A panel unit root test in the presence of cross-section dependence

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1 A panel unit root test in the presence of cross-section dependence Mehdi Hosseinkouchack Department of Economics, Goethe University Frankfurt, Grueneburgplatz, (House of Finance, 60323, Frankfurt am Main, Germany Summary A panel unit root test is derived based on a Lagrangian Multiplier for panels with a cross-section dependence modeled using factor models. he test statistic is shown to be different from square of test statistic of Pesaran (2007 even for a case of no constant and no trend, hence the test statistic is not simply a different calculation of the suggestion made in Pesaran (2007. Implementation of the test is very simple and it is easy to understand its mechanics. Furthermore, the limiting distribution of of individual cross-sectionally augmented test statistics which appear to be functions of standard wiener processes are shown to be free of nuisance parameters. he critical values of the proposed test statistics are tabulated. he small sample size properties of the proposed test statistics are shown to outperform the suggestion of Pesaran (2007 in terms of power, in many cases, i.e. the proposed test statistic is more powerful, in particular, for alternatives involving some individuals with unit root and some with stationary behaviors, which makes it quite appealing for practical purposes. he asymptotics are established under N/ δ > 0 with N being the number of cross-sections and the length of time series. Keywords: Panel Data, Unit Root, Cross-section Dependence, Factor Models, Lagrangian Multiplier.. INRODUCION It is a well established fact that leaving the cross-sectional dependence out would lead to serious size distortions and power loss. Chang (2002, Choi (2002, Phillips and Sul (2003, Bai and Ng (2004, Moon and Perron (2004, Smith et al. (2004, Breitung and Das (2005, Choi and Chue (2007, Pesaran (2007, Demetrescu et al. (2009 and Pesaran et al. (2009 propose different panel unit root tests allowing for cross-section dependence. Chang (2002 uses a nonlinear instrumental approach to deal with the cross-sectional dependence. Choi (2002 proposes a special form of covariance matrix to model the crosssectional dependence. his is while Bai and Ng (2004, Moon and Perron (2004, and Phillips and Sul (2003 turn to residual factor models to capture the cross-sectional dependence. Phillips and Sul (2003 propose an orthogonalization procedure that asymptotically eliminates the common factors. Bai and Ng (2004 and Moon and Perron (2004 use similar approaches in the sense that they estimate the factors and work with de-factored time series. Pesaran (2006 proposed that cross-sectional means of differenced data, and cross-section mean of lagged data are good proxies for unknown factors. his fact is used in Pesaran (2007 and Pesaran et al. (2009 to proxy for unobserved factors instead of using factor estimation which involves estimating the number of factors then the factors themselves. One important distinguishing feature of these tests is the asymptotic relationship between dimensions of panel under consideration. For example Phillips and Sul (2003,

2 2 M. Hosseinkouchack Moon and Perron (2004, and Bai and Ng (2004 require a fairly big time series dimension ( compared to cross-section dimension (N, i.e. they need N/ 0. Chang (2002 has introduced a test which is valid for fixed N as gets big (cf. Pesaran (2007 and the test of Smith et al. (2004 has the same restriction. Pesaran (2007 introduces a framework in which he allows for N/ δ for a positive real valued δ. he small sample size properties of existing tests have been examined extensively. Furthermore, test of Pesaran (2007 is extremely simple to implement. Following the approach to Pesaran (2007, this paper proposes a single factor structure for the error term to tackle the cross-sectional dependence, while the test statistic proposed here is base on Lagrangian Multiplier. he proposed test has the same asymptotic behavior for N and as the proposed test of Pesaran (2007 but in the contrary to time series literature on testing for unit roots, it is not equivalent to a t-statistic based test. In testing for the unit root in panels, cross-sectional units are all nonstationary under the null while the alternative could take two possible forms. 2 First, all of the crosssectional units are assumed to be stationary. Secondly some of the cross-sectional units are allowed to be nonstationary. As mentioned in Pesaran (2007 a panel unit root test can only be consistent whose alternative is of the second form. 3 his note seems, however, to have remained just as an important hint to econometricians, while they tend to address on the power features of their proposed tests for simulated panels whose all of cross-sections are stationary. In the current work we address this issue and observe that the test of Pesaran (2007 is not always desirable when the fraction of stationary cross-sectional units is away from one. Nevertheless, our proposed test outperforms a t-statistic based test as the alternative gets closer to the null, i.e. as the number of nonstationary units in the alternative grows. he structure of the paper is as follows. he panel data model considered in this paper is discussed in the next section and the proposed test statistic is also presented. Section 3 states the asymptotic results and provides some discussion on how the proposed test differs from the test proposed by Pesaran (2007. Simulation studies and size and power features of the proposed test are contrasted with the test of Pesaran (2007 in section 4. Section 5 concludes. Appendix A and B contain the mathematical proofs and tables, respectively. 2. MODEL Consider a simple linear heterogeneous panel data model of the following form y it = ( φ i µ i + φ i y i,t + u it, i =, 2,..., N; t =, 2,..., he error term has a single-factor structure, i.e. u it = γ i f t + ε it, for which ε it is the idiosyncratic error term and f t is the unobserved common factor associated with the factor loading γ i. With α i = ( φ i µ i and β i = ( φ i rewrite the model as y it = α i + β i y i,t + γ i f t + ε it (2. for which we are interested to introduce a test for the following hypothesis H 0 : β i = 0 i (2.2 cf. Breitung and Pesran (2007 and Choi ( Except Demetrescu et al. (2009 that use a KPSS approach. 3 his is a note mentioned in an unpublished work by Im K. S. and M. H. Pesaran (2003: On the Panel Unit Root ests Using Nonlinear Instrumental Variable.

3 against an alternative of the form A panel unit root test 3 H a : β i < 0, i =, 2,..., N, β i = 0, i = N +, N + 2,..., N Where N /N tends to a nonzero fraction 0 < δ as N. Under the null we have that y it = γ i f t + ε it (2.3 Using Pesaran (2007 one can write this equation in a matrix format as y i = δ i y + ξ it for which y i = ( y i, y i,..., y i, y = ( y, y 2,..., y, δ i = γi γ, and ξ it = ε it δ i ε t. ( Let M w = I W W ( W W with W = τ, y, y, y i, where y = ( y 0, y,..., y and y i, = (y i,0, y i,,..., y i, and M δ = I y ( y y y. hen and M δ y i = M δ ξ i M w y i = M w ξ i where ξ i = (ξ i, ξ i2,..., ξ i ( 0,ωi 2I ( ωi 2 = σi 2 2δ ( i + δ2 i N N σ2 = σi 2 + O N and σ 2 is the mean of σi 2 over i. he proposed test statistics is based on the following which is clearly and LM type test. ( LM i = y ( im δ M w M δ y i = y im w y i (2.4 y i M δ y i y i M δ y i which using (7 and (8 can be written as ( LM i = ξ i M wξ i ξ i M δξ i and with ν i = ξ i /ω i (0, I we have LM i = ( ν i M wν i ν i M δν i (2.5 he following assumption are considered all through this paper. Assumption. Idiosyncratic error term, ε it, is independently distributed across time and cross sections, and has finite fourth moment, and is a mean zero random variable with variance σi 2. Assumption 2. unobserved common factor, f t, does not exhibit serial correlation has mean zero and variance σf 2 with finite fourth moment. Without loss of generality we can assume the variance of the common factor to be unity. Assumption 3. ε it, f t, and γ i are independently distributed for all i.

4 4 M. Hosseinkouchack LM i ε i εi N σi 2 3. LM BASED UNI ROO ES heorem 3.. Given assumptions and 2, for the process defined in (2. and generated under (2.2, for a fixed, we have ( ( ( Γ q ( i + q(2 i Γ Ψ f Γ + I r Γ 2Ψ ( fi Γ 2 + Γ 3Ψ (2 fi Γ q ( i ε i εi σ 2 i ε i f(f f f ε i σ 2 i (3. where LM i is defined in (2.5, and all the matrices in the limit expression can be found in the Appendix A. heorem 3.2. heorem 2. Given assumptions -3, and lim N γ = γ 0, if N is followed by, then LM i N l if (3.2 where l if is defined in Appendix A and is free from nuisance parameters. o test for a unit root in a time series setting it is quite straightforward to see that for a no intercept no trend case an LM test statistic is the same as the square of a DF test statistic. However, this is not the case for the panel model considered here. heorem 2 establishes asymptotic results for an intercept case. In (2. considering α i = 0 for all i, the limiting distribution of cross-sectionally augmented Lagrangian multiplier test can be derived and shown to be ( LM i = 2HID BI 2 D 2 H 2 E BE with the B, D, E, H, and I defined in Appendix A. On the other hand a cross-sectionally augmented DF test statistics, t DFi, proposed in Pesaran (2007 has the following limiting distribution for a no intercept and no trend case. t DFi = I HD B [ ] E D B For these two limiting distributions one might observe that + q(2 i LM i = [t DFi ] 2 + H2 B for which the additional term is in fact [ H 2 B = 0 W f (r dw i (r 0 W f 2 (r dr ] 2 (3.3 which originates from the factors present in the model. 4 In this respect one would not expect the same relationship between t DFi and LM i as established for the simple timeseries case for a no intercept and no trend case. In other words, factor augmentation step makes the two test statistics not to be equivalent in their traditional way. H2 B has a χ2 ( 4 Wf is a wiener process driving the common factor and W i is a wiener process driving the idiosyncratic error term for cross-section i.

5 A panel unit root test 5 distribution which means that the cross-section dependence makes an LM type test to be more variable than a case where the cross-sectional units are independent. Equation (2.5 sketches a Lagrangian Multiplier test which in time series analysis is known to be asymptotically equivalent to other types of tests. In a pure time series analysis framework and under some conventional assumptions, one would expect that an LM test statistic would not be different from testing procedures based on other possible definitions of the distance between the null and the alternative. But as we see from (3.3 this is not asymptotically true for the panel model considered here. he literature on panel econometrics distinguishes between 2 types of cross-sectional dependence: strong and weak. A strong cross-sectional dependence is when the covariance of y it possesses at least one eigen value that explodes as the cross-section dimension goes to infinity. One way to allow for a strong dependence is to consider a common factor in the error term. heorem and 2, do not allow for serial correlation in the error term which is important in practice. Serial correlation could come through the factor, f t, or through the remainder of the error term after removing the common factor. Considering an AR representation for the factor would only require to correct the underlying Wiener process for f t. Considering the serial correlation to come through the idiosyncratic error term would however involve a different asymptotic result which in fact could be written in the same fashion as Pesaran (2007 while augmenting more lags in the regression would help removing such an effect. 4. SIMULAION AND COMPARISON SUDIES he computation of the critical values were based on 0000 regressions and calculating the value of the proposed test statistic in (2.4, while a correction for small sample sizes was always considered. In generating the individual series for generating the all of the critical values, time is considered to start from 50 with y i, 50 = 0. he panel dimensions are also spanning a wide range of possibilities i.e. N, {0, 5, 20, 30, 50, 70, 00, 200}. ables (-(3 and (3-(5 tabulated the critical values of the proposed test for the average of cross-sectionally augmented LM statistic and for the individual cross-sectionally augmented cases, where 3 common model specifications are considered. he Date Generating Process (DGP considered to examine the small sample size of the proposed test takes into account this possibility that the alternative, as sketched in the introduction, can include cases in which only a subset of the cross-sections are stationary. his point has been set off in simulation studies done by Pesaran (2007 where cross-sectional augmentation in conducting a panel unit root test is discussed. As we will see test of Pesaran (2007 looses its power the more one deviates from a panel with purely stationary cross-sections. he simulation study for analyzing small sample size properties of the proposed test is contrasted with the tests proposed by Pesaran (2007. he proposed test is called MH, the test of Pesaran (2007 is tagged CIPS and CIPS represents the truncated version of CIPS. he simulation studies for sketching on the power and size comparison of MH and CIPS are carried out for N, {0, 20, 30, 50, 00}. ables (4-(6 tabulate the size and power of the proposed test in contrast with the est of Pesaran (2007. For these tables the power is calculated for a simulated panel data set with one third of its individuals being I( and the rest being I(0. able (4 pertains to a model where there is no intercept and no trend, and ables (5 and (6

6 6 M. Hosseinkouchack correspond respectively to cases of including an intercept and an intercept and a time trend, respectively. In table (4 both tests are always correctly sized, while both almost have no power for N and equal to 0. For the no intercept and no trend case, CIPS has relatively better power for N = 0, while as N increases, MH tends to slightly gain power. For the cases of N and equal to 50, and N = 00 and = 50 MH exhibits a substantially higher power. he results shown in table (5 are always in favor of CIPS as it always exhibits a slightly higher power, this is while the size is correct for both tests. able (6 shows a substantially higher power for the proposed test compared to the power of CIPS. ables (7-(9, tabulate power and size results for in the same fashion as tables (4-(6 except that the power has been calculated for simulated panel data sets with half of its individuals being I( and the rest being I(0. able (7 and (8 show results pretty much the same as the results of tables (4-(5, while MH has a substantially higher power for the intercept and trend case tabulated in table (9. In tables (0-(2, power has been calculated for the case where two third of the unit in the simulated panel data are considered I( and the rest to be I(0. As this table shows, CIPS exhibits a lower relative power compared to MH in almost all the cases. his is while both tests exhibit lower power profile for an intercept case compared to the case of no intercept and the case of an intercept and a trend. For the intercept and trend case, shown in table (2, MH likewise tables (6 and (9 exhibits a substantially higher power compared to CIPS. 5. CONCLUSION A Lagrangian Multiplier test is proposed for unit root testing in panels with crosssectionally dependent units modeled through considering a simple factor structure. he test is applicable to test the existence of a unit root in each individual separately while averaging the test statistics across the units provides a tool for testing the existence of a unit root in the panel data. he asymptotics of the proposed test are derived and show to be independent of the nuisance parameters. For the no intercept case it is also shown that the proposed test is not the square of the test proposed by Pesaran (2007. he power and size of the proposed test is contrasted to those of CIPS. Power of the test is calculated or cases in which the panel data set is simulated with a one third, half, and two third of the units being I( and the rest being I(0. As the results show, in the intercept and trend case the proposed test has a drastically higher power compared to that CIPS. Furthermore, as the fraction of I( individuals increase to two third, the result show that MH has a relatively higher power for the no intercept case and the intercept case compared to that of CIPS. he size of both tests is always correct for all the cases. As the simulation results show, the proposed test deserves more theoretical investigations. ACKNOWLEDGEMENS he author is grateful to Uwe Hassler and Michael Binder for their helpful comments. REFERENCES Bai, J. and S. Ng (2004. A PANIC attack on unit roots and cointegration. Econometrica 72,

7 A panel unit root test 7 Baltagi, B., G. Bresson and A. Pirotte (2007. Panel Unit Root ests and Spatial Dependence. Journal of Applied Econometrics 22, Breitung, J., and S. Das (2005. Panel unit root tests under cross sectional dependence Statistica Neerlandica 59, Breitung, J. and M. H. Pesaran (2007. Unit roots and cointegration in panels. In Ma tya s L. and Sevestre P. (Eds, he Econometrics of Panel Data: Fundamentals and Recent Developments in heory and Practice, Kluwer: Dordrecht. Choi, I. (2002. Combination unit root tests for cross-sectionally correlated panels. Mimeo, Hong Kong University of Science and echnology. Choi, I. (2006. Nonstationary panels. In Mills. C. and K. Patterson (EdsPalgrave Handbooks of Econometrics, Volume, Palgrave Macmillan: Basingstoke. Choi I. and. K. Chue (2007. Subsampling hypothesis tests for nonstationary panels with applications to exchange rates and stock prices. Journal of Applied Econometrics 22, Chang Y. (2002. Nonlinear IV unit root tests in panels with cross-sectional dependency. Journal of Econometrics 00, Demetrescu M., U. Hassler and A. I. arcolea (2009. esting for Stationarity in Large Panels with Cross-Dependence, and US Evidence on Unit Labor Cost Forthcoming in Journal of Applied Statistics. Hamilton J. D. (994. ime Series Analysis New Jersey: Princeton. Moon, H. R. and B. Perron (2004. esting for a unit root in panels with dynamic factors. Journal of Econometrics 22, Pesaran, M. H. (2006. Estimation and inference in large heterogeneous panels with cross section dependence. Econometrica 74, Pesaran, M. H. (2007. A simple panel unit root test in the presence of cross-section dependence. Journal of Applied Econometrics 22, Pesaran M. H., L. V. Smith and. Yamagata (2009 A Panel Unit Root est in the Presence of a Multifactor Error Structure. Working paper, Cambridge University. Phillips P. C. B. and D. Sul (2003 Dynamic panel estimation and homogeneity testing under cross section dependence. Econometrics Journal 6, Smith L. V., S. Leybourne S,. K. Kim and P. Newbold (2004 More powerful panel data unit root tests with an application to mean reversion in real exchange rates. Journal of Applied Econometrics 9,

8 8 M. Hosseinkouchack APPENDIX A: PROOFS OF RESULS Proof of heorem. In (2.4 first we consider ν i M wν i. ν im w ν i = ν iν i ( ν iwd ( DW WD (DW ν i where and DW WD = D = / / y τ y τ y y y τ τ y i, 3 2 DW ν i = Using (2.3 the following equations follow. y τ τ y i, y y 3 2 y y 3 i, 2 y y y y 2 y y 2 i, y y i, 2 y y i, 2 y i, y i, y = y 0 τ + γs f, + s τ ν i y ν i y ν i y i, ν i y = γf +ε y i, = y i0 τ + γ i s f, + s i, (A. (A.2 (A.3 (A.4 (A.5 (A.6 with τ = (,,..., s i, = (0, s i,, s i,2,..., s i,, s = (0, s, s 2,..., s, s i,t = t j= ε ij, s t = N N j= s it, s f, = (s f0, s f,..., s f,, and s ft = t j= f j. Equation (A.6 can be used to eliminate the partial sum of unobserved factors from equation (A.5, namely y i, = (y i0 δ i y 0 τ δ i y + s i, δ i s Let s i, = (s i, δ i s /ω i and rewrite as y i, = (y i0 δ i y 0 τ δ i y + ω i s i, (A.7 s i, is a random walk associated with ν i = ξ i /ω i (0, I. Now using (A.4-(A.7, along with some results from Pesaran (2006 we can simplify the elements of (A.2 and (A.3 as follows y τ =γ y τ =y 0 ( + γ ( f ( τ f ε + f, τ ( ( s τ y i, = y i0 f, τ γ i τ 3 2 i, τ 3 2

9 A panel unit root test 9 ( f y y = γ 2 ( f f ( ε ε ε + 2γ + y y = γy 0 ( f τ y 0 ( ε τ ( f s +γ + γ ( f y τ y i, = γy i y y = y2 0 + γ2 2 y y i, = y 0 y i0 2 y i, y i, = y i0 +y i0 ( ε τ ( ε s f, + γ 2 ( f s f, ( ε s 3 2 ( f ( s f, f s i, + γγ i + γ ( ε ( s f, ε s i, + γ i f, s f, +2γy 0 f, τ s 2 + 2y 0 τ 2 ( ( τ s f, + y 0 γ i 2 +γy i0 f, τ 2 +y i0 τ 2 ( + γi 2 +2y i0 γ i ( τ s f, 2 f, s f, + γγ i 2 + γ s f, i 2 + f, s f, 2 + 2γ + y 0 γ i ( τ s i, + + 2y i0 ( τ s i, 2 + γ f, s 2 2 s i, 2 i, s i, 2 f, s i, 2 + 2γ i f, s i, 2 Now, noting that all the cross sectional means go to zero as N goes to infinity we can write where DW WD N Γ Ψ f Γ + I r Γ 2Ψ ( fi Γ 2 + Γ 3Ψ (2 fi (A.8 Γ = γ γ γ i, Ψ f = f τ f τ f f s f, τ f s f, 3 2 s f, τ f s f, 3 2 s f, τ 3 2 f s f, 3 2 s f, s f, 2 s f, s f, 2 s f, τ 3 2 f s f, 3 2 s f, s f, 2 s f, s f, 2

10 0 M. Hosseinkouchack I r = Γ 3 =, Γ 2 = γ i γ γ , Ψ(2 fi =, Ψ( fi = τs i, f s i, 3 2 s i, s f, s f, si, 2 Now we consider the elements of equation (A.3. First recall that δ i = γi γ, and ξ it = ε it δ i ε t with ωi 2 = σ2 i +O ( N and si, = (s i, δ i s /ω i with s i = s i, +ν i. hen we can simplify the elements of (A.3 as follows. τ ν i = ( τ ε i ω i δ i ( τ ε ω i ( f y ( ν i ε ν ν i = γ + i ( f ε i = γ ω i + ( ε ε i ω i γγ i ( δ i ( f ε ω i ε ε ω i ( τ ( ν i s ( f, ν i s y ν i = y 0 + γ + ν i ( τ ( ε i τ ε = y 0 y ω i 0 δ i ω i ( f, ε i s f, ε +γ γδ i ω i ω i ( + ε i s δ ε i i ω i ω i y i, ν i = y i0 ( τ ν i = y i0 ( τ ε i ω i f, ε i +γ i ω i i, ε i + ω i f, ν i + γ i ( τ ε y i0 δ i ω i f, ε γ i δ i ω i δ i i, ε i ω i i, ν i + Hence equation (A.3 has the following limit as N goes to infinity: DW ν i N Γ q ( i + q(2 i (A.9 s i, s i, 2 f s i, 3 2 s i, s f, 2 τs i, 3 2

11 where q ( i = A panel unit root test τ ε i σ i f ε i σ i s f, εi σ i s f, εi σ i, q(2 i = s i, εi σ i As N goes to infinity the limit of first term in equation (A., ν i ν i can simply be writen as ν iν i = ε i ε i + δ 2 i ε iε i 2δ i ε i ε i ω 2 i N ε i ε i σ 2 i For the denominator, ν i M δν i with M δ = I y ( y y y, we have for which it can easily be shown that ν im δ ν i = ν iν i ν i y ( y y y ν i ν im δ ν i N ε i ε i σ 2 i ε i f (f f f ε i σ 2 i (A.0 (A. Putting (A.8, (A.9, (A.0 and (A. together proves the result. Proof of heorem 2. he only thing we should do is to let to go to infinity for the results derived in heorem. Using Hamilton (994 and doing similar derivation the following preliminary results would be straightforward. B, f τ 0, s f, τ τs i, σ i W f (r dr(= A, f f 0 W i (r dr(= C, f s i, s i, si, σ 2 i 2 0 W 2 i (r dr(= E, τ ε i σ i 3 2, f s f, 0, 2 3 s i, s f, σ i 2 0, s f, s f, 2 Wi ( (= F, s f, εi σ i 0 W f 2 (r dr(= 0 W f (r W i (r dr(= D, 0 W f (r dw i (r (= H, s i, εi σ i 2 0 W i (r dw i (r = /2 [ Wi 2( ] (= I and f ε i σ i Wif ( which is a brownian motion. Using the preliminary results, it would be obvious that for (A.8 we have the following, 0 0 W f (r dr 0 W f (r dr Ψ f 0 W f (r dr 0 0 W f 2 (r dr 0 W f 2 (r dr 0 W f (r dr 0 0 W f 2 (r dr (r dr Ψ ( fi 0 W 2 f 0 W i (r dr 0 σ i 0 W f (r W i (r dr σ 2 i 0 W i 2 (r dr σ i 0 W f (r W i (r dr W i (r dr Ψ (2 fi σ i 0 W f (r W i (r dr

12 2 M. Hosseinkouchack Now consider (A.9 and observe that, W i ( q ( W i if ( 0 W f (r dw i (r 0 W f (r dw i (r, q(2 i /2 [ Wi 2 ( ] Hence the limit of (A.2 and (A.3 are known when N is followed by. Further consider that results, we see that l if = ε i εi σ 2 i ε i f(f f f ε i σ 2 i as ε i f(f f f ε i σ 2 i Wfi 2 (. Using these 2ACHI 2BCF I 2HID 2AF HE + 2CF HD + 2AF ID +BI 2 + H 2 E + BF 2 E C 2 H 2 A 2 I 2 F 2 D 2 BE + 2ACD D 2 BC 2 A 2 E for which the letters are defined in the preliminary results mentioned above.

13 A panel unit root test 3 APPENDIX B: ABLES able. Critical Values. N Percent Percent Percent Note: Critical Values for average of individual cross-sectionally augmented Dickey-Fuller χ 2 distribution (no intercept no trend case.

14 4 M. Hosseinkouchack able 2. Critical Values. N Percent Percent Percent Note: Critical Values for average of individual cross-sectionally augmented Dickey-Fuller χ 2 distribution (intercept case.

15 A panel unit root test 5 able 3. Critical Values. N Percent Percent Percent Note: Critical Values for average of individual cross-sectionally augmented Dickey-Fuller χ 2 distribution (intercept and trend case.

16 6 M. Hosseinkouchack able 4. Size and power. N est Size Power CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH Note: Size and power for average of individual cross-sectionally augmented Dickey-Fuller χ 2 distribution (no intercept and no trend. Power is calculated for the case when one-third of the units are I( and the rest I(0. able 5. Size and power. N est Size Power CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH Note: Size and power for average of individual cross-sectionally augmented Dickey-Fuller χ 2 distribution (intercept. Power is calculated for the case when one-third of the units are I( and the rest I(0.

17 A panel unit root test 7 able 6. Size and power. N est Size Power CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH Note: Size and power for average of individual cross-sectionally augmented Dickey-Fuller χ 2 distribution (intercept and trend. Power is calculated for the case when one-third of the units are I( and the rest I(0. able 7. Size and power. N est Size Power CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH Note: Size and power for average of individual cross-sectionally augmented Dickey-Fuller χ 2 distribution (no intercept no trend. Power is calculated for the case when half of the units are I( and the rest I(0.

18 8 M. Hosseinkouchack able 8. Size and power. N est Size Power CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH Note: Size and power for average of individual cross-sectionally augmented Dickey-Fuller χ 2 distribution (intercept. Power is calculated for the case when half of the units are I( and the rest I(0. able 9. Size and power. N est Size Power CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH Note: Size and power for average of individual cross-sectionally augmented Dickey-Fuller χ 2 distribution (intercept and trend. Power is calculated for the case when half of the units are I( and the rest I(0.

19 A panel unit root test 9 able 0. Size and power. N est Size Power CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH Note: Size and power for average of individual cross-sectionally augmented Dickey-Fuller χ 2 distribution (no intercept no trend. Power is calculated for the case when two-third of the units are I( and the rest I(0. able. Size and power. N est Size Power CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH Note: Size and power for average of individual cross-sectionally augmented Dickey-Fuller χ 2 distribution (intercept. Power is calculated for the case when two-third of the units are I( and the rest I(0.

20 20 M. Hosseinkouchack able 2. Size and power. N est Size Power CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH CIPS CIPS* MH Note: Size and power for average of individual cross-sectionally augmented Dickey-Fuller χ 2 distribution (intercept and trend. Power is calculated for the case when two-third of the units are I(0 and the rest I(.

21 A panel unit root test 2 able 3. Critical values. N Percent Percent Percent Note: Critical Values for individual cross-sectionally augmented Dickey-Fuller χ 2 distribution (no intercept no trend.

22 22 M. Hosseinkouchack able 4. Critical values. N Percent Percent Percent Note: Critical Values for individual cross-sectionally augmented Dickey-Fuller χ 2 distribution (intercept.

23 A panel unit root test 23 able 5. Critical values. N Percent Percent Percent Note: Critical Values for individual cross-sectionally augmented Dickey-Fuller χ 2 distribution (intercept and trend.