Nonparametric rank tests for non-stationary panels
Introduction
This paper develops rank tests for the number of common stochastic trends present in a time series panel. The tests are designed to perform well in situations where the cross-sectional dimension of the panel is too large for traditional multivariate cointegration methods to be used successfully. The paper also investigates the relationships between these rank tests and other key tests in the non-stationary panel literature, such as panel unit root tests and tests for the fraction of individual series which are versus processes.
Much of the recent non-stationary panel literature has focused on permitting increasingly general forms of cross-sectional dependencies among members of the panel (see, for example, Breitung and Pesaran, 2008, Banerjee and Wagner, 2009, for recent overviews). However, as we discuss in Section 2 of this paper, the extent of the cross-sectional dependency that one permits under the data generating process (DGP) is inherently tied to the types of hypotheses that one can successfully test with asymptotic size control. In particular, as we will see, the absence or presence of (cross unit) cointegration among the series is often a key feature in this regard.1 This is particularly important in relation to the ability to determine the overall number of individual series that are versus , as well as the ability to determine which particular series are versus . Granted, in situations where the time series dimension is large enough, one might consider using time series methods alone rather than panel methods to determine the number of individual series which follow versus processes. However, often one is interested not only in the individual series properties, but also the implications of the linkages among the individual series, and most importantly the cross-sectional dependencies that are driven by the common stochastic trends.
An important component of the non-stationary panel literature has been the literature on testing for unit roots in panels. A popular approach to accommodating what have been considered fairly general forms of cross-sectional dependence within this literature has been the factor model approach. An underlying assumption of this approach is the decomposition of the series of the panel into what are assumed to be independent common and idiosyncratic components. The idiosyncratic components are then tested for unit roots and the common components are tested either for unit roots in the single factor case, or cointegration in the multiple factor case. However, in many applications, such as the income convergence illustration we provide in Section 5, one is not interested to know the unit root versus stationarity properties of these separate components, but rather one is interested to know these properties about the individual raw series. For such cases we argue that our rank test approach is the best suited and most general approach available for panels with moderate to large cross-sectional dimensions.
In this regard, our tests do not impose any restrictions on the cross-sectional dependencies of the series. The only restriction on the series’ behavior is that a joint functional central limit theorem must hold for the first differences of the -vector of series. In such a general setup, except for the extreme cases when the rank is either full (so that all series are and not cointegrated), or zero (so that all series are ), all series will in general be and cointegrated. In particular, our computationally simple tests are based on multivariate variance ratios computed from tuning parameter free estimates of the respective components that do not require, and are thus not affected by, choices with respect to kernel and bandwidth, lag augmentation or the number of factors to be extracted. These estimators are based on advances in long-run variance estimation pioneered by Kiefer and Vogelsang, 2002a, Kiefer and Vogelsang, 2002b.
Because the tests are cointegration rank tests, they can be used to infer any rank, and not just the null hypothesis of full rank, which is standard in the literature on non-stationary panels. In fact, the forms of hypotheses considered within this literature are very limited. Specifically, while the null hypothesis is almost always taken to be that all series are , the alternative hypothesis is usually formulated as that at least some series are . This leaves a rejection of the null somewhat uninformative as it does not indicate how many series there are. This issue is discussed to some extent by Pesaran (2012), who recommends “the (panel unit root) test outcome to be augmented with an estimate of the proportion of the cross-section units for which the individual unit root tests are rejected” (see page 545). Motivated in part by this recommendation, we also propose a sequential rank testing procedures that compare favorably with for example the Johansen (1995) vector autoregression (VAR) based approach for moderate values of , and increasingly so for larger values of . We also show that for the special case of panels without the presence of cross-unit cointegration, our rank tests can be used to test the same null hypothesis usually tested by conventional panel unit root tests,2 with the additional advantage that any rank can be tested. Accordingly, in this setting our tests allow one to determine the fractions of versus series present in the panel.3
The new tests have good small-sample properties, which is demonstrated in a series of Monte Carlo simulation experiments. The proposed sequential rank test procedure is also shown to compare well to the Johansen (1995) trace test. In terms of sample size, the comparative advantage of our test occurs when is moderately sized, in that it is smaller than the dimension, but larger than one can handle well with parametric based multivariate cointegration methods. Similarly, we also show that when used in place of panel unit root tests the new tests outperform widely-used first-generation as well as state-of-the-art second generation tests under the conditions for which these other tests were designed.4
The remainder of the paper is organized as follows. In Section 2 we first discuss the DGPs and assumptions used in our approach and then discuss the relationships of our setup to the assumptions and DGPs used in the existing non-stationary panel literature. Section 3 presents the rank tests, provides critical values and discusses the local asymptotic power (LAP) properties of our tests for the special case of cross-sectionally independent panels. In this way, by comparing the LAP of our tests with the LAP of two widely-used first-generation panel unit roots, we seek to demonstrate that there is no cost to the generality of our approach even when the more restrictive assumption of cross-section independence is true. Next, in Section 4 we study the small-sample performance of our tests and compare our tests with several second-generation tests under the conditions for which these tests were designed. Finally, we also compare the small-sample performance of our sequential rank test procedure with the Johansen trace test in this section. Section 5 in turn contains a brief empirical illustration of the rank tests taken from the growth and convergence literature. Section 6 offers concluding remarks.
Section snippets
Assumptions
The DGP is a stated in terms of the -dimensional vector of time series , and is given by with observations available for . Here , for , is a polynomial trend function (with ) and is the associated matrix of trend coefficients.5 The typical specifications considered for include a constant () or a
The rank tests
In this section we develop the tests which are designed to test any null hypothesis versus any alternative hypothesis .
Small-sample performance
In this section, we use Monte Carlo simulations to evaluate the small-sample properties of the new tests. The results are organized in two subsections. In Section 4.2, we investigate size and power, as well as the accuracy of the estimated ranks based on the sequential test. In Section 4.3, we compare the performance of the new tests with that of the second-generation tests of Ng (2008) when one wishes to test for the fraction of series that follow processes, and with Bai and Ng
Empirical illustration
In this section we pursue a brief illustration taken from the empirical growth and convergence literature. Our analysis follows in large part the interpretation of Evans (1998) (see Banerjee and Wagner, 2009, for a more detailed discussion, including the role of deterministic components). Specifically, suppose that , log per capita GDP, which we refer to as income, in country at time , is . Then the income panel is said to exhibit absolute convergence if, for any pair of countries
Conclusions
We have introduced new cointegration rank tests for time series panels. The tests have several important and partly distinctive features: First, conceptually, the extent of cross-sectional dependencies is not limited or restricted in shape, apart from some standard technical assumptions that lead to limiting distributions that are functionals of Wiener processes. Second, computationally, the tests do not require the estimation of any nuisance parameters and are free of choices concerning
Acknowledgments
We thank seminar participants at Brown University, Cornell University, Maastricht University, Texas AM, University of Montreal and Williams College, and conference participants at the Econometric Society World Congress, the Midwest Econometrics Group at the University of Chicago, the Unit Root and Cointegration Conference in Faro, the Conference on Factor Models in Panels at Goethe University Frankfurt, the 11th International Panel Data Conference, the NY Econometrics Camp, and in particular
References (44)
Nonparametric tests for unit root and cointegration
J.~Econometrics
(2002)Bootstrap unit root tests in panels with cross-sectional dependency
J. Econometrics
(2004)Unit root tests for panel data
J. Internat. Money Financ.
(2001)- et al.
Getting PPP right: Identifying mean-reverting real exchange rates in panels
J. Bank. Finance
(2009) - et al.
Testing for unit roots in heterogeneous panels
J. Econometrics
(2003) - et al.
Beyond panel unit root tests: Using multiple testing to determine the non stationarity properties of individual series in a panel
J.~Econometrics
(2012) - et al.
Testing for a unit root in panels with dynamic factors
J. Econometrics
(2004) - et al.
Incidental trends and the power of panel unit root tests
J. Econometrics
(2007) - et al.
Cross-sectional dependence robust block bootstrap panel unit root tests
J. Econometrics
(2011) On the interpretation of panel unit root tests
Econom. Lett.
(2012)
Testing for unit roots in a panel random coefficient model
J. Econometrics
Determining the number of factors in approximate factor models
Econometrica
A PANIC attack on unit roots and cointegration
Econometrica
Panel unit root tests with cross-section dependence: A further investigation
Econometric Theory
Panel methods to test for unit roots and cointegration
Unit roots and cointegration in panels
Using panel data to evaluate growth theories
Internat. Econom. Rev.
For which countries did PPP hold? A multiple testing approach
Emp. Econom.
Local asymptotic power of the Im-Pesaran-Shin panel unit root test and the impact of initial observations
Econometric Theory
Finite sample correction factors for panel cointegration tests
Oxf. Bull. Econ. Stat.
The performance of panel unit root and stationarity tests: Results from a large scale simulation study
Econometric Rev.
Consistent covariance matrix estimation for linear processes
Econometric Theory
Cited by (12)
Statistical disclosure and economic growth: What is the nexus?
2022, World DevelopmentA nonparametric analysis of energy environmental Kuznets Curve in Chinese Provinces
2020, Energy EconomicsCitation Excerpt :Therefore, provinces are bound to improve the quality of economic development by adjusting and optimizing the industrial structure and improving energy efficiency, thereby achieving the convergence of energy consumption amongst Chinese provinces. Panel-B (Table 6) provides the Pedroni et al. (2015) estimates for provincial economic growth (lnYt). Here too, the test statistic is greater than the corresponding 5% level critical value, thereby rejecting the null of full cross-unit cointegration (c = 30).
Extreme canonical correlations and high-dimensional cointegration analysis
2019, Journal of EconometricsCitation Excerpt :Analysis of cointegration between a large number of time series is a challenging but useful exercise. Its applications include high-dimensional vector error correction modelingfor forecasting purposes (Engel et al. (2015)), inference in nonstationary panel data models (Banerjee et al. (2004), Pedroni et al. (2015)), and verification of the assumptions under which composite commodity price indexes satisfy microeconomic laws of demand (Lewbel (1996), Davis (2003)). A central role in the likelihood-based cointegration analysis is played by the squared sample canonical correlation coefficients between a simple transformation of the levels and the first differences of the data.
Panel cointegration techniques and open challenges
2019, Panel Data Econometrics: TheoryINFERENCE ON THE DIMENSION OF THE NONSTATIONARY SUBSPACE IN FUNCTIONAL TIME SERIES
2023, Econometric Theory