An instrumental variable approach for panel unit root tests under cross-sectional dependence

For dynamic panel models with cross-sectional dependence, several unit root tests are constructed using a Huber-type instrument, whose null asymptotics are standard Gaussian and do not depend on nuisance parameters. A Monte-Carlo simulation shows that the proposed tests have better sizes and comparable powers relative to other two existing tests developed for cross-sectionally dependent dynamic panel models.     

A nonlinear panel data model of cross-sectional dependence

This paper proposes a nonlinear panel data model which can endogenously generate both ‘weak’ and ‘strong’ cross-sectional dependence. The model’s distinguishing characteristic is that a given agent’s behaviour is influenced by an aggregation of the views or actions of those around them. The model allows for considerable flexibility in terms of the genesis of this herding or clustering type behaviour. At an econometric level, the model is shown to nest various extant dynamic panel data models. These include panel AR models, spatial models, which accommodate weak dependence only, and panel models where cross-sectional averages or factors exogenously generate strong, but not weak, cross sectional dependence. An important implication is that the appropriate model for the aggregate series becomes intrinsically nonlinear, due to the clustering behaviour, and thus requires the disaggregates to be simultaneously considered with the aggregate. We provide the associated asymptotic theory for estimation and inference. This is supplemented with Monte Carlo studies and two empirical applications which indicate the utility of our proposed model as a vehicle to model different types of cross-sectional dependence.     

A Lagrange Multiplier test for cross-sectional dependence in a fixed effects panel data model

It is well known that the standard Breusch and Pagan (1980) LM test for cross-equation correlation in a SUR model is not appropriate for testing cross-sectional dependence in panel data models when the number of cross-sectional units (n)$\left(n\right)$ is large and the number of time periods (T)$\left(T\right)$ is small. In fact, a scaled version of this LM test was proposed by Pesaran (2004) and its finite sample bias was corrected by Pesaran et al. (2008). This was done in the context of a heterogeneous panel data model. This paper derives the asymptotic bias of this scaled version of the LM test in the context of a fixed effects homogeneous panel data model. This asymptotic bias is found to be a constant related to n$n$ and T$T$, which suggests a simple bias corrected LM test for the null hypothesis. Additionally, the paper carries out some Monte Carlo experiments to compare the finite sample properties of this proposed test with existing tests for cross-sectional dependence.     

A test of cross section dependence for a linear dynamic panel model with regressors

This paper proposes a new testing procedure for detecting error cross section dependence after estimating a linear dynamic panel data model with regressors using the generalised method of moments (GMM). The test is valid when the cross-sectional dimension of the panel is large relative to the time series dimension. Importantly, our approach allows one to examine whether any error cross section dependence remains after including time dummies (or after transforming the data in terms of deviations from time-specific averages), which will be the case under heterogeneous error cross section dependence. Finite sample simulation-based results suggest that our tests perform well, particularly the version based on the [Blundell, R., Bond, S., 1998. Initial conditions and moment restrictions in dynamic panel data models. Journal of Econometrics 87, 115–143] system GMM estimator. In addition, it is shown that the system GMM estimator, based only on partial instruments consisting of the regressors, can be a reliable alternative to the standard GMM estimators under heterogeneous error cross section dependence. The proposed tests are applied to employment equations using UK firm data and the results show little evidence of heterogeneous error cross section dependence.     

Testing for time-invariant unobserved heterogeneity in generalized linear models for panel data

Recent literature on panel data emphasizes the importance of accounting for time-varying unobservable individual effects, which may stem from either omitted individual characteristics or macro-level shocks that affect each individual unit differently. In this paper, we propose a simple specification test of the null hypothesis that the individual effects are time-invariant against the alternative that they are time-varying. Our test is an application of Hausman (1978) testing procedure and can be used for any generalized linear model for panel data that admits a sufficient statistic for the individual effect. This is a wide class of models which includes the Gaussian linear model and a variety of nonlinear models typically employed for discrete or categorical outcomes. The basic idea of the test is to compare two alternative estimators of the model parameters based on two different formulations of the conditional maximum likelihood method. Our approach does not require assumptions on the distribution of unobserved heterogeneity, nor it requires the latter to be independent of the regressors in the model. We investigate the finite sample properties of the test through a set of Monte Carlo experiments. Our results show that the test performs well, with small size distortions and good power properties. We use a health economics example based on data from the Health and Retirement Study to illustrate the proposed test.     

Testing for a unit root in a random coefficient panel data model

This paper proposes new unit root tests in the context of a random autoregressive coefficient panel data model, in which the null of a unit root corresponds to the joint restriction that the autoregressive coefficient has unit mean and zero variance. The asymptotic distributions of the test statistics are derived and simulation results are provided to suggest that they perform very well in small samples.     

Taking a new contour: A novel approach to panel unit root tests

The paper introduces a novel approach to testing for unit roots in panels, which takes a new contour that is drawn along the line given by the equi-squared-sum instead of the traditional one given by the equi-sample-size. We show in the paper that the distributions of the unit root tests are asymptotically normal along the new contour under both the null and the local-to-unity alternatives. Subsequently, we demonstrate that this startling finding may be exploited constructively to invent tools and methodologies for effective inferences in panel unit root models. Simulations show that our approach works quite well in finite samples.     

Sieve estimation of panel data models with cross section dependence

In this paper we consider the problem of estimating semiparametric panel data models with cross section dependence, where the individual-specific regressors enter the model nonparametrically whereas the common factors enter the model linearly. We consider both heterogeneous and homogeneous regression relationships when both the time and cross-section dimensions are large. We propose sieve estimators for the nonparametric regression functions by extending Pesaran’s (2006) common correlated effect (CCE) estimator to our semiparametric framework. Asymptotic normal distributions for the proposed estimators are derived and asymptotic variance estimators are provided. Monte Carlo simulations indicate that our estimators perform well in finite samples.     

Cross-sectional dependence robust block bootstrap panel unit root tests

In this paper we consider the issue of unit root testing in cross-sectionally dependent panels. We consider panels that may be characterized by various forms of cross-sectional dependence including (but not exclusive to) the popular common factor framework. We consider block bootstrap versions of the group-mean (Im et al., 2003) and the pooled (Levin et al., 2002) unit root coefficient DF tests for panel data, originally proposed for a setting of no cross-sectional dependence beyond a common time effect. The tests, suited for testing for unit roots in the observed data, can be easily implemented as no specification or estimation of the dependence structure is required. Asymptotic properties of the tests are derived for T going to infinity and N finite. Asymptotic validity of the bootstrap tests is established in very general settings, including the presence of common factors and cointegration across units. Properties under the alternative hypothesis are also considered. In a Monte Carlo simulation, the bootstrap tests are found to have rejection frequencies that are much closer to nominal size than the rejection frequencies for the corresponding asymptotic tests. The power properties of the bootstrap tests appear to be similar to those of the asymptotic tests.     

Beyond panel unit root tests: Using multiple testing to determine the nonstationarity properties of individual series in a panel

Most panel unit root tests are designed to test the joint null hypothesis of a unit root for each individual series in a panel. After a rejection, it will often be of interest to identify which series can be deemed to be stationary and which series can be deemed nonstationary. Researchers will sometimes carry out this classification on the basis of n$n$ individual (univariate) unit root tests based on some ad hoc significance level. In this paper, we suggest and demonstrate how to use the false discovery rate (FDR)$\left(FDR\right)$ in evaluating I(1)/I(0)$I\left(1\right)/I\left(0\right)$ classifications.     

Maximum likelihood estimation and inference methods for the covariance stationary panel AR(1)/unit root model

This paper considers Maximum Likelihood (ML) based estimation and inference procedures for linear dynamic panel data models with fixed effects.     

Bias in dynamic panel estimation with fixed effects,incidental trends and cross section dependence

Explicit asymptotic bias formulae are given for dynamic panel regression estimators as the cross section sample size N→∞$N\to \infty$. The results extend earlier work by Nickell [1981. Biases in dynamic models with fixed effects. Econometrica 49, 1417–1426] and later authors in several directions that are relevant for practical work, including models with unit roots, deterministic trends, predetermined and exogenous regressors, and errors that may be cross sectionally dependent. The asymptotic bias is found to be so large when incidental linear trends are fitted and the time series sample size is small that it changes the sign of the autoregressive coefficient. Another finding of interest is that, when there is cross section error dependence, the probability limit of the dynamic panel regression estimator is a random variable rather than a constant, which helps to explain the substantial variability observed in dynamic panel estimates when there is cross section dependence even in situations where N is very large. Some proposals for bias correction are suggested and finite sample performance is analyzed in simulations.     

Common breaks in means and variances for panel data

This paper establishes the consistency of the estimated common break point in panel data. Consistency is obtainable even when a regime contains a single observation, making it possible to quickly identify the onset of a new regime. We also propose a new framework for developing the limiting distribution for the estimated break point, and show how to construct confidence intervals. The least squares method is used for estimating breaks in means and the quasi-maximum likelihood (QML) method is used to estimate breaks in means and in variances. QML is shown to be more efficient than the least squares even if there is no change in the variances.     

Panel unit root tests under cross-sectional dependence

In this paper alternative approaches for testing the unit root hypothesis in panel data are considered. First, a robust version of the Dickey-Fuller t -statistic under contemporaneous correlated errors is suggested. Second, the GLS t -statistic is considered, which is based on the t -statistic of the transformed model. The asymptotic power of both tests is compared against a sequence of local alternatives. To adjust for short-run serial correlation of the errors, we propose a pre-whitening procedure that yields a test statistic with a standard normal limiting distribution as N and T tends to infinity. The test procedure is further generalized to accommodate individual specific intercepts or linear time trends. From our Monte Carlo simulations it turns out that the robust OLS t -statistic performs well with respect to size and power, whereas the GLS t -statistic may suffer from severe size distortions in small and moderate sample sizes. The tests are applied to test for a unit root in real exchange rates.     

The effect of data transformation on common cycle,cointegration, and unit root tests: Monte Carlo results and a simple test

Cointegration, common cycle, and related tests statistics are often constructed using logged data, even without clear reason why logs should be used rather than levels. Unfortunately, it is also the case that standard data transformation tests, such as those based on Box–Cox transformations, cannot be shown to be consistent unless assumptions concerning whether variables I(0)$I(0)$ or I(1)$I(1)$ are made. In this paper, we propose a simple randomized procedure for choosing between levels and log-levels specifications in the (possible) presence of deterministic and/or stochastic trends, and discuss the impact of incorrect data transformation on common cycle, cointegration and unit root tests.     

A class of simple distribution-free rank-based unit root tests

We propose a class of distribution-free rank-based tests for the null hypothesis of a unit root. This class is indexed by the choice of a reference densityg, which need not coincide with the unknown actual innovation density f. The validity of these tests, in terms of exact finite-sample size, is guaranteed, irrespective of the actual underlying density, by distribution-freeness. Those tests are locally and asymptotically optimal under a particular asymptotic scheme, for which we provide a complete analysis of asymptotic relative efficiencies. Rather than stressing asymptotic optimality, however, we emphasize finite-sample performances, which also depend, quite heavily, on initial values. It appears that our rank-based tests significantly outperform the traditional Dickey-Fuller tests, as well as the more recent procedures proposed by Elliott et al. (1996), Ng and Perron (2001), and Elliott and Müller (2006), for a broad range of initial values and for heavy-tailed innovation densities. Thus, they provide a useful complement to existing techniques.     

Nonresponse in dynamic panel data models

To verify whether data are missing at random (MAR) we need to observe the missing data. There are only two exceptions: when the relationship between the probability of responding and the missing variables is either imposed by introducing untestable assumptions or recovered using additional data sources. In this paper, we briefly review the estimation and test procedures for selectivity in panel data. Furthermore, by extending the MAR definition from a static setting to the case of dynamic panel data models, we prove that some tests for selectivity are not verifying the MAR condition.     

Nonlinear IV unit root tests in panels with cross-sectional dependency

We propose a unit root test for panels with cross-sectional dependency. We allow general dependency structure among the innovations that generate data for each of the cross-sectional units. Each unit may have different sample size, and therefore unbalanced panels are also permitted in our framework. Yet, the test is asymptotically normal, and does not require any tabulation of the critical values. Our test is based on nonlinear IV estimation of the usual augmented Dickey–Fuller type regression for each cross-sectional unit, using as instruments nonlinear transformations of the lagged levels. The actual test statistic is simply defined as a standardized sum of individual IV t-ratios. We show in the paper that such a standardized sum of individual IV t-ratios has limit normal distribution as long as the panels have large individual time series observations and are asymptotically balanced in a very weak sense. We may have the number of cross-sectional units arbitrarily small or large. In particular, the usual sequential asymptotics, upon which most of the available asymptotic theories for panel unit root models heavily rely, are not required. Finite sample performance of our test is examined via a set of simulations, and compared with those of other commonly used panel unit root tests. Our test generally performs better than the existing tests in terms of both finite sample sizes and powers. We apply our nonlinear IV method to test for the purchasing power parity hypothesis in panels.