Computation and analysis of multiple structural change models

In a recent paper, Bai and Perron ( 1998 ) considered theoretical issues related to the limiting distribution of estimators and test statistics in the linear model with multiple structural changes. In this companion paper, we consider practical issues for the empirical applications of the procedures. We first address the problem of estimation of the break dates and present an efficient algorithm to obtain global minimizers of the sum of squared residuals. This algorithm is based on the principle of dynamic programming and requires at most least‐squares operations of order O(T²) for any number of breaks. Our method can be applied to both pure and partial structural change models. Second, we consider the problem of forming confidence intervals for the break dates under various hypotheses about the structure of the data and the errors across segments. Third, we address the issue of testing for structural changes under very general conditions on the data and the errors. Fourth, we address the issue of estimating the number of breaks. Finally, a few empirical applications are presented to illustrate the usefulness of the procedures. All methods discussed are implemented in a GAUSS program. Copyright © 2002 John Wiley & Sons, Ltd.     

Panel Stationarity Test with Structural Breaks*

In this paper, we extend the heterogeneous panel data stationarity test of Hadri [Econometrics Journal, Vol. 3 (2000) pp. 148–161] to the cases where breaks are taken into account. Four models with different patterns of breaks under the null hypothesis are specified. Two of the models have been already proposed by Carrion‐i‐Silvestre et al. [Econometrics Journal, Vol. 8 (2005) pp. 159–175]. The moments of the statistics corresponding to the four models are derived in closed form via characteristic functions. We also provide the exact moments of a modified statistic that do not asymptotically depend on the location of the break point under the null hypothesis. The cases where the break point is unknown are also considered. For the model with breaks in the level and no time trend and for the model with breaks in the level and in the time trend, Carrion‐i‐Silvestre et al. [Econometrics Journal, Vol. 8 (2005) pp. 159–175] showed that the number of breaks and their positions may be allowed to differ across individuals for cases with known and unknown breaks. Their results can easily be extended to the proposed modified statistic. The asymptotic distributions of all the statistics proposed are derived under the null hypothesis and are shown to be normally distributed. We show by simulations that our suggested tests have in general good performance in finite samples except the modified test. In an empirical application to the consumer prices of 22 OECD countries during the period from 1953 to 2003, we found evidence of stationarity once a structural break and cross‐sectional dependence are accommodated.     

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. Copyright © 2013 John Wiley & Sons, Ltd.     

Testing for unit roots in the possible presence of multiple trend breaks using minimum Dickey–Fuller statistics

Trend breaks appear to be prevalent in macroeconomic time series, and unit root tests therefore need to make allowance for these if they are to avoid the serious effects that unmodelled trend breaks have on power. Carrion-i-Silvestre et al. (2009) propose a pre-test-based approach which delivers near asymptotically efficient unit root inference both when breaks do not occur and where multiple breaks occur, provided the break magnitudes are fixed. Unfortunately, however, the fixed magnitude trend break asymptotic theory does not predict well the finite sample power functions of these tests, and power can be very low for the magnitudes of trend breaks typically observed in practice. In response to this problem we propose a unit root test that allows for multiple breaks in trend, obtained by taking the infimum of the sequence (across all candidate break points in a trimmed range) of local GLS detrended augmented Dickey–Fuller-type statistics. We show that this procedure has power that is robust to the magnitude of any trend breaks, thereby retaining good finite sample power in the presence of plausibly-sized breaks. We also demonstrate that, unlike the OLS detrended infimum tests of Zivot and Andrews (1992), these tests display no tendency to spuriously reject in the limit when fixed magnitude trend breaks occur under the unit root null.     

Learning,forecasting and structural breaks

We provide a general methodology for forecasting in the presence of structural breaks induced by unpredictable changes to model parameters. Bayesian methods of learning and model comparison are used to derive a predictive density that takes into account the possibility that a break will occur before the next observation. Estimates for the posterior distribution of the most recent break are generated as a by‐product of our procedure. We discuss the importance of using priors that accurately reflect the econometrician's opinions as to what constitutes a plausible forecast. Several applications to macroeconomic time‐series data demonstrate the usefulness of our procedure. Copyright © 2008 John Wiley & Sons, Ltd.     

The limit distribution of the estimates in cointegrated regression models with multiple structural changes

We study estimation and inference in cointegrated regression models with multiple structural changes allowing both stationary and integrated regressors. Both pure and partial structural change models are analyzed. We derive the consistency, rate of convergence and the limit distribution of the estimated break fractions. Our technical conditions are considerably less restrictive than those in Bai et al. [Bai, J., Lumsdaine, R.L., Stock, J.H., 1998. Testing for and dating breaks in multivariate time series. Review of Economic Studies 65, 395–432] who considered the single break case in a multi-equations system, and permit a wide class of practically relevant models. Our analysis is, however, restricted to a single equation framework. We show that if the coefficients of the integrated regressors are allowed to change, the estimated break fractions are asymptotically dependent so that confidence intervals need to be constructed jointly. If, however, only the intercept and/or the coefficients of the stationary regressors are allowed to change, the estimates of the break dates are asymptotically independent as in the stationary case analyzed by Bai and Perron [Bai, J., Perron, P., 1998. Estimating and testing linear models with multiple structural changes. Econometrica 66, 47–78]. We also show that our results remain valid, under very weak conditions, when the potential endogeneity of the non-stationary regressors is accounted for via an increasing sequence of leads and lags of their first-differences as additional regressors. Simulation evidence is presented to assess the adequacy of the asymptotic approximations in finite samples.     

The effects of external and internal shocks on total factor productivity

This paper examines structural changes that occur in the total factor productivity (TFP) within countries. It is possible that some episodes of high economic growth or economic decline are associated with permanent productivity shocks; therefore, this research has two objectives. The first one is to estimate the structural changes present in TFP for a sample of 77 countries between 1950 (1960) and 2000. The second one is to identify possible explanations for breaks. Two sources were analyzed: (i) episodes in political and economic history; (ii) changes in international trade – a measure of absorption of technology. The results suggest that about one-third of the TFP time-series present at least one structural break. Downwards breaks are more common, indicating that after a break the TFP has much difficulty to recover. When we investigated factors related with structural change, developed countries presented a break near the first oil shock while the developing countries’ breaks are more spread along the decades. Thus, external strikes seem to be more relevant for developed countries. However, for each country and break date, it was possible to find an event close to the break date endogenously detected. Last, the relevance of international trade, measured by trade share percentage of GDP, seems to be limited to explain abrupt changes in TFP.     

The Contribution of Structural Break Models to Forecasting Macroeconomic Series

This paper compares the forecasting performance of models that have been proposed for forecasting in the presence of structural breaks. They differ in their treatment of the break process, the model applied in each regime and the out‐of‐sample probability of a break. In an extensive empirical evaluation, we demonstrate the presence of breaks and their importance for forecasting. We find no single model that consistently works best in the presence of breaks. In many cases, the formal modeling of the break process is important in achieving a good forecast performance. However, there are also many cases where rolling window forecasts perform well. Copyright © 2014 John Wiley & Sons, Ltd.     

Structural changes in heterogeneous panels with endogenous regressors

This paper extends Pesaran's (Econometrica, 2006, 74, 967–1012) common correlated effects (CCE) by allowing for endogenous regressors in large heterogeneous panels with unknown common structural changes in slopes and error factor structure. Since endogenous regressors and structural breaks are often encountered in empirical studies with large panels, this extension makes Pesaran's CCE approach empirically more appealing. In addition to allowing for slope heterogeneity and cross‐sectional dependence, we find that Pesaran's CCE approach is also valid when dealing with unobservable factors in the presence of endogenous regressors and structural changes in slopes and error factor loadings. This is supported by Monte Carlo experiments.     

Exogenous uncertainty and the identification of structural vector autoregressions with external instruments

We provide necessary and sufficient conditions for the identification (point‐identification) of structural vector autoregressions (SVARs) with external instruments considering the case in which r instruments are used to identify g structural shocks of interest, r ≥ g ≥ 1. Novel frequentist estimation methods are discussed by considering both a “partial shocks” identification strategy, where only g structural shocks are of interest and are instrumented, and a “full shocks” identification strategy, where despite g structural shocks being instrumented, all n=g+(n?g) structural shocks of the system can be identified under certain conditions. The suggested approach is applied to investigate empirically whether financial and macroeconomic uncertainty can be approximated as exogenous drivers of US real economic activity, or rather as endogenous responses to first moment shocks, or both. We analyze whether the dynamic causal effects of nonuncertainty shocks on macroeconomic and financial uncertainty are significant in the period after the global financial crisis.     

Estimating restricted structural change models

This paper considers issues related to multiple structural changes, occurring at unknown dates, in the linear regression model when restrictions are imposed on the parameters. This includes, for example, the important special case where different nonadjacent regimes are the same. The estimates are constructed as global minimizers of the restricted sum of squared residuals and we provide an extension of the algorithm discussed in Bai and Perron [2003b, Computation and analysis of multiple structural change models. Journal of Applied Econometrics 18, 1–22] to efficiently compute them. We show that the estimates of the break dates have the same asymptotic properties with or without the restrictions imposed; that is, in large samples, there is no efficiency gain from imposing valid restrictions as far as the estimates of the break dates are concerned. Of course, efficiency gains occur for the other parameters of the model. Simulations show that in small samples, all parameters are more efficiently estimated using the restrictions. We also consider tests of the null hypothesis of no structural change. These are also more powerful when the restrictions are imposed. A Gauss code for all the procedures discussed in this paper is available from the authors.     

An efficient Bayesian approach to multiple structural change in multivariate time series

This paper provides a feasible approach to estimation and forecasting of multiple structural breaks for vector autoregressions and other multivariate models. Owing to conjugate prior assumptions we obtain a very efficient sampler for the regime allocation variable. A new hierarchical prior is introduced to allow for learning over different structural breaks. The model is extended to independent breaks in regression coefficients and the volatility parameters. Two empirical applications show the improvements the model has over benchmarks. In a macro application with seven variables we empirically demonstrate the benefits from moving from a multivariate structural break model to a set of univariate structural break models to account for heterogeneous break patterns across data series.     

Timing structural change: a conditional probabilistic approach

We propose a strategy for assessing structural stability in time‐series frameworks when potential change dates are unknown. Existing stability tests are effective in detecting structural change, but procedures for identifying timing are imprecise, especially in assessing the stability of variance parameters. We present a likelihood‐based procedure for assigning conditional probabilities to the occurrence of structural breaks at alternative dates. The procedure is effective in improving the precision with which inferences regarding timing can be made. We illustrate parametric and non‐parametric implementations of the procedure through Monte Carlo experiments, and an assessment of the volatility reduction in the growth rate of US GDP. Copyright © 2006 John Wiley & Sons, Ltd.     

A general test for time dependence in parameters

A new test for time‐dependent parameters is proposed. The Trig‐test is based on a trigonometric expansion to approximate the unknown functional form of the variation in the parameters concerned. It is shown to have the correct empirical size and excellent power to detect structural breaks and stochastic parameter variation. The appropriate use of the Trig‐test is demonstrated by testing for structural breaks in the US inflation rate. The test detects a statistically significant increase in the US inflation rate beginning in the early 1970s and lasting through to the early 1980s. Copyright © 2004 John Wiley & Sons, Ltd.     

MULTIVARIATE METHODS FOR MONITORING STRUCTURAL CHANGE

Detection of structural change is a critical empirical activity, but continuous ‘monitoring’ for changes in real time raises well‐known econometric issues that have been explored in a single series context. If multiple series co‐break then it is possible that simultaneous examination of a set of series helps identify changes with higher probability or more rapidly than when series are examined on a case‐by‐case basis. Some asymptotic theory is developed for maximum and average CUSUM detection tests. Monte Carlo experiments suggest that these both provide an improvement in detection relative to a univariate detector over a wide range of experimental parameters, given a sufficiently large number of co‐breaking series. This is robust to a cross‐sectional correlation in the errors (a factor structure) and heterogeneity in the break dates. We apply the test to a panel of UK price indices. Copyright © 2011 John Wiley & Sons, Ltd.     

A flexible approach to parametric inference in nonlinear and time varying time series models

Many structural break and regime-switching models have been used with macroeconomic and financial data. In this paper, we develop an extremely flexible modeling approach which can accommodate virtually any of these specifications. We build on earlier work showing the relationship between flexible functional forms and random variation in parameters. Our contribution is based around the use of priors on the time variation that is developed from considering a hypothetical reordering of the data and distance between neighboring (reordered) observations. The range of priors produced in this way can accommodate a wide variety of nonlinear time series models, including those with regime-switching and structural breaks. By allowing the amount of random variation in parameters to depend on the distance between (reordered) observations, the parameters can evolve in a wide variety of ways, allowing for everything from models exhibiting abrupt change (e.g. threshold autoregressive models or standard structural break models) to those which allow for a gradual evolution of parameters (e.g. smooth transition autoregressive models or time varying parameter models). Bayesian econometric methods for inference are developed for estimating the distance function and types of hypothetical reordering. Conditional on a hypothetical reordering and distance function, a simple reordering of the actual data allows us to estimate our models with standard state space methods by a simple adjustment to the measurement equation. We use artificial data to show the advantages of our approach, before providing two empirical illustrations involving the modeling of real GDP growth.     

Greek sovereign bond index,volatility, and structural breaks

This article investigates volatility changes in the 10-year Greek sovereign bond index returns using the multiple structural break test developed by Bai and Perron (Econometrica 66:47–78, 1998, J Appl Econ 18:1–22, 2003), which allows for endogenous identification of break dates. We find that there exists one break date in volatility, April 2010, when the European debt crisis worsened and the Greek sovereign bond was downgraded to junk status. We also obtain evidence of performance improvement in our modeling by including structural break dummies into the variance equation. We observe sharp drops in a measure of volatility persistence after incorporating the structural change. Our findings are important for not only investors who assess the volatility of sovereign bonds for portfolio risk management, but also for policy makers who wish to understand and minimize the impacts of excess volatility on the financial system in government bond markets.     

Structural breaks and GARCH models of exchange rate volatility

We investigate the empirical relevance of structural breaks for GARCH models of exchange rate volatility using both in‐sample and out‐of‐sample tests. We find significant evidence of structural breaks in the unconditional variance of seven of eight US dollar exchange rate return series over the 1980–2005 period—implying unstable GARCH processes for these exchange rates—and GARCH(1,1) parameter estimates often vary substantially across the subsamples defined by the structural breaks. We also find that it almost always pays to allow for structural breaks when forecasting exchange rate return volatility in real time. Combining forecasts from different models that accommodate structural breaks in volatility in various ways appears to offer a reliable method for improving volatility forecast accuracy given the uncertainty surrounding the timing and size of the structural breaks. Copyright © 2008 John Wiley & Sons, Ltd.     

Identification in Structural Vector Autoregressive Models with Structural Changes,with an Application to US Monetary Policy

A growing line of research makes use of structural changes and different volatility regimes found in the data in a constructive manner to improve the identification of structural parameters in structural vector autoregressions (SVARs). A standard assumption made in the literature is that the reduced form unconditional error covariance matrix varies while the structural parameters remain constant. Under this hypothesis, it is possible to identify the SVAR without needing to resort to additional restrictions. With macroeconomic data, the assumption that the transmission mechanism of the shocks does not vary across volatility regimes is debatable. We derive novel necessary and sufficient rank conditions for local identification of SVARs, where both the error covariance matrix and the structural parameters are allowed to change across volatility regimes. Our approach generalizes the existing literature on ‘identification through changes in volatility’ to a broader framework and opens up interesting possibilities for practitioners. An empirical illustration focuses on a small monetary policy SVAR of the US economy and suggests that monetary policy has become more effective at stabilizing the economy since the 1980s.     

Cointegration in Panel Data with Structural Breaks and Cross‐Section Dependence

The power of standard panel cointegration statistics may be affected by misspecification errors if structural breaks in the parameters generating the process are not considered. In addition, the presence of cross‐section dependence among the panel units can distort the empirical size of the statistics. We therefore design a testing procedure that allows for both structural breaks and cross‐section dependence when testing the null hypothesis of no cointegration. The paper proposes test statistics that can be used when one or both features are present. We illustrate our proposal by analysing the pass‐through of import prices on a sample of European countries. Copyright © 2013 John Wiley & Sons, Ltd.