Test of Hypotheses in Panel Data Models When the Regressor and Disturbances are Possibly Nonstationary

This paper considers the problem of hypotheses testing in a simple panel data regression model with random individual effects and serially correlated disturbances. Following Baltagi, Kao and Liu (2008), we allow for the possibility of non-stationarity in the regressor and/or the disturbance term. While Baltagi et al. (2008) focus on the asymptotic properties and distributions of the standard panel data estimators, this paper focuses on test of hypotheses in this setting. One important finding is that unlike the time series case, one does not necessarily need to rely on the “super-efficient” type AR estimator by Perron and Yabu (2009) to make inference in panel data. In fact, we show that the simple t-ratio always converges to the standard normal distribution regardless of whether the disturbances and/or the regressor are stationary.

1 least squares (OLS), …xed e¤ects (FE), …rst-di¤erence (FD), and generalized least squares (GLS) estimators when both the time-series length (T ) and the number of cross-sections (n) are large. They show that these estimators have asymptotic normal distributions and have di¤erent convergence rates dependent on the nonstationarity of the regressor and the remainder disturbances. Some of their important …ndings include the following: (i) When the disturbance term is I(0) and the regressor is I(1), the FE estimator is asymptotically equivalent to the GLS estimator and OLS is less e¢ cient than GLS; (ii) When the disturbance term and the regressor are I(1), GLS is more e¢ cient than the FE estimator since GLS is p nT consistent, while FE is p n consistent. As a result, they recommend the GLS estimator as the preferred estimator, and they show using Monte Carlo experiments that the loss in e¢ ciency of the OLS, FE, and FD estimators relative to true GLS can be substantial. This paper is a follow up paper which is concerned with test of hypotheses using these standard panel data estimators. One important …nding, is that unlike the time series setting, one does not necessarily need to rely on the "super-e¢ cient"type AR estimator by Perron and Yabu (2009) to make inference in panel data. In fact, we show that the simple t-ratio based on the FGLS estimator of Baltagi and Li (1991), will always converge to the standard normal distribution regardless of whether the disturbances and/or the regressor are stationary or not. We also show using Monte Carlo experiments that inference based on the OLS, FE, and FD estimators could be misleading relative to that based on feasible GLS. The outline of the paper is as follows: Section 2 considers a simple panel data regression model with unobserved individual e¤ects and AR(1) remainder disturbances and derives the asymptotic distributions of the t statistics of the standard FE and FD estimators, respectively. This is done for four cases, corresponding to whether the remainder disturbances and/or the regressor are stationary or not. In Section 3, we derive the corresponding asymptotic distributions of the t statistic for the FGLS estimator under these four cases.
Section 4 reports the …nite sample properties of the proposed tests using Monte Carlo experiments. Section 5 concludes. All proofs are given in the appendix.
Unless otherwise speci…ed, for all the asymptotic results in this paper, we let n and T go to in…nity simultaneously (i.e., (n; T ) ! 1), see Phillips and Moon (1999). We require n T ! 0 in some cases. We write the integral R 1 0 W (s)ds as R W and W as W R W when there is no ambiguity over limits. We use p ! to denote convergence in probability, d ! to denote convergence in distribution, to denote Kronecker product, and [x] to denote the largest integer x.

The Model and Assumptions
Consider the following panel data regression model: y it = + x it + u it ; i = 1; : : : ; n; t = 1; : : : ; T 2 where u it = i + it ; and and are scalars. 1 We assume that the individual e¤ect i is random with i iid(0; 2 ) and f it g is an AR(1) it = it 1 + e it ; j j 1 (2) where e it is a white noise process with variance 2 e . The i is independent of the it for all i and t. 2 Let fx it g be also an AR (1) such that where " it is a white noise process with variance 2 " . In this paper we assume that The initialization of this system is y i1 = x i1 = O p (1) for all i. large. They …nd that, when it is I(0) (i.e., < 1), the FE 3 and the GLS estimators are both p nT consistent and (asymptotically) equivalent. However, this asymptotic equivalence breaks down when it is I(1) (i.e., = 1). In this case, the GLS and the FD 4 estimators are both p nT consistent and more e¢ cient than the FE estimator (which is p n consistent).
De…ne the innovation vector w it = (e it ; " it ) 0 . We assume that w it is a linear process that satis…es the following assumptions: Assumption 1 For each i, we assume: 1. w it = (L) it = P 1 j=0 j it j ; P 1 j=0 j a k j k < 1; j (1)j 6 = 0 for some a > 1: 2. For a given i, it is i.i.d. with zero mean and variance-covariance matrix ; and …nite fourth order cumulants.
Assumption 2 We assume it and jt are independent for i 6 = j: That is, we assume cross-sectional independence. 1 For simplicity, we consider the case of one regressor, but our results can be extended to the multiple regressors case. In fact, we assume that for the multiple regressors case, X 0 X is of full rank to avoid the complexity from possible cointegration. 2 This model was studied by Baltagi and Li (1991) under stationarity of the regressors and the disturbances. 3 The …xed e¤ects estimator of is given by, where x i: = 1 T P T t=1 x it and y i: = 1 T P T t=1 y it , see Hsiao (2003). 4 The FD estimator is the OLS estimator of a …rst-di¤erenced regression, see Hsiao (2003). That is, where The long-run variance covariance matrix of fw it g with Assumption 3 is given by

The Fixed E¤ects and the First Di¤erence Estimators
In this paper, we focus on testing the common slope , We start by investigating the asymptotic distributions of the t-statistics for H 0 based on the FE and FD estimators. Let us denote these by t F E and t F D , respectively. We derive these asymptotic distributions under four scenarios where the disturbances and the regressor are allowed to be I(0) or I(1).
If v it is known to be I (0), 5 the corresponding t-test for H 0 using the FE estimator b F E , is given by: where s F E = is the estimator of suggested by Baltagi and Li (1991). Next, we derive the limiting distribution of b when j j < 1 as well as when = 1. 5 Note that the FE and the GLS estimators are asymptotically equivalent for this case, see Baltagi et al. (2008). Theorem 1 Assume (n; T ) ! 1: 1. If j j < 1; j j < 1 with n T ! 0; t F E d ! N 0; 1 + 1 : 2. If = 1; j j < 1; t F E cannot be obtained.
The results of Theorem 1 show that, under the null, t F E has a normal distribution if the disturbance term is I(0) regardless of the stationarity or non-stationarity of the regressor. Note that we cannot even implement the t test when the error term is I(1). In fact, one cannot compute the standard error s F E in this setting as shown in the Appendix.
Next, we turn to the case of the FD estimator, b F D . 6 The corresponding t-test for H 0 using the FD estimator b F D , is given by: 6 Note that if v it is known to be I(1), the FD and the GLS estimators are asymptotically equivalent, see Baltagi et al. (2008). 5 where Theorem 2 Assume (n; T ) ! 1 and n T ! 0: 1. If j j < 1; j j < 1; The results of Theorem 2 show that, under the null, t F D has a normal distribution regardless of the stationarity or non-stationarity of the regressor and/or the disturbance term.

The Feasible GLS Estimator
We rewrite equation (1) in vector form where y is nT 1, x is a vector of x it of dimension nT 1, nT is a vector of ones of dimension nT , u is nT 1, is a vector of i , is a vector of it and Z = I n T . By the partitioned inverse rule, it can be shown, see Baltagi et al. (2008) Substituting (9), one gets: where G 1 and G 2 are de…ned accordingly, see also the Appendix. The variance-covariance matrix is given by: where T is a vector of ones of dimension T . A is the variance-covariance matrix of it , which for the AR (1) is given by: where = 0 T A 1 T . When j j < 1, this estimation is equivalent to the Prais-Winsten transformation method suggested by Baltagi and Li (1991) for the panel data model. One can easily verify that A 1 = C 0 C, where C = is the well known Prais-Winsten transformation for the AR(1) model. Baltagi and Li (1991) suggest premultiplying the panel model (9) by (I n C) to get rid of serial correlation in the remainder term, and then performing a Fuller and Battese (1973) transformation in the second step to take care of the random e¤ects.
In order to obtain the FGLS estimator, b F GLS , we use an estimate of suggested by Baltagi and Li (1991) based on FE residuals given below equation (7). The asymptotic distribution of b was derived in where T 1 is a vector of ones of dimension T 1: Using a trick by Wansbeek and Kapteyn (1983) Estimates for 2 e and 2 can be obtained from^ whereû are the Prais-Winsten transformed residuals (see Baltagi and Li (1991) for more details). Hence, 2 can be estimated as^ Substituting^ 2 e ,^ 2 , and b into equation (14), one obtains b F GLS . The corresponding t-test for H 0 using the FGLS estimator b F GLS , is given by: where b G 1 and b G 2 are given as equation (11) with the replacement of by^ .

Case 1: Without Individual E¤ects
We begin with a simple case where i = 0. That is, the individual e¤ects are not included in the true model, but there is …rst order serial correlation. This is not realistic in panel data economic models, but we study it as a base case. The variance-covariance matrix given in (12) reduces to In this case, the FGLS estimator, b F GLS , will be based on e and e 2 e given by, and u it denotes the OLS residual. 7 7 Note that we use the OLS residuals instead of the FE residuals in this case. That is, b Lemma 2 Assume (n; T ) ! 1 and n T ! 0: : As shown above, we have the same rate of converging speed as that assuming individual e¤ects except for case (3). That is, in the panel cointegration case, we have the convergence rate p nT which is the same as that of the GLS estimator and the FE estimator. However, note that once we add the individual e¤ects, the OLS estimator has the slower convergence rate p nT rather than p nT because i dominates v i . 8 Lemma 3 Assume (n; T ) ! 1: As can be seen in Lemma 3, we …nd that the limiting distribution of e is the same as that of b using the FE residuals, when j j < 1 with n T ! 0. However, this limiting distribution is di¤erent when = 1. Compare, T (e 1) p ! 0 without individual e¤ects with T (b 1) p ! 3 with individual e¤ects. We also …nd that, in both cases, the consistency of e 2 e can be achieved. Based on the above results, one can derive the asymptotic distribution of the t-ratio for each case. 8 The limiting distribution of the OLS estimator with individual e¤ects is given by " ! e.g., see Baltagi et al. (2008) for details.
Theorem 3 Assume (n; T ) ! 1 and n T ! 0. Without individual e¤ ects, e always leads to t F GLS d ! N (0; 1) regardless of the stationarity or non-stationarity of the regressor and/or the disturbance term.
Theorem 3 shows that t F GLS always converges to the standard normal case whether the disturbance term is I(0) or I(1) and whether the regressor is I(0) or I(1). That is, without individual e¤ects, the t-ratio based on the FGLS, can be used for inference using the standard normal distribution. Hence, in this case, one does not have to consider the "super-e¢ cient" type estimator by Perron and Yabu (2009) which is designed to bridge the gap between I(0) and I(1). 9

Case 2: With Individual E¤ects
This section derives the asymptotic distribution of t F GLS given in (16) and discussed in Section 3.
One can de…ne the "super-e¢ cient" estimator b s as for some 2 (0; 1) and " > 0. Hence, when b is in a T neighborhood of 1, it is assigned a value of 1. For details, see Perron and Yabu (2009).
3. If j j < 1, = 1 Theorem 4 implies that the t-ratio based on b by Baltagi and Li (1991) asymptotically leads to the standard normal distribution regardless of the stationarity or non-stationarity of the regressor and/or the disturbance term. This is an interesting …nding because despite the fact that we do not have a consistent estimate of 2 when = 1, we can still obtain t F GLS converging to N (0; 1). Accordingly, we have a similar result to that of Theorem 3 except that one cannot expect consistent estimates for all the variance components when = 1.
For each experiment, we perform 10; 000 replications and compute the t-statistics using OLS, FE, FD, GLS-CO, GLS-PW, and true GLS. With this design we have 900 experiments. GAUSS 7:0:6 is used to perform the simulations. Random numbers for e it , i , and " it are generated by the GAUSS procedure RNDNS. We generate n(T + 1000) random numbers and then split them into n series so that each series has the same mean and variance. The …rst 1; 000 observations are discarded for each series.
Tables 1 to 4 report the empirical size of these various t-statistics, when the true size is 5%, for ( ; ) =  Table 1, the size of tOLS varies between 10 and 18%, while the size of tFE varies between 9 and 11%. This gets worse for the non-stationary disturbances case in Table 2, where the size of tOLS and tFE varies between 18 and 20%.
For the non-stationary regressor case in Table 3, the size of tOLS varies between 24 and 80%, while the size of tFE varies between 17 and 20%. The spurious regression case in Table 4 gives the worst performance for tOLS with size varying between 59 and 83%. The size for tFE is also bad varying between 51 and 78%. (ii) In all cases, except case 1, tFD performs well with empirical size close to 5%. For case 1, tFD is slightly over-sized at 7 to 9%. (iii) tGLS gives the best performance, with empirical size not statistically di¤erent from 5%, for all cases considered. (iv) Both tGLSPW and tGLSCO perform well across experiments. In fact, for small sample sizes such as (n; T ) = (20; 20); they are undersized in case 2, and oversized in cases 3 and 4. However, as n and/or T increase, the empirical size of tGLSPW and tGLSCO improves considerably.
We also note that the size of tOLS gets worse as the percentage of heterogeneity across individuals ( ) increases. However, this heterogeneity measure does not a¤ect the performance of tFE and tFD, since both estimators wipe out the individual e¤ects. Theorems 3 and 4 also imply that the t-ratio using FGLS should 1 0 Note that Baltagi and Li (1997) …x 2 + 2 across experiments. Here, one cannot obtain 2 in the nonstationary case.
Instead we …x 2 + 2 e and our results are not sensitive to the choice of this sum. In fact, we tried 5, 10, and 20, and the results are similar. In conclusion, we note that tGLS gives the best performance, but it is infeasible. We recommend tFGLS for testing H 0 : = 0 when the researcher has no perfect foresight on stationarity of the regressor and/or the error term. tFD is a viable alternative to tFGLS if either the regressor or the error is nonstationary. tOLS and tFE are not recommended in these cases.

Robustness to Heterogeneous AR Parameters and Heteroskedasticity 11
In this section we check the robustness of our results to (i) heterogeneity in the AR parameters in both the regressor and the error term and also to (ii) heteroskedasticity in the error terms. To accomplish this we run two sets of Monte Carlo experiments. The …rst set of experiments allow the AR parameters to vary across individuals. More speci…cally, i (for the regressor) and i (for the error term) are allowed to be uniformly distributed, i.e., IIDU (0; 1). The estimation and test procedure are the same as before while the Data Generating Process is di¤erent. Table 5 reports the empirical size of these new experiments. Interestingly, the t-statistics using FGLS turn out to be robust across these experiments. In fact, tGLSPW and tGLSCO have empirical size that varies between 4 5%. tOLS and tFE perform badly again. In fact, tOLS has empirical size that varies between 19% and 67%, while tFE has empirical size that varies between 16% and 34%. tFD is slightly oversized with empirical size that varies between 6% and 7%.
As for the presence of heteroskedasticity in the error terms, we generate the error terms using the following design: where 1 it and 2 it are generated from N (0; 1), respectively. To incorporate heteroskedasticity, i are generated as follows: tFGLS are slightly oversized. In fact, the size for tGLSPW and tGLSCO varies between 6 and 7% for various sample sizes. tOLS and tFE are bad with size varying between 12 to 18% and 11 to 12%, respectively. tFD is also oversized at 9-10%. (ii) Panel B presents the results under a higher degree of heteroskedasticity (c = 10). In this case all the t-statistics are way oversized. The size for tGLSPW and tGLSCO varies 1 1 We would like to thank the referee for this suggestion. 13 between 36 and 40%. 12 Hence, we conclude that tFGLS is robust to heterogeneous AR parameters, but not to heteroskedasticity in the error terms.

Conclusion
This paper derived the limiting distribution of the t-statistic for H 0 : = 0 ; using di¤erent panel data estimators including FE, FD, and FGLS. This is done in the context of a linear panel data regression model with possible nonstationarity in the regressor and/or the error term. We showed that one can use t statistics based on the FGLS estimator regardless of the nonstationarity of the regressor and/or the disturbance term.
This is unlike the time-series case, where one has to consider a "super-e¢ cient"type AR estimator of Perron and Yabu (2009) to achieve the normal limiting distribution of the t-ratio. One caveat is that this may not be robust to heteroskedasticity of the error terms, but it is robust to heterogenous AR parameters across individuals. 1 2 The case of heteroskedastic error terms remains to be studied in the future. For possible ideas on how to handle this problem, see, Baltagi and Kao (2000).
For the denominator, by, e.g., Baltagi et al. (2008). Similarly, we have since it 1 and x it 1 are uncorrelated. Hence, we have (1) and accordingly we have Finally, we conclude that if (n; T ) ! 1 and n T ! 0, then One can show that all the rest of terms except I is o p (1). For example, consider II.
Similarly, it can be easily shown that Using a similar argument as in Phillips and Moon (1999) and Baltagi et al. (2008), it can be shown that Also note that as (n; T ) ! 1, we have where v it and x it are not correlated. Hence, and by using a similar argument, we get which is the same result as in Lemma 1.(1).
and it can be shown that b 2 e = I + o p (1): Let us consider II; for example.
n p nT (^ ) as (n; T ) ! 1 with the joint limit argument. By a similar process, we have as (n; T ) ! 1.
For the denominator, Consider II …rst. Using a similar argument as in Phillips and Moon (1999) and Baltagi et al.
Also as (n; T ) ! 1, we have Consider III. It can be shown that as (n; T ) ! 1 using Hence, we have Consider II. It can be easily shown that Consider III. It is also easy to see that We conclude that using equation (C.5) in Kao (1999). Hence, = 3: From above, it can be shown that as (n; T ) ! 1. We illustrate II only as an example. It can be shown that Also as (n; T ) ! 1, we have Kao (1999). Next consider III.
where it and x it are not correlated. Hence, we have by equation (C.3) in Kao (1999).
For the numerator, b it b it 1 = e it (^ F E )" it , and it can be shown that Consider II. It can be shown that Consider III and IV . In a similar vein as II, one can see that We conclude that using equation (C.5) in Kao (1999). Combining these results, we get 36 which is the required result. Next we consider b 2 e . Note that After some tedious algebra, it can be easily shown that B Proof of Theorem 1 Proof. Now we are ready to prove Theorem 1.
Therefore, if (n; T ) ! 1 and n T ! 0, then From the construction of S F E , it is obvious that we cannot obtain s F E because we have a complex number problem. Accordingly we cannot have t F E either.
(1), one can easily verify that if (n; T ) ! 1 and n T ! 0, then we cannot obtain s F E or t F E either.

C Proof of Theorem 2
Proof. Now we prove Theorem 2.
= I + II + III: Consider I. It can be shown that as (n; T ) ! 1 using For II and III, one can verify that II = O p 1 nT and III = O p 1 nT using the fact that if (n; T ) ! 1 and n T ! 0, then This uses a similar argument as in Phillips and Moon (1999), also Corollary 5.1 in Baltagi et al. (2008).
Hence, we have^ Now recall that Here one can easily see that 1 + 39 as (n; T ) ! 1. We conclude that if (n; T ) ! 1 and n T ! 0, then = I + II + III: Also note that For III, it is easy to see that (1): Also recall that Hence, we conclude that if (n; T ) ! 1 and n T ! 0, then ) r 3. j j < 1; = 1 case By a similar argument as above, it can be easily shown that This is because if (n; T ) ! 1 and n T ! 0, then p nT (^ F D Also recall that Hence, if (n; T ) ! 1 and n T ! 0, we have ) r  Hence, we conclude that if (n; T ) ! 1 and n T ! 0, then = N (0; 1):

D Proof of Lemma 2
The OLS estimator of is given by Proof. We consider the denominator …rst and then move to the numerator to prove Lemma 2. 1

The numerator
(a) If j j < 1; j j < 1, it can be shown that as (n; T ) ! 1.
1 Note that i is not included in error term here.
Using the results above, the proof of Lemma 2 is straightforward E Proof of Lemma 3 In this section, we consider the limiting distribution of using OLS residuals and we check the consistency of 2 e under nonstationarity of both the error term and the regressor. Proof. Assume (n; T ) ! 1: For the denominator, Consider II …rst. It is easy to see that II = O p 1 nT because if (n; T ) ! 1 and n T ! 0, then : Also, by Lemma 2.(1), we have since it and x it are not correlated. Hence, Let us look at the numerator.
Consider I. One can see that Consider III. Using a similar argument, it can be shown that as (n; T ) ! 1. Hence, we have 1 1 2 as (n; T ) ! 1. Therefore, we conclude that if (n; T ) ! 1 and n T ! 0, then Next we show e 2 e is a consistent estimator. Note that where E nT = I nT J nT and J nT = nT 0 nT =nT . Hence, To rearrange the terms, note that and accordingly Consider I.
(1 e ) i: Now it is easy to see that In a similar vein as I, it is easy to see that Expanding this equation, one can show that (1 e )x i: as (n; T ) ! 1: This is because in the …rst term and p nT (e ) = O p (1). Also note that 1 and accordingly 1 nT Consider III.
as (n; T ) ! 1 using the Cauchy-Schwarz inequality. Summarizing, we proved that e 2 e p ! 2 e : (b) j j < 1, = 1 case This is the panel cointegration case. Consider e = For the denominator, Consider II …rst. With the joint limit, one can verify II = O p 1 nT using that if (n; T ) ! 1 and by Lemma 2.(3). Also note that as (n; T ) ! 1, we have since it and x it are not correlated. Hence, we have as (n; T ) ! 1.

50
Consider III: Hence, one can see that and we conclude that if (n; T ) ! 1 and n T ! 0, then For the denominator, Consider II …rst. It is easy to see II = O p (2) and because as (n; T ) ! 1, we have Next consider III. One can show that since it and x it are not correlated and accordingly 1 Hence, we have as (n; T ) ! 1 by, e.g., equation (C.3) in Kao (1999).
as (n; T ) ! 1 with the joint CLT.
Consider IV . After some algebra, it can be shown that Lastly, consider I.
We …nally have = 0: Next we show e 2 e is a consistent estimator. Again we have x 0 E nT I n Ĉ 0Ĉ E nT = I + II + III:
Now it is easy to see that I ! 2 e as (n; T ) ! 1 using 1 , and T (e 1) = o p (1) with the joint limit. Consider II. After some tedious algebra, it can be shown that as (n; T ) ! 1: Now one can see that using the fact that if (n; T ) ! 1 and n T ! 0, then p n ^ (b) = 1, = 1 case Note that :

55
For the denominator, Consider II …rst. With the joint limit, one can see that as (n; T ) ! 1 using the fact that if (n; T ) ! 1 and n T ! 0, then where it and x it are not correlated. Accordingly, we conclude that by equation (C.3) in Kao (1999).
Consider I. One can verify that as (n; T ) ! 1. Consider III and IV . In a similar vein as II, it is easy to see that Summarizing, we have = 0: Next we show e 2 e is a consistent estimator. It is clear that I ! 2 e as (n; T ) ! 1 as shown already. Consider II. In a similar process as above, one can show that as (n; T ) ! 1. Hence we have since if (n; T ) ! 1 and n T ! 0, then p n ^ Preparation: Note that from equation (9), we have where y is nT 1, x is a vector of x it of dimension nT 1, nT is a vector of ones of dimension nT , u is nT 1, is a vector of i , is a vector of it and Z = I n T . Also recall from equation (13) that and Proof. Following Baltagi et al. (2008), we …rst de…ne matricesÂ andĈ which replace in the matrix A and C in equation (12) and (14) with e given by, where b u it denotes the it-th OLS residual. Using the de…nition of 1 in equation (13) and e 2 e given by, where b u = (I n Ĉ )b u and b u denotes the nT 1 vector of the OLS residuals, one obtains: x 0 iÂ 1 i ; 59 and 0 nT^ ( it e it 1 ) ; In this section, we assume that (n; T ) ! 1 and n T ! 0 unless otherwise speci…ed.
1. j j < 1, j j < 1 case (a) De…ne First we consider Expanding this equation, we will show that 60 as (n; T ) ! 1. Consider I.
Consider II.
Consider III.
Hence, we have one concludes that Next consider it is easy to see that and (1 e ) p nT (e ) p nT Also recall that 1 as shown above.
(c) We conclude that Using a similar argument as above, we …rst consider Expanding this equation, we get x 2 it 1 + I + II + III + IV: Consider I. With the joint limit, we have and (e 1) = o p 1 T : Consider II. In a similar vein as I, Finally, because we know 1 nT (1 + ) 2 e as (n; T ) ! 1. Next, it can be shown that and accordingly We also know that Also note that 3. j j < 1, = 1 case (a) This is the panel cointegration case. Note that Expanding this equation, we get x 2 it 1 + I + II + III + IV + V: Consider I.
Consider IV and V . It is easy to see that Finally, because we know as (n; T ) ! 1. Next note that = I + II: For I, one can see that Therefore, we have 1 Also note that Hence, we have as (n; T ) ! 1.
We …rst consider 1 p nT (1 )x it 1 e it + I + II + III + IV + V: Consider I. One can see that as (n; T ) ! 1.
(c) Finally, we conclude that In a similar process as above, we consider …rst Expanding this equation, we get Consider I. It is easy to see that For II, as (n; T ) ! 1. Next note that as shown in 2.(a). Hence, as (n; T ) ! 1.
First, we consider 1 p nT " it e it + I + II + III: Consider I.
as (n; T ) ! 1. Hence, the second term of 1 p nT b G 2 is o p (1) and we conclude that

Also recall that
as (n; T ) ! 1: (c) Finally, we have

G Proof of Theorem 4
We study the following lemmas before proving Theorem 4.

Lemma 1 (B)^
and III p I II.
Proof. Note thatû The …rst term in I is The second term in I is Hence, we have Consider II. In a similar vein as I, we get II = 1 n (T 1) Consider III. By the Cauchy-Schwarz inequality, we have as (n; T ) ! 1. This is because in the …rst term, Let us look at the second term. One can verify that Also note that T d 2 = T 2^ 1 ^ +T p ! 1 as T ! 1.
Consider II. It can be also shown that One concludes that (c) Let us calculate the term b G 1 in equation (14) …rst.
We investigate from Lemma 3 (B). As shown in Theorem 3.1.(a), one can see that Next, it can be shown that = I + II + III + IV + V + V I: Consider I and II. One can see that Consider III. Consider II. From Lemma 1 (B), we have as (n; T ) ! 1. This is because 1 ^ OLS 0 I nT J nT I n Ĉ 0 b J TĈ I nT J nT x = I + II + III + IV + V + V I: In a similar process as in II, one can verify that III = O p ( 1 n ) as (n; T ) ! 1 using the fact that if (n; T ) ! 1 and n T ! 0, then p n ^ This follows because, if (n; T ) ! 1 and n T ! 0, we get from Lemma 1. This is because 1