Unit roots and structural breaks

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Single-equation tests for cointegration with GLS detrended data

We provide GLS-based versions of two widely used approaches for testing whether or not non-stationary economic time series are cointegrated: single-equation static re- gression or residual-based tests and single-equation conditional error correction model (ECM) based tests. Our approach is to consider nearly optimal tests for unit roots and apply them in the cointegration context. Our GLS versions of the tests do in- deed provide substantial improvements over their OLS counterparts. We derive the local asymptotic power functions of all tests considered for a DGP with weakly ex- ogenous regressors. This allows obtaining the relevant non-centrality parameter to quasi-di§erence the data. We investigate the e§ect of non-weakly exogenous regressors via simulations. With weakly exogenous regressors strongly correlated with the depen- dent variable, the ECM tests are clearly superior. When the regressors are potentially non-weakly exogenous, the residuals-based tests are clearly preferred

Inference on locally ordered breaks in multiple regressions

We consider issues related to inference about locally ordered breaks in a system of equations, as originally proposed by Qu and Perron (2007 Qu, Z., Perron, P. (2007). Estimating and testing structural changes in multivariate regressions. Econometrica 75:459–502.[Crossref], [Web of Science ®], [Google Scholar]). These apply when break dates in different equations within the system are not separated by a positive fraction of the sample size. This allows constructing joint confidence intervals of all such locally ordered break dates. We extend the results of Qu and Perron (2007 Qu, Z., Perron, P. (2007). Estimating and testing structural changes in multivariate regressions. Econometrica 75:459–502.[Crossref], [Web of Science ®], [Google Scholar]) in several directions. First, we allow the covariates to be any mix of trends and stationary or integrated regressors. Second, we allow for breaks in the variance-covariance matrix of the errors. Third, we allow for multiple locally ordered breaks, each occurring in a different equation within a subset of equations in the system. Via some simulation experiments, we show first that the limit distributions derived provide good approximations to the finite sample distributions. Second, we show that forming confidence intervals in such a joint fashion allows more precision (tighter intervals) compared to the standard approach of forming confidence intervals using the method of Bai and Perron (1998 Bai, J., Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica 66:47–78.[Crossref], [Web of Science ®], [Google Scholar]) applied to a single equation. Simulations also indicate that using the locally ordered break confidence intervals yields better coverage rates than using the framework for globally distinct breaks when the break dates are separated by roughly 10% of the total sample size

Testing for common breaks in a multiple equations system

The issue addressed in this paper is that of testing for common breaks across or within equations. Our framework is very general and allows integrated regressors and trends as well as stationary regressors. The null hypothesis is that some subsets of the parameters (either regression coe cients or elements of the covariance matrix of the errors) share one or more common break dates, with the break dates in the system asymptotically distinct so that each regime is separated by some positive fraction of the sample size. Under the alternative hypothesis, the break dates are not the same and also need not be separated by a positive fraction of the sample size. The test con- sidered is the quasi-likelihood ratio test assuming normal errors, though as usual the limit distribution of the test remains valid with non-normal errors. Also of indepen- dent interest, we provide results about the consistency and rate of convergence when searching over all possible partitions subject only to the requirement that each regime contains at least as many observations as the number of parameters in the model. Sim- ulation results show that the test has good nite sample properties. We also provide an application to various measures of in ation to illustrate its usefulness

Extracting and analyzing the warming trend in global and hemispheric temperatures

This article offers an updated and extended attribution analysis based on recently published versions of temperature and forcing datasets. It shows that both temperature and radiative forcing variables can be best represented as trend stationary processes with structural changes occurring in the slope of their trend functions and that they share a common secular trend and common breaks, largely determined by the anthropogenic radiative forcing. The common nonlinear trend is isolated, and further evidence on the possible causes of the current slowdown in warming is presented. Our analysis offers interesting results in relation to the recent literature. Changes in the anthropogenic forcings are directly responsible for the hiatus, while natural variability modes such as the Atlantic Multidecadal Oscillation, as well as new temperature adjustments, contribute to weaken the signal. In other words, natural variability and data adjustments do not explain in any way the hiatus; they simply mask its presence

Residuals-based tests for cointegration with generalized least-squares detrended data

We provide generalized least-squares (GLS) detrended versions of single-equation static regression or residuals-based tests for testing whether or not non-stationary time series are cointegrated. Our approach is to consider nearly optimal tests for unit roots and to apply them in the cointegration context. We derive the local asymptotic power functions of all tests considered for a triangular data-generating process, imposing a directional restriction such that the regressors are pure integrated processes. Our GLS versions of the tests do indeed provide substantial power improvements over their ordinary least-squares counterparts. Simulations show that the gains in power are important and stable across various configurations

A note on estimating and testing for multiple structural changes in models with endogenous regressors via 2SLS

This note provides a simple proof for the problem of estimating and testing for multiple breaks in a single equation framework with regressors that are endogenous. We show based on standard assumptions about the regressors, instruments, and errors that the second-stage regression of the instrumental variable procedure involves regressors and errors that satisfy all the assumptions in Perron and Qu (2006, Journal of Econometrics 134, 373–399) so that the results about consistency, rate of convergence and limit distributions of the estimates of the break dates, in addition to the limit distributions of the tests, are obtained as simple consequences. The results are obtained within a unified framework for various cases about the nature of the reduced form: stable, no structural changes but time variations in the parameters, structural changes at dates that are common to those of the structural form, and structural changes occurring at arbitrary dates.This is a revised version of parts of a paper previously circulated under the title "Estimating and Testing Multiple Structural Changes in Models with Endogenous Regressors." Perron acknowledges financial support for this work from the National Science Foundation under grant SES-0649350. We are grateful to Zhongjun Qu, two referees, the co-editor Robert Taylor and the editor Peter C. B. Phillips for useful comments. Address correspondence to Pierre Perron, Department of Economics, Boston University, 270 Bay State Rd., Boston, MA, 02215, USA; e-mail: ([email protected]). (SES-0649350 - National Science Foundation

Testing for changes in forecasting performance

We consider the issue of forecast failure (or breakdown) and propose methods to assess retrospectively whether a given forecasting model provides forecasts which show evidence of changes with respect to some loss function. We adapt the classical structural change tests to the forecast failure context. First, we recommend that all tests should be carried with a fixed scheme to have best power. This ensures a maximum difference between the fitted in and out-of-sample means of the losses and avoids contamination issues under the rolling and recursive schemes. With a fixed scheme, Giacomini and Rossi’s (2009) (GR) test is simply a Wald test for a one-time change in the mean of the total (the in-sample plus out-of-sample) losses at a known break date, say m, the value that separates the in and out-of-sample periods. To alleviate this problem, we consider a variety of tests: maximizing the GR test over values of m within a pre-specified range; a Double sup-Wald (DSW) test which for each m performs a sup-Wald test for a change in the mean of the out-of-sample losses and takes the maximum of such tests over some range; we also propose to work directly with the total loss series to define the Total Loss sup-Wald (TLSW) and Total Loss UDmax (TLUD) tests. Using theoretical analyses and simulations, we show that with forecasting models potentially involving lagged dependent variables, the only tests having a monotonic power function for all data-generating processes considered are the DSW and TLUD tests, constructed with a fixed forecasting window scheme. Some explanations are provided and empirical applications illustrate the relevance of our findings in practice.First author draf

Continuous record Laplace-based inference about the break date in structural change models

Building upon the continuous record asymptotic framework recently introduced by Casini and Perron (2018a) for inference in structural change models, we propose a Laplace-based (Quasi-Bayes) procedure for the construction of the estimate and confidence set for the date of a structural change. It is defined by an integration rather than an optimization-based method.A transformation of the least-squares criterion function is evaluated in order to derive a proper distribution, referred to as the Quasi-posterior. For a given choice of a loss function, the Laplace-type estimator is the minimizer of the expected risk with the expectation taken under the Quasi-posterior. Besides providing an alternative estimate that is more precise—lower mean absolute error (MAE) and lower root-mean squared error (RMSE)—than the usual least-squares one, the Quasi-posterior distribution can be used to construct asymptotically valid inference using the concept of Highest Density Region. The resulting Laplace-based inferential procedure is shown to have lower MAE and RMSE, and the confidence sets strike the best balance between empirical coverage rates and average lengths of the confidence sets relative to traditional long-span methods, whether the break size is small or large.First author draf

Using OLS to estimate and test for structural changes in models with endogenous regressors

We consider the problem of estimating and testing for multiple breaks in a single-equation framework with regressors that are endogenous, i.e. correlated with the errors. We show that even in the presence of endogenous regressors it is still preferable, in most cases, to simply estimate the break dates and test for structural change using the usual ordinary least squares (OLS) framework. Except for some knife-edge cases, it delivers estimates of the break dates with higher precision and tests with higher power compared to those obtained using an instrumental variable (IV) method. Also, the OLS method avoids potential weak identification problems caused by weak instruments. To illustrate the relevance of our theoretical results, we consider the stability of the New Keynesian hybrid Phillips curve. IV-based methods only provide weak evidence of instability. On the other hand, OLS-based ones strongly indicate a change in 1991:Q1 and that after this date the model loses all explanatory power