Jörg Breitung

The local power of some unit root tests for panel data

To test the hypothesis of a difference stationary time series against a trend stationary alternative, Levin & Lin (1993) and Im, Pesaran & Shin (1997) suggest bias adjusted t-statistics. Such corrections are necessary to account for the nonzero mean of the t-statistic in the case of an OLS detrending method. In this chapter the local power of panel unit root statistics against a sequence of local alternatives is studied. It is shown that the local power of the test statistics is affected by two different terms. The first term represents the asymptotic effect on the bias due to the detrending method and the second term is the usual location parameter of the limiting distribution under the sequence of local alternatives. It is argued that both terms can offset each other so that the test has no power against the sequence of local alternatives. These results suggest to construct test statistics based on alternative detrending methods. We consider a class of t-statistics that do not require a bias correction. The results of a Monte Carlo experiment suggest that avoiding the bias can improve the power of the test substantially.

Nonparametric tests for unit roots and cointegration

Following Bierens (1997a,b) and Vogelsang (1998a,b), unit root tests can be constructed which are asymptotically invariant to parameters involved by the short run dynamics of the process. Such an approach is called nonparametric by Bierens (1997b) and can be used to test a wide range of nonlinear models. We consider three different versions of such a test. However, simulation results suggest that only the variance ratio statistic is able to compete with the traditional augmented Dickey-Fuller test. A straightforward generalization of the variance ratio statistic is suggested, which can be used to test the cointegration rank in the spirit of Johansen (1988).

A Parametric approach to the Estimation of Cointegration Vectors in Panel Data

ABSTRACT In this article, a parametric framework for estimation and inference in cointegrated panel data models is considered that is based on a cointegrated VAR(p) model. A convenient two-step estimator is suggested where, in the first step, all individual specific parameters are estimated, and in the second step, the long-run parameters are estimated from a pooled least-squares regression. The two-step estimator and related test procedures can easily be modified to account for contemporaneously correlated errors, a feature that is often encountered in multi-country studies. Monte Carlo simulations suggest that the two-step estimator and related test procedures outperform semiparametric alternatives such as the fully modified OLS approach, especially if the number of time periods is small.

Real-Time Forecasting of GDP Based on a Large Factor Model with Monthly and Quarterly Data

This paper discusses a factor model for estimating monthly GDP using a large number of monthly and quarterly time series in real-time. To take into account the different periodicities of the data and missing observations at the end of the sample, the factors are estimated by applying an EM algorithm combined with a principal components estimator. We discuss the in-sample properties of the estimator in real-time environments and methods for out-of-sample forecasting. As an empirical application, we estimate monthly German GDP in real-time, discuss the nowcast and forecast accuracy of the model and the role of revisions. Furthermore, we assess the contribution of timely monthly data to the forecast performance.

Rank Tests for Nonlinear Cointegration

A test procedure based on ranks is suggested to test for nonlinear cointegration. For two (or more) time series it is assumed that monotonic transformations exist such that the normalized series can asymptotically be represented as Wiener processes. Rank-test procedures based on the difference between the sequences of ranks are suggested. If there is no cointegration between the time series, the sequences of ranks tend to diverge, whereas under cointegration the sequences of ranks evolve similarly. Monte Carlo simulations suggest that for a wide range of nonlinear models the rank tests perform better than their parametric competitors. To test for nonlinear cointegration, a variable addition test based on ranks is suggested. In an empirical illustration, the rank statistics are applied to test the relationship between bond yields with different times to maturity.

Inference on the Cointegration Rank in Fractionally Integrated Processes

For univariate time series we suggest a new variant of efficient score tests against fractional alternatives. This test has three important merits. First, by means of simulations we observe that it is superior in terms of size and power in some situations of practical interest. Second, it is easily understood and implemented as a slight modification of the Dickey-Fuller test, although our score test has a limiting normal distribution. Third and most important, our test generalizes to multivariate cointegration tests just as the Dickey-Fuller test does. Thus it allows to determine the cointegration rank of fractionally integrated time series. It does so by solving a generalized eigenvalue problem of the type proposed by Johansen (1988). However, the limiting distribution of the corresponding trace statistic is X2 , where the degrees of freedom depend only on the cointegration rank under the null hypothesis. The usefulness of the asymptotic theory for finite samples is established in a Monte Carlo experiment.

Applied Time Series Econometrics: Structural Vector Autoregressive Modeling and Impulse Responses

Introduction In the previous chapter we have seen how a model for the DGP of a set of economic time series variables can be constructed. When such a model is available, it can be used for analyzing the dynamic interactions between the variables. This kind of analysis is usually done by tracing the effect of an impulse in one of the variables through the system. In other words, an impulse response analysis is performed. Although this is technically straightforward, some problems related to impulse response analysis exist that have been the subject of considerable discussion in the literature. As argued forcefully by Cooley & LeRoy (1985), vector autoregressions have the status of “reduced form” models and therefore are merely vehicles to summarize the dynamic properties of the data. Without reference to a specific economic structure, such reduced-formVAR models are difficult to understand. For example, it is often difficult to draw any conclusion from the large number of coefficient estimates in a VAR system. As long as such parameters are not related to “deep” structural parameters characterizing preferences, technologies, and optimization behavior, the parameters do not have an economic meaning and are subject to the so-called Lucas critique. Sims (1981, 1986), Bernanke (1986), and Shapiro & Watson (1988) put forward a new class of econometric models that is now known as structural vector autoregression (SVAR) or identified VAR . Instead of identifying the (autoregressive) coefficients, identification focuses on the errors of the system, which are interpreted as (linear combinations of) exogenous shocks. In the early applications of Sargent (1978) and Sims (1980), the innovations of the VAR were orthogonalized using a Choleski decomposition of the covariance matrix.

Dynamic factor models

SummaryFactor models can cope with many variables without running into scarce degrees of freedom problems often faced in a regression-based analysis. In this article we review recent work on dynamic factor models that have become popular in macroeconomic policy analysis and forecasting. By means of an empirical application we demonstrate that these models turn out to be usefu in investigating macroeconomic problems.

Temporal Aggregation and Spurious Instantaneous Causality in Multiple Time Series Models

Large aggregation interval asymptotics are used to investigate the relation between Granger causality in disaggregated vector autoregressions (VARs) and associated contemporaneous correlation among innovations of the aggregated system. One of our main contributions is that we outline various conditions under which the informational content of error covariance matrices yields insight into the causal structure of the VAR. Monte Carlo results suggest that our asymptotic findings are applicable even when the aggregation interval is small, as long as the time series are not characterized by high levels of persistence.

TESTING FOR UNIT ROOTS IN PANELS WITH A FACTOR STRUCTURE

This paper considers various tests of the unit root hypothesis in panels where the cross-section dependence is due to common dynamic factors. Three situations are studied. First, the common factors and idiosyncratic components may both be nonstationary. In this case test statistics based on generalized least squares (GLS) possess a standard normal limiting distribution, whereas test statistics based on ordinary least squares (OLS) are invalid. Second, if the common component is I(1) and the idiosyncratic component is stationary (the case of cross-unit cointegration), then both the OLS and the GLS statistics fail. Finally, if the idiosyncratic components are I(1) but the common factors are stationary, then the OLS-based test statistics are severely biased, whereas the GLS-based test statistics are asymptotically valid in this situation. A Monte Carlo study is conducted to verify the asymptotic results.The research for this paper was carried out within research project “Unit roots and cointegration in panel data†financed by the German Research Association (DFG). We thank Paulo Rodrigues and two anonymous referees for helpful comments and suggestions.