alessio Moro, Charles Nelson, Irina Panovska, Jeremy Piger, Werner Ploberger, Barbara Rossi, Rami Tabri, Farshid Vahid, Tao Zha, and seminar and conference participants at Monash University, University of Alabama, Universitat Pompeu Fabra, University of Western Australia, University of Wollongong, the North American Summer Meeting of the Econometric Society, the NBER-NSF Time Series Conference, the Midwest Econometrics Group, and the Society for Nonlinear Dynamics and Econometrics Symposium also provided useful feedback. We acknowledge the use of GAUSS code by Zhongjun Qu and Pierre Perron, and can provide compatible code upon request that calculates likelihood-ratio-based confidence sets for structural breaks within their modeling framework. The usual disclaimers apply.

A comparison of alternative methods to construct confidence intervals for the estimate of a break date in linear regression models

“…Lemma 1 (Eo and Morley, 2015, Proposition 1) With W ( ) a standard Wiener process, the likelihood ratio statistic for the break date satis…es…”

“…The parameters of interest (T 0 1 ; ; ) are estimated by maximizing the quasi-likelihood function, i.e., (T 1 ;^ ;^ ) = arg max (T 1 ; ; ) l T (T 1 ; ; ). Following the notation in Eo and Morley (2015), let l T (T j ) denote the logarithm of the pro…le likelihood function for the break date:…”

“…Recently, Eo and Morley (2015) generalized Siegmund's (1988) method to a system of multivariate regressions, based on inverting the likelihood ratio test to obtain a con…dence set, labelled ILR. Using results from Qu and Perron (2007), they considered the problem in the context of a system of multivariate equations.…”

Statistical Inference for Structurally Changed Threshold Autoregressive Models

“…At the same time, the confidence interval based on the approximating method in Section 4 has a coverage rate below the nominal level when the threshold/structural change effect is large, see the discussion in Hansen (2000) for threshold scenario, and Elliott and Müller (2007), Eo and Morley (2015) and Elliott et al (2015) for change-point scenario. Furthermore, they commented that the likelihood-ratio test is asymptotically pivotal when the threshold/structural change effect is small and the confidence region based on the inverted likelihood-ratio test is asymptotically valid, even if the threshold/structural change effect is relatively large.…”

“…see, for example, Eo and Morley (2015). Then, we use (5.12) to solve for the critical valuec 1−α and a 1 − α likelihood-ratio based confidence set for k 0 is given by…”