Comparison of Panel Cointegration Tests

Transcription

1 Comparison of Panel Cointegration Tests M.Sc. Deniz Dilan Karaman Humboldt University Berlin Institute for Statistics and Econometrics Spanduerstr Berlin karaman@wiwi.hu-berlin.de 27 May 2004 Abstract The main aim of this paper is the comparison of size and power properties of four residual-based and two maximum-likelihood-based panel cointegration tests with a DGP consisting of three variables. In addition to this, size-adjusted power results are also presented. In this study Pedroni (1999) s panel-ρ, group-ρ, panel-t, group-t statistics and Larsson, Lyhagen and Løthgren (2001) s standardized LR-bar statistic will be considered. A new panel cointegration test statistic analogous to the test statistic of Larsson et al. (2001) is introduced as well, which is based on maximum-eigenvalue test statistic. The simulation results indicate that the panel-t and maximum-likelihood-based statistics have better size and power properties among the six panel cointegration test statistics discussed in this paper. Finally, the Fisher Hypothesis is tested with two different data sets for OECD countries. The results point out the existence of the Fisher relation for the residual-based panel cointegration tests. 1 Introduction For the last two decades in the empirical literature cointegration techniques have been a widely used method. But it was sometimes difficult to find large time series for some empirical problems. To solve this limited time observations problem and to make use of the advantage of the growing multiple cross-sectional dimension, it was necessary to develop unit root and cointegration tests for the pooled time series panels. One other reason for the development of these techniques for the panel data, was the low power of the ADF and DF unit root tests for the univariate case against near unit root alternatives. Following this necessity, in recent years after the extension of the univariate unit root tests to the panel data by Levin and Lin (1992), Quah (1994), Breitung and Meyer (1994) and Im et al. (1997), the extension of the cointegration tests to the panel data has grasped a wide interest in the literature. There are mainly two different approaches for the panel cointegration tests, residual-based and maximum-likelihood-based. McCoskey and Kao (1998), Kao (1999), Pedroni (1995, 1997, 1999) propose residualbased, while Groen and Kleibergen (1999), Larsson and Lyhagen (1999) and Larsson, Lyhagen and Løthgren (2001) propose maximum-likelihood-based panel cointegration test statistics. Preliminary version. Do not copy without author s permission. 1

2 McCoskey and Kao (1998) derive a panel cointegration test for the null of cointegration which is an extension of the LM test and the locally best unbiased invariant (LBUI) test for an MA root. They take Harris and Inder (1994) and Shin (1994) as a basis for their research. Kao (1999) considers the spurious regression for the panel data and introduces two types of panel cointegration tests, the Dickey-Fuller (DF) and augmented Dickey-Fuller (ADF) type tests. He proposes four different DF type tests, and makes use of the sequential limit theory of Phillips and Moon (1999) for the asymptotic distributions of these tests. Groen and Kleibergen (1999) present within maximum-likelihood framework how homogenous and heterogeneous cointegration vectors are estimated using the GMM estimator, and propose a likelihood ratio test for the common cointegration rank, which is based on these GMM estimators and the cross-dependence. Larsson et al. (2001), on the other hand, propose panel cointegration test statistic based on cross-independence. The tests of Pedroni (1995, 1999) and Larsson, Lyhagen and Løthgren(2001) will be introduced in the next section. In this paper the properties of the residual-based panel cointegration tests of Pedroni (1999) will be compared to the properties of the maximum-likelihood-based panel cointegration rank test of Larsson et al. (2001). In addition to this, the approach in Larsson et al. (2001) will be extended to a maximum eigenvalue based test for the panel cointegration. In the simulation study we concentrate our attention to the changes in size, power and sizeadjusted power of the panel cointegration tests when time and cross-section dimensions and various parameters in the data generating process vary, e.g. the correlation parameter between the disturbances to stationary and non-stationary part of the DGP for each cross-section. The results of the simulation study, which depend on a DGP with three variables, illustrate that panel-t statistic has the best size and power properties. The sizes of the panel-t and maximumlikelihood-based tests are around the nominal size of 5% when T and N increases. While the power of panel-t statistic is near unity when N is large also when there is high correlation. The power of the standardized LR-bar statistic is higher than the standardized LR-max bar statistic when the cointegration rank is close to zero (ψ is near unity) and the correlation is high. Lastly, we test the Fisher relation among the OECD countries for different time spans using the tests considered in this paper. The second part of the paper covers the panel cointegration tests of Pedroni (1999), Larsson et al. (2001) and the maximum eigenvalue-based panel cointegration test. In the third section we present the way how the DGP of Toda (1995) is modified for the panel data, and the fourth section gives a description of the simulation study. The fifth section is devoted to the interpretation of the simulation results. The validity of the Fisher relation is be the subject of the sixth section. The seventh section concludes. The simulation results are presented in Appendix A, B and C. 2

3 2 Panel Cointegration Tests 2.1 Pedroni (1999) Pedroni (1999) extends the procedure of residual-based panel cointegration tests that he introduced in Pedroni (1995) for the models, where there are more than one independent variable. He proposes several residual-based null of no cointegration panel cointegration test statistics. In this study two within-dimension-based 1 (panel-ρ and panel-t) and two betweendimension-based 2 (group-ρ and group-t) panel cointegration statistics of Pedroni (1999) will be compared with the maximum-likelihood-based panel cointegration statistics. Panel-ρ statistic is an extension of the non-parametric Phillips-Perron ρ-statistic, and parametric panel-t statistic is an extension of the ADF t-statistic. Between-dimension-based statistics are just the group mean approach extensions of the within-dimension-based ones. Group-ρ statistic is chosen, because Gutierrez (2002) has found out that this test statistic has the best power among the test statistics of Pedroni (1999), Larsson et al. (2001) and Kao (1999). Group-t statistic is considered, because the data generating process is appropriate for parametric ADFtype tests. And the within-dimension versions of these statistics are considered in order to be able to compare. The starting point of the residual-based panel cointegration test statistics of Pedroni (1999) is the computation of the residuals of the hypothesized cointegrating regression 3, y i,t = α i + β 1i x 1i,t + β 2i x 2i,t β Mi x Mi,t + e i,t (1) t = 1,..., T ; i = 1,..., N; m = 1,..., M where T is the number of observations over time, N denotes the number of individual members in the panel, and M is the number of independent variables. It is assumed here that the slope coefficients β 1i,..., β Mi, and the member specific intercept α i can vary across each crosssection. To compute the relevant panel cointegration test statistics the panel cointegration regression in (1) should be estimated first. For the computation of the panel-ρ and panel-t statistics take the first-difference of the original series and estimate the residuals of the following regression: y i,t = b 1i x 1i,t + b 2i x 2i,t b Mi x Mi,t + π i,t Using the residuals from the differenced regression, with a Newey-West (1987) estimator calculate the long run variance of ˆπ i,t, which is symbolized as ˆL 2 11i. ˆL 2 11i = 1 T T ˆπ i,t T t=1 k i s=1 ( 1 s ) T ˆπ i,tˆπ i,t s k i + 1 t=s+1 1 Within-dimension-based statistics are calculated by summing the numerator and the denominator over N cross-sections separately. 2 Between-dimension-based statistics are calculated by dividing the numerator and the denominator before summing over N cross-sections. 3 In this study the regression equation with an heterogeneous intercept will be considered. Note that it could also be estimated without a heterogeneous intercept, or with time trend and/or common time dummies. 3

4 For panel-ρ and group-ρ statistics estimate the regression ê i,t = ˆγ i ê i,t 1 + û i,t using the residuals ê i,t from the cointegration regression (1). Then compute the long-run variance (ˆσ 2 i ) and the contemporaneous variance (ŝ 2 i ) of û i,t, where T ŝ 2 i = 1 T ˆσ 2 i = 1 T t=1 û i,t T û it + 2 T t=1 k i s=1 ( 1 s ) T û i,t û i,t s k i + 1 t=s+1 and k 4 i is the lag length. is the lag length. In addition to this calculate also the term λ i = 1 ) 2 (ˆσ 2 i ŝ 2 i. On the other side for panel-t and group-t statistics using again the residuals ê i,t of cointegration regression (1) estimate ê i,t = ˆγ i ê i,t 1 + K i t=1 ˆγ i,k ê i,t k +û i,t and compute the variance of û i,t, which is denoted as ŝ 2 i. In this study to determine the lag truncation order of the ADF t-statistics the step-down procedure and the Schwarz lag order selection criterion are used. ŝ 2 i = 1 T T t=1 û 2 i,t, s 2 N,T 1 N N i=1 ŝ 2 i. The next step is the calculation of the relevant panel cointegration statistics using the following expressions. - Panel-ρ statistic T NZˆρN,T 1 T N ( N i=1 T t=1 ˆL 2 11iê2 i,t 1 ) 1 N i=1 T t=1 ˆL 2 11i (ê i,t 1 ê i,t ˆλ ) i (2) - Panel-t statistic Z t N,T ( s 2 N,T N i=1 T t=1 ˆL 2 11iê2 i,t 1 ) 1/2 N i=1 T t=1 ˆL 2 11iêi,t 1 ê i,t (3) - Group-ρ statistic T N 1/2 ZˆρN,T 1 T N 1/2 ( T N ê 2 i,t 1 i=1 t=1 t=1 ) 1 T (ê i,t 1 ê i,t ˆλ ) i (4) 4 Pedroni (1995) used k i = 4 ( T 100) 2/9 as lag truncation function for the Newey-West kernel estimation as recommended in Newey and West (1994). The nearest integer is taken as the lag length for different T dimensions. 4

5 - Group-t statistic N ( T N 1/2 Z tn,t N 1/2 i=1 t=1 ŝ 2 i ê 2 i,t 1 ) 1/2 T ê i,t 1 ê i,t (5) Lastly, apply the appropriate mean and variance adjustment terms to each panel cointegration test statistic, so that the test statistics are standard normally distributed. κ N,T µ N ν = N(0, 1), where κ N,T is the appropriately standardized form of the test statistic, µ and ν are the functions of moments of the underlying Brownian motion functionals. The appropriate mean and variance adjustment terms for different number of regressors (m is the number of regressors without taking the intercept into account) and different panel cointegration test statistics are given in Table 2 in Pedroni (1999) 5. The null hypothesis of no cointegration for the panel cointegration test is the same for each statistic, H 0 : γ i = 1 for all i whereas the alternative hypothesis for the between-dimension-based and within-dimensionbased panel cointegration tests differs. The alternative hypothesis for the between-dimensionbased statistics is H 1 : γ i < 1 for all i, where a common value for γ i = γ is not required. For within-dimension-based statistics the alternative hypothesis H 1 : γ = γ i < 1 for all i assumes a common value for γ i = γ. Under the alternative hypothesis, all the panel cointegration test statistics considered in this paper diverge to negative infinity. Thus, the left tail of the standard normal distribution is used to reject the null hypothesis. t=1 2.2 Larsson, Lyhagen and Løthgren (2001) Larsson et al. (2001) present a maximum-likelihood-based panel test for the cointegrating rank in heterogeneous panels. They propose a standardized LR-bar test based on the mean of the individual rank trace statistic of Johansen (1995). The panel data set consists of N cross-sections observed over T time periods, where i is the index for the cross-section, t represents the index for the time dimension and j = 1,..., p 5 This table contains the mean and variance values for the cases when there is no heterogeneous intercept, or when there is a heterogeneous intercept or/and a time trend in the heterogeneous regression equation. 5

6 is the number of variables in each cross-section. The following heterogeneous V AR(k i ) model, k i Y it = A ik Y i,t k + ε it i = 1,..., N, (6) k=1 is considered for each cross-section under the assumptions that ε it is Gaussian white noise with a nonsingular covariance matrix ε it N p (0, Ω i ), and the initial conditions Y i, ki +1,..., Y i,0 are fixed. One shortcoming of this model is that it allows neither an intercept nor a time trend in the VAR model. The error correction representation for Equation (6) is Y it = Π i Y i,t 1 + k i 1 k=1 Π i = (I p A i1 A i2... A i,ki ) Γ ik Y i,t k + ε it i = 1,..., N (7) Γ ik = (A i,k A i,ki ) k = 1,..., k i 1, where Π i is of order (p p). In the reduced rank form it is possible to write Π i = α i β i, where α i and β i are of order p r i and have full column rank. Larsson et al. (2001) consider the null hypothesis that all of the N cross-sections have at most r cointegrating relationships among the p variables. Then the null hypothesis for the panel cointegration test looks like H 0 : rank(π i ) = r i r for all i = 1,..., N, where H 1 : rank(π i ) = p for all i = 1,..., N. The starting point for the standardized LR-bar statistic of Larsson et al. (2001) is the computation of the trace statistic for each cross-section i, which is denoted as LR it {H(r) H(p)} = 2 ln Q it {H(r) H(p)} = T p j=r i +1 ln(1 ˆλ i,j ), (8) where ˆλ i,j is the jth eigenvalue of the ith cross-section to the eigenvalue problem given in Johansen (1995). The average of the N individual trace statistics is used in order to calculate the standardized LR-bar statistic: LR NT {H(r) H(p)} = 1 N LRiT {H(r) H(p)} (9) The standardized LR-bar statistic for the panel cointegration rank test is defined as ( N LRNT {H(r) H(p)} E(Z k ) ) γ LR {H(r) H(p)} =, (10) V ar(zk ) 6

7 where E(Z k ) is the mean and V ar(z k ) is the variance of the asymptotic trace statistic Z k. { 1 ( 1 ) 1 } 1 Z k tr (dw )W W W W (dw ), where 2 ln Q it {H(r) H(p)} ω Z k and W is a k = (p r) dimensional Brownian motion. Larsson et al. (2001) have simulated the mean and variance of the asymptotic trace statistic using the simulation procedure described in Johansen (1995) for different k values 6. Under the assumptions that a.) one deals with variables that are integrated at most of order one; 7 b.) E(ε it ) = 0, E(ε it ε jt ) = Ω i for i = j and E(ε it ε jt ) = 0 otherwise 8 ; c.) the first two moments of Z k exists; d.) E( 2 ln Q T ) = E(Z k ) + O(T 1 ) and V ar( 2 ln Q T ) = V ar(z k ) + O(T 1 ) 9 ; e.) the short term dynamics varies over individuals, but the long term dynamics is assumed to be constant 10, the standardized LR-bar statistic is standard normally distributed as N and T in such a way that NT 1 0. The panel cointegration rank test of Larsson et al. (2001) is a one-sided test H 0 : rank(π i ) = r i r, which is rejected for all i, if the standardized LR-bar statistic is bigger than the (1 α) standard normal quantile, where α is the significance level of the test. The sequential procedure of Johansen (1988) is used as the testing procedure. First H 0 : r = 0 is tested, and if r = 0 is rejected H 0 : r = 1 is tested. The procedure continues until the null hypothesis is not rejected or the H 0 : p 1 is rejected. 2.3 Standardized LR-max Bar Statistic In this subsection using the same procedure defined in Larsson et al. (2001), a panel cointegration test, which is an extension of the maximum eigenvalue statistic defined by Johansen (1991,1995), is proposed. The starting point of the standardized LR-max bar statistic is again the heterogeneous VAR model defined in (6). VECM representation of this VAR model will 6 Simulated mean and variance values of the asymptotic trace statistic can be found in Table 1 of Larsson et al. (2001) 7 Assumption 1 in Larsson et al. (2001). 8 Necessary in order to establish the asymptotic distribution of the panel cointegration test. 9 Necessary in order to find the joint convergence rate of N and T. 10 Assumption 3 and Assumption 3 in Larsson et al. (2001). These assumptions are made in order to assure the standard normal distribution of standardized LR-bar statistic as N. 7

8 help us to calculate the maximum eigenvalue statistic for each cross-section. However, this time the null hypothesis that all of the N groups in the panel have at most r cointegrating relations among the p variables, against the alternative hypothesis that all of the N groups in the panel have at most one rank greater than r is tested. The null hypothesis for the standardized LR-max bar statistic can be formulated as, H 0 : rank(π i ) = r i r for all i = 1,..., N and the alternative hypothesis is, H 1 : rank(π i ) = r i r + 1 for all i = 1,..., N. The maximum eigenvalue statistic for each cross-section i is computed as LR max it {H(r) H(r + 1)} = 2 ln Q it {H(r) H(r + 1)} = T ln(1 ˆλ i,r+1 ), (11) where ˆλ i,r+1 is the (r + 1)th eigenvalue of the ith cross-section. Analogous to the LR-bar statistic, LR-max bar statistic is defined as the average of N individual maximum eigenvalue statistics, LR max NT {H(r) H(r + 1) = 1 N N i=1 LR max it {H(r) H(r + 1)}. (12) And the standardized LR-max bar statistic will be ( ) N LR max NT {H(r) H(r + 1) E(Zmax k ) γ LR max {H(r) H(r + 1)} =. (13) V ar(z max k ) Z max k is the asymptotic maximum eigenvalue statistic, which is defined as { 1 ( 1 ) 1 } 1 Zk max λ max (dw )W W W W (dw ), 0 where W is again a k = (p r) dimensional Brownian motion, E(Zk max ) and V ar(zk max ) are the simulated mean and variance of the asymptotic maximum eigenvalue statistic. The appropriate mean and variance of the asymptotic maximum eigenvalue statistic for different values of k are simulated using the procedure in Johansen (1995, Chapter 15) and presented in Table I. As the simulation values of Larsson et al. (2001) will be slightly different from the ones in Table I, the mean and variance of the asymptotic trace statistic are simulated one more time in order to sustain the equality of the first (k = 1) mean and variance values for Z k and Zk max. Under the same assumptions like the standardized LR-bar statistic and taking the proof of the asymptotic distribution of the standardized panel trace statistic analogous for the standardized panel maximum eigenvalue statistic, standardized LR-max bar is also standard normally distributed as N and T This conclusion is valid, since Z k and Z max k γ LR max = N(0, 1) 0 have the same Brownian motion functionals. 8 0

9 The testing procedure of the standardized LR-max bar statistic is again the sequential testing procedure of Johansen (1988). The null hypothesis of standardized panel maximum eigenvalue test is rejected like the standardized panel trace statistic, when γ LR max > z 1 α. k = (p r) E(Z k ) V ar(z k ) k = (p r) E(Zk max ) V ar(zk max ) Table I: Simulated moments for Z max k and Z k. 3 Data Generating Process In this research the Monte Carlo study is based on the Toda s (1995) data generating process, as it has been used in several papers in the literature 12. The canonical form of the Toda process allows us to see the dependence of the test performance on some key parameters. In the following lines you can find the modified data generating process of Toda for the panel data. Let y i,t be a p-dimensional vector, where i is again the index for the cross-section, t is the index for the time dimension and p denotes the number of variables in the model. Data generating process has the form of a VAR(1) process. The general form of the modified Toda process for a system of three variables and in the absence of a linear trend in the data looks like then, ψ a 0 0 y i,t = 0 ψ b 0 y i,t 1 + e i,t (14) 0 0 ψ c t = 1,..., T ; i = 1,..., N, where the initial values of y i,t, which can be represented as y i,0 are zero. The error terms for 12 Lütkepohl and Saikkonen (2000), Saikkonen and Lütkepohl (1999,2000), Hubrich, Lütkepohl and Saikkonen (2001) etc. 9

10 each cross-section has the following structure. ( ) ɛ1i,t ɛ i,t = i.i.d.n ɛ 2i,t ( ( Ir Θ 0, Θ I p r In this framework the true cointegrating rank of the process is denoted by r, whereas ɛ 1i,t and ɛ 2i,t are the disturbances to the stationary and non-stationary parts of the data generating process, respectively, while Θ represents the vector of instantaneous correlation between the stationary and non-stationary components of the relevant cross-section. Taking Equation (14) into account, when ψ a = ψ b = ψ c = 1 a cointegrating rank of r = 0 is obtained. Thus, the data generating process becomes, )) y i,t = I 3 y i,t 1 + ɛ i,t, (15) where ɛ i,t i.i.d.n(0, I 3 ), which means that the process consists of three non-stationary components and these components are instantaneously uncorrelated. This can also be illustrated as, y it = Π i,t y i,t 1 + ɛ i,t y it = ɛ i,t where Π i,t = (I A i1 ) and A i1 represents the coefficient matrix of the VAR(1) process. With ψ a < 1 and ψ b = ψ c = 1 the true cointegrating rank of the DGP is 1, and the it is composed of one stationary and two non-stationary components, which can be formulated as. ( ) ψa 0 y i,t = y 0 I i,t 1 + ɛ i,t (16) 2 with ɛ i,t i.i.d.n ( ( 1 Θ 0, Θ I 2 )) where Θ = (θ a, θ b ) and θ a, θ b < 1. The cointegrating rank of the process is r = 2 when ψ a and ψ b are less than unity in absolute value and ψ c = 1. This can be represented in matrix form as, ψ a 0 0 y i,t = 0 ψ b 0 y i,t 1 + ɛ i,t (17) with ɛ i,t i.i.d.n ( ( I2 Θ 0, Θ 1 )) where Θ = (θ a, θ b ) and θ a, θ b are less than unity in absolute value. This process consists of one non-stationary and two stationary components and these components are correlated, when θ a and θ b 0. 10

11 When ψ a, ψ b and ψ c < 1 the DGP is a I(0) process, and the cointegrating rank is r = 3, which can be represented as, y i,t = Ψy i,t 1 + ɛ i,t, (18) where Ψ = diag(ψ a, ψ b, ψ c ) and ɛ i,t i.i.d.n(0, I 3 ). 4 Simulation Study In order the see the performance of the tests to the changes in some key parameters, throughout the simulation study the time and cross-section dimensions, ψ a, ψ b, ψ c parameters and also the correlation parameters between the disturbances to the non-stationary and stationary part of the DGP for each cross-section, which are denoted as θ a and θ b will vary. The correlation parameters θ a and θ b take the values {0.0, 0.4, 0.7} and ψ parameters take the values {0.5, 0.8, 0.95, 1}. Mainly these values are similar to the values considered by Toda (1995). The only difference is Toda has taken 0.8 and 0.90 as the upper bound for θ a, θ b and ψ a, ψ b, ψ c, respectively. The value 0.95 for ψ parameters will help us to see how the tests react when the cointegrating rank of the process is near zero. The lower bounds for θ a, θ b and ψ a, ψ b, ψ c are assumed to be the same as the values considered by Toda (1995). The performance of the tests under the assumption of no instantaneous correlation between the disturbances is checked by θ a = θ b = 0. In order to be able to compare the results with Larsson et al. (2001), for the cross section dimension N = {1, 5, 10, 25, 50} and for the time dimension T = {10, 25, 50, 100, 200} are taken into account, The total number of replications is While generating the random error terms, seeded values are used and the first 100 observations are deleted, so that the starting values are not anymore zero. The tests were programmed in GAUSS 5.0. The maximum lag order for the panel-t and group-t statistics is limited to 3, because this was the maximum lag order allowing an efficient estimation for small time dimension case, e.g T = 10. In addition to this, a kernel estimator is used to select the lag order for the non-parametric panel-ρ and group-ρ statistics, as explained in Section 2. For the maximum-likelihood-based test statistics a VAR model lag order selection criterion was not used, because the data was generated under the assumption of a V AR(1) process. Only the null of no cointegration hypothesis is tested for both residual-based and maximum-likelihoodbased panel cointegration tests, because the residual-based tests cannot test for the order of panel cointegrating rank. In the next sections the simulation results for the empirical size and power and the sizeadjusted power of the panel cointegration tests will be discussed. The tables for the simulation results are presented in the Appendix B and C. 11

12 5 Interpretation of the Simulation Results: The importance of this simulation study lies in the DGP. In order to see the size and power properties of his tests Pedroni (1995) based his Monte Carlo study on MA processes and the size and power results for the system consisting of more than one independent variable was missing. In this study the DGP is based on AR process and study covers the small sample properties of the residual-based tests when there is more than one independent variable in the DGP. Size-adjusted power of the panel cointegration tests is presented for the first time in the literature. 5.1 Empirical Size and Power Properties The most interesting results for empirical size and power properties of the panel cointegration tests are presented in Appendix B 13. The graphs for the empirical size properties can be found in Figures 1-5, on the other hand Figures 6-26 illustrate the empirical power results Size properties Figures 1 and 2 show us that the empirical sizes of group-ρ and panel-ρ statistics are always zero for T = 10, 25 and N 1, which means that the true hypothesis of no cointegration can never be rejected. The severe size distortions for the other test statistics when T is small and N is large, can easily be recognized from Figures 1 and 2, e.g. the empirical sizes of the test statistics except for the panel-and group ρ statistics are unity when T = 10 and N 25. This points out the fact that these tests are not appropriate if the time dimension is much smaller than the cross-section dimension. However, in Figures 4 and 5 it is obvious that when T and N dimensions grow the empirical sizes of maximum-likelihood-based and panel-t tests approach to the nominal size of 5% level, especially for T = 200 and N 5. On the other hand, the empirical size of the group-ρ statistics is around 5% when T = 100 and N 5, and for panel-ρ when T = 50 and N 5. The size distortions of group-t, panel-t and maximum-likelihood-based test statistics also decrease for fixed N when T increases The power properties As there is not much space to present all the results regarding the empirical power properties, we discuss here only the cases where ψ parameters are 0.5, 0.95, 1 and where the disturbances are either uncorrelated or strongly correlated. For those who are also interest in other cases, the results can be provided by the author. First of all the empirical power results for ψ a = ψ b = ψ c = 0.5, 1 without correlation are 13 In the figures, Larsson(CW) represents the standardized LR-bar test statistic calculated with the simulated mean and variance values in this paper. Std. LR-bar is for the standardized LR-bar test statistic calculated with the simulated mean and variance values of Larsson et al. (2001). sc is the abbreviation for Schwarz lag selection Criterion, whereas sd denotes the step-down lag selection method. 12

13 examined. What Figures 6, 8, and 10 in Appendix B exhibit is, the rejection rate of the null of no cointegration for panel-ρ and group-ρ statistics when the true rank is bigger than zero, is always 0 for T = 10 and N 1. This is also valid when we alter the parameters for θ s and/or ψ s. This outcome is represented partly in the Appendix B due to the lack of space. When N is large the powers of the test statistics (except for panel-ρ and group-ρ statistics) approach unity even for small T dimensions, e.g. T = 10. For group-ρ and panel-ρ test statistics the power converges to unity first when T = 25, N 10 and N 25, respectively. The results for T = 50, 100 and 200 are neglected, because the powers of all the test statistics are unity afterwards. In general the power of choosing a higher rank than the hypothesized rank (H 0 : r i = 0) when the true rank is bigger than zero, goes to unity for all the test statistics, when T and Nincrease. If we increase ψ parameters to 0.8 from 0.5 for different rank assumptions there won t be severe changes in the results, so we do not consider them here. With the increase in ψ parameters to 0.95, some noticeable changes in the power properties of the panel cointegration tests occur. For example, large T and N dimensions are necessary, so that the test statistics have high power. With the comparison of Figures 7, 9 and 11 with Figures 12, 15 and 18, correspondingly it can be concluded that in particular the powers of the standardized LR-bar and LR-max bar statistics do not approach unity so fast. In addition to this, the powers of the group-ρ and panel-ρ statistics are around zero also for T = 25. For moderate panel datas the powers of maximum-likelihood-based test statistics are distinct from unity when the true cointegrating rank is 1, which is illustrated by Figure 13. If the true rank is bigger than 1, especially when r = 3, the difference between the powers of standardized LR-bar and LR-max bar statistics increases. This difference can be observed in Figures 16, 17 and 19. Last but not least, the powers of the panel-t statistics converge unity for r = 1, 2 and 3 when cross-section dimension is large enough, e.g. 50. The power results do not change drastically if the correlation between the error terms is not high or ψ parameters are low. So we just take the cases where θ a = θ b = 0.7 and ψ parameters are near unity into account. Figures 21 and 24 demonstrate the convergence of the power of all the panel cointegration tests to unity for T = 10. For large T dimensions only the results for group-ρ, panel-ρ and group-t statistics are presented, because the power of maximum-likelihood-based and the panel-t tests statistics are near unity for almost all T and N dimensions. Form Figures 22 and 25 for r = 1 and 2, respectively, it is obvious that the powers of four test statistics are near zero. When the true cointegrating rank is r = 1 the powers of the group-ρ and group-t statistics do not approach unity, even for high T and N dimensions. This is not anymore valid for the true cointegrating rank of 2. In general the power results of the panel-t and group-t statistics for Schwarz and stepdown methods of lag selection converge to each other, if T and N increases. Standardized LR-bar statistics do not exhibit different results in powerwhen the statistics are calculated either with the simulated mean and variance values from this paper or with the simulated values from Larsson et al. (2001). 13

14 5.2 Size-Adjusted Power Properties In section 5.1 it was pointed out that the empirical power results of the panel-ρ and group-ρ statistics are always zero and the power results of the other test statistics approach unity for small T dimensions, even if the θ and ψ parameters change. In order to have a better view about the small sample properties of the tests statistics, it is a good idea to adjust the power for size. The relevant graphs for the size-adjusted power results are demonstrated in Appendix C starting from Figure 27. When ψ a = ψ b = ψ c = 0.5, 1 and there is no correlation, just the graphs for T = 10 will be discussed, because the powers of all the test statistics approach unity for T 25 and N 10. If T = 10, maximum-likelihood-based and group-t statistics have the lowest power for the true cointegrating ranks of 1 and 2. In Figures 27 and 28 it is clear that the panel-ρ statistic has the highest power, which reaches and 0.681, respectively for r = 1 and 2. If the true cointegrating rank is 3 like in Figure 29, the power of rejecting the null-hypothesis of no cointegration is the highest for the standardized LR-bar statistics with 0.534, whereas the group-t statistics have the lowest power with As there is not much difference in the size-adjusted power results when ψ increases to 0.8, we just do not discuss this case in this paper. If the ψ parameters are near unity with 0.95 and T = 10, the powers of all the test statistics are at most for r = 1, 2 and 3, which is also obvious from Figures 30, 34 and 38. Figures 31 and 32 show us that the maximum-likelihood-based test statistics have the lowest power and the panel-ρ and panel-t test have the highest power. With the true cointegrating assumption of r = 2, Figures 35 and 36 present that the maximum-likelihood-based test statistics have the lowest power again. The powers of all the test statistics converge to unity for high T and N dimensions, which proves what the theory assumes. One interesting outcome of the Monte Carlo study belongs to the case where T = 100 and r = 3. For this case the standardized LR-bar test statistics have the highest power among all the test statistics, whereas the power of the standardized LR-max bar test statistic is the lowest. This eye-catching difference can be observed in Figure 40. In order to understand how the test statistics behave under the assumption of correlated error terms, only the case with the highest correlation parameters is discussed because of the same reasons we have stressed in section 5.1. For ψ a = ψ b = ψ c = 0.95, 1 the powers of the maximum-likelihood-based test statistics and the panel-t statistics approach to one even if T = 10. On the other hand, the powers of the other test statistics are near zero for small T dimensions. (Figures 42 and 45) For T = 100 and 200 only the group-ρ, panel-ρ and group-t statistics are illustrated again, because the other test statistics converge to unity faster. The power of rejecting the cointegrating rank of zero when the true rank is 1 for group-ρ and panel-ρ statistics cannot go to unity even if T and N dimensions are high. The power can converge to unity for the panel-ρ statistic if the true cointegrating rank is 2. (Figure 47) The powers of group-t and panel-t test statistics are again not much different for Schwarz and step-down lag selection methods, when T and N increase. Also the powers of the standardized LR-bar test statistics are the same when the statistics are either calculated with the simulated mean and variance values in this paper or with the simulated values in Larsson et 14

15 al. (2001). 6 Empirical Example: Fisher Hypothesis In this section we try to find out whether the panel cointegration analysis give different results than the results of the usual cointegration techniques to an empirical example. For this purpose Fisher Hypothesis is considered, which is a widely tested economic relation in the macroeconomic literature. There is a mixture of conclusions in the empirical literature for Fisher effect. The nonstationarity of the nominal interest and inflation rates made the application of the cointegration techniques possible in order to test for the long-run relation between the nominal interest and inflation rates. The studies which find evidence for Fisher relation using the unit root and cointegration techniques are: Atkins (1989), Evans and Lewis (1995), Crowder and Hoffman (1996), Crowder (1997), whereas the studies of Rose (1988), MacDonald and Murphy (1989), Mishkin (1992) and Dutt and Ghosh (1995) cannot find any evidence for the Fisher effect. As an alternative method to the cointegration techniques the structural VAR methodology of King and Watson (1997) is applied by Koustas (1998) and Koustas and Serletis (1999), where they couldn t find any evidence for Fisher relation either. With the application of the panel unit root and cointegration tests the recent panel data study for 9 industrialized countries by Crowder (2003) concludes that the Fisher effect exists. Before interpreting the results of the panel cointegration tests to the Fisher relation, using the data set considered in this paper, it maybe better to give a short description of the Fisher relation. The Fisher Hypothesis states that the real interest rate (r t ) is the difference between the nominal interest rate (i t ) and the expected inflation rate (π e t ), r t = i t π e t which means that no one will lend at a nominal rate lower than the expected inflation, and the nominal interest rate will be equal to the cost of borrowing plus the expected inflation. i t = r t + π e t (19) Another aspect of the Fisher relation is that the real interest rates are constant or show little trend in the long run. This can be explained with the phenomena that the nominal interest rate absorbs all the changes in the expected inflation rate when the change in the growth rate of the money supply alter the inflation rate. If the real interest rate changes with a change in the expected inflation, then the Fisher Hypothesis will not hold. When stationarity of the real interest rates with a positive constant (r ) and a normally distributed error term (u t N(0, σu)) 2 is assumed, the equation for r t becomes, r t = r + u t. (20) In addition to this, with the assumption that the agents do not make systematic errors and the actual inflation rate (π t ) differs from the expected inflation rate (π e t ) with a stationary 15

16 process ( ξ t N(0, σ 2 ξ )), the equation for π t is, π t = π e t + ξ t. (21) When we insert (20) and (21) into (19), the Fisher equation for the cointegration analysis looks like, i t = a + bπ t + ε t, where a = r, ε t = u t ξ t and according to the theory b = 1. We search for the existence of a cointegrating relation between the nominal interest rate and the inflation rate in the panel data, in order to see if the Fisher relation holds. To test the Fisher Hypothesis two different data sets consisting of quarterly nominal interest-and inflation rates are considered. The first data set is the monthly data for 19 OECD countries 14 from 1986:06 to 1998:12. The second data set consists of monthly data for 11 OECD countries 15 from 1991:02 to 2002:12. The results of the panel cointegration tests for the Fisher Hypothesis are presented in Appendix A. While testing the Fisher Hypothesis with the panel cointegration tests, we face a problem. Standardized LR-bar and LR-max bar tests cannot be applied to the VAR models with an intercept. Therefore we have to limit our attention to the case where there is no intercept in the VAR model for the maximum-likelihood-based panel cointegration tests. In order to standardize the test statistics of Pedroni, mean and variance values for the case when there is one independent variable in the system (m = 1) are necessary. These values can be found in Pedroni (1995). The lag selection criterion for the maximum-likelihood-based panel cointegration tests is the Schwarz Criterion and the maximum lag order is set equal to 6. For the ADF t-statistic based panel cointegration tests of Pedroni, two lag selection methods will be considered: Step-down method and Schwarz Criterion. The maximum lag order for these methods is limited to 12, because the data sets consist of monthly data. When the first data set (1989 : : 12) is considered, country-by-country trace tests point out the existence of the Fisher relation, except for Austria and UK. On the other hand, when the whole panel data is tested, standardized LR-bar test cannot reject the null hypothesis of all the countries having at most cointegrating rank of 2. This means that the underlying heterogeneous VAR model is stable. This result is also valid for the cases when the standardized LR-bar statistic is applied to the data set without Austria or to the data set without Austria and UK. The results of the standardized LR-max bar statistic are not different from the ones for the standardized LR-bar statistic 16. Standardized panel maximum eigenvalue test also cannot reject the null hypothesis of the r = r i 2, which points out the stability of the variables. Country-by-country test results for the second data set (1991 : : 12) cannot reject the null hypothesis of r i = 1, except for Japan. On the other hand, the results show that 14 Germany, France, Italy, Netherlands, Spain, Finland, Austria, Ireland, Portugal, Belgium, US, Japan, UK, Denmark, Mexico, Norway, Iceland, Sweden, Canada. 15 US, Korea, Japan, UK, Denmark, Mexico, Norway, Iceland, Hungary, Sweden, Canada. 16 Results are presented in Table II and Table III of Appendix A 16

17 the Fisher Hypothesis does not hold for Sweden. This outcome for Sweden is different from the country-by-country for the first data set. Standardized LR-bar test accepts the hypothesis of cointegrating rank of two, even when we test the relation without taking Japan into account. The results for the standardized LR-max bar test do not give also a different outcome for the second data set 17. The residual-based panel cointegration tests allows us to consider two different cases, the case where there is a heterogeneous intercept and the case where there is no heterogeneous intercept in the regression equation. The results of Pedroni s tests are presented in the Appendix A starting with Table VI to Table IX. The outcomes of the residual-based panel cointegration tests are different from the maximum-likelihood-based tests. For the residual-based panel cointegration tests the rank order of the cointegrating matrix cannot be tested. By testing the null hypothesis of no cointegration it can just be determined whether there is a cointegration relation. All of the residual-based panel cointegrating tests reject the null of no cointegration for both data sets, when there is no heterogeneous intercept in the panel regression equation. This is also the same case when we exclude Austria and UK from the first data set, and Japan from the second data set. However, some of the test statistics give different results for both data sets with the assumption of a heterogeneous intercept in the panel regression equation. The panel-ρ test statistic cannot reject the null of no cointegration, if the first data set is considered as a whole, which means that the Fisher Hypothesis does not hold. This is also valid for the panel-t statistics if the tests are undertaken for the second data set excluding Japan. 7 Conclusion With the extensive simulation study in Section 5, which covers the empirical size, power and size-adjusted power of six panel cointegration tests, it can be concluded that the panel-t test statistic has the best size and power properties. We found out that the power of the panel-t statistic approaches unity for small T and N dimensions, even when there is strong correlation between the innovations to the non-stationary and stationary part of the data generating process, while the empirical size of it is around the empirical nominal size of 5% when T = 200 and N 5. On the other hand, the other three residual-based panel cointegration test statistics; group-ρ, panel-ρ and group-t have really poor power results if the correlation parameters and ψ parameters are high (e.g. when θ a = θ b = 0.7 and ψ a = ψ b = 0.95 respectively). The second test statistics which have the best size and power properties are the maximumlikelihood-based panel cointegration test statistics namely, the standardized LR-bar and LRmax bar statistics. They have better power if the correlation parameter is high and the ψ parameter is around unity. The empirical size of the standardized LR-bar and LR-max bar statistics is around 5% like the panel-t test statistic when both T and N grow, especially if time dimension grows faster than the cross-section dimension just as the theory points out. 17 Results are presented in Table IV and Table V of Appendix A, respectively. 17

18 The power difference between the standardized LR-bar and LR-max bar statistics is large, for some combinations of θ a = θ b = 0.0, 0.4, 0.7 (e.g. θ a = θ b = 0.4), if the true cointegrating rank is r 2 and the ψ parameters are For such cases it may be better to apply standardized LR-bar statistic, instead of standardized LR-max bar, because the standardized LR-bar statistic has higher power, especially when T is large. It should be also emphasized that the size and power results of the residual-based panel cointegration tests can depend on the choice of the dependent variable. In this paper the first variable of the DGP has been taken as the dependent variable for the residual-based panel cointegration tests. In Section 6 while we were testing the Fisher hypothesis with the panel cointegration test statistics, we were able to present the results of the residual-based panel cointegration tests under the assumption of a heterogeneous intercept in the panel regression equation, whereas maximum-likelihood-based statistics had to be considered without a heterogeneous intercept in the VAR model. For a future study the procedure in Larsson et al. (2001) can be extended for a maximum-likelihood-based panel cointegration test statistic with a constant and a linear trend in the data. Residual-based panel cointegration tests of Pedroni pointed out the existence of the Fisher relation for two different data set consisting of OECD countries. However, the maximum-likelihood based test statistics failed to find any evidence. References [1] Atkins, Frank J. (1989) Cointegration, error correction and the Fisher Effect, Applied Economics, 21(16), [2] Baltagi, Badi and Chihwa Kao (2000) Nonstationary Panels, Cointegration in Panels and Dynamic Panels: A Survey, Center for Policy Research Working Paper No.16. [3] Banerjee, Anindya (1999) Panel Data Unit Roots and Cointegration: An Overview, Oxford Bulletin of Economics and Statistics, 61 Special Issue, [4] Breitung, Jörg and Wolfgang Meyer (1994) Testing for Unit Roots in Panel Data: Are Wages on Different Bargaining Levels Cointegrated?, Applied Economics, 26, [5] Crowder, William J. (1997) The long-run Fisher Relation in Canada, Canadian Journal of Economics, Vol. 30(4), [6] Crowder, William J. (2003) Panel Estimates of the Fisher Effect, unpublished manuscript University of Texas at Airlington Department of Economics. [7] Crowder, William J. and Dennis L. Hoffman (1996) The long-run Relationship Between Nominal Interest Rates and Inflation: The Fisher Equation Revisited, Journal of Money, Credit and Banking, 28(1), [8] Dutt, Swarna D. and Dipak Gosh (1995) The Fisher Hypothesis: Examining the Canadian Experience, Applied Economics, 27(11), [9] Evans, Martin and Karen Lewis (1995) Do Expected Shifts in Inflation Affect Estimates of the long-run Fisher Relation?, Journal of Finance, 50(1),

19 [10] Gutierrez, Luciano (2002) On the Power of Panel Cointegration Tets: A Monte Carlo Comparison, Economics Letters, 80(1), [11] Groen, Jan J. J. and Frank Kleibergen (1999) Likelihood-based Cointegration Analysis in Panels of Vector Error Correction Models, Discussion Paper /4, Tinbergen Institute, The Netherlands. [12] Harris, David and Brett Inder (1994) A Test of the Null Hypothesis of Cointegration, In Nonstationary Time Series Analysis and Cointegration, edited by Hargreaves, Colin P. New York: Oxford Univeristy Press, pp [13] Hubrich, Kirstin, Helmut Lütkepohl and Pentti Saikkonen (2001) A Review of Systems of Cointegration Tests, Econometric Reviews, 20(3), [14] Im, Kyung So, M. Hashem Pesaran and Yongcheol Shin (1997) Testing for Unit Roots in Heterogeneous Panels, Manuscript, Department of Applied Economics, University of Cambridge, United Kingdom. [15] Johansen, Søren (1988) Statistical Analysis of Cointegrating Vectors, Journal of Economic Dynamics and Control 12, [16] Johansen, Søren (1991) Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models, Econometrica 59, [17] Johansen, Søren (1995) Likelihood-Based Inference in Cointegrated Vector Autoregressive Models, Oxford, Oxford University Press. [18] Kao, Chihwa (1999) Spurious Regression and Residual-Based Tests for Cointegration in Panel Data, Journal of Econometrics 90, [19] Kao, Chihwa (2000) On the Estimation and Inference of a Cointegrated Regression in Panel Data, Advances of Econometrics 15, [20] Karlsson, Sune and Mickael Løthgren (2000) On the Power and Interpretation of Panel Unit Root Tests, Economics Letters 66, [21] Larsson, Rolf and Johan Lyhagen (1999) Likelihood-Based Inference in Multivariate Panel Cointegration Models, Stockholm School of Economics Working Paper Series in Economics and Finance, No [22] Larsson, Rolf, Johan Lyhagen, and Mickael Løthgren (2001) Likelihood-based Cointegration Tests in Heterogeneous Panels, Econometric Journal, 4, [23] Levine, Andrew and Chien-Fu Lin (1992) Unit Root Test in Panel Data: Asymptotic and Finite Sample Properties, Discussion Paper No: 92-93, University of California at San Diego. [24] Lütkepohl, Helmut and Pentti Saikkonen (2000) Testing for the Cointegrating Rank of a VAR Process with a Time Trend, Journal of Econometrics, 95,

20 [25] MacDonald, Ronald and P. D. Murphy (1989) Testing the long-run Relationship Between Nominal Interest Rates and Inflation Using Cointegrating Techniques, Applied Economics, 21, [26] McCoskey, Suzanne and Chihwa Kao (1998) A Residual-based Test of the Null of Cointegration in Panel Data, Econometric Reviews, 17, [27] McCoskey, Suzanne and Chihwa Kao (1999) Comparing Panel Data Cointegration Tests with an Application to the Twin Deficits Problem, Working Paper, Center for Policy Research, Syracuse University, New York. [28] Mishkin, Frederic S. (1992) Is the Fisher Effect for Real? A Reexamination of the Relationship Between Inflation and Interest Rates, Journal of Monetary Economics, 30(2), [29] Newey, Whitney K. and Kenneth D. West (1987) A Simple Positive Semi-Definite, Heteroscedasticity and Autocorrelation Consistent Covariance Matrix, Econometrica, 50, [30] Newey, Whitney K. and Kenneth D. West (1994) Autocovariance Lag Selection in Covariance Matrix Estimation, Review of Economic Studies, 61, [31] Pedroni, Peter (1995) Panel Cointegration; Asymptotic and Finite Sample Properties of Pooled Time Series Tests, with an Application to the PPP Hypothesis, Indiana University Working Papers in Economics, No , June. [32] Pedroni, Peter (1996) Fully-Modified OLS for Heterogeneous Cointegrated Panels and the Case of Purchasing Power Parity, Indiana University Working Papers in Economics No , June. [33] Pedroni, Peter (1997) On the Role of Cross-Sectional Dependency in Panel Unit Root and Panel Cointegration Exchange Rate Studies, Working Paper, Indiana University. [34] Pedroni, Peter (1999) Critical Values For Cointegration Tests in Heterogeneous Panels With Multiple Regressors, Oxford Bulletin of Economics and Statistics 61, [35] Pedroni, Peter (2001) Panel Cointegration; Asymptotic and Finite Sample Properties of Pooled Time Series Tests, with an Application to the PPP Hypothesis:, Working Paper, Indiana University. [36] Phillips, Peter C. B. and Hyungsik R. Moon (1999) Linear Regression Limit Theory for Nonstationary Panel Data, Econometrica, 67, [37] Quah, Danny (1994) Exploiting Cross Section Variation for Unit Root Inference in Dynamic Data, Economics Letters, 44, [38] Rose, Andrew K. (1988) Is the Real Interest Rate Stable, Journal of Finance, 43(5), [39] Saikkonen, Pentti and Helmut Lütkepohl (1999) Local Power of Likelihood Ratio Tests for the Cointegrating Rank of a VAR Process, Econometric Theory, 15,