DYNAMIC PANEL GMM WITH NEAR UNITY. Peter C. B. Phillips. December 2014 COWLES FOUNDATION DISCUSSION PAPER NO. 1962

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1 DYNAMIC PANEL GMM WIH NEAR UNIY By Peter C. B. Phillips December 04 COWLES FOUNDAION DISCUSSION PAPER NO. 96 COWLES FOUNDAION FOR RESEARCH IN ECONOMICS YALE UNIVERSIY Box 088 New Haven, Connecticut

2 Dynamic Panel GMM with Roots Near Unity Peter C. B. Phillips Yale University, University of Auckland, Singapore Management University & University of Southampton August 5, 04 Abstract Limit theory is developed for the dynamic panel GMM estimator in the presence of an autoregressive root near unity. In the unit root case, Anderson-Hsiao lagged variable instruments satisfy orthogonality conditions but are well-known to be irrelevant. For a fixed time series sample size GMM is inconsistent and approaches a shifted Cauchydistributed random variate as the cross section sample size n. But when, either for fixed n or as n, GMM is consistent and its limit distribution is a ratio of random variables that converges to twice a standard Cauchy as n. In this case, the usual instruments are uncorrelated with the regressor but irrelevance does not prevent consistent estimation. he same Cauchy limit theory holds sequentially and jointly as n, with no restriction on the divergence rates of n and. When the common autoregressive root ρ = + c/ the panel comprises a collection of mildly integrated time series. In this case, the GMM estimator is n consistent for fixed and n consistent with limit distribution N 0, 4 when n, sequentially or jointly. hese results are robust for common roots of the form ρ = + c/ γ for all γ 0, and joint convergence holds. Limit normality holds but the variance changes when γ =. When γ > joint convergence fails and sequential limits differ with different rates of convergence. hese findings reveal the fragility of conventional Gaussian GMM asymptotics to persistence in dynamic panel regressions. Keywords: Cauchy limit theory, Dynamic panel, GMM estimation, Instrumental variable, Irrelevant instruments, Panel unit roots, Persistence. JEL classification: C30, C360 his paper originated in a Yale take home examination Phillips, 03. Some of the results were presented at a Conference in honor of Richard J. Smith s 65th birthday at Cambridge University in May, 04. he author acknowledges support from the NSF under Grant No. SES

3 Introduction he use of instrumental variables IV in dynamic panel estimation was suggested by Anderson and Hsiao 98, 98 and has led to a substantial theoretical and applied literature on the use of IV and generalized method of moment GMM estimation techniques in dynamic panels. he unit root case is well-known to present diffi culties for IV/GMM methods because lagged variable instruments satisfy the required orthogonality conditions but fail the relevance condition. he problem was discussed in Blundell and Bond 998 and Moon and Phillips 004. It is easy to dismiss the unit root case as unidentified by IV/GMM formulations involving lagged level instruments. As a result there are few analyses of GMM asymptotics in this apparently unidentified case. An important exception is Kruiniger 009 who considered dynamic panel estimation with persistent data when the cross section sample size n and the time series sample size is fixed, showing inconsistency of the GMM estimator of the autoregressive parameter. he existence of other techniques that do deliver consistent estimation in the unit root case has partly diverted attention from the GMM approach, although these alternative methods also present diffi culties such as bias and bias discontinuities in the case of level maximum likelihood Hahn and Kuersteiner, 003, likelihood function anomalies in the case of first difference maximum likelihood Han and Phillips, 03, and sensitivity to departures from stationary errors under X-differencing Han, Phillips, and Sul, 04. In view of these diffi culties as well as the convenience of standard software implementation, GMM and its many variants are still heavily used in empirical work with dynamic panels. In such applications, conventional GMM Gaussian asymptotic theory is typically assumed to apply when either or both the cross section sample size n and time series sample size tend to infinity. When the autoregressive root lies in the vicinity of unity, these Gaussian asymptotics are inevitably fragile because of failing instrument relevance. he present paper completes existing theory by providing an asymptotic analysis of GMM in the unit root panel AR model using large n, large, and joint n, asymptotics. For fixed, we show that GMM is inconsistent and approaches a shifted and scaled Cauchy distributed random limit variate as n, which corresponds to the finding in Kruiniger 009. For fixed n, GMM is consistent as and has a limit distribution that involves a ratio of random variables which depends on the distribution of the data, so no invariance principle applies. When as n, GMM is consistent and its limit distribution is two times a standard Cauchy. he same limit the-

4 ory holds both sequentially, irrespective of the order of divergence of n,, and jointly as n,, irrespective of the relative rates of divergence of n and. Importantly, the usual instruments are uncorrelated with the regressor in this case, but this irrelevance does not prevent consistent estimation at least as. In nonstationary data models even orthogonal instruments can be effective in delivering consistent estimation, as was pointed out in early nonstationary time series work Phillips and Hansen, 990. Similar effects arise with panel data in the unit root case even though the model is differenced to remove fixed effects prior to regression. In this event, the differenced regressor is itself stationary and so the relevance effect arises from a sample covariance between a stationary and unit root process giving a random limit with zero mean and positive variance, thereby helping to explain the well-known dispersion of the GMM estimator which applies here even in the limit in the unit root case. he Cauchy form of the asymptotics and uncertainty reflected in the heavy tailed distribution is reminiscent of and related to the limit theory that applies in unidentified simultaneous equations models when estimated by instrumental variables under conditions of apparent identification Phillips, 989. he paper further investigates near unit root cases where the common autoregressive coeffi cient lies in the vicinity of unity. We focus primarily on cases where ρ = + c/, consonant with the convergence rate of GMM when ρ =. Results for large n, large, sequential, and joint asymptotics are provided. he limit theory leads to a correction of the asymptotic variance reported in Anderson and Hsaio 98. Extensions of these results are given for common roots of the form ρ = + c/ γ for all γ 0,, γ =, and γ >. he remainder of the paper is organized as follows. Succeeding sections give the limit theory for the panel unit root model under fixed n, fixed, sequential n,, sequential, n, and joint n, asymptotics. Later sections examine the impact of local to unity parameterizations on the asymptotic theory. Extensions to the multiple instrument and differenced instrument cases are considered in the penultimate section. Section 5 concludes with some further discussion. Proofs and derivations are given in the Appendix. hroughout the paper, we use the notation n, seq to signify followed by n ; correspondingly,, n seq signifies n followed by ; n, denotes joint asymptotics where there is no restriction on the passage of n and to infinity; and j = j for all integer j. 3

5 Model Preliminaries In the dynamic panel regression model y it = α i ρ + ρy it + u it, i =,..., n; t =,.... the α i are fixed effects for which σ α = lim n n n i= α i <, the errors u it are iid 0, σ with finite fourth moment across all i and over all t, and the initial conditions y i0 = O p for all i and are independent of the u it for all i and t. Heterogeneity over i may be introduced without disturbing some of the results given below provided large n limit theory applies and uniformity conditions continue to hold for joint n, asymptotics. In order to deliver quick results we will maintain the iid assumption for u it in what follows, while pointing out some of the extensions that apply. We define u is = 0 for all s 0 and we often assume for simplicity that y i0 = 0, a.s., although calculations are usually shown for the more general case. We start by studying the simple linear IV/GMM estimator Anderson and Hsiao, 98 which uses instruments y it in the differenced regression y it = ρ y it + u it,. leading to the estimator ρ gmm = n i= y ity it / n i= y it y it. When the true autoregressive coeffi cient in. is ρ = we have ρ gmm = n i= u ity it n i= y..3 it y it With ρ = we have y it = u it whose partial sum solution is y it = t s= u is + y i0 up to the initial condition y i0 and since E u it y it = E u it y it = 0,.4 the instrument y it satisfies the orthogonality condition in both. and.. So instrument orthogonality to the regression error in. holds. However, orthogonality is generally insuffi cient for identification and consistent estimation, for which relevance of the instrument to use the terminology of Phillips, 989 is typically needed. In the present 4

6 case, we have E y it y it = E u it y it = 0, for all t and all i.5 so the instrument y it is actually orthogonal to the regressor y it in. and relevance fails. In this event, the moment conditions.4 do not identify the unit root Kruiniger, 009. As is well-known, therefore, the GMM estimator.3 is expected to perform poorly in finite samples and to be inconsistent in the limit, as the instrument y it is irrelevant for the regressor y it in.. Similar properties of orthogonality and irrelevance hold for all instrumental variables that take the form of lagged variables y is : s =,,...t. 3 Asymptotics when ρ = 3. Large n Asymptotics Start with the case where is fixed and n. Consider n standardized forms of the numerator and denominator of.3, viz., N n = n i= u it y it, D n = n i= y it y it. Observe that u it y it = u it y it + u it y it = u it y it + u it u it under ρ =, so by partial summation u it y it = u i y i u i y i0 Adding u i y i0 = u i y i0 u i y i0 to each side gives u it y it = u i y i u i y i0 u it u it. 3. u it u it. 3. hen, N n = n n i= u i y i u i y i0 n n i= u it u it, and D n = n n i= u it y it, for which we have the following limit behavior as n when is fixed. heorem For fixed as n 5

7 i N n, D n N 0, V, V = σ 4 n ii ρ gmm n + / / /, where j = j; C, where C is a standard Cauchy variate. hus, when is fixed and n, ρ gmm is inconsistent and converges weakly to a Cauchy distribution centred on, a result that was earlier obtained in Kruiniger 009, theorem i for the random coeffi cient case with = 3. he heavy tailed limit distribution arises because the denominator D n has a random limit and its Gaussian distribution is symmetrically distributed with a positive density at zero, which ensures that no integer moments exist. he random limiting denominator reflects the presence of random information in the GMM signal in the limit. Next consider sequential asymptotics in which n is followed by. From heorem ii we deduce directly that giving ρ gmm + / C C, 3.3 n ρgmm C. 3.4,n seq Evidently, the GMM estimator ρ gmm is consistent as, even though the instrumental variable y it is irrelevant in the panel regression for all t. he rate of convergence is, which is slower than the usual rate for unit root nonstationary data in time series regression. he explanation for the large consistency of ρ gmm is that, although the relevance condition fails for all t and E y it y it = 0, the sample covariance moment condition does not have a zero limit as. Instead, y it y it = u it y it 0 B i db i 0 a.s. 3.5 where B i is Brownian motion with variance σ for all i Phillips, 987a. On the other hand, as n with fixed, the sample covariance n y it y it p 0, 3.6 i= 6

8 in view of.5. he nonzero limit 3.5 ensures some relevance as in the nonstationary instrument y it in spite of the fact that E y it y it = 0. But since the limit 3.5 is random, there is inevitably high variability in the GMM estimate. he variability is sustained in the Cauchy distribution limit for which there are heavy tails and no finite sample integer moments, just as in the fixed case. he convergence rate is because the IV regression signal is y it y it = O p, the order of a sample covariance between integrated and stationary processes. his O rate is slower than the usual O convergence rate for unit root and IV unit root regressions where, with both instrument and regressor integrated, the signal is of order O not O. 3. Large Asymptotics Start with the case where n is fixed. As the following result shows, the limit theory for does not obey an invariance principle and is dependent on the distribution of the data. But when is followed by n, an invariance principle holds and we again have a Cauchy distribution limit. heorem i As with n fixed ρgmm n i= u i B i G i. 3.7 n i= B i σ where B i r n i= are a family of iid Brownian motions with variance σ that are independent of the family of iid Gaussian variates G i n i= each with zero mean and variance σ 4 and all independent of the variate u i which is an identically distributed copy of u it. ii When followed by n ρgmm C. 3.8 n, seq Hence, ρ gmm is consistent as when the cross section sample size n is fixed. he explanation is the same as that given above concerning the relevance of the nonstationary instrument y it. Observe that 3.7 is a ratio of two random variables each of which is 7

9 centred on the origin and the denominator has positive probability density at the origin, which ensures that the ratio 3.7 has no finite sample integer moments.. Importantly, as shown in the proof of the theorem, the limit 3.7 involves only a partial application of an invariance principle. he component u i in the numerator of 3.7 is not the outcome of an invariance principle but is instead distribution dependent since u i = d u it for all t in view of the identical distribution assumption concerning u it. Evidently, when is followed by n the sequential limit distribution is identical to the limit distribution with the reverse order of sequential limits i.e., n followed by as given in Joint Limit heory as n, he equivalence of the sequential limit results 3.4 and 3.8 suggests that the limit theory is robust to the path of divergence of the respective cross section and time series sample sizes or the relative rates at which n,. he limit theory under joint sample size expansion n, is proved in the following result using the criteria for joint convergence given in Phillips and Moon 999. heorem 3 ρ gmm n, C and joint convergence applies as n, irrespective of the order and rates of expansion of the respective sample sizes. he heavy tailedness property of the GMM estimator ρ gmm manifested in the joint limit theory to a Cauchy variate applies irrespective of the manner in which the cross section and time series sample sizes diverge to infinity. he rate of convergence is, as in both forms of sequential asymptotics, and is slower than the usual O rate associated with unit root time series because of the diminished signal from the apparently irrelevant instrument y it used in the GMM regression. 4 Local Unit Root Asymptotics when ρ = + c/ γ, c < 0 here are several local unit root LUR cases that may be considered. For large n fixed asymptotics it is possible to consider deviations from unity of the form ρ = + c/n γ for γ 0, as in Kruiniger 009. his formulation is largely for mathematical convenience in analyzing the effects of local departures from unity in large n asymptotics. Importantly, the autoregressive parameter ρ measures time series dependence in the panel data y it. It 8

10 is therefore more diffi cult to justify modeling time series dependence through a parameter whose value ρ = ρ n = + c/n γ depends on the number of cross section observations. In particular, the dependence ρ n = + c/n γ implies that the AR coeffi cient of an individual time series like y t in the panel will approach unity simply by increasing the number of panel observations. Given cross section independence in the panel, it seems hard to justify such dependence of ρ n on n other than for the mathematical convenience of more closely studying limit behavior in the vicinity of unity. One possible justification is that the commonality of the AR parameter ρ across individual time series in the panel y it provides a linkage across the panel that rationalizes formulations such as ρ n = + c/n γ. hen, raising the number of cross section observations n enables us to model phenomena with common AR time dependence that is increasingly close to unity, even in spite of the cross section independence in the panel. In this case, in view of the commonality of ρ across section, more cross section information may reasonably be expected to enable us to model phenomena with AR time dependence closer to persistence. By contrast, time series sample size dependences of ρ on, such as ρ = ρ = + c/ γ, are already commonplace in the time series literature. he classifications used in that literature for measuring departures from unity apply in the same way for panels. hus, when γ = the departures are deemed to be local to unity LUR concordant with a Pitman drift when the estimation convergence rate is O, as is typical in time series regression. When γ 0,, the departures are said to constitute a mild unit root MUR and lead to mildly integrated time series in the sense of Phillips and Magdalinos 007. In both cases, as the time series sample size increases, the triangular array model formulation allows us to model time series phenomena with AR time dependence that approaches persistence ρ = and differentiates the effects of such parameterizations on the limit theory, thereby bridging part of the large limit theory gap between fixed stationary and unit root cases. he justification for using such LUR and MUR formulations of ρ as is now well established in the time series literature. Accordingly, in view of the convergence rate of the GMM estimator when ρ =, this section concentrates largely on MUR asymptotic theory for localizing sequences that are of the form ρ = +c/. As earlier, we will consider large n and large asymptotics, sequential limits, and joint convergence. We also look at cases where ρ = + c/ γ and develop large n, large asymptotics that cover the implied wider and narrower vicinities of unity that occur for more general parameterizations with γ > 0. 9

11 4. Large n and Sequential, n seq Asymptotics It is natural to start with the case where ρ = + c with fixed c < 0 and fixed as n. his localization seems appropriate given that asymptotics apply when ρ =, but more general cases may be considered and these are discussed below. Fixing the parameters c, implies a fixed ρ < and Gaussian asymptotics apply. Anderson and Hsaio 98, AH gave results for the fixed stationary case in a model with random effects as n,, but their expression for the asymptotic variance is incorrect. In the fixed effects case, which is closely related, the limit theory is as follows. heorem 4 In model. with ρ = + i n ρ gmm ρ ii n ρ gmm ρ N 0, ω n,,n seq = +ρ N 0, 4, c for fixed c < 0, we have. Explicit expressions for ω N, ω D are given in where ω = ω N ω D + O 6. and 6. in the Appendix. Parts i and ii continue to hold when ρ = + c with γ the same convergence rates n and n and the same limit variances for all γ 0,. iii When γ =, the Gaussian limit theory i still applies but the limit variance ω has the alternate form and ω 8c c e c + c e c + o, as 4. n ρgmm ρ iv When γ >, n 3 γ ρ gmm ρ N,n seq c e c 0, 8c + c e c N 0, 8,n seq c.. 4. In the stationary case with fixed c < 0, fixed, and the stationary initial condition y i0 = α i + j=0 ρj u i, j, we have y it = α i + j=0 ρj u t j and the asymptotic variance when he Anderson-Hsaio 98 formula given by equation 8.4 in their paper assumes, incorrectly, that u ity it is a martingale difference, so the formula omits cross product terms in the expression. he correct limiting variance of n ρ gmm ρ is given by + ρ, as shown in theorem 4 and 4.3. his result applies also in the stationary random effects case as discussed below. 0

12 n then has the simpler explicit form ω = + ρ + ρ σ α /σ + + ρ ρ ρ + ρ for large. 4.3 he same limit theory applies in the stationary, random effects case where α i iid 0, σ α, which was studied in AH 98. Expression 4.3 corrects the formula given in AH equation 8.4 for the limiting variance in the random effects model. See Phillips and Han 04 for further details. Different localization rates may be studied in the same way. Importantly, whatever rate γ 0, is used for ρ = + c to approach unity, the limit variance ω γ continues to apply for all fixed and fixed c. Correspondingly, since ω +ρ for large, we still get convergence and a normal limit theory for these localization coeffi cients, irrespective of how close to unity + c is. Sequential asymptotics n ρ γ gmm ρ N 0, 4,n seq then hold whenever n followed by. So, theorem 4 continues to apply for ρ = + c and all γ 0,. γ Moreover, as indicated in part iii of the theorem, when γ =, sequential Gaussian asymptotics hold but the variance of the limiting distribution changes to 8c c ec Observe that 8c c e c +c e c. + c e c 8 as c c so the Gaussian limit theory changes when closer approaches to the unit root occur. Moreover, as shown in the proof of heorem 4iv, when ρ = + c γ with c < 0 fixed and γ > the limit distribution is given by n 3 γ ρ gmm ρ 0, N 8c,n seq, 4.5 where the variance corresponds with 4.4 and the rate of convergence depends on γ, changes from n to n 3 γ, and reduces as γ increases. hus, when γ 3 the rate of convergence approaches n and when γ > 3 there is divergence because the limit variance when n is ω = 8 + o and the variance diverges as. c γ So sequential, n seq asymptotics fail and the distribution diverges as. In he error in AH 98, equation 8.4 was noted independently by Yinja Jeff Qiu in his take home examination solution 04 to Phillips 03.

13 that case, the convergence rate is effectively slower than n and the limit theory is not captured by sequential asymptotics where, n seq. Instead, as shown below, unit root asymptotics apply when n, seq because ρ = + c is in close proximity γ to ρ = when γ > and first. 4. Large and Sequential n, seq Asymptotics We now consider limits in which and ρ = + c differs moderately from unity. In a time series framework, this formulation is a special case of moderate integration in the sense of Phillips and Magdalinos 007. Again, more general cases where ρ = + c are γ considered below. he panel asymptotics are given in the following result. heorem 5 In model. with ρ = + i ρ gmm ρ ii n ρ gmm ρ c for fixed c < 0, we have n n i= ζ i, where ζ i iid N 0,, N 0, 4. n, seq It follows that ρ gmm is consistent and ρ gmm ρ is asymptotically Gaussian N 0, 4/n when. In sequential limits as n, seq, the limit distribution is Gaussian N 0, 4 after rescaling, just as in heorem 4 above when, n seq. Joint convergence to N 0, 4 then follows in the same manner as heorem 3 and is given in the following result. heorem 6 When ρ = + c with fixed c < 0, we have ρ gmm ρ n, N 0, 4 and joint convergence applies as n, irrespective of the order and rates of expansion of the respective sample sizes. Next consider the case where ρ = + c γ with fixed c < 0. heorem 7 Let ρ = + c γ with fixed c < 0 and γ > 0. i When γ 0,, ρ gmm ρ n ρgmm ρ n n i= ζ i, where ζ i iid N 0,, and N 0, n, seq

14 n i=σ u i J ci ζ i ii When γ =, ρ gmm ρ n i=c 0 J cir dr+, where J 0 J cirdw i ci r = r 0 ecr s dw i s, W i are standard Brownian motions, and the ζ i iid N 0, and independent of W i for all i. Further when γ = and followed by n, we have n ρgmm ρ iii When γ >, ρ gmm ρ 0, N 8c n, seq ρgmm ρ c ec e c c n i= ζ i n i= 0 W idw i and then. 4.7 C 4.8 n, seq he N 0, 4 sequential limit theory given in 4.6 mirrors heorem 5ii, showing that this limit result is robust for all moderately integrated panels with mild integration parameter γ 0,. Joint limit theory applies in this case, precisely as in the proof of heorem 6, so the details are omitted here. When γ =, the large limit theory under ii involves the standardized diffusion processes J ci, as is usual in local to unity cases. he corresponding sequential n, seq limit theory in 4.7 retains the n convergence rate and has a limit variance that depends on the localizing coeffi cient c, again as may be expected in the LUR case. Moreover, this limit theory is the same as the, n seq sequential asymptotics given in 4. of heorem 4 and both n convergence and limiting normality continue to hold. Again, the limit theory is independent of the direction of the asymptotics and joint convergence holds in the same way as heorem 6.. When γ >, the sequential n, seq limit theory 4.8 corresponds exactly to the panel unit root limit Cauchy distribution since the panel autoregressive root ρ = + c is closer γ to unity than the local to unity coeffi cient ρ = + c and first. It is this close proximity of ρ = + c to unity as that ensures that the panel unit root limit theory γ obtains when γ >. Importantly, joint convergence no longer holds in this case. Instead, directional asymptotics occur and the limit distribution depends on the nature of the sample size expansion. For when, n seq we have n 3 γ ρ gmm ρ N 0, 8,n seq c as obtained in 4.5, whereas when n, seq we have ρ gmm ρ as given in 4.8. n, seq C In effect, the non-gaussian Cauchy limit theory cannot be captured in, n seq directional sequential asymptotics where the limit theory is Gaussian because 3

15 ρ < as n. 5 Further Discussion Anderson and Hsiao 98 also suggested using the lagged differences y it rather than the lagged levels y it as an instrumental variable. his estimator has the form ρ gmm = n i= y it y it n i= y = it y it n i= u itu it n i= u, when ρ =. it u it Calculations similar to those given in Section 3 now lead to n u it u it, i= u it u it N n 0, σ ], which is invariant to after rescaling by. It follows that ρ gmm C, showing that n ρ gmm is inconsistent, miscentred around the origin, with a random variable limit that has heavy tails like ρ gmm and is invariant to the time series sample size. In consequence, sequential asymptotics give the same limit, viz., ρ gmm C. Similarly,,n seq u it u it, i= u it u it N i= 0, σ 4 n 0 0 ] which is invariant to n after scaling by n. hen ρ gmm C, leading directly to the sequential asymptotics ρ gmm C. Similar arguments to those given earlier show n, seq that this limit theory applies jointly as n, irrespective of the rates of divergence of the sample sizes. Use of lagged differences y it as instruments therefore leads to an inconsistent estimator of ρ in the unit root case for fixed, fixed n, and joint asymptotics. In this case, both the regressors y it and the instruments y it are stationary with covariance E y it y it = 0. So the sample signal n n i= y it y it has zero expectation and zero limit in probability, thereby providing no leverage for consistent estimation. Mildly integrated cases with ρ = + c/ γ may also be examined using these methods, as may GMM estimates with more instruments, but they are not considered in the present work and will be reported elsewhere. 4

16 6 Appendix Proof of heorem. Part i follows by the Lindeberg Lévy CL with n i= u ity it y it y it N 0, V, 6. E V = u ity it E u ity it y it y it E u ity it y. it y it E y it y it o evaluate, note by partial summation as indicated in 3. and 3., we have u it y it = u i y i u i y i0 o compute V, note that E u it y it = E u i y i u i y i0 6. u it y it 6.3 u it u it = E u i y i u i y i0 E u i y i u i y i0 u it u it + E u it u it = Eu i y i + Eu iy i0 + = σ 4 + σ Ey i0 + σ 4 = σ 4, E u it u it the final line following if the initial condition y i0 = 0, which will be assumed in the calculations below. he large n asymptotic results will continue to hold for y i0 = O p even for finite with some obvious minor adjustments to the variance matrix expressions involving 5

17 quantities of O in. Next E y it y it = E u it y it = σ Ey it = σ 4 t = σ 4 /, and, with y i0 = 0 or up to O in if y i0 0 E u it y it u it y it = E u i y i u i y i0 u it u it u it y it = E u it u it u it y it E u i y i0 = E u it u it u it y it + = Eu it u it = σ 4 s,;s t = σ 4. u it u it u is y is hen V = σ 4 σ 4 σ 4 σ 4 / = σ 4 /, as stated. For Part ii, simply write ρ gmm = N n D n, and note from i that N n, D n ] n σ / ξn,, ξ D,, where ξn,, ξ D, is bivariate N 0,. Next, decompose ξ N, as ξ N, = ξ N.D, + / ξ D, where ξ N.D, N / 0, / = N 0, is independent of ξ D,, so that ξ N.D, ξ D, N 0, 0 0 / Combining these results, we have by joint weak convergence and continuous mapping that ]. 6

18 as n with fixed, ρ gmm = N D ξ N, = ξ N.D, ξ D, n ξ D, = + ξ N.D, = + ξ D, / 6.4 ξ D, / ζ N ζ D / + / C, 6.5 where ζ N, ζ D N 0, I and C is a standard Cauchy variate. hus / ρ gmm + n / C, 6.6 yielding the stated result. Proof of heorem. From.3 and 3. we have ρ gmm = = n i= u ity it n i= y it y it n i= u i y i u i y i0 u it u it n i= u it y it, and rescaling gives ρgmm = n i= u i y i u i y i0 u it u it n i= u it y it. 6.7 By partial summation u it y it = t= t u it u is + y i0 = u it t= s= t= t= u it + u it y i0. Using the fact that E u it u is u is = 0 for all t, s, we have by standard functional limit t= 7

19 theory for r 0, ] r / t= u it u it u it ] B i r G i r ] BM σ 0 0 σ 4 ], where B i and G i are independent Brownian motions for all i. hen, since y i0 = O p and t= u it = o p, we deduce the joint weak convergence / t= u it t= u ity it t= u itu it B i B i σ G i = B i 0 B idb i G i. 6.8 Since u it is iid over t and i, it follows that u i u i as, where the limit variates u i are independent over i and have the same distribution as u it. Note that u i is independent of / t= u it, t= u ity it, t= u itu it and, hence, asymptotically independent of / t= u it, t= u ity it, t= u itu it. It follows that u i is independent of the vector of limit variates 6.8. We therefore have the combined weak convergence / t= u it t= u ity it / t= u itu it u i Setting G i = G i, the stated result 3.7 ρgmm B i B i σ G i = u i B i 0 B idb i G i u i. 6.9 n i= u i B i G i 6.0 n i= B i σ follows from 6.7 and 6.9 by continuous mapping. For part ii we consider sequential asymptotics in which is followed by n. Observe that u i B i G i is iid over i with zero mean and variance E u i B i G i = E u i E B i + E G i = σ 4, 8

20 and is uncorrelated with B i. Since B i σ is iid with zero mean and variance σ 4, application of the Lindeberg Lévy CL as n gives n / n i= u ] i B i G i n / n i= B i σ where ζ N, ζ D N 0, I. Hence, giving the required result. Proof of heorem 3. ρgmm n σ 4 ] / ζn σ 4 / /, 6. ζd C, 6. n, seq We proceed by examining a set of suffi cient conditions for joint convergence limit theory developed in Phillips and Moon 999. In particular, we consider conditions that suffi ce to ensure that sequential convergence as n, seq i.e., followed by n implies joint convergence n, where there is no restriction on the diagonal path in which n and pass to infinity. We start by defining the vector of standardized components appearing in the numerator and denominator of ρ gmm X n = n / i= u i y i u i y i0 u it u it, n / From 6.9 and 6. we have the sequential convergence X n X n : = X : = n n / i= σ 4 / ζn, which in turn implies the sequential limit ρ gmm Lemma 6b of Phillips and Moon 999, when X n i= u it y it. 6.3 u i B i G i, n / B i σ i= σ 4 / ζ D, 6.4 C given in 6.. By n, seq X n X sequentially, joint n 9

21 weak convergence X n X as n, holds if and only if for all bounded, continuous real functions f on R. lim sup Ef X n Ef X n = n, Simple primitive conditions suffi cient for 6.5 to hold are available in the case where the components of the random quantity X n involve averages of iid random variables as in the present case where we have X n = n / n i= Y i with the Y i independent over i. Component-wise we have X n = X n, X n := where Y i = Y i, Y i with Y i = u i y i u i y i0 Y i = u it y it Y i := n / Y i, n / i= Y i, i= u it u it Y i := u i B i G i, B i σ, for all i. he working probability space can be expanded as needed to ensure that the limit random quantities Y i := Y i, Y i are defined in the same space for all i so that averages involving n i= Y i are meaningful. In this framework we can use a result on joint convergence by Phillips and Moon 999 see lemma PM below to verify condition 6.5. In what follows we use the notation of lemma PM. We proceed to verify these conditions for Y i and Y i. First, Y i is integrable since E u i E y i / <, E u i y i E u it u it E u it u it E u it <, E u it y it E u it y it Eu it Eyit / <. 0

22 o show i holds, observe that E Y i = EY i + EY i = E u i y i = σ 4 + σ 4 u it u it + E u it y it t <, 6.6 when y i0 = 0, with obviously valid extension to the case where y i0 = O p with finite second moments. hen lim sup n, n i= E Y i = lim sup E Y i lim sup E Y i / <, as required. o show ii holds, simply observe that EY i = EY i = 0. o show iii holds, note that lim sup n, n i= E Y i Y i > nɛ = lim supe Y i Y i > nɛ = 0, for all ɛ > 0, since sup E Y i < by virtue of 6.6. Finally, note that lim sup n n i= E Y i Y i > nɛ = lim supe Y i Y i > nɛ = 0, n since E Y i <, proving iv. Hence, condition 6.5 holds and we have joint weak convergence X n = n / i= Y i X := n, σ 4 / σ 4 / ζn, ζ D, irrespective of the divergence rates of n and to infinity. By continuous mapping, the required result follows for the GMM estimator so that ρ gmm C jointly as n, n, irrespective of the order and rates of divergence of the respective sample sizes. Lemma PM Phillips and Moon, 999, theorem Suppose the m random vec-

23 tors Y i are independent across i for all and integrable. Assume that Y i Y i as for all i. hen, condition 6.5 holds if the following hold: i lim sup n n i= E Y i <, n, ii lim sup n n i= EY i EY i <, n, iii lim sup n n i= E Y i Y i > nɛ = 0, for all ɛ > 0 n, iv lim sup n Proof of heorem 4. c n n i= E Y i Y i > nɛ = 0, for all ɛ > 0 In case i is fixed as well as c < 0, which implies that ρ = + is fixed. So large n asymptotics follow as in the asymptotically stationary case. From. we have y it = α i ρ + ρy it + u it = α ic + + c y it + u it and y it = ρ y it + u it so that y it = α i ρ+ρ y it +u it = α ic + c y it +u it. hen, as usual, E u it y it = E u it y it = 0 and orthogonality holds. When y i0 = 0, back substitution gives y it = α i ρ t t + ρ j u it j, and E y it = α i ρ t, Var y it = σ t j=0 ρj = σ ρt, and E y ρ it = σ ρ t ρ t. Instrument relevance is determined by the magnitude of the moment α i E y it y it = E α i ρ + ρ y it + u it y it j=0 = α i ρ ρ t + ρ σ ρt ρ + α i ρ + ρ t = σ ρt + ρ α i ρ ρ t ρ t 6.7 which is non zero for c < 0 and zero when c = 0, corresponding to the unit root case ρ = considered earlier. Note that in the fully stationary case where initial conditions are in the infinite past so that y i0 = α i + j=0 ρj u i, j and y it = α i + j=0 ρj u t j we have E y it y it = α i ρ + ρ E yit = α σ i ρ ρ ρ + α i = σ + ρ,

24 which corresponds with the leading term of 6.7 when t with ρ <. Now consider the numerator and denominator of the centred and scaled GMM estimate n ρgmm ρ = n n i= u ity it n n i= y it y it =: N n D n. 6.8 First, noting that y it y it is quadratic in α i, and using j = j and σ α = lim n n n i= α i, the denominator of 6.8 takes the following form as n n i= = lim n n = σ + ρ = σ + ρ = σ + ρ y it y it p i= ρ ρ lim n n E y it y it i= σ ρt + α i ρ ρ t ρ t ] + ρ ] + σ α ρ ρ ρ ρ ρ ρ ] + σ α ρ ρ ρ ρ ρ ρ ] + σ α ρ ρ + ρ ] σ α ρ ρ ] ρ + ] ρ ρ ], 6.9 which is again zero when c = 0 ρ =. urning to the numerator, we have E u it y it = 0 by orthogonality and by a standard CL argument for fixed as n with n u it y it N 0, v i= v = lim n n E u it y it We evaluate the above variance as follows. Using 3. and y i0 = 0, we have u it y it = u i y i u i y i0 i= u it y it = u i y i u it y it, 6.0 3

25 with variance E u i y i u it y it = σ E y i + σ E y it = σ 4 ρ ρ + α i σ ρ + σ E y it. Using E y it = α i ρ t, Var y it = σ t j=0 ρj = σ ρt, E y ρ it = σ ρ t + ρ α i ρ t, and yit = α i ρ + ρ y it + u it, the final term E y it above is α i ρ + ρ E yit 3 + σ α i ρ ρ t 3 = σ α i ρ + α i ρ ρ + ρ ρ E yit 3 ] ρ ρ = σ α i ρ + α i ρ ρ + ρ σ ρ +α i ρ ρ ρ + ] ρ ρ ] = σ ρ + σ ρ + ρ + ρ + α i ρ ρ + α i ρ = σ + ρ σ ρ + ρ + α i ρ ρ + ρ. hen ρ ρ + ] ρ ρ E u i y i u it y it = σ 4 ρ ρ + α i σ ρ + σ E y it = σ 4 ρ ρ + α i σ ρ σ ρ σ4 ρ + ρ + α i σ ρ ρ + ρ = σ4 ρ ρ + σ4 + ρ ρ + ρ + α i σ ρ + α ρ i σ ρ + ρ = σ4 ρ ρ + σ4 + ρ ρ + ρ + α i σ ρ ρ + ρ + ρ ], + ρ 4

26 and ω N = lim n n = σ4 + ρ From 6.9 we have E u it y it i= ρ ρ + σ4 ρ + ρ + σ ασ ρ ρ + ρ + ρ ] + ρ ω D = which leads to the asymptotic variance ω = ω N ω D = = + ρ 6.. σ ] ρ + ρ ρ + σ α ρ ] ρ 6. + ρ σ 4 +ρ + σ4 ρ ρ ρ +ρ + σ ασ ρ ] σ +ρ ρ + σ ρ α + O 3/ ] ρ + ρ+ρ +ρ ] ρ ρ +ρ, 6.3 giving the stated result for i. he error magnitude as in the asymptotic expansion 6.3 is justified as follows. Since c < 0 is fixed we have ρ = c c and ρ + c ρ = c c hen, by direct calculation as ω N ω D = = σ 4 ρ +ρ + σ4 +ρ c σ +ρ + ρ + O ] ec c c c σ ασ e c / e c / c +ec / ] +ρ e c / ] + σ α e c / ec / ] +ρ 6.5 c 3/ he sequential limit theory ii follows directly from i and the asymptotic expansion 6.3 of ω. 5

27 If ρ = + c with γ 0,, it is clear that the above fixed, c limit theory as γ n continues to hold. hen, as, we have in place of 6.4 ρ ρ = c + c γ ] γ c γ ec γ c γ c γ γ c leading to ω N + ρ = + O ω D γ 4 + O γ It follows that ii continues to hold with the same convergence rate n and same limit variance 4 for all γ 0,. When γ =, the sequential normal limit theory in ii still holds but the variance of the limiting distribution changes. Observe that in this case ρ ρ = ] + c c c ec c c e c. c Using 6. we then have the following limit behavior as ω N σ4 + σ 4 ω D σ e c ec c c + O ] = 4 + ec c ] + o, + O ec c so that Hence ω = 8c n ρgmm ρ c e c + c e c + o. N,n seq c e c 0, 8c + c e c, 6.6 so the n Gaussian limit theory holds but with a different variance when ρ = + c. Observe that 8c c e c + c e c 8 as c 0, c indicating that the variance in 6.6 diverges and the n convergence rate fails as the unit root is approached via c 0. 6

28 Next, examine the case where ρ = + c with γ > and c < 0, so that ρ is in γ the immediate vicinity of unity, closer than the LUR case but still satisfying ρ < for fixed. In that case, we still have Gaussian limit theory as n because ρ <. o find the limit theory as, n seq we consider the behavior of the numerator and denominator of ω. First, note that log + c ] = c γ c + O γ γ so that ] 3γ + c = + c γ c + c γ γ + O γ giving γ ρ ρ = ] + c + c c + c ] γ c = γ γ + O γ 3γ γ c c γ γ c γ c c + c + o = γ γ γ c + c + O = + c γ + O γ + c γ γ + O γ γ = + c γ + O γ. Using this result, and letting ω D = vd with v D = σ +ρ ρ ρ ]+σ α we have v D = σ ] c γ + o + σ α + ρ + O c γ c γ + c + o γ = cσ + ρ γ + o + σ α c + γ + c 4 γ = cσ + ρ γ + o + σ α = cσ + ρ γ cσ c α γ 3γ c γ + O c γ c γ + c γ 3γ 4 c γ + O + c + O c γ 3γ γ 3γ + o = cσ γ + o, ρ ρ +ρ + O + o ] 3γ + o ] ], 7

29 so that the denominator is ω D = c σ γ + o. he numerator ω N is ρ ρ + σ4 ρ + ρ + σ ασ ρ ρ + ρ + ρ ] + ρ c + γ + o + σ 4 c γ + c + c γ c γ + O γ +σ ασ c c 3 γ + O γ + o ] σ 4 + ρ = σ 4 = c γ + c γ σ 4 + σ ασ c c γ + o ] = σ 4 + o ]. Combining these results we obtain ω = ω N ω D = σ 4 + o ] = + o c σ 4 4 γ + o It now follows that for ρ = + c γ with c < 0 fixed and γ > n 3 γ ρ gmm ρ 0, N 8c,n seq. γ c + c + c γ γ γ + c γ + o 8 c γ + o. Hence when ρ is closer to unity than a local unit root, the n rate of convergence is reduced to n 3 γ. When γ = 3 the rate of convergence is simply n and for γ > 3 the large n Gaussian asymptotic distribution N 0, ω diverges as because ω = 8 + o diverges with. In this event, sequential, n c γ seq asymptotics fail. In effect, the convergence rate is slower than n and the non-gaussian Cauchy limit theory cannot be captured in these, n seq directional sequential asymptotics even though ρ = + c with γ > is closer proximity to a unit root than than the usual local unit γ root case with γ =. Proof of heorem 5. In the mildly integrated case where ρ = + c we have y it = α ic + + c y it + u it and y it = ρ y it + u it so that y it = α ic + ] + o 8

30 c y it + u it = α i ρ + ρ y it + u it. By partial summation, as shown above in 3., we have u ity it = u i y i u i y i0 u it y it, so that ρ gmm ρ = n i= u i y i u i y i0 u it Rescaling and using y i0 = 0 gives ρgmm ρ = n i= α ic + n i= u i y i u it n i= α ic + α ic + ] c y it 3 + u it. 6.7 c y it + u it y it α ic + ] c y it 3 + u it. 6.8 c y it + u it y it Since u it u it G i N 0, σ 4, u it = o p, and u it y it 3 = o p see 6.30 below the numerator is i= = u i y i i= u it α ic + c ] y it 3 + u it u it u it + o p Using Phillips and Magdalinos 007, theorem 3. we find that 3/ y it p σ c, 3/4 y it u it N he denominator of 6.8 therefore satisfies i= 0, G i. 6.9 i= σ 4 c α ic + c y it + u it y it p Hence, using 6.9 and 6.3 we have ρ gmm ρ where ζ i iid N 0,. hen n ρgmm ρ n i= ζ i, and 3/ y it = o p. i= 6.30 c σ. 6.3 c n nσ i= G i = n n i= ζ i N 0, 4, 6.3 n 9

31 which gives i and then leads directly to the sequential limit n ρ gmm ρ N 0, 4. Proof of heorem 6. n, seq he proof follows the same lines as the proof of heorem 3 above. As before, we define the vector of standardized components appearing in the numerator and denominator of ρ gmm ρ in 6.8 X n = X n, X n := where Y i = Y i, Y i with Y i = u i y i Y i = n / u it α ic + Y i, n i= Y i, i= c ] y it 3 + u it α ic + c y it + u it y it Y i := c σ c. From 6.9 and 6.3 we have the sequential convergence X n X n : = X : = n n / σ ζ, σ G i, n which in turn implies the sequential limit n ρ gmm ρ i= i= Y i := G i, c σ c, where ζ = N 0,, 6.33 n, N 0, 4 given in 6.. Since X n X n X sequentially, joint weak convergence X n X as n n, holds in the same manner as heorem 3 with only minor definitional changes. First, Y i is integrable just as before. o show Lemma Ai holds, observe that E Y i = EY i + EY i = E u i y i + E u it α ic + c y it 3 + u it α ic + c y it + u it y it 30

32 = σ Ey i + E = σ Ey i + E α ic 3/ E u it s=3 = σ4 c + σ4 + O = σ 4 + o, u it α ic + u it u it + c α i E u is + c σ c y it 3 + u it α ic E u it s=3 Eyit 3 + O u it + c E u it y it 3 y is 3 + c 3/ E u it u it + O + O since from 6.4 E yit = σ ρ t +α ρ i ρ t = σ t c + o. hen E Y i < and we deduce that lim sup n, n i= E Y i = lim sup E Y i lim sup E Y i / <, as required. Condition ii holds, as we again have EY i = EY i = 0; and condition iii and iv hold because sup E Y i < and E Y i <. We then have joint weak convergence X n = n / i= Y i X := n, σ ζ, σ, irrespective of the divergence rates of n and to infinity. By continuous mapping, the required result follows for the GMM estimator so that ρ gmm ρ N 0, 4 holds n, jointly as n, irrespective of the order and rates of divergence. Proof of heorem 7. this case, y it = α ic γ α ic γ We have ρ = + c γ for some fixed c < 0 and let. In + + c γ yit + u it and y it = ρ y it + u it so that y it = + c γ y it + u it. As before, we have ρgmm ρ = n i= u i y i u i y i0 n i= α i c γ u it α i c γ + c γ y it 3 + u it ] + c γ y it + u it yit s=3 y is 3 3

33 We use the following results from Phillips and Magdalinos 007 and Magdalinos and Phillips 009, which hold for all γ 0,, γ y it p σ c, +γ/ y it u it N 0, σ 4 c, and / γ y it = O p hen, since u it u it G i N 0, σ 4, /+γ u it = o p, and /+γ u it y it 3 = o p when γ 0,, the numerator of 6.34 is i= = u i y i i= u it α ic γ + c ] γ y it 3 + u it u it u it + o p Using 6.35, we find that the denominator of 6.34 satisfies i= Hence, as ρgmm ρ α ic γ + c γ y it + u it y it p n i= G i σ n = σ n hen, as is followed by n, we have n ρgmm ρ n i= ζ i i= ζ i G i. i= i= c σ = σ n c. σ n, where ζ i iid N 0,. N 0, 4, for all γ 0,. n Next consider the case γ =. he numerator of 6.34 is then = i= i= u i y i u i y i u it α ic + c ] y it 3 + u it u it u it + o p u i K ci G i, i= 3

34 since by standard functional limit theory for near integrated processes Phillips, 987b we have / y i, y it u it K ci r, 0 K ci db i, where B i r =: σw i r are iid Brownian motions with common variance σ, and K ci r = r 0 ecr s db i s =: σj ci r is a linear diffusion. he denominator of 6.34 satisfies i= Hence ρgmm ρ α ic + c y it + u it y it i= c n i= u i K ci n i= G i n i= c 0 K ci r dr + 0 K ci r db i = 0 K ci r dr + 0 K ci db i. n i= σ u i Jci ζ i n i= c 0 J ci r dr + 0 J cidw i, 6.36 where the ζ i iid N 0, and are independent of the W i and u i for all i. his gives the first part of ii. Scaling the numerator and denominator of 6.36, noting that 0 J ci r dw i has zero mean and finite variance, and using the independence of ζ i, u i, and W i, we obtain n ρgmm ρ n n n i= N 0, c ec c = N 0, 8c ce 0 J ci r dr n n i= σ u i Jci ζ i c 0 J ci r dr + 0 J ci r dw i c ec e c c since, using results in Phillips 987b, we have E 0 J ci r dr = ec c c, and E σ u i Jci ζ i = E σ u i EJci + Eζ i = + ec c Hence, when γ =, we have n ρgmm ρ N 0, 8c n, c e c e c c = c ec. c

35 From Lemma of Phillips 987b we have c 0 J ci r dr, c / J ci r dw i, Z i, Z i iid N 0,, c 0 0 and c ec c = + o as c 0, so that 8c c e c e c c 8c 4c c = 8 c for small c which explodes as c 0, consonant with the unit root case where we only have convergence. Observe that both 6.37 and 6.38 correspond to earlier results with the reverse order of sequential convergence, n seq. Next suppose γ > so that ρ = + c γ is closer to unity than the LUR case with γ =. In this case, the numerator and denominator of 6.34 have the same limits as in the unit root case, viz., and hen = i= i= i= ρgmm ρ u i y i u i y i u it α ic γ + c ] γ y it 3 + u it u it u it + o p α ic γ + c γ y it + u it y it n i= u i B i G i n i= 0 B = idb i u i B i G i, i= i= 0 B i db i. n n i= σ u i Wi ζ i n n i= 0 W C, idw n i ] since n n i= σ u i Wi ζ i, n n i= 0 W idw i N 0 0,. n 0 / 34

36 7 References Anderson,. W. and C. Hsiao 98, Estimation of dynamic models with error components, Journal of the American Statistical Association, 76, Anderson,. W. and C. Hsiao 98, Formulation and estimation of dynamic models using panel data, Journal of Econometrics, 8, Blundell, R. and S. Bond 998. Initial conditions and moment restrictions in dynamic panel data models, Journal of Econometrics, 87, Hahn, J. and G. Kuersteiner 00 Asymptotically unbiased inference for a dynamic panel model with fixed effects when both N and are large, Econometrica, 70, Han, C. and P. C. B. Phillips 03. First Difference MLE and Dynamic Panel Estimation, Journal of Econometrics, 75, Han, C., P. C. B. Phillips and D. Sul, 04. X-Differencing and Dynamic Panel Model Estimation, Econometric heory, 30, pp 0-5. Kruiniger, H. 009: GMM Estimation of Dynamic Panel Data Models with Persistent Data, Econometric heory, 5, Moon, H. R. and P. C. B. Phillips, 004. GMM Estimation of Autoregressive Roots Near Unity with Panel Data, Econometrica, 7, Phillips, P. C. B. 987a. ime Series Regression with a Unit Root, Econometrica, 55, Phillips, P. C. B. 987b. owards a Unified Asymptotic heory for Autoregression, Biometrika 74, Phillips, P. C. B Partially identified econometric models, Econometric heory 5, Phillips, P. C. B. 03. Econometrics IV: ake Home Examination, Yale University 35

37 Phillips, P. C. B. and C. Han 04. On the Limit Distributions of FDLS and Anderson Hsiao Estimators, working paper. Phillips, P. C. B. and B. E. Hansen 990. Statistical inference in instrumental variables regression with I processes, Review of Economic Studies 57, Phillips, P. C. B.,. Magdalinos 007, "Limit heory for Moderate Deviations from a Unit Root," Journal of Econometrics 36, Phillips, P.C.B. and H.R. Moon 999 : Linear Regression Limit heory for Nonstationary Panel Data, Econometrica, 67, Qiu, Yinjia Jeff. 04, ake Home Examination Solution to Phillips