Variance ratio tests of the seasonal unit root hypothesis

In this paper we develop tests of the seasonal (quarterly) unit root null hypothesis which reject in favour of stationarity for small values of certain variance ratio statistics, similar to those used by Canova and Hansen (J. Bus. Econom. Statist. 13 (1995) 237) in a different testing context. We demonstrate that our proposed statistics have pivotal limiting distributions under both the null and near seasonally integrated alternatives even when we allow for the possibility of both weak dependence and periodically heteroscedastic behaviour in the driving shocks. This is in contrast to the popular regression-based lag-augmented seasonal unit root tests of Hylleberg et al. (J. Econometrics 44 (1990) 215). A simulation study into the finite sample size and power properties of our proposed tests suggests that they display far superior size properties and, overall, broadly comparable power properties to the corresponding tests of Hylleberg et al. (J. Econometrics 44 (1990) 215), implemented with data-based lag augmentation. The results for the variance ratio tests at the seasonal harmonic frequency are particularly encouraging.     

Numerical distribution functions for unit root and cointegration tests

This paper employs response surface regressions based on simulation experiments to calculate distribution functions for some well-known unit root and cointegration test statistics. The principal contributions of the paper are a set of data files that contain estimated response surface coefficients and a computer program for utilizing them. This program, which is freely available via the Internet, can easily be used to calculate both asymptotic and finite-sample critical values and P-values for any of the tests. Graphs of some of the tabulated distribution functions are provided. An empirical example deals with interest rates and inflation rates in Canada.     

Measurement errors and outliers in seasonal unit root testing

Seasonal and non-seasonal data are frequently observed with noise. For instance, the time series can have irregular abrupt changes and interruptions following as a result of additive or temporary change outliers caused by external circumstances. Equally, the time series can have measurement errors. In this paper we analyse the above types of data irregularities on the behavior of seasonal unit root tests. Outliers and measurement errors can seriously affect seasonal unit root inference and it is shown how the distortion of the tests will depend upon the frequency, magnitude, and persistence of the outliers as well as on the signal to noise ratio associated with measurement errors. Some solutions to the implied inference problems are suggested and shown to work in practice.     

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.     

Unit root tests in panel data: asymptotic and finite-sample properties

We consider pooling cross-section time series data for testing the unit root hypothesis. The degree of persistence in individual regression error, the intercept and trend coefficient are allowed to vary freely across individuals. As both the cross-section and time series dimensions of the panel grow large, the pooled t-statistic has a limiting normal distribution that depends on the regression specification but is free from nuisance parameters. Monte Carlo simulations indicate that the asymptotic results provide a good approximation to the test statistics in panels of moderate size, and that the power of the panel-based unit root test is dramatically higher, compared to performing a separate unit root test for each individual time series.     

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.     

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.     

Bootstrap unit root tests in panels with cross-sectional dependency

We apply bootstrap methodology to unit root tests for dependent panels with N cross-sectional units and T time series observations. More specifically, we let each panel be driven by a general linear process which may be different across cross-sectional units, and approximate it by a finite order autoregressive integrated process of order increasing with T. As we allow the dependency among the innovations generating the individual series, we construct our unit root tests from the estimation of the system of the entire N cross-sectional units. The limit distributions of the tests are derived by passing T to infinity, with N fixed. We then apply bootstrap method to the approximated autoregressions to obtain critical values for the panel unit root tests, and establish the asymptotic validity of such bootstrap panel unit root tests under general conditions. The proposed bootstrap tests are indeed quite general covering a wide class of panel models. They in particular allow for very general dynamic structures which may vary across individual units, and more importantly for the presence of arbitrary cross-sectional dependency. The finite sample performance of the bootstrap tests is examined via simulations, and compared to that of commonly used panel unit root tests. We find that our bootstrap tests perform relatively well, especially when N is small.     

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.     

LM tests of spatial dependence based on bootstrap critical values

To test the existence of spatial dependence in an econometric model, a convenient test is the Lagrange Multiplier (LM) test. However, evidence shows that, in finite samples, the LM test referring to asymptotic critical values may suffer from the problems of size distortion and low power, which become worse with a denser spatial weight matrix. In this paper, residual-based bootstrap methods are introduced for asymptotically refined approximations to the finite sample critical values of the LM statistics. Conditions for their validity are clearly laid out and formal justifications are given in general, and in detail under several popular spatial LM tests using Edgeworth expansions. Monte Carlo results show that when the conditions are not fully met, bootstrap may lead to unstable critical values that change significantly with the alternative, whereas when all conditions are met, bootstrap critical values are very stable, approximate much better the finite sample critical values than those based on asymptotics, and lead to significantly improved size and power. The methods are further demonstrated using more general spatial LM tests, in connection with local misspecification and unknown heteroskedasticity.     

The relationship between power and level for generic unit root tests in finite samples

We show that the maximum power of a generic unit root test against any stationary alternative is equal to the true level of the test. We then use Monte Carlo methods to investigate the implications for several such tests. We show patterns of rejection probabilities over a variety of unit root and stationary processes. We discuss the implications of these results for some of the uses of unit root tests in applied work.     

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.     

A review of Phillips-type right-tailed unit root bubble detection tests

Recent developments on the right-tailed unit root tests of Phillips et al., which are used to date stamp the origination and collapse dates of asset price bubbles, have generated considerable interest. This paper provides a review for both empirical researchers that adopt these new econometric tools to detect the presence of asset price bubbles, and theoretical papers that extend these testing procedures. This paper also uses the psymonitor package in R to demonstrate the practical use of such tests using an example based on data for British Railway Mania of the 1840s.     

‘Objective’ bayesian unit root tests

Due to weaknesses in traditional tests, a Bayesian approach is developed to investigate whether unit roots exist in macroeconomic time-series. Bayesian posterior odds comparing unit root models to stationary and trend-stationary alternatives are calculated using informative priors. Two classes of reference priors which are informative but require minimal subjective prior input are used. In this sense the Bayesian unit root tests developed here are objective. Bayesian procedures are carried out on the Nelson–Plosser and Shiller data sets as well as on generated data. The conclusion is that the failure of classical procedures to reject the unit root hypothesis is not necessarily proof that a unit root is present with high probability.     

Panel unit root tests in the presence of a multifactor error structure

This paper extends the cross-sectionally augmented panel unit root test (CIPS$CIPS$) proposed by Pesaran (2007) to the case of a multifactor error structure, and proposes a new panel unit root test based on a simple average of cross-sectionally augmented Sargan–Bhargava statistics (CSB$CSB$). The basic idea is to exploit information regarding the m$m$ unobserved factors that are shared by k$k$ observed time series in addition to the series under consideration. Initially, we develop the tests assuming that m⁰$m^0$, the true number of factors, is known and show that the limit distribution of the tests does not depend on any nuisance parameters, so long as k≥m⁰−1$k\ge m^0-1$. Small sample properties of the tests are investigated by Monte Carlo experiments and are shown to be satisfactory. Particularly, the proposed CIPS$CIPS$ and CSB$CSB$ tests have the correct size for all   combinations of the cross section (N$N$) and time series (T$T$) dimensions considered. The power of both tests rises with N$N$ and T$T$, although the CSB$CSB$ test performs better than the CIPS$CIPS$ test for smaller sample sizes. The various testing procedures are illustrated with empirical applications to real interest rates and real equity prices across countries.     

On the convergence of the critical values of uniformly most powerful unbiased tests for two-sided hypotheses

Summary The behavior of the critical values of two-sided uniformly most powerful unbiased tests is studied under the condition of weak convergence of the distribution of the test statistic.     

Asymptotic and bootstrap tests for linearity in a TAR-GARCH(1,1) model with a unit root

This paper derives the limiting distribution of the Lagrange Multiplier (LM) test for threshold nonlinearity in a TAR model with GARCH errors when one of the regimes contains a unit root. It is shown that the asymptotic distribution is nonstandard and depends on nuisance parameters that capture the degree of conditional heteroskedasticity and non-Gaussian nature of the process. We propose a bootstrap procedure for approximating the exact finite-sample distribution of the test for linearity and establish its asymptotic validity.     

The power of bootstrap and asymptotic tests

The bootstrap discrepancy measures the difference in rejection probabilities between a bootstrap test and one based on the true distribution. The order of magnitude of the bootstrap discrepancy is the same under the null hypothesis and under non-null processes described by Pitman drift. If the test statistic is not an exact pivot, critical values depend on which data-generating process (DGP) is used to determine the null distribution. We propose using the DGP which minimizes the bootstrap discrepancy. We also show that, under an asymptotic independence condition, the power of both bootstrap and asymptotic tests can be estimated cheaply by simulation.     

Unemployment in Greece: Evidence from Greek regions using panel unit root tests

The purpose of the paper is to examine the nature of Greek regional unemployment. The paper contributes to the literature assessing the stochastic properties of Greek unemployment rate in the context of the Greek regions by relying on various univariate and panel unit root tests. In particular, recently developed and more powerful panel unit-root tests that control for structural breaks, heterogeneity and cross-sectional dependence in the panel are employed. The results show that in all cases, after taking into account the fact that regional unemployment rates in Greece are subject to a structural break, the null hypothesis of a unit root is not rejected, indicating that the Greek regional unemployment series are non-stationary with the presence of a structural break.     

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.