Testing for Error Correction in Panel Data*

This paper proposes new error correction‐based cointegration tests for panel data. The limiting distributions of the tests are derived and critical values provided. Our simulation results suggest that the tests have good small‐sample properties with small size distortions and high power relative to other popular residual‐based panel cointegration tests. In our empirical application, we present evidence suggesting that international healthcare expenditures and GDP are cointegrated once the possibility of an invalid common factor restriction has been accounted for.     

Testing for Fractional Integration Versus Short Memory with Structural Breaks*

Although it is commonly accepted that most macroeconomic variables are non‐stationary, it is often difficult to identify the source of the non‐stationarity. Integrated processes and short‐memory models with trending components, possibly affected by structural breaks, imply similar features in the data and, accordingly, are hard to distinguish. The goal of this article is to extend the classical testing framework of I(1) versus I(0) + trends and/or breaks by considering a more general class of models under the null hypothesis: fractionally integrated (FI) processes. The asymptotic properties of the proposed tests are derived and it is shown that they are very well‐behaved in finite samples. An illustration using US inflation data is also provided.     

Testing the Null of Cointegration with Structural Breaks*

We propose a Lagrange Multiplier‐type statistic to test the null hypothesis of cointegration allowing for the possibility of a structural break, in both the deterministic and the cointegration vectors. Our proposal focuses on the presence of endogenous regressors. The test complements the usual non‐cointegration tests so as to obtain stronger evidence of cointegration. We consider the cases of known and unknown dates of the break. In the latter case, we show that minimizing the Sum of Squared Residuals results in a super‐consistent estimator of the break fraction. Finally, the behaviour of the tests is studied through Monte Carlo experiments.     

Testing for Panel Cointegration with Multiple Structural Breaks*

This paper proposes a Lagrange multiplier (LM) test for the null hypothesis of cointegration that allows for the possibility of multiple structural breaks in both the level and trend of a cointegrated panel regression. The test is general enough to allow for endogenous regressors, serial correlation and an unknown number of breaks that may be located at different dates for different individuals. We derive the limiting distribution of the test and conduct a small Monte Carlo study to investigate its finite sample properties. In our empirical application to the solvency of the current account, we find evidence of cointegration between saving and investment once a level break is accommodated.     

Panel LM Unit‐root Tests with Level Shifts*

This paper proposes a new panel unit‐root test based on the Lagrangian multiplier (LM) principle. We show that the asymptotic distribution of the new panel LM test is not affected by the presence of structural shifts. This result holds under a mild condition that N/T→k, where k is any finite constant. Our simulation study shows that the panel LM unit‐root test is not only robust to the presence of structural shifts, but is more powerful than the popular Im, Pesaran and Shin (IPS) test. We apply our new test to the purchasing power parity (PPP) hypothesis and find strong evidence for PPP.     

Cointegration Testing in Panels with Common Factors*

Panel unit‐root and no‐cointegration tests that rely on cross‐sectional independence of the panel unit experience severe size distortions when this assumption is violated, as has, for example, been shown by Banerjee, Marcellino and Osbat [Econometrics Journal (2004), Vol. 7, pp. 322–340; Empirical Economics (2005), Vol. 30, pp. 77–91] via Monte Carlo simulations. Several studies have recently addressed this issue for panel unit‐root tests using a common factor structure to model the cross‐sectional dependence, but not much work has been done yet for panel no‐cointegration tests. This paper proposes a model for panel no‐cointegration using an unobserved common factor structure, following the study by Bai and Ng [Econometrica (2004), Vol. 72, pp. 1127–1177] for panel unit roots. We distinguish two important cases: (i) the case when the non‐stationarity in the data is driven by a reduced number of common stochastic trends, and (ii) the case where we have common and idiosyncratic stochastic trends present in the data. We discuss the homogeneity restrictions on the cointegrating vectors resulting from the presence of common factor cointegration. Furthermore, we study the asymptotic behaviour of some existing residual‐based panel no‐cointegration tests, as suggested by Kao [Journal of Econometrics (1999), Vol. 90, pp. 1–44] and Pedroni [Econometric Theory (2004a), Vol. 20, pp. 597–625]. Under the data‐generating processes (DGP) used, the test statistics are no longer asymptotically normal, and convergence occurs at rate T rather than as for independent panels. We then examine the possibilities of testing for various forms of no‐cointegration by extracting the common factors and individual components from the observed data directly and then testing for no‐cointegration using residual‐based panel tests applied to the defactored data.     

Finite Sample Correction Factors for Panel Cointegration Tests*

In this paper we present finite T mean and variance correction factors and corresponding response surface regressions for the panel cointegration tests presented in Pedroni (1999, 2004) , Westerlund (2005) , Larsson et al. (2001) and Breitung (2005) . For the single equation tests, we consider up to 12 regressors and for the system tests vector autoregression dimensions up to 12 variables. All commonly used specifications for the deterministic components are considered. The sample sizes considered are T ∈ {10,20,30,40,50,60,70,80,90,100,200,500}.     

Combining Significance of Correlated Statistics with Application to Panel Data*

The inverse normal method, which is used to combine P‐values from a series of statistical tests, requires independence of single test statistics in order to obtain asymptotic normality of the joint test statistic. The paper discusses the modification by Hartung (1999, Biometrical Journal, Vol. 41, pp. 849–855) , which is designed to allow for a certain correlation matrix of the transformed P‐values. First, the modified inverse normal method is shown here to be valid with more general correlation matrices. Secondly, a necessary and sufficient condition for (asymptotic) normality is provided, using the copula approach. Thirdly, applications to panels of cross‐correlated time series, stationary as well as integrated, are considered. The behaviour of the modified inverse normal method is quantified by means of Monte Carlo experiments.     

Testing for Hysteresis in Unemployment in OECD Countries: New Evidence using Stationarity Panel Tests with Breaks*

This paper tests hysteresis effects in unemployment using panel data for 19 Organization for Economic Co‐operation and Development (OECD) countries covering the period 1956–2001. The tests exploit the cross‐sectional variations of the series, and additionally, allow for a different number of endogenous breakpoints in the unemployment series. The critical values are simulated based on our specific panel sizes and time periods. The findings stress the importance of accounting for exogenous shocks in the series and support the natural‐rate hypothesis of unemployment for the majority of the countries analysed.     

Tests for Stationarity in Series with Endogenously Determined Structural Change*

We consider tests of the null hypothesis of stationarity against a unit root alternative, when the series is subject to structural change at an unknown point in time. Three extant tests are reviewed which allow for an endogenously determined instantaneous structural break, and a related fourth procedure is introduced. We further propose tests which permit the structural change to be gradual rather than instantaneous, allowing the null hypothesis to be stationarity about a smooth transition in linear trend. The size and power properties of the tests are investigated, and the tests are applied to four economic time series.     

Specification Testing of Markov Switching Models*

This paper proposes a set of formal tests to address the goodness‐of‐fit of Markov switching models. These formal tests are constructed as tests of model consistency and of both parametric and non‐parametric encompassing. The formal tests are then combined with informal tests using simulation in combination with non‐parametric density and conditional mean estimation. The informal tests are shown to be useful in shedding light on the failure (or success) of the encompassing tests. Several examples are provided.     

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.     

Regression‐based Tests for a Change in Persistence*

We show that the minimal forward (reverse) recursive unit tests of Banerjee, Lumsdaine and Stock [Journal of Business and Economics Statistics (1992) Vol. 10, pp. 271–288] are consistent against the alternative of a change in persistence from I(0) to I(1) [I(1) to I(0)]. However, these statistics are also shown to diverge for series which are I(0) throughout. Consequently, a rejection by these tests does not necessarily imply a change in persistence. We propose a further test, based on the ratio of these statistics, which is consistent against changes either from I(0) to I(1), or vice versa, yet does not over‐reject against constant I(0) series. Consistent breakpoint estimators are proposed.     

Bootstrap HAC Tests for Ordinary Least Squares Regression*

There is a need for tests that are derived from the ordinary least squares (OLS) estimators of regression coefficients and are useful in the presence of unspecified forms of heteroskedasticity and autocorrelation. A method that uses the moving block bootstrap and quasi‐estimators in order to derive a consistent estimator of the asymptotic covariance matrix for the OLS estimators and robust significance tests is proposed. The method is shown to be asymptotically valid and Monte Carlo evidence indicates that it is capable of providing good control of significance levels in finite samples and good power compared with two other bootstrap tests.     

Diagnostic Tests of Cross‐section Independence for Limited Dependent Variable Panel Data Models*

This article considers the problem of testing for cross‐section independence in limited dependent variable panel data models. It derives a Lagrangian multiplier (LM) test and shows that in terms of generalized residuals of Gourieroux et al. (1987) it reduces to the LM test of Breusch and Pagan (1980) . Because of the tendency of the LM test to over‐reject in panels with large N (cross‐section dimension), we also consider the application of the cross‐section dependence test (CD) proposed by Pesaran (2004) . In Monte Carlo experiments it emerges that for most combinations of N and T the CD test is correctly sized, whereas the validity of the LM test requires T (time series dimension) to be quite large relative to N. We illustrate the cross‐sectional independence tests with an application to a probit panel data model of roll‐call votes in the US Congress and find that the votes display a significant degree of cross‐section dependence.     

Power of Tests for Unit Roots in the Presence of a Linear Trend*

Dickey and Fuller [Econometrica (1981) Vol. 49, pp. 1057–1072] suggested unit‐root tests for an autoregressive model with a linear trend conditional on an initial observation. TPower of tests for unit roots in the presence of a linear trendightly different model with a random initial value in which nuisance parameters can easily be eliminated by an invariant reduction of the model. We show that invariance arguments can also be used when comparing power within a conditional model. In the context of the conditional model, the Dickey–Fuller test is shown to be more stringent than a number of unit‐root tests motivated by models with random initial value. The power of the Dickey–Fuller test can be improved by making assumptions to the initial value. The practitioner therefore has to trade‐off robustness and power, as assumptions about initial values are hard to test, but can give more power.     

Tests for a Break in Level when the Order of Integration is Unknown*

In this paper, we consider tests for a break in the level of a series at an unknown point in time. It is often the case that uncertainty exists concerning the order of integration of the series; consequently, we focus on tests that are applicable when the order of integration is not known. The size and power of existing tests are analysed, and a modification to one of the established sets of tests is proposed which offers improved performance in certain circumstances.     

Truncated Product Methods for Panel Unit Root Tests*

This paper proposes two new panel unit root tests based on Zaykin et al. (2002) ’s truncated product method. The first one assumes constant correlation between P‐values and the second one uses sieve bootstrap to allow for general forms of cross‐section dependence in the panel units. Monte Carlo simulation shows that both tests have reasonably good size and are powerful in cases of some very large P‐values. The proposed tests are applied to a panel of real GDP and inflation density forecasts, resulting in evidence that professional forecasters may not update their forecast precision in an optimal Bayesian way.     

Testing for Misspecification in Binary Response Models with Competing Distributions*

In empirical studies, the probit and logit models are often used without checks for their competing distributional specifications. It is also rare for econometric tests to be focused on this issue. Santos Silva [Journal of Applied Econometrics (2001 ), Vol. 16, pp. 577–597] is an important recent exception. By using the conditional moment test principle, we discuss a wide class of non‐nested tests that can easily be applied to detect the competing distributions for the binary response models. This class of tests includes the test of Santos Silva (2001 ) for the same task as a particular example and provides other useful alternatives. We also compare the performance of these tests by a Monte Carlo simulation.     

Time Series Concepts for Conditional Distributions*

The paper asks the question – as time series analysis moves from consideration of conditional mean values and variances to unconditional distributions, do some of the familiar concepts devised for the first two moments continue to be helpful in the more general area? Most seem to generalize fairly easy, such as the concepts of breaks, seasonality, trends and regime switching. Forecasting is more difficult, as forecasts become distributions, as do forecast errors. Persistence can be defined and also common factors by using the idea of a copula. Aggregation is more difficult but causality and controllability can be defined. The study of the time series of quantiles becomes more relevant.