Test of Hypotheses in a Time Trend Panel Data Model with Serially Correlated Error Component Disturbances

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1 HE UNIVERSIY OF EXAS A SAN ANONIO, COLLEGE OF BUSINESS Working Paper SERIES Date September 25, 205 WP # 000ECO est of Hypotheses in a ime ren Panel Data Moel with Serially Correlate Error Component Disturbances Bai H. Baltagi Syracuse University bbaltagi@maxwell.syr.eu Chihwa Kao Syracuse University ckao@maxwell.syr.eu Long Liu University of exas at San Antonio Long.liu@utsa.eu Copyright 205, by the authors. Please o not quote, cite, or reprouce without permission from the authors. ONE USA CIRCLE SAN ANONIO, EXAS BUSINESS.USA.EDU

2 est of Hypotheses in a ime ren Panel Data Moel with Serially Correlate Error Component Disturbances Bai H. Baltagi, Chihwa Kao, Long Liu Abstract his paper stuies test of hypotheses for the slope parameter in a linear time tren panel ata moel with serially correlate error component isturbances. We propose a test statistic that uses a bias correcte estimator of the serial correlation parameter. he propose test statistic which is base on the corresponing fixe effects feasible generalize least squares FE-FGLS estimator of the slope parameter has the stanar normal limiting istribution which is vali whether the remainer error is I0 or I. his performs well in Monte Carlo experiments an is recommene. Keywors: Panel Data, Generalize Least Squares, ime ren Moel, Fixe Effects, First Difference, Nonstationarity. JEL Classification: C23, C33. Introuction Panel ata regression moels with large cross-sectional an time-series imensions have attracte much attention in recent years, e.g., see the surveys by Baltagi an Kao 2000, Phillips an Moon 2000, Choi 2006 an Breitung an Pesaran 2008 to mention a few. Phillips an Moon 999 provie joint asymptotic analysis of poole estimators in panel regressions with non-stationary regressors when the unerlying regression isturbances follow stationary processes. Uner the aitional conition, n/ 0, they show that sequential asymptotic results for their poole estimators woul be equivalent to the joint ones. Kao an Emerson 2004 an Baltagi, Kao an We eicate this paper in honour of Peter C.B. Phillips s many contributions to econometrics an in particular non-stationary time series analysis an panel ata. We woul like to thank an anonymous referee an the eitor om Fomby for their helpful suggestions. Aress corresponence to: Bai H. Baltagi, Department of Economics an Center for Policy Research, 426 Eggers Hall, Syracuse University, Syracuse, NY ; tel: ; fax: ; bbaltagi@maxwell.syr.eu. Chihwa Kao: Department of Economics, Center for Policy Research, 426 Eggers Hall, Syracuse University, Syracuse, NY ; tel: ; fax: ; ckao@maxwell.syr.eu. Long Liu: Department of Economics, College of Business, University of exas at San Antonio, One USA Circle, X ; tel: ; fax: ; long.liu@utsa.eu. See also chapter 2 of Baltagi 2008 for a textbook treatment of this subject.

3 Liu 2008 show that the asymptotics of the stanar panel ata estimators, like the fixe effects FE, first-ifference FD an generalize least squares GLS estimators of the slope coeffi cient epen crucially upon whether the error term is I0 or I. For example, when the error term is I0, the FE an GLS estimators are asymptotically equivalent. However, when the error term is I, this asymptotic equivalence breaks own an the GLS estimator is more effi cient than the FE estimator. his paper consiers fixe effects GLS FE-GLS base test statistics to test hypotheses regaring the slope parameter of a panel ata time tren moel where apriori knowlege as to whether the errors are I0 or I is not available. 2 We iscuss the asymptotic properties of estimators of the autoregressive parameter an the corresponing fixe effects feasible GLS FE- FGLS estimators of the slope parameter. he main contribution of this paper is to introuce a FE-FGLS base test statistic which is robust when the error term is either I0 or I. he paper is organize as follows: Section 2 presents the panel ata time tren moel with an AR error term. In Section 3, we iscuss the FE-GLS estimator for this moel an propose a test statistic that uses a bias correcte estimator of the serial correlation parameter an the corresponing FE-FGLS estimator of the regression parameter. Monte Carlo simulations are given in Section 4, while Section 5 provies the concluing remarks. All the proofs are given in the Appenix. A few wors on notation. All limits are taken sequentially as an n unless otherwise specifie. We use n, to enote the sequential limit. Convergence in probability an istribution are enote by p an, respectively. he limiting istribution of ouble inexe integrate processes has been extensively stuie by Phillips an Moon 999, he Moel Consier the following panel ata time tren moel: y it δ + βt + u it, i,..., n, t,...,, where u it µ i + ν it, an δ an β are scalars. We assume that the iniviual effects µ i are ranom with µ i ii0, σ 2 µ an ν it following an AR process which may or may not be stationary ν it ρν it + e it 2 2 he results in this paper make use of the asymptotic results for a panel ata time tren regression moel stuie by Kao an Emerson

4 with ρ, where e it is a white noise process with variance σ 2 e. he µ i s are inepenent of the ν it s for all i an t. his moel has been stuie by Baltagi an Krämer 997 an Kao an Emerson In fact, Baltagi an Krämer 997 showe the equivalence of OLS, GLS an FE estimators for the panel ata time tren moel, but without serial correlation. Baltagi an Krämer 997 also investigate the relative effi ciency of the FD estimator with respect to the other estimators of β as. Kao an Emerson 2004 extene Baltagi an Krämer to moel with serially correlate remainer errors 2. 3 hey showe that the FE estimator is asymptotically equivalent to GLS when the error term is I0 but that GLS is more effi cient than FE when the error term is I. Kao an Emerson show that the properties of the stanar panel ata estimators, like the FE, FD, an GLS estimators of β epen crucially upon the value of ρ. 4 When ν it is I0, i.e., ρ <, the FE an GLS estimators are both 3/2 consistent an asymptotically equivalent. However, when ν it is I, i.e., ρ, this asymptotic equivalence breaks own an the GLS estimator is more effi cient than the FE estimator. his has serious implications for applie research when ν it is serially correlate an it is unknown whether the remainer isturbances are I0 or I. 5 In this paper, we are intereste in testing H 0 : β β 0 without assuming knowlege of whether v it is I 0 or I. 6 Hypothesis testing on the slope of the tren has been stuie in the econometric time series literature, e.g., Canjels an Watson 3 One can exten the simple time tren moel in this paper to a polynomial tren moel by following similar steps as in section 6 of Emerson an Kao Baltagi, Kao an Liu 2008 stuy the asymptotic properties of OLS, FE, FD an GLS in the ranom effects error components regression moel with an autocorrelate regressor an an autocorrelate remainer error both of which can be stationary or nonstationary. hey show that when the error term is I0 an the regressor is I, the FE estimator is asymptotically equivalent to the GLS estimator an OLS is less effi cient than GLS ue to a slower convergence spee. However, when the error term an the regressor are I, GLS is more effi cient than the FE estimator since GLS is consistent, while FE is n consistent. his implies that GLS is the preferre estimator uner both cases i.e., regression error is either I0 or I. 5 One referee suggest testing the joint hypothesis: H 0 : β 0 an ρ 0. Alternatively, testing the joint hypothesis: H 0 : β 0 an ρ. Extening the results of this paper to these joint hypotheses is beyon the scope of this paper an shoul be subject for future research. 6 Baltagi, Kao an Na 20 also consier hypotheses testing in an I0 or I regressor case. However, the results in this paper for a time tren panel ata moel with serially correlate error component isturbances are ifferent. For example, we show that the GLS base t-statistics with iniviual fixe effects have a ifferent limiting istribution compare to that without the fixe effects. 3

5 997, Vogelsang an Fomby 2002, Bunzel an Vogelsang 2005, Roy, Falk an Fuller 2004, an Perron an Yabu he focus of this paper is on the corresponing test in panel ata. Consier the FE estimator of β, which is given by n t ˆβ F E t t y it y i n t t t 2, 3 where t t t an y i t y it. 7 If v it is known to be I 0, the t-statistic for the null hypothesis H 0 can be constructe using the FE estimator as follows: where V ar βf E ˆσ 2 v n t F E ˆβ F E β 0 V ar ˆβF, 4 E with ˆσ 2 tt t 2 v n ˆν2 it an ˆν it are the within resiuals from, i.e., ˆν it y it ȳ i. ˆβ F E t t. he next theorem erives the limiting istribution of t F E when the error term is I0 or I. heorem Assume n,,. If ρ <, t F E N 0,. 2. If ρ, t F E N 0, 3. 5 From heorem, we note that t F E in 4 will converge to a stanar normal only when v it is I0, an t F E will iverge when the error term is I. his is not surprising since ˆσ 2 v O p, i.e., σ 2 v is not ientifie when v it is I. If v it is known to be I, then the optimal test for testing the null hypothesis H 0 is base on the t-statistic using the FD estimator, ˆβ F D, which is given by ˆβ F D t where y it y it y i,t. he corresponing t-statistic is given by: y it, 5 t F D ˆβ F D β 0 V ar ˆβF, 6 D 7 Note that the FE an GLS estimators are asymptotically equivalent for this case, see Kao an Emerson

6 where V ar ˆβF D ˆσ2 e with ˆσ2 e n y it ˆβ 2. F D he next theorem erives the limiting istribution of t F D when the error term is I0 or I. heorem 2 Assume n,,. If ρ <, tf D N 0,. ρ 2. If ρ, t F D N 0,. he results of heorem 2 show that t F D N 0, when vit is in fact I, uner the null. On the other han, t F D 0 if vit is I 0, uner the null. In view of this an given that the orer of integration of v it is not known in practice, it is natural to consier alternative robust test proceures. 3 he FE-FGLS Estimator Rewrite equation in matrix form as y δι + βx + u 7 with u Z µ µ + ν, where u u,..., u, u 2,..., u 2,..., u n,..., u with the observations stacke such that the slower inex is over iniviuals an the faster inex is over time. µ is an n vector with typical element µ i, ν is an vector with typical element ν it, an Z µ I n ι, where I n is an ientity matrix of imension n, ι is a vector of ones of imension, an enotes the Kronecker prouct. y is an vector with typical element y it, x ι n x i, where ι n is a vector of ones of imension n an x i is a vector inicating a time tren with elements, 2,...,. ι is a vector of ones of imension. As shown in Baltagi an Li 99, one can write the variance-covariance matrix as Φ E uu σ 2 µ In ι ι + σ 2 e I n A, 8 5

7 where A is the variance-covariance matrix of v it, i.e., A ρ 2 ρ ρ 2 ρ ρ ρ ρ 2 ρ 2 ρ ρ ρ ρ 2 ρ 3 9 when ρ < an A when ρ. When ρ <, one can easily verify that A C C, where ρ ρ ρ 0 0 C ρ ρ 0 is the Prais-Winsten transformation matrix as in Baltagi an Li 99. As suggeste in Baltagi an Li 99, one can apply the Prais-Winsten transformation matrix C to transform the remainer AR isturbances into serially uncorrelate classical errors: y δι + βx + u, where y I n C y, x I n C x ι n x i with x i Cx i an ι I n C ι ρ ι n ι α using the fact that Cι ρ ι α, where ια α, ι an α + ρ / ρ. he transforme regression isturbances are in vector form u I n C u I n Cι µ + I n C v ρ I n ι α µ + v, 2 6

8 where v I n C v. As shown in Baltagi an Li 99, the variance-covariance matrix of the transforme isturbance is Φ E u u σ 2 µ ρ 2 I n ι α ι α + σ 2 ε I n I 3 an σ ε Φ /2 σ ε In σ J α + In E α, 4 α where E α I J α, J α ι α ια /2, 2 ι α ια α2 +, σ 2 α σ 2 e + θσ 2 µ an θ 2 ρ 2. Premultiplying the PW transforme observations in Equation by σ ε Φ /2, one gets σ ε Φ /2 y σ ε Φ /2 δι + σ ε Φ /2 βx + σ ε Φ /2 u. 5 he least squares estimator of the transforme equation yiels the GLS estimator ˆβ GLS. As shown in Baltagi, Kao an Liu 2008, ˆβ GLS has a faster converging spee than both the FE an FD estimators as n,. his is true whether v it an x it are I or I0. Baltagi, Kao an Na 20 further showe that the t-test statistic for H 0 : β β 0 base on ˆβ GLS is always N 0, as n,. A critical assumption for the GLS estimator is that E µ i x it 0. It is well known that when there is correlation between the regressors an the iniviual effects, GLS suffers from omitte variable bias, while FD an FE wipe out this source of enogeneity an remain consistent. In case of serial correlation, Baltagi, Kao an Liu 2008 suggest a FE-GLS estimator that uses the within transformation to wipe out the µ is an then runs GLS estimation to account for the serial correlation in the remainer error. Premultiplying Equation by I n E α, one gets I n E α y I n E α x β + I n E α v 6 using E α ια 0. he least squares estimator of the transforme equation gives us the FE-GLS estimator, given by ˆβ F E GLS x I n E α y x I n E α x. 7 It is worth pointing out that the FE-GLS encompasses both the within an first-ifference estimators. o see this, note that i if ρ 0, an there is no serial correlation in the remainer error, we have C I, x x, α, ι α ι an hence E α E, where E I J an J is 7

9 a matrix of /. he FE-GLS estimator in Equation 7 reuces to the within estimator x I n E y x I n E x. ii Note also that J α can be rewritten as J α l l l l, where l ρι α ρ α, ι + ρ, ρι 2,. If ρ, we have l 0 an hence J α iag, 0,, 0 an Eα I J α 0 0. Also, if ρ 0, C 0 0 I D with D, which is the well-known first ifference matrix. Hence x I n 0 x an [0 D D D. he FE-GLS estimator in Equa- D 0 I D tion 7 reuces to the first-ifference estimator x I n D Dy x I n D Dx. From Equation 7, we have where F x V ar ˆβF E GLS i x i ια x i 2 2 σ2 e nf. herefore, an F 2 n ˆβ F E GLS β F 2, 8 F n x i u i ια x i n 2 n ια u i. It is easy to show that ˆβ t F E GLS F E GLS β V ar ˆβF F 2 /F σ 2 e / nf nf2. 9 σ 2 e F E GLS Note that both the FE-GLS estimator of β an its corresponing t-statistic o not epen on σ 2 µ or σ 2 α. With a consistent estimator ˆρ, the corresponing FE-FGLS estimator ˆβ F E F GLS is obtaine by replacing C an E α by their corresponing estimators Ĉ an Êα. As suggeste by Baltagi an Li 99, an estimator of σ 2 e can be obtaine as ˆσ 2 e I û n Êα û, where û is an vector of OLS resiuals from the Prais-Winsten transforme regression using ˆρ. he corresponing t-statistic base on the FE-FGLS estimator can be obtaine from equation 9. he asymptotic properties are summarize in the following heorem: heorem 3 Assume n,,. When ρ <, if ˆρ p ρ, we have t F E F GLS N 0,. 8

10 2. When ρ, if ˆρ p κ, we have κ t F E F GLS N 0, 2 3κ + 3 κ 4 0κ κ 2 20κ κ 4 9κ κ 2. 54κ + 36 heorem 3 implies that we nee κ 0 when ρ. Otherwise t F E F GLS oes not converge to a N0,. Baltagi an Li 99 suggest estimating ρ using n ˆρ ˆν itˆν i,t, 20 t2 ˆν2 i,t n where ˆν it enotes the FE resiual from equation which is efine in Section 2. It can be obtaine from a regression of ˆν it on ˆν i,t. he asymptotics for ˆρ are given in the following theorem: heorem 4 Assume n,,. If ρ <, 2. If ρ, ˆρ ρ + + ρ N 0, ρ 2. ˆρ + 3 N 0, 5. 5 he asymptotic istribution of ˆρ in heorem 4 is actually the same as heorems 2 an 4 in Hahn an Kuersteiner 2002 that iscuss a ynamic panel ata moel. Also, the result for the case where ρ in heorem 4 is the same as heorem 2 in Kao 999 which iscusses the spurious panel ata moel. As we can see, the results from heorem 4 : When ρ <, there is a bias of + ρ / in ˆρ, but it still implies ˆρ p ρ as n,. When ρ, ˆρ oes not converge to zero in probability if there are iniviual effects in the moel. Substituting κ 3 into heorem 3, one can verify that ˆρ suggeste by Baltagi an Li 99 leas to t F E F GLS N 0, when ρ. his limits the usefulness of the FE-GLS estimator when the error term is I an there are iniviual effects in the panel moel. his ifference is ue to the fact that µ i can not be consistently estimate when the error term is I, see Kao an Emerson o achieve t F E F GLS N 0,, we nee κ 0. herefore, when ρ <, a bias-correcte estimator of ρ is ˆρ+ +ˆρ. When ρ, a bias-correcte estimator of ρ is ˆρ + 3. Combining the two cases, we suggest a bias-correcte estimator of ρ as follows: ρ ˆρ + +ˆρ if ˆρ > 3 if ˆρ 3 9.

11 he asymptotics for ρ are given in the following theorem: heorem 5 Assume n,,. If ρ <, ρ ρ N 0, ρ If ρ, ρ N 0, 5. 5 herefore, we have t F E F GLS N 0, using ρ for both ρ < an ρ. In this section, we showe that the t-statistic base on FE-GLS is no longer robust if there are iniviual effects. Extra steps nee to be taken to achieve the robustness when equation inclues iniviual effects. 3. he Moel Without Iniviual Effects Let us stuy the case where there are no iniviual effects, i.e., µ i 0 for all i. he variancecovariance matrix in equation 8 reuces to Φ σ 2 e I n A an hence Φ σ 2 e I n A. Equation reuces to y δι + βx + v, 2 where the variance-covariance matrix of the transforme isturbance is E v v σ 2 ε I n I. he least squares estimator of the transforme equation yiels the GLS estimator: where M ι ˆβ GLS x M ι y x M ι x, 22 I ι ι ι ι I J n J α using the fact that ι ρ ι n ι α. It is easy to see that M ι x I J n J α ιn x i ι n x i J α x i In E α x. his proves that ˆβ GLS an ˆβ F E GLS are the same if there are no iniviual effects in the moel. he t-statistic base on ˆβ GLS is in turn the same as the one base on ˆβ F E GLS in equation 9, i.e., t GLS nf2 σ 2 e F, 23 0

12 where F an F 2 are efine in Equation 8. Similar to equation 20 for the general moel with iniviual effects, let n ˆρ ûitû i,t, 24 n û2 i,t where û it is the OLS resiual, i.e., û it y it y ˆβ OLS t t with y n t y it an t t t. Define û as an vector of OLS resiuals from the Prais-Winsten transforme regression using ˆρ. An estimator of σ 2 e is ˆσ 2 e û û. Substituting ˆρ an ˆσ 2 e, the t-statistic corresponing to the FGLS estimator can be obtaine from equation 23. he asymptotic properties are summarize in the following heorem: heorem 6 Assume n,,. If ρ <, ˆρ ρ N 0, ρ 2, 2. If ρ, ˆρ p N 0, 3. herefore, we have t F GLS N 0, using ˆρ for both ρ < an ρ. heorem 6 shows that t F GLS converges to N0,, whether the error term is I0 or I, when there are no iniviual effects in the moel. his is an interesting result, i.e., the t-ratio base on FGLS effectively briges the gap between the I0 an I error terms if there are no iniviual effects in the moel. his implies that inference on the slope parameter can be performe using the stanar normal istribution if there are no iniviual effects. his is ifferent from the pure time series moel as in Perron an Yabu 2009 which requires a super-effi cient estimate in orer to achieve this goal. We know that this will change if there are iniviual effects in the moel. his is the more likely case in panel ata with heterogeneity across iniviuals. 4 Monte Carlo Results his section reports the results of Monte Carlo experiments esigne to investigate the finite sample properties of the FE-FGLS base t F E F GLS. he moel is generate by y it α + βt + µ i + v it, i,..., n, t,...,, 25

13 with α 5, β 0, µ i ii N 0, 5, an ν it ρν it + e it, with ρ varying over the range 0, 0.2, 0.4, 0.6, 0.8, 0.9,, v i0 0, e it ii N 0, σ 2 e, an σ 2 e 5. he sample sizes n, are 500,20, 500,50, 50,500, 0,00, 0,50, 50,0 an 20,0, respectively. For each experiment, we perform, 000 replications. For each replication we estimate the moel using: i FD: first-ifference ignoring serial correlation; ii FE: fixe-effects ignoring serial correlation; iii FE- GLS: FE-GLS estimator using the true value of ρ ; iv FE-FGLS : FE-FGLS estimator using ρ calculate by the metho suggeste in Baltagi an Li 99; an v FE-FGLS 2 : FE-FGLS estimator using a bias-correcte estimator ρ. able reports the meian, interquantile range an root mean square error RMSE of estimators of ˆρ an ρ. Following Kelejian an Prucha 999, [ RMSE is efine as bias 2 + IQR/.35 2 /2, where bias is the ifference between the meian an the true parameter value an IQR is the interquantile range. hat is IQR c c 2, where c an c 2 are the 0.75 an 0.25 quantiles respectively. As explaine in Kelejian an Prucha 999, these characteristics are closely relate to the stanar measures of bias an root mean square error RMSE but, unlike these measures, are assure to exist. When the true ρ is larger than 0.4, ρ has smaller RMSE than ˆρ. his is especially true when ρ is close to. For the first two n, combinations, ables 2 an 3 report the size an power of the t-test for H 0 : β 0 corresponing to each estimator of β. ables 4 an 5 report the size-ajuste power. Several conclusions emerge from these results. For the FD estimator, if ρ, the size of the corresponing t-test is when n 500 an 20 an when n 500 an 50. However, the size is always zero for other values of ρ. For the FE estimator, if ρ 0, the size of the corresponing t-test is when n 500 an 20 an when n 500 an 50. he size increases with ρ. his verifies the asymptotic results in heorem. he stanar eviation increases with ρ an ecreases with. his is consistent with the asymptotic results in heorem 2. For the FE-FGLS estimators, the size of the corresponing t-test is too large for FE-FGLS, especially if ρ > 0.4. However, the t-test corresponing to FE-FGLS 2 has reasonable size an power, compare to FE-FGLS. Our simulation results confirm the robustness of t F E F GLS2 using the bias-correcte estimator ρ. For the other n, combinations, ables 6 reports the size of the t-test for H 0 : β 0 corresponing to each estimator. We can see that the results are robust to small samples an ifferent ratios of n/. 2

14 5 Conclusion In this paper, we iscuss test of hypotheses in a linear time tren panel ata moel with serially correlate error component isturbances. he error term coul be either stationary or nonstationary. We consier estimation an testing using the FE, FD an FE-GLS estimators. Different from the results in the pure time series case, the t-test base on FGLS always converges to N0, no matter whether the error term is I0 or I, when there are no iniviual effects in the moel. When there are iniviual effects in the moel, the t-statistic base on FE-GLS is no longer robust. We suggest a bias-correcte estimator of ρ to achieve robustness. We show that it performs well in Monte Carlo experiments an is recommene. While the focus of this paper is test of hypothesis in a simple linear tren panel ata moel with error components an serial correlation, it is important to exten this work to ynamic panel ata moels with cross-section epenence across the units. his shoul be the focus of future research. References [ Baltagi, B. H. 2008, Econometric Analysis of Panel Data, Wiley. [2 Baltagi, B.H. an Kao, C. 2000, Nonstationary Panels, Cointegration in Panels an Dynamic Panels: A survey, Avances in Econometrics 5, 7 5. [3 Baltagi, B. H., Kao, C., an Liu, L. 2008, Asymptotic Properties of Estimators for the Linear Panel Regression Moel with Ranom Iniviual Effects an Serially Correlate Errors: he Case of Stationary an Non-Stationary Regressors an Resiuals, Econometrics Journal,, [4 Baltagi, B. H., Kao, C., an Na, S. 20, est of Hypotheses in Panel Data Moels When the Regressor an Disturbances are Possibly Nonstationary, Avances in Statistical Analysis, 95, [5 Baltagi, B.H., an W. Krämer. 997, A Simple Linear ren Moel with Error Components, Problem 97.2., Econometric heory 3, 463. [6 Baltagi, B. H., an Li, Q. 99, A ransformation hat Will Circumvent the Problem of Autocorrelation in an Error Component Moel, Journal of Econometrics, 52,

15 [7 Breitung J., an Pesaran M.H. 2008, Unit Roots an Cointegration in panels, Chapter 9 in L. Matyas an P. Sevestre es. he Econometrics of Panel Data: Funamentals an Recent Developments in heory an Practice, Springer, Berlin, [8 Bunzel, H., an Vogelsang,. J. 2005, Powerful ren Function ests hat are Robust to Strong Serial Correlation with an Application to the Prebish Singer Hypothesis, Journal of Business an Economic Statistics, 23, [9 Canjels, E., an Watson, M. W. 997, Estimating Deterministic rens in the Presence of Serially Correlate Errors, Review of Economics an Statistics, 79, [0 Choi, I. 2006, Nonstationary panels, Chapter 3 in.c. Mills an K. Patterson es., Palgrave Hanbooks of Econometrics, Volume, pp , Palgrave, Macmillan. [ Emerson, J. an Kao, C. 2000, esting for Structural Change of a ime ren Regression in Panel Data, Center for Policy Research Working Papers 5, Center for Policy Research, Maxwell School, Syracuse University. [2 Hahn, J., an Kuersteiner, G. 2002, Asymptotically Unbiase Inference for a Dynamic Panel Moel with Fixe Effects when Both n an are Large, Econometrica, 704, [3 Kao, C. 999, Spurious Regression an Resiual-Base ests for Cointegration in Panel Data, Journal of Econometrics, 90, -44. [4 Kao, C., an Emerson, J. 2004, On the Estimation of a Linear ime ren Regression with a One-Way Error Component Moel in the Presence of Serially Correlate Errors: Part I, Journal of Probability an Statistical Science, 2, [5 Kelejian, H. H., an Prucha, I. R. 999, A Generalize Moments Estimator for the Autoregressive Parameter in Spatial Moel, International Economic Review, 40, [6 Phillips, P.C.B., an Moon, M. 999, Linear Regression Limit heory for Nonstationary Panel Data, Econometrica 67, 057. [7 Phillips, P.C.B., an Moon, M. 2000, Nonstationary Panel Data Analysis: An overview of Some Recent Developments, Econometric Reviews 9,

16 [8 Perron, P., an Yabu,. 2009, Estimating Deterministic rens with an Integrate or Stationary Noise Component, Journal of Econometrics, 5, [9 Roy, A., Falk, B., an Fuller, W. A. 2004, esting for ren in the Presence of Autoregressive Error, Journal of the American Statistical Association, 99, [20 Vogelsang,. J. 998, ren Function Hypothesis esting in the Presence of Serial Correlation, Econometrica, 66, [2 Vogelsang,. J., an Fomby,. B. 2002, he Application of Size Robust ren Analysis to Global Warming emperature Series, Journal of Climate, 5,

17 able : Meian, IQR an RMSE of Estimators of ρ ˆρ ρ n ρ Meian IQR RMSE Meian IQR RMSE

18 able 2: Size an Power of the t-test for H 0 : β 0 n 500, 20 ρ β FD FE FE-GLS FE-FGLS FE-FGLS

19 able 3: Size an Power of the t-test for H 0 : β 0 n 500, 50 ρ β FD FE FE-GLS FE-FGLS FE-FGLS Notes: Cases with power of.000 have been omitte after their first occurrence. 8

20 able 4: Size-ajuste Power of the t-test for H 0 : β 0 n 500, 20 ρ β FD FE FE-GLS FE-FGLS FE-FGLS Notes: Cases with power of.000 have been omitte after their first occurrence. 9

21 able 5: Size-ajuste Power of the t-test for H 0 : β 0 n 500, 50 ρ β FD FE FE-GLS FE-FGLS FE-FGLS Notes: Cases with power of.000 have been omitte after their first occurrence. 20

22 able 6: Size of the t-test for H 0 : β 0 n ρ FD FE FE-GLS FE-FGLS FE-FGLS

23 Appenix A Proof of heorem Proof. he proof of is a textbook result an hence omitte here. Consier 2. Recall ˆσ 2 v n ˆν2 it, where ˆν it y it ȳ i. ˆβ F E t t v it v i. ˆβF E β t t. Hence ˆσ2 v Notice that 2 2 n ˆν 2 it 2 [ ˆβF E β by equation C3 in Kao 999, { 5/2 2 { v it v i. 2 + [ 2 ˆβF 2 E β } v it v i. t t. v it v i. 2 p σ 2 e 6 ˆβF E β N 0, 6 5 σ2 e } t t 2 by heorem 4 in Kao an Emerson 2004, an 5/2 v it v i. t t 3N by an equation on page 23 in Kao an Emerson , σ2 e 20 Hence we have an therefore 2 V ar ˆβF E t F E ˆσ2 p v σ2 e 6, ˆσ2 v 3 n t t t 2 p σ 2 e/6 /2 2σ2 e ˆβF E β N 0, 6 5 σ2 e N 2σ 2 2 V ar βf e E 0,

24 B Proof of heorem 2 Proof. Consier. Recall y it β + ν it an hence y it ˆβ F D ν it ˆβF D β. We have ˆσ 2 e Notice that an y it ˆβ F D 2 [ 2 ν it ˆβF D β ν it 2 + [ 2 2 [ ˆβF 2 D β ˆβF D β ν it 2 ρ 2 ν 2 i,t + p ρ 2 σ 2 e ρ 2 + σ2 e + 0 2σ2 e + ρ, by heorem 3 in Kao an Emerson an Hence we have tf D p ν it 0, 2σ ˆβF 2 D β N e 0, ρ 2 V ar ˆβF D ˆσ 2 p e 2σ2 e + ρ, ˆβF D β 0 V ar ˆβF D 2 N 0, ρ 2 σ 2 e 2σ 2 e +ρ e 2 it + 2 ρ N 0,. ρ Part 2 can be shown easily following Kao an Emerson 2004 an hence omitte. ν it. ν i,t e it 23

25 Lemma x i x i ˆρ 2 ˆι α x i ˆρ t 2 + 2ˆρ ˆρ t2 t + ˆρ 2 + ˆρ 2, t2 t + ˆρ + + ˆρ, t2 x i v i ˆρ 2 ν i + ˆρ ρ ˆρ ˆι α v i + ˆρ v i + ρ ˆρ tν i,t + ˆρ ρ ˆρ t2 ν i,t + t2 v i v i ˆρ 2 ν 2 i + ρ ˆρ 2 e it, t2 ν 2 i,t + 2 ρ ˆρ t2 ν i,t + ˆρ t2 ν i,t e it + t2 t2 e 2 it. te it + ˆρ t2 e it, Proof. Note that x i Cx i ˆρ 2, 2 ˆρ, 3 2ˆρ,, ˆρ an ˆι α ˆα,,,, with ˆα + ˆρ / ˆρ. We have x i x i ˆρ 2 + [t ˆρ t 2 ˆρ 2 + t2 ˆρ 2 t 2 + 2ˆρ ˆρ t2 ˆι α x i + ˆρ + ˆρ [ ˆρ t + ˆρ 2 t2 t + ˆρ 2 + ˆρ 2, t2 t ˆρ t + ˆρ + t2 t + ˆρ + + ˆρ, t2 [ ˆρ t + ˆρ t2 t2 x i v i ˆρ 2 v i + ˆρ 2 ν i + t ˆρ t v it ˆρv i,t t2 [ ˆρ t + ˆρ [ρ ˆρ ν i,t + e it t2 ˆρ 2 ν i + ˆρ ρ ˆρ ˆι α v i + ˆρ v i + tν i,t + ˆρ ρ ˆρ t2 ν i,t + ˆρ t2 v it ˆρv i,t + ˆρ v i + t2 + ˆρ v i + ρ ˆρ ν i,t + t2 24 e it, t2 te it + ˆρ t2 [ρ ˆρ ν i,t + e it t2 e it, t2

26 an v i v i ˆρ 2 v 2 i + v it ˆρv i,t 2 ˆρ 2 ν 2 i + t2 ˆρ 2 ν 2 i + ρ ˆρ 2 ν 2 i,t + 2 ρ ˆρ t2 [ρ ˆρ ν i,t + e it 2 t2 ν i,t e it + t2 t2 e 2 it. C Proof of heorem 3 Proof. Consier part. For a fixe n, from Lemma, we have 3 x i x i ˆρ 2 3 t ˆρ ˆρ 2 t t t + 2 ˆρ 2 + ˆρ 2 p ρ 2 3, 3 x 3/2 i vi an 2ˆια x i ˆρ 2 t + t ˆρ 2 3/2 ν i + ˆρ ρ ˆρ 3/2 tν i,t t + ˆρ 3/2 te it + ˆρ e it t ˆι α v i + ˆρ ν i + ρ ˆρ v i vi ˆρ 2 ν2 i + ρ ˆρ 2 t t ˆρ ˆρ p ρ 2, ν i,t ν 2 i,t + ρ ˆρ 2 t2 ˆ ˆρ ˆρ p + ˆρ ρ ˆρ ρ σ e + t ν i,t e it + t2 rw i, e it σe W i, t2 e 2 it t σ 2 e, as, using 3 t t2 p 3, 2 t t p 2, 3/2 t tν i,t σ v rwi, t ν i,t σ v W i, 3/2 t te it n, we have σ e rwi, t e it ν i,t σ e W i an ˆρ p ρ. herefore, for a fixe ˆF 3 3 x i x i Ṱ 2 2 2ˆια x p ρ 2 i 3 ρ 2 2 ρ2, 2 25

27 an n 3/2 ˆF 2 3/2 n α x i vi 2ˆι x i ˆ 2 ρ [ ρ σ e rw i 2 [ rw i 2 W i. ρ σ e n ˆι α v i n [σ e W i Since rw i 2 W i N 0, 2, we have n ˆF 3/2 2 N 0, ρ2 σ 2 e 2 as n,. It is easy to show that ˆσ 2 e is consistent for σ 2 e. Finally, we have n ˆF t F E F GLS 3/2 2 N ˆσ 2 e ˆF 3 as n,. 0, ρ2 σ 2 e 2 ρ 2 σ 2 e 2 N 0, Consier part 2. For a fixe n, from Lemma, we have x i x i [ ˆρ 2 3 t 2 + 2ˆρ [ ˆρ 2 t t2 t2 +ˆρ 2 + [ ˆρ 2 p 2 3 κ2 κ +, ˆια x i [ ˆρ 2 t + ˆρ + + ˆρ p 2 κ +, x i vi +ˆρ [ ˆρ [ σ e κ 2 3/2 t [ ˆρ + ˆρ ν i + [ ˆρ 2 3/2 t rw i κ ν i,t + [ ˆρ W i κ rw i + W i, 3/2 t 5/2 t te it + ˆρ tν i,t t e it ˆι α vi + ˆρ ν i +[ ˆρ 3/2 ν i,t t + t e it σe [ κ W i + W i, v i vi + ˆρ [ ˆρ ν2 i + 3/2 [ ˆρ2 3/2 + [ ˆρ 2 ν i,t e it + t2 26 t2 e 2 it σ 2 e, t2 ν 2 i,t

28 an ˆ 2 + ρ ρ + ρ 2 ρ + ρ p κ 2 κ as, using 3 t t2 p 3, 2 t t p 2, 5/2 t tν i,t rw i, 3/2 t ν i,t Wi, 3/2 t te it rw i, t e it W i an ˆρ p κ. herefore, for a fixe n, we have ˆF x an n ˆF 2 i x i [ ˆρ 2 ˆια x i ˆρ ˆ 2 n p 3 κ2 κ + κ 2 κ κ x i vi [ ˆρ ˆια x i ˆρ ˆ ˆι α 2 vi [ σ e κ 2 rw i κ W i κ rw i + W i κ 2 κ + 2 κ n σ e [κ 2 rw i κ n σ e [ κ W i κ W i + W i rw i + W i + κ 2 as. Since rw i W i W i, W i W i, rw i 2 2 r 2 W i, we know that κ 2 rw i κ W i κ rw i + W i + κ κ 2 κ 2 κ W i + W i rw i + κ W i + κ 2 [ 2 κ2 r κ2 r + 2 κ W i, κ3 + 8κ 2 24κ + 24, 2 2 κ κ Wi r 2 2 W i + W i W i + W i [ Wi r 2 W i which follows a normal istribution with zero mean an variance [ 2 κ2 r κ2 r + 2 κ 2 r 20 κ4 2 κ κ2 κ +. Hence n ˆF 2 σe N N 0, 0, 20 κ4 2 κ κ2 κ + as n,. It is easy to show that ˆσ 2 e is consistent for σ 2 e. Finally, we have n ˆF 2 t F E F GLS σ 2 e ˆF σ en 0, 20 κ4 2 κ κ2 κ + σ κ 4 9κ 3 +33κ 2 54κ+36 e 2κ 2 3κ+3 κ 2 3κ + 3 κ 4 0κ κ 2 20κ κ 4 9κ κ 2 54κ

29 as n,. Lemma 2 an ˆν 2 i,t [ 2 v i,t v i. 2 + ˆβF E β 2 ˆβF E β ˆν it ρˆν i,t ˆν i,t e it v i,t v i. + ρ ρ ˆβF E β [ ρ [ ˆβF E β When ρ, it reuces to [ 2 + ˆβF E β n ρ ˆν itˆν i,t v i,t v i. t t v i. v i v i. v i,t v i. t t 2ρ e it t t + 2 ρ v i. t t n. t2 t t 2 v i v i. [ e it v i,t v i. ˆβF E β v i v i. [ ˆβF E β e it t t + Proof. Note that ˆν it y it ȳ i. ˆβ F E t t v it v i. ˆβF E β n 2 2 ˆβF E β. t t an hence Also, we have ˆν 2 i,t 2 v i,t v i. 2 + ˆβF E β 2 ˆβF E β v i,t v i. t t. t t 2 ˆν it ρˆν i,t [ v it v i. ˆβF E β e it ρ v i. ˆβF E β t t [ ρ v i,t v i. ˆβF E β t t [ ρ t t + ρ 28

30 an hence ˆν it ρˆν i,t ˆν i,t [ [ e it ρ v i. ˆβF E β ρ t t + ρ v i,t v i. ˆβF E β t t e it ρ v i. v i,t v i. ˆβF E β ˆβF E β I + II + III + IV. 2 e it ρ v i. t t + ˆβF E β v i,t v i. ρ t t + ρ ρ t t + ρ t t It is easy to see that I using t2 v i,t v i. ρ ρ For term III, we have e it ρ v i. v i,t v i. e it v i,t v i. ρ e it v i,t v i. + ρ v i. v i,t v i. v i. v i v i., [ t v it v i. v i v i. v i v i.. For term II, we have v i,t v i. ρ t t + ρ v i,t v i. t t + 2ρ v i,t v i. t t 2ρ v i,t v i. v i v i.. e it ρ v i. t t e it t t ρ v i. t t e it t t + 2 ρ v i., 29

31 since [ t2 t t t t t herefore, n t t 2. For term IV, we have ρ t t + ρ t t ρ t t + t t t2 n ρ n ρ t t 2 + n t2 t2 t t t2 t t 2 + n. 2 [ ˆν it ρˆν i,t ˆν i,t e it v i,t v i. + ρ ρ ˆβF E β [ ρ [ ˆβF E β [ 2 + ˆβF E β n ρ v i. v i v i. v i,t v i. t t 2ρ e it t t + 2 ρ v i. t t n. t2 v i v i. D Proof of heorem 4 Proof. Note that n ˆρ ρ ˆν it ρˆν i,t ˆν i,t n. ˆν2 i,t Consier. First, from Lemma 2, we have ˆν 2 i,t 2 v i,t v i. 2 + [ 3/2 ˆβF E β [ [ 3/2 ˆβF 2 E β 3 [ 3/2 t t 2 t2 v i,t v i. t t. 30

32 Notice that v i,t v i. 2 p σ 2 e ρ 2 by Lemma 3. in Baltagi, Kao an Liu 2008, 3/2 ˆβF 2σ 2 E β N e 0, ρ 2 by heorem 3 in Kao an Emerson 2004 an 3/2 v i,t v i. t t N by an equation on page 7 in Kao an Emerson , σ 2 e 2 ρ 2 Hence we have Also, we have ˆν 2 i,t p σ2 e ρ 2. ˆν it ρˆν i,t ˆν i,t e it v i,t v i. + ρ v i. v i v i. ρ [ 3/2 ˆβF [ ρ E β 3/2 [ 3/2 ˆβF [ E β 3/2 [ + [ 3/2 ˆβF 2 E β ρ I + II + III + IV. Consier I: Notice that e it v i,t v i. + n [ 3 t2 σ 2 e ρ σ 2 en v i,t v i. t t 2ρ 3/2 e it t t + ρ 2 t t , e it v i,t ρ 2, n n v i v i. n v i. n e it v i. σ2 e ρ using e it v i,t σ 2 e N 0, ρ 2 3

33 an n n e it v i. σ2 e ρ N by the Lineberg-Feller central limit theorem an E e it v i. E v it e is t2 ogether with the fact that we have t s2 v i. v i v i. I + n σ 2 e ρ t s2 σ 2 en 0, 0, lim n, V ar t E v it e is v i. v i ρ 2. Consier II: By heorem 3 in Kao an Emerson 2000, we have Also it is easy to show that an 3/2 Hence the term II o p. 3/2 ˆβF 2σ 2 E β N e 0, ρ 2. v i,t v i. t t N n 0, v i v i. O p. e it v i. t2 t ρ j σ 2 e t j0 v 2 i. p 0, σ 2 e 2 ρ 2, Consier III: By an equation on page 7 in Kao an Emerson 2004, we have Also it is easy to show that Hence the term III o p. 3/2 Consier IV : It is easy to show that 3 e it t t N n v i. O p. t t 2 p 2, 0, σ2 e, 2 σ2 e ρ. 32

34 an hence the term IV o p. Hence we have herefore, ˆν it ρˆν i,t ˆν i,t + n σ 2 e ρ σ 2 en 0, ρ 2. ˆρ ρ + + ρ as n,. n t2 ˆν it ρˆν i,t ˆν i,t + n n ˆν2 i,t σ 2 en 0, ρ 2 + o σ 2 p N 0, ρ 2 e ρ 2 σ 2 e ρ + o p Next we show part 2. First, from Lemma 2, we have n [ ˆν 2 i,t v i,t v i. 2 2 [ ˆβF n E β ˆβF E β 2 3 t t 2. t2 5/2 [t t v it v i. Notice that ˆβF E β N 0, 6 5 σ2 e by heorem 4 in Kao an Emerson 2004, 5/2 [t t v i,t v i. 5/2 by an equation on page 23 in Kao an Emerson 2004, an Hence we have 2 3 ˆν 2 i,t 2 t t 2 2. [t t v i,t v i,t v i. 2 + o p. N 0, σ2 e O p 20 33

35 Next, from Lemma 2, we have ˆν itˆν i,t e it v i,t v i. [ [ ˆβF E β n [ [ ˆβF n E β 3/2 Notice that by equation C5 in Kao 999, n v i v i. n 3/2 e it t t e it v i,t v i. p σ2 e 2 vi n e it t t N by an equation on page 7 in Kao an Emerson 2004, an ˆβF E β N 0, 6 5 σ2 e by heorem 4 in Kao an Emerson Hence we have herefore, ˆν itˆν i,t ˆρ n t2 ˆν itˆν i,t 2 n t2 ˆν2 i,t + 2 v i v i. 3/2 v it 0, σ2 e 2 t2 e it v i,t v i. + o p. [ 2 ˆβF E β. p 0, n e it v i,t v i. t2 v i,t v i. 2 + o p 2 n as n,. As shown in equation C5 in Kao 999, e it v i,t v i. ζ 3i, t2 with E ζ 3i σ2 e 2 an V ar ζ 3i σ4 e 2. Also, as shown in equation C2 in Kao 999, 2 v i,t v i. 2 ζ4i, t2 34

36 with E ζ 4i σ2 e 6 an V ar ζ 4i σ4 e 45. By equation C0 in Kao 999, we know σ2 e σ ˆρ n 2 4 σ 2 2 e e 2 2 σ 4 N σ 0, 2 e σ e e σ 2 4. e It simplifies to ˆρ 3 N 0, 5 5 as n,. E Proof of heorem 5 Proof. Let S { ˆρ > 3} an S { ˆρ 3}. Consier. When ρ <, it suffi ces to show that ρ ρ ˆρ ρ + +ˆρ ρ ˆρ +ˆρ p 0. We have lim Pr ρ ˆρ + ˆρ > ɛ n, lim Pr ρ ˆρ + ˆρ n, + lim n, Pr ρ ˆρ + ˆρ > ɛ S Pr S > ɛ S Pr S. he first term is zero given that, if S is true, we have ρ ˆρ+ +ˆρ so that Pr ρ ρ +ˆρ > ɛ S 0. he secon term is zero since ˆρ ρ + +ρ O p implies ˆρ 3 ρ+ n [ ˆρ ρ + + ρ 3 + ρ ρ+o p +o p n an hence Pr S 0 as. herefore, Pr ρ ˆρ +ˆρ > ɛ 0 as n,. Consier 2. When ρ <, we have lim Pr ρ > ɛ lim Pr ρ > ɛ S Pr S n, n, + lim n, Pr ρ > ɛ S Pr S. Now the fact that ˆρ + 3 Op implies Pr S 0 as n,, so that the first term is zero. For the secon term, if S is true, ρ so that Pr ρ > ɛ S 0. hus, Pr ρ > ɛ 0 as n,. 35

37 Lemma 3 an û 2 i,t û it ρû i,t û i,t When ρ, it reuces to û it û i,t where û it û it û i,t. vi,t v ˆβOLS β 2 ˆβOLS β t t v i,t v [ eit ρ v v i,t v + ˆβOLS β + ˆβOLS β 2 + ˆβOLS β t t 2 [ t t e it ρ v [ ρ t t + ρ v i,t v [ eit vi,t v + ˆβOLS β + ˆβOLS β [t t ρ t t + ρ. t t e it vi,t v 2 + ˆβOLS β t t, Proof. Since û it y it y ˆβ OLS t t v it v ˆβOLS β t t, where y n t y it an v n t v it an hence û it ρû i,t [ vit v ˆβOLS β e it ρ v ˆβOLS β e it ρ v ˆβOLS β t t Results in Lemma 3 can be easily obtaine. [ vi,t ρ v ˆβOLS β t t [t ρ t ρ t [ ρ t t + ρ. Lemma 4 We have 3/2 ˆβOLS 2σ 2 β N e 0, ρ 2 36

38 when ρ < an ˆβOLS β N 0, 6 5 σ2 e when ρ. Proof. Using u it µ i + v it, we have n t ˆβ OLS β t t u n it t n t t t 2 t t v it n t t t 2 since n t t t µ i n µ i t t t 0. By heorems 3 an 4 in Kao an Emerson 2004, we have the asymptotic property of ˆβ OLS in Lemma 4. F Proof of heorem 6 Proof. Consier. Note that n ˆρ ρ û it ρû i,t û i,t n. û2 i,t From Lemma 3, we have Notice that û 2 i,t 2 vi,t v 2 + [ 3/2 ˆβOLS β from Lemma 3. of Baltagi, Kao an Liu 2008, 3/2 an by Lemma 4. 3 t t v it v [ [ 3/2 ˆβOLS 2 β 3 [ 3/2 vi,t v 2 p σ 2 e ρ 2 3/2 t t 2 p 2, 3/2 ˆβOLS 2σ 2 β N e 0, ρ 2 37 t t v it v. t t 2 t t v it + o p σ2 e ρ 2 N 0,, 2

39 Hence we have as n,. Also, from Lemma 3, we have Notice that 3/2 an 3 û it ρû i,t û i,t û 2 i,t p [ eit ρ v v i,t v + [ 3/2 ˆβOLS β + [ 3/2 ˆβOLS β 3/2 3/2 + [ 3/2 ˆβOLS 2 β 3 [ eit ρ v v i,t v + 2 ρ v σ 2 en 0, by Lemma 4. Hence we have e it v i,t v [ ρ 2 + o, [ t t e it ρ v 3/2 [t t ρ t t + ρ ρ σ2 e ρ 2. [ t t e it ρ v [ ρ t t + ρ v i,t v [t t ρ t t + ρ. e it + ρ 3 v i,t [t t e it +o p σ 2 en 3/2 ˆβOLS 2σ 2 β N e 0, ρ 2 û it ρû i,t û i,t σ 2 e N 38 0, t t 2 + o p p ρ 2, ρ 2 0,, 2

40 as n,. herefore, ˆρ ρ n t2 û it ρû i,t û i,t as n,. n t2 û2 i,t σ 2 en 0, ρ 2 N σ 0, ρ 2 2 e ρ 2 Consier 2. Note that n ˆρ û itû i,t. n û2 i,t From Lemma 3, we have 2 Notice that û 2 i,t 2 2 n by equation C3 in Kao 999, vi,t v [ 2 [ 2 + ˆβOLS β n [ ˆβOLS β 2 [ 5/2 vi,t v 2 p σ 2 e 2 ˆβOLS β N 0, 6 5 σ2 e 3 t t v i,t v. t t 2 by Lemma 4, 5/2 t t v i,t v 5/2 by an equation on page 23 in Kao an Emerson 2004 an 3 t t 2 2. [t t v i,t + o p N 0, σ2 e 20 2 Hence we have û 2 i,t 2 2 n vi,t v [ 2 [ 2 + ˆβOLS β n [ ˆβOLS β [ 5/2 3 t t 2 t t v i,t v p σ 2 e 6 39

41 as n,. Also, from Lemma 3, û it û i,t [ eit vi,t v + [ [ ˆβOLS β n 3/2 + [ [ ˆβOLS 3/2 β 3/2 [ + [ 2 ˆβOLS β vi,t v t t. t t e it Notice that [ eit vi,t v N 0, σ2 e 2 e it v it n, e i,t 3/2 v it an 3/2 3/2 t t e it N 0, σ2 e, 2 vi,t v N 0, σ2 e, 3 ˆβOLS β N 0, 6 5 σ2 e by Lemma 4 an Hence we have as n,. herefore, as n,. t t p 2. ˆρ n t2 û itû i,t 2 n t2 û2 i,t û it û i,t N 0, σ2 e 2 σ 2 e/6 N 0, σ2 e N 0,