panel unit root

This is an important issue since the application of tests belonging to the …rst generation to series that are characterized by cross-sectional dependencies leads to size distortions and low power (Banerjee, Marcellino and Osbat, 2000, Strauss and Yigit, 2003). In response to the need for panel unit root tests that allows for cross-sectional correlations, various tests have been proposed belonging to what we call the class of the second generation tests. Rather than considering correlations across units as nuisance parameters, this new category of tests aims at exploiting these co-movements in order to de…ne new test statistics. As argued by Quah (1994), the modelling of cross-sectional dependencies is a di¢ cult task since no natural ordering exists in unit observations. This is why various tests have been proposed including the works of Bai and Ng (2001), Phillips and Sul (2003a), Moon and Perron (2004a), Choi (2002), Ploberger and Phillips (2002), Moon, Perron and Phillips (2003), Chang (2002) and Pesaran (2003). Two main approaches can be distinguished. The …rst one relies on the factor structure approach and includes the contributions of Bai and Ng (2001), Phillips and Sul (2003a), Moon and Perron (2004a), Choi (2002) and Pesaran (2003). The second approach consists in imposing few or none restrictions on the residuals covariance matrix. This approach has been adopted by Chang (2002) among others, who proposed the use of instrumental variables in order to solve the nuisance parameter problem due to cross-sectional dependency.

4. Monte Carlo analysis In this Section we carry out a small Monte Carlo analysis to compare the size and power properties of the the different unit root tests implemented (as of today) in punitroots. We use a slightly simplified version of a data generating process (DGP) that was originally pro- posed in Costantini and Lupi (2011). This DGP encompasses other DGPs commonly used in panel unit root Monte Carlo analyses. For brevity we will consider only three paradigmatic experiments. In the first, all the series are cross-sectionally independent. Given that cross- section independence can hardly be seen as a realistic feature of macro-panels, in the other two experiments the time series are generate more realistically in such a way that they are cross-sectionally dependent.

1 Introduction In recent years, the issue of testing for unit root in panel data has been a much debated topic. The literature about the development of such tests was initially based upon the assumption of cross-sectional independence between the units and it produced the so called ”first generation panel unit root tests”. However, in several empirical applications, this assumption is likely to be violated and O’Connell (1998) showed that not considering the possible dependence between units could introduce severe bias in the first generation panel unit root tests. Hence researchers were interested in developing tests invariant with respect to the cross-sectional dependence, the so called ”second generation unit root tests”.

Abstract This paper extends the cross sectionally augmented panel unit root test proposed by Pesaran (2007) to the case of a multifactor error structure. The basic idea is to exploit in- formation regarding the unobserved factors that are shared by other time series in addition to the variable under consideration. Importantly, our test procedure only requires speci…- cation of the maximum number of factors, in contrast to other panel unit root tests based on principal components that require in addition the estimation of the number of factors as well as the factors themselves. Small sample properties of the proposed test are investigated by Monte Carlo experiments, which suggest that it controls well for size in almost all cases, especially in the presence of serial correlation in the error term, contrary to alternative test statistics. Empirical applications to Fisher’s in‡ation parity and real equity prices across di¤erent markets illustrate how the proposed test works in practice.

There is now a sizeable literature on testing for unit roots in panels where both cross section (N ) and time (T ) dimensions are relatively large. Reviews of this literature are provided in Banerjee (1999), Baltagi and Kao (2000), Choi (2004), and more recently in Breitung and Pesaran (2007). The so called …rst generation panel unit root tests pioneered by Levin, Lin and Chu (2002) and Im, Pesaran and Shin (2003) focussed on panels where the idiosyncratic errors were cross sectionally uncorrelated. More recently, to deal with a number of applications such as testing for purchasing power parity or output convergence, the interest has shifted to the case where the errors are allowed to be cross sectionally correlated using a residual factor structure. 1 These second generation tests include the contributions of Moon and Perron (2004), Bai and Ng (2004, 2007) and Pesaran (2007). 2 The tests proposed by Moon and Perron (2004) and Pesaran (2007) assume that under the null of unit roots the common factor components have the same order of integration as the idiosyncratic components, whilst the test procedures of Bai and Ng (2004, 2007) allow the order of integration of the factors to di¤er from that of the idiosyncratic components, by assuming di¤erent processes generating the two. A small sample comparison of some of these tests is provided in Gengenbach, Palm and Urbain (2006).

The literature already provides several more powerful alternatives to the ADF unit root test. However, all of them demonstrate limited ability to reject correctly the unit root hypothesis when applied to highly persistent time series with a limited span. This paper attempts to produce a more e¢ - cient panel unit root test allowing a more reliable analysis of such data sets. Our new test, the DF-GLS-SUR test, is an extension of Elliott, Rothenberg, and Stock’s (1996) GLS-transformation to a version of the Levin, Lin and Chu’s (2002) test. The use of Monte Carlo simulations allow us to show the interesting behavior of this new test. For both the demeaned and de- trended cases, the DF-GLS-SUR test o¤ers a uniformly higher …nite-sample power than the ADF-SUR test. Furthermore, the DF-GLS-SUR-test perfor- mance remains attractive when studying a data with heterogeneous rates of convergence across the series.

University of Southern California and Cambridge University October 25, 2003 Abstract This paper re-examines the panel unit root tests proposed by Chang (2002). She establishes asymptotic independence of the t-statistics when integrable functions of lagged dependent variable are used as instruments even if the original series are cross sectionally dependent. From this rather remarkable result she claims that her non-linear instrumental variable (NIV) panel unit root test is valid under general error cross correlations for any N (the cross section dimension) as T (the time dimension of the panel) tends to infinity. We show that her claim is valid only if N ln T / √ T → 0, as N and T → ∞, and this condition is unlikely to hold in practice, unless N is very small. The favourable simulation results reported by Chang are largely due to her particular choice of the error correlation ma- trix, which results in weak cross section dependence. Also, the asymptotic independence property of the t -statistics disappears when Chang’s modi- fied instruments are used. Using a common factor model with a sizeable degree of cross section correlations, we are able to show that Chang’s NIV panel unit root test suffers from gross size distortions, even when N is small relative to T (for example N = 5, T = 100).

root tests asymptotically depend on nuisance parameters in the presence of per- manent variance shifts. Hence, seriously distorted empirical type one errors and deceptive inference are the consequences of a violation of the implicit assumption of time invariant volatility. The magnitude of size distortions is shown to depend on specific break patterns. Generally, largest (positive) size distortions are observed for early negative and late positive shifts in the level of the process’ unconditional variance. So far, only Hanck (2009b) attempts to generalize these results to the field of panel unit root testing. However, while he considers intersection tests for heterogenous panels which are constructed by combining the p-values obtained from volatility break robust univariate tests, this paper concentrates on the class of class of homogenous PURTs based on a pooled DF regression. We show that the sec- ond generation ’White-type’ corrected PURT proposed in Herwartz and Siedenburg (2008) retains a Gaussian limiting distribution under discrete shifts of the innova- tion variance. In contrast, the first generation test of Levin et al. (2002) and the second generation test of Breitung and Das (2005) do not converge to a nuisance free limiting distribution in this case. Moreover, the local asymptotic power function of the test statistic is derived. It turns out that in absence of volatility breaks, its local asymptotic power equals those of the statistic proposed by Breitung and Das (2005), while in the presence of a volatility break, the power of depends on the timing and direction of the break. Deterministic terms and residual serial correlation are ac- counted for by detrending and prewhitening schemes proposed in Breitung (2000) and Breitung and Das (2005), respectively. While the prewhitening scheme works well even under second order moment instability, the detrending scheme invokes serious deviations of empirical type one errors from the nominal significance level if there is a break in the innovation variance.

The efficient normalization of the individual tests is much more difficult than one might think. As is well known, the individual unit root tests have null distributions that are nonstandard and nonnormal. Their time T -asymptotics yield distributions commonly rep- resented by various functionals of Brownian motions, and in particular, known to be asym- metric and skewed. See, e.g., Fuller (1996) for the tabulations of them. Consequently, the standardization through the mean and variance adjustment or the p-value transformation, which are two most frequently used methods for normalization, often works poorly even when T is relatively large. Worse, the errors made in the normalizations for individual tests are accumulated as N of them are combined to compute the panel unit root test. Obviously, the problem gets worse as N increases. We require, however, that N tend to infinity to obtain the normal N -asymptotics. This is a serious dilemma. It is well known that all the existing panel unit root tests suffer from rather serious size distortions when N is large compared to T .

6. Conclusion A number of panel unit root tests allowing for cross-section dependence have been proposed in the literature. In this paper we propose a nonlinear heterogeneous panel unit root test for testing the null hypothesis of unit-root processes against the alternative that allows a proportion of units to be generated by globally stationary ESTAR processes and a remaining non-zero proportion to be generated by unit root processes. The proposed test is simple to apply and accommodates both nonlinearity and cross sectional dependence. Our test is compared to Pesaran’s (2005) linear test via Monte Carlo simulation exercises, and it is found that our test holds correct size and under the hypothesis that data are generated by globally stationary ESTAR processes has a better power than the Pesaran test. We also calculate critical values for varying cross section and time dimensions which can be used in future applications of our test.

Several panel unit root tests based on different ways to account for cross-unit dependence are reviewed. The note then illustrates the tests by checking whether the martingale difference hypothesis is appropriate for stock prices on the German stock market: according to the martingale difference hypothesis, logarithmized stock prices follow an integrated process without short-run dynamics. Compared with usual tests for no autocorrelation, unit root tests do not require strong moment conditions and can cope with stock returns series exhibiting infinite kurtosis. Evidence against the martingale difference hypothesis is found in a panel of 30 DAX stocks observed daily between 2004 and 2007.

Δ , , ∆ , , ,⋯ , ,⋯ 9 Finally, we are able to tabulate critical values at 1%, 5%, and 10% significance levels for the above series-specific nonlinear panel unit root test with smooth breaks via bootstrapping simulations under the null of a random walk for the optimal frequency obtained in step 1 for each ASEAN country. The critical values obtained from 5000 replications are reported in Table 2, and the t-statistic estimated under the null hypothesis of a unit root is plotted in Figure 1. As shown in Table 2, the results indicate that relative PPP holds with regard to China.

Abstract This paper proposes a new testing approach for panel unit roots that is, unlike previ- ously suggested tests, robust to nonstationarity in the volatility process of the innovations of the time series in the panel. Nonstationarity volatility arises for instance when there are structural breaks in the innovation variances. A prominent example is the reduction in GDP growth variances enjoyed by many industrialized countries, known as the ‘Great Moderation.’ The panel test is based on Simes’ [Biometrika 1986, “An Improved Bonfer- roni Procedure for Multiple Tests of Significance”] classical multiple test, which combines evidence from time series unit root tests of the series in the panel. As time series unit root tests, we employ recently proposed tests of Cavaliere and Taylor [Journal of Time Series Analysis, “Time-Transformed Unit Root Tests for Models with Non-Stationary Volatil- ity”]. The panel test is robust to general patterns of cross-sectional dependence and yet straightforward to implement, only requiring valid p-values of time series unit root tests, and no resampling. Monte Carlo experiments show that other panel unit root tests suffer from sometimes severe size distortions in the presence of nonstationary volatility, and that this defect can be remedied using the test proposed here. The new test is applied to test for a unit root in an OECD panel of gross domestic products, yielding inference robust to the ‘Great Moderation.’ We find little evidence of trend stationarity.

Abstract This paper proposes a new testing approach for panel unit roots that is, unlike previ- ously suggested tests, robust to nonstationarity in the volatility process of the innovations of the time series in the panel. Nonstationarity volatility arises for instance when there are structural breaks in the innovation variances. A prominent example is the reduction in GDP growth variances enjoyed by many industrialized countries, known as the ‘Great Moderation.’ The panel test is based on Simes’ [Biometrika 1986, “An Improved Bonfer- roni Procedure for Multiple Tests of Significance”] classical multiple test, which combines evidence from time series unit root tests of the series in the panel. As time series unit root tests, we employ recently proposed tests of Cavaliere and Taylor [Journal of Time Series Analysis, “Time-Transformed Unit Root Tests for Models with Non-Stationary Volatil- ity”]. The panel test is robust to general patterns of cross-sectional dependence and yet straightforward to implement, only requiring valid p-values of time series unit root tests, and no resampling. Monte Carlo experiments show that other panel unit root tests suffer from sometimes severe size distortions in the presence of nonstationary volatility, and that this defect can be remedied using the test proposed here. The new test is applied to test for a unit root in an OECD panel of gross domestic products, yielding inference robust to the ‘Great Moderation.’ We find little evidence of trend stationarity.

This paper evaluates whether the exchange rate of Bangladesh is mean reverting or not by applying first-generation and second-generation panel unit root test approaches. The paper considers annual data from 1986 to 2011 of major twenty-two trading partners of Bangladesh. Although some inconclusive outcomes emerge from the first-generation tests, the second-generation test reconciles the controversy and confirms that the weak form of PPP is relevant for Bangladesh. Consequently, the bilateral exchange rate of Bangladesh is mean reverting and the PPP hypothesis can be considered as an exchange rate determinant in the long-run.

This paper proposes a reassessment to the hypothesis that the persistence of current earnings performance is decreasing in the magnitude of the accrual component of earnings and increasing in the magnitude of the cash flow component of earnings. For this purpose, a threshold autoregressive panel unit root approach is proposed using a Fisher-type. This approach allowed us to distinguish between unconditioned and conditioned measures of persistence, making it possible to infer whether the earnings components condition its persistence. The approach was applied to a sample of 126 Brazilian firms in the period from 1995 to 2007. Our main results are the finding of relevant earnings persistence heterogeneity between the firms in the sample, a relatively lower unconditioned measure of earnings persistence, and a partial rejection of the hypothesis afore mentioned, specifically about the effects of the accruals components over the earnings persistence.

Therefore testing PPP using panel unit root tests are widely used, however, this paper shows that the inference used in these applications are likely to be wrong, i.e. the actual size may be very far from the nominal one. The base currency used introduces a common stochastic trend which is not accounted for in the distribution of the test statistics. Our paper analytically derives some useful expressions which help to understand the consequences of the common stochastic trend. E.g. the LL test is shown to diverge with the number of cross- sections. A Monte Carlo simulation is carried out to analyze the consequences for two panel unit root tests. The tests investigated in the Monte Carlo are the IPS and Hadri (2000), H, which is the panel unit roots version of the panel coin- tegration test of McCoskey and Kao (1998). The IPS test has a null hypothesis of unit root therefore we also have a look at the H test which has stationarity as null hypothesis. Strictly, the H test is a test of stationarity but, for simplicity, through out the paper the term testing for unit roots is used for all tests. The result is that for very small panels, N = 2, the size is approximately correct but for larger panels, N 10; the size can be signi…cantly distorted, i.e. the empirical size is much too large.

The investigation of the stationary in panel data has received great attention in panel analysis for the past few decades. It is an important issue in modeling the panel with the involvement of times series dimension in this study. This investigation can be done via unit root test. The panel unit root tests can be found in Im et al. ( 2003 ), Levin and Lin ( 1992 , 1993 ), Levin et al. ( 2002 ), Bai and Ng ( 2004 ), Philips and Sul ( 2003 ), Moon and Perron ( 2004 ), Pesaran ( 2007 ) and Choi ( 2001 , 2002 ). Hurlin ( 2010 ) distinguished two generations of unit root tests on which the first generation tests relied on the assumption that all cross sectional units are independent. The first generation of unit root tests were those proposed by Quah ( 1994 ), Breitung and Meyer ( 1994 ) and Levin and Lin ( 1992 , 1993 ).

1 Introduction Over the past decade the problem of testing for unit roots in heterogeneous panels has at- tracted a great of deal attention. See, for example, Bowman (1999), Choi (2001), Hadri (2000), Im, Pesaran and Shin (1995, 2003), Levin, Lee, and Lu (2002), Maddala and Wu (1999), and Shin and Snell (2000). Baltagi and Kao (2000) provide an early review. This lit- erature, however, assumed that the individual time series in the panel were cross-sectionally independently distributed. While it was recognized that this was a rather restrictive assump- tion, particularly in the context of cross country (region) regressions, it was thought that cross-sectionally de-meaning the series before application of the panel unit root test could partly deal with the problem. (see Im, Pesaran and Shin (1995)). However, it was clear that cross-section de-meaning could not work in general where pair-wise cross-section covariances of the error terms differed across the individual series. Recognizing this deficiency new panel unit root tests have been proposed in the literature by Bai and Ng (2002), Chang (2002), Choi (2002), Harvey and Bates (2002), Moon and Perron (2003), and Phillips and Sul (2002). Chang (2002) proposes a non-linear instrumental variable approach to deal with the cross section dependence of a general form and establishes that individual Dickey-Fuller (DF) or the Augmented DF (ADF) statistics are asymptotically independent when an integrable function of the lagged dependent variables are used as instruments. From this she concludes that her test is valid for both T (the time series dimension) and N (the cross section dimen- sion) are large. However, as shown by Im and Pesaran (2003), her test is valid only if N is fixed as T → ∞. Using Monte Carlo techniques, Im and Pesaran show that Chang’s test is grossly over-sized for moderate degrees of cross section dependence, even for relatively small values of N . 1

6 Conclusion A number of panel unit root tests allowing for cross section dependence have been pro- posed in the literature. In this paper we propose a nonlinear heterogeneous panel unit root test for testing the null hypothesis of unit-root processes against the alternative that allows a proportion of units to be generated by globally stationary ESTAR processes and a remaining non-zero proportion to be generated by unit root processes. The proposed test is simple to apply and accommodates both nonlinearity and cross sectional depen- dence. Our test is compared to the Pesaran [ 2007 ] linear test via Monte Carlo simulation exercises, and it is found that our test holds correct size and under the hypothesis that data are generated by globally stationary ESTAR processes has a better power than the Pesaran test. We also calculate critical values for varying cross section and time dimensions which can be used in future applications of our test.