Bias Reduction in Dynamic Panel Data Models by Common Recursive Mean Adjustment

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1 Bias Reduction in Dynamic Panel Data Models by Common Recursive Mean Adjustment Chi-Young Choi University of exas at Arlington elson C. Mark University of otre Dame and BER Donggyu Sul University of exas at Dallas May 25, 2009 Abstract he within-group estimator (same as the least squares dummy variable estimator) of the dominant root in dynamic panel regression is known to be biased downwards. his paper studies recursive mean adjustment (RMA) as a strategy to reducethisbiasforar(p) processesthatmay exhibit cross-sectional dependence. Asymptotic properties for, jointly are developed. When log 2 (/) ζ where ζ is a non-zero constant, the estimator exhibits nearly negligible inconsistency. Simulation experiments demonstrate that the RMA estimator performs well in terms of reducing bias, variance and mean square error both when error terms are cross-sectionally independent and when they are not. RMA dominates comparable estimators when is small and/or when the underlying process is persistent. Keywords: Recursive Mean Adjustment, Fixed Effects, Cross-sectional Dependence. JEL Classification umbers: C33 Corresponding author. d.sul@utdallas.edu. We are grateful to two anonymous referees for constructive comments that helped to improve the paper. We also thank Peter C.B. Phillips for suggesting the topic and for helpful comments on an earlier draft and Chirok Han for useful and thoughtful comments.

2 Introduction In small samples, the least squares estimator of the autocorrelation coefficient ρ [0, ), of a stationary but persistent first-order autoregressive time series y t = α + ρy t + t, t 0,σ 2, is biased downwards when the constant is also estimated. o illustrate the source of the bias, suppose one runs least squares without a constant but on deviations from the sample mean. hat is, regress (y t ȳ) on (y t ȳ). he regression error t is correlated with current and future values of y t, and these current and future values of y t are embedded in the ȳ component of the explanatory variable. It follows that the error term is correlated with the regressor (y t ȳ). In the panel data context, this small bias is also present in fixed-effects estimators. For fixed as, ickell (98) shows that the within group (WG, which is equivalent to the least squares dummy variable method) estimator for the dynamic panel regression model is substantially inconsistent. his paper studies and applies the recursive-mean adjustment (RMA, henceforth) technique to reduce bias in the estimation of linear dynamic panel data models when the dominant root is homogeneous across individuals. he paper builds on work by So and Shin (999a) who show that the RMA strategy is useful in reducing bias in univariate regression and in the context of unit-root testing [So and Shin (999b, 2002)]. In RMA, the constant is dealt with by adjusting observations with the common recursive mean ȳ t =(t ) P t s= y s, instead of the sample mean. As a result, the adjusted regressor (y t ȳ t ) is orthogonal to the regression error t because the recursive mean does not contain any future values of y t. he RMA strategy, however, does not completely eliminate the bias because the error term after RMA adjustment contains ( ρ)ȳ t which is correlated with the adjusted regressor. But when ρ =intheunivariatetimeseriescontext,thermamethod completely eliminates the small sample bias which explains why it has been used in unit-root testing by aylor (2002), Phillips, Park and Chang (2004), and Sul (2008). We find that the RMA estimator effectively alleviates the bias problem in a finite sample compared to the WG estimator. Similar results are obtained in a more general AR(p) model using a simple two-step approach to reducing the bias. he performance of the RMA method is notable particularly when ρ is near unity in which alternative bias reduction methods based on GMM/IV estimators may not properly work due to the weak moment conditions [Brundell and Bond (998)]. he context in which this study is conducted is relevant to the empirical studies based on a data set with larger and small,suchasfirm level analyses on the estimation of dynamic labor demand equation [e.g. Blundell and Bond (998)] or production function [e.g., Blundell, Bond and Windmeijer (2000)] or dynamics of macroeconomic variables at the regional level [e.g. Campbell and Lapham (2004), Bun and Carree (2005)]. he remainder of the paper is organized as follows. In the next section, we begin by discussing the If there is no constant term, the bias is 2ρ/ which is trivial for even moderate. Mariott and Pope (954) and Kendall (954) discuss and characterize the first-order approximation of this bias. Several bias correction strategies have been suggested in the literature, such as median unbiased estimation [Andrews (993)], approximately median unbiased estimation [Andrews and Chen (994)], and mean unbiased estimation [Phillips and Sul (2007)].

3 asymptotic properties of the panel RMA estimator for a general AR(p) model under cross-sectionally independent observations. his section also discusses extensions of the framework to environments with incidental trends, local-to-unity observations, and issues involved when the dominant root is heterogeneous across individuals. Section 3 considers an environment where the observations are cross-sectionally correlated and can be represented by a common factor structure. As in Alvarez and Arellano (2003), Bai (2003), and Hahn and Kuersteiner (2002), our asymptotic analysis here is based on large and large. Section 4 reports results of Monte Carlo experiments to evaluate the precision and effectiveness of the RMA estimator in reducing bias and the accuracy of the asymptotic theory for small and moderate sample sizes. Section 5 concludes. An appendix contains proofs and details of many arguments made in the text. Before proceeding, a few words on the notations might be helpful. hroughout the paper, denotes the span of time-series and 0 is written as the cross-section dimension of panel, and the symbols and plim d respectively indicate the weak convergence in distribution and probability limit as. Also, an estimator is said to be inconsistent if the probability limit of an estimator is not equal to its true value as with fixed. For example, the WG estimator for the first autocorrelation coefficient is inconsistent as with fixed, whereas it becomes consistent as regardless of. An estimator is asymptotically biased when we compare the magnitude of the inconsistency of an estimator for small with that for large. 2 Asymptotic properties when the observations are cross-sectionally independent he data are assumed to be generated by the following latent model. Assumption. structure For i =,... and t =,..., the observations {y it } have the latent model y it = μ i + βw it + z it, () px z it = ρ j z it j + it, (2) j= where w it is strictly exogenous, Ew it z is =0for all t and s, and all roots of lag polynomial of z it lie outside the unit circle, μ i μ, σμ 2,zi 0,σi 2/ ρ 2 for ρ = P p j= ρ j, and it 0,σi 2 is independent of μ i and y i, and has finite moments up to the fourth order. 2 2 Assumption admits the level of an exogenous regressor w it. In their study on natural gas demand, Balestra and erlove (966) distinguish between level and quasi-difference exogenous regressors. hey argue that per capita income affects gas demand in levels but relative price affects gas demand in the quasi-difference. In their regression model, y it is the quantity of natural gas, w it is the per capita income, and q it is the relative price of natural gas. he 2

4 his data generating process (DGP) assumes that the initial condition is stationary, as in Blundell and Bond (998). 3 he latent model ()-(2) is observationally equivalent to the dynamic panel regression representation p X y it = a i + ρy it + φ j y it j + βw it + j= px κ l w it l + it. (3) where a i =( ρ) μ i,ρ= P p j= ρ j,φ j = ρ j + φ j for j =2,...,p with φ = P p k=2 ρ k,and κ l = βρ l for l =,...,p. ote that so long as w it is strictly exogenous, both the WG estimator for β in()-whichisthestaticversion-andin(3)-whichisthedynamicversion-areconsistent,but the latter is more efficient than the former for a large. he RMA estimator for this model is obtained from the following steps. Step : (RMA Estimator for ρ) Estimate (3) with the WG estimator. Let ˆx it = P p ˆφ j= j,wg y it j + ˆβ WG w it + P p j= ˆκ j,wgw it j, and y it + = y it ˆx it. his allows (3) to be rewritten as l= y + it = a i + ρy it + + it where + it = it +(x it ˆx it ). hen using pooled least squares, regress y + it ȳ it = ρ (yit ȳ it )+e it (4) where ȳ it =(t ) P t s= y is,e it = ( ρ) z it + it +(x it ˆx it ),and z it =(t ) P t his gives ˆρ p RMA, the RMA estimator of ρ. s= z is. Step 2: (RMA Estimator for φ j and κ l ) For the other coefficients, ˆβ WG is consistent but ˆφ j,wg and ˆκ l,wg are not. he inconsistency of ˆφ j,wg and ˆκ l,wg, however, can be reduced by running an additional WG regression, ³y it ˆρ p RMA y it ˆβ Xp WG w it = a i + φ j y it j + j= px κ l w it l + e p it, (5) quasi-difference exogenous regressor can be introduced by letting it = γq it + o it in (3). According to Phillips and Sul (2007), the WG estimator for β in () is consistent, while the WG estimator for γ is inconsistent for fixed as. In this case, we can utilize the simple bias correction method for γ provided by Phillips and Sul (2007) for a moderately large but small. 3 Assumption does not include the case of unit-root because RMA estimator becomes consistent when ρ =.For the unit-root case, however, the initial condition can be set as z i = O p (). See Kiviet (995) for the impact of a nonstationary initial condition on the inconsistency. l= 3

5 where e p it = it + ρ ˆρ p RMA yit + ³β WG ˆβ w it. Let us call the resulting estimator ˆφ p j,rma and ˆκ p l,rma.4 We have the following asymptotic properties of the RMA estimators for ρ and φ. Proposition : (Asymptotic Properties of the RMA Estimators). generated by Assumption. Let the observations be (i) For fixed, as, ˆρ p RMA and ˆφ p j,rma are inconsistent where p plim ˆρ RMA ρ = B (ρ, )+O 2, (6) plim ³ˆφp j,rma φ j = 2 B (ρ, )+O 2, (7) B (ρ, ) ( ρ) C (ρ, ) µ ln D (ρ, ) = O > 0, ( ) tx tx C (ρ, )= t 2t hγ (z) h γ (z) h, t= D (ρ, )=( )γ (z) 0 h= and γ (z) h is the covariance between z it and z it+h. X t= h= t {γ (z) Xt 0 2t h= hγ (z) h }, (ii) If log 2 (/) ζ where ζ is a non-zero constant, then as,, ˆρ p RMA ρ B (ρ, ) d 0, ρ 2. (iii)if log 2 (/) 0 as,, then ˆρ p RMA is asymptotically distributed as ˆρ p RMA ρ d 0, ρ 2. (8) he proof is given in Appendix A. As mentioned earlier, the WG estimator for β is consistent and thus its asymptotic properties are omitted from Proposition. 4 In the case of quasi-difference exogenous regressor, we need an additional step to reduce the asymptotic bias for γ. o be specific, let y ++ it = y it ˆρ p RMA yit p ˆφ j= j,rma y it j + ˆβ WG w it + p l= ˆκ l,rmaw it l and running WG on y ++ it = a i + γq it + p it gives ˆγ RMA, the RMA estimator for γ. SeeCase3inSection4fortherelevantMonteCarlo simulation results. 4

6 Remark (Exact Inconsistency Formula for AR() Case) inconsistency for fixed as is given as he explicit formula of the B(ρ, ) plim (ˆρ RMA ρ) = ρ log + B, ( ρ)b 2, ( ) log + ρ 2 B = ρ log, B 2, where B, = t= X t ρt = O (), B 2, =2ρ t= ρ t t 2 ( ρ) 2 = O (). + O 2, (9) See Appendix A for the detailed proof. o evaluate the inconsistency exactly, we plot the inconsistency in Figure. A couple of interesting features emerge from the plot. First, the maximum inconsistency is which occurs when =3and ρ =0.46. Second, as predicted, the inconsistency diminishes as ρ gets closer to unity. However, it is important to note that this small inconsistency would affect the statistical inference when >. Asymptotic Bias Figure : Inconsistency of RMA estimator under AR() and independence Remark 2 (Consistent Cases) here are two special cases when ˆρ RMA becomes consistent for fixed with. he first case is when ρ =, which implies e it = it in (4) and hence E(y it e it )=E(ȳ it e it )=0. he second case is when =2,or =3, for any ρ [0, ). In this 5

7 case, B (ρ, 3) = ( ρ) E (y i y i ) y i + y i2 2 {y i + y i2 } 2 {yi + y i2 } E h(y i y i ) 2 yi 2 + y i2 2 {y i + y i2 } i 2 E yi2 2 = ( ρ) y2 i =0, E h(y i2 y i ) 2i because E y 2 i2 =E y 2 i by the covariance stationarity of yit. Remark 3 (Incidental Linear rends) When the DGP in ()-(2) contains an incidental linear trend, the suggested RMA procedure is no longer valid. 5 While the case of incidental trend is not much studied in the literature of dynamic panel regressions, it is an important issue in the literature of panel unit-root testing. 6 In our treatment of the trend, we consider the latent AR() model of y it = a i + b i t + z it, z it = ρz it + it, which is observationally equivalent to y it = α i + β i t + ρy it + it, where α i = a i ( ρ)+b i ρ and β i = b i ( ρ). Here we briefly review Sul s (2008) detrending method before suggesting a new detrending approach to reducing the bias in finite sample. Let 2ȳ it be the common mean adjustment component. hen, 2ȳ it =2a i + b i t +2 z it so that we have y it 2ȳ it = a i + 2 b i + ρ (y it 2ȳ it )+e τ it, (0) where e τ it = 2( ρ) z it + it. he result is a trendless regression with fixed-effects of a i + 2 b i. he RMA estimator for ρ, ˆρ τ RMA, is obtained from running WG in (0) and its inconsistency is given by plim (ˆρ τ RMA ρ) =G (ρ, )=O ln, () 5 According to the exact inconsistency formula of the WG estimator shown by Phillips and Sul (2007), the first order inconsistency for panel AR() under incidental trend case is 2(+ρ), which is twice as large as the inconsistency under fixed effects, ( + ρ). 6 Shin and So (999a,999b), for instance, suggest a recursive detrending method. heir method, however, does not completely eliminate the incidental trend components in univariate context as noted by Sul, Phillips and Choi (2005). aylor (2002) and Phillips, Park and Chang (2004) propose alternative detrending methods that are effective under unit-root case but are substantially upward biased when ρ<. Sul (2008) also suggests a double recursive mean adjustment method that yields much smaller bias when ρ<. 6

8 where G (ρ, ) is presented in Appendix B2. Although this double recursive demeaning method produces smaller upward bias than the alternative methods adopted in the panel unit-root literature, the bias is still non-negligible when ρ<. Alternatively, we suggest the following bias reduction method. First, use the WG estimator to obtain an initial estimate of the trend coefficient ˆb i from the latent model representation, and ³ construct the detrended ³ observations y it = y it ˆb i t. ext, use pooled least squares to regress y it ȳ it on y it ȳ it, where ȳ it =(t P ) t s= y is and the regression error e it = ( ρ) z it + it b i ρ 2 b i ( ρ) t where b i = ˆb i b i. Call the estimate ˆρ IRD. otice that the trend is not yet completely removed because the remnants of the trend now sit in the regression error. o deal with this, update the trend coefficients by WG estimation of y it ˆρ IRD y it = a i + β i t + ε it and call the estimated trend coefficient ˆβ i,wg 0. Update the trend estimate ˆb 0 i = ˆβ i,wg 0 / ( ˆρ IRD). If ˆρ IRD, set ˆb 0 i = ˆb i. Continue until convergence. For ˆρ IRD, we have the weak convergence result plim (ˆρ IRD ρ) =B (ρ, )+ ( 2ρ) ρ 2 + O 2, (2) 2 where B (ρ, ) is given in Proposition. he proof is presented in Appendix B. In (2), the second term has a faster diminishing speed than that of B (ρ, ), but its magnitude is larger than that of B (ρ, ) even for a moderately large. his inconsistency, however, is far smaller than that of the WG estimator, as illustrated in Figure RMAtau WG IRD RMA without trend Figure 2: Comparison of the absolute asymptotic biases for incidental linear trend case ( =50, = 000) Figure 2 plots the absolute asymptotic bias of ˆρ IRD, ˆρ τ RMA and ˆρ WG for a large under incidental 7

9 linear trends. As mentioned earlier, ˆρ τ RMA produces no asymptotic bias when ρ =, but significantly upward biased when ρ<. Consequently it is even dominated by ˆρ WG when ρ<0.8 as displayed in Figure 2. By contrast, the asymptotic bias of ˆρ IRD is rather modest over the entire range of ρ. Whilst the asymptotic bias of ˆρ IRD varies slightly with and ρ, the size of the asymptotic bias is far smaller than those of ˆρ τ RMA and ˆρ WG except when ρ =. his is what the asymptotic theory in (2) predicts. ote that the asymptotic bias of ˆρ IRD declines with ρ until it is kinked when ρ is near unity. his kink is due to the truncation of ˆρ IRD in the iterative procedure. he figure also compares the asymptotic bias of ˆρ IRD with that of ˆρ RMA when there is no trend. ˆρ IRD also outperforms ˆρ RMA in the no trend case in most regions of ρ. Remark 4 (Weakly integrated and local-to-unity processes) Since the RMA estimator becomes consistent when ρ as shown above, one should anticipate that the estimator exhibits only modest inconsistency under local-to-unity. o confirm this guess, let ρ = c/ α for 0 <α and some c>0. When0 <α<, the process is said to be nearly stationary (Giraitis and Phillips, 2006) or weakly integrated (Park, 2003). When α =, the process is said to be local-to-unity. In either case, we have plim (ˆρ RMA ρ) ln µ + O +α + O 2. It can be seen that the inconsistency is of order O (ln /) so that the first order inconsistency is larger than that of WG estimator. However, as we will show shortly, the overall inconsistency of the RMA estimator is much smaller than that of WG estimator. Remark 5 (GLS Demeaning Procedure) One might wonder how the generalized least squares (GLS) adjustment proposed by Elliott, Rothenberg and Stock (996), originally designed for efficient unit-root tests, would compare to the RMA estimator if it were employed to reduce the bias in the local-to-unity environment. o explore this, let us assume ρ = c. he GLS correction requires a quasi-demeaning of the observations using the factor of 7/. Define ( y g it = y it 7 ( yit if t> y i if t =, Z 7 t = if t> if t =. ext, let P u it = y it yg it Z t. P Z2 t 8

10 hen the estimator ˆρ GLS is obtained by running pooled least squares on u it = ρu it + ε it, where ˆρ GLS =ˆρ long + O p and ˆρ long is the pooled least-squares estimator from regressing (y it y i ) on (y it y i ). For a local parameter value of c =7, the inconsistency in the GLS-corrected estimator becomes plim (ˆρ GLS ρ) = 35 + O 2. (3) As shown in Appendix C, the second order term in (3) includes a large component such as 49/ 2, which causes ˆρ GLS to be inconsistent for moderately as large as 00. A quick evaluation of the asymptotic bias in ˆρ RMA in comparison to ˆρ GLS and ˆρ long is presented in Figure 3 under the simulation environment of = 2000 and c =[0, 4] for {00, 500}. As can be seen from Panel A, the inconsistency of ˆρ RMA is in general smaller than that of ˆρ GLS except when <c<5. Moreover,the dominance of ˆρ RMA over ˆρ GLS gets stronger with larger. As exhibited in Panel B, when =500, the simulated inconsistency of ˆρ RMA is much smaller than that of ˆρ GLS for all c values. In sum, this simulation experiment suggests that the bias reduction provided by RMA is superior to that of GLS estimator when ρ is near unity RMA GLS Long Differencing 45 degree line RMA GLS Long Differencing 45 degree line c=4 c=3 c=2 c= c=0 c=9 c=8 c=7 c=6 c=5 c=4 c=3 c=2 c= c= c=4 c=3 c=2 c= c=0 c=9 c=8 c=7 c=6 c=5 c=4 c=3 c=2 c= c=0 Panel A: = 00, = 2000 Panel B: = 500, = 2000 Figure 3: Comparison of Inconsistencies among ˆρ RMA, ˆρ GLS, and ˆρ long Remark 6 (Heterogeneous Panels) In the current study, the dominant root, ρ, isassumed to be homogeneous across the cross-section. In practice, the homogeneity restriction can be tested using formal inference techniques [e.g. Pesaran and Yamagata (2008)]. If the homogeneity restriction 7 More results on the simulation experiment are available at 9

11 is rejected, one may consider the pooled mean group RMA estimator. Pesaran and Smith (995) study the pooled mean group (PMG) estimator for nonstationary panel data, and Pesaran, Shin and Smith (999) extend the analysis to dynamic panel regressions under cross-sectional independence. Pesaran (2006) also develops the PMG estimator under cross-sectional dependence. In general, the consistency of the PMG estimator requires the sequential limit of first, and next. For fixed and, however, the PMG estimator is inconsistent just like the WG estimator. Since both univariate and pooled RMA estimators are consistent as, the asymptotic property of the PMG-RMA estimator can be derived under sequential limits. A more interesting but also more challenging environment is the case for fixed and in which the asymptotic properties of the PMG-RMA estimator are yet known. We leave this important issue for future research. 3 Asymptotic Properties with Cross-sectionally Correlated Observations In this section we adapt the RMA estimator to dynamic panel data models with cross-sectionally dependent observations where the dependence arises from a common factor specification. he environment under consideration is again the one in which observations are weakly stationary and are generated by the following assumption. Assumption 2. model For i =,... and t =,..., the observations {y it } are generated by the latent y it = μ i + z it, (4) px z it = ρ j z i,t j + u it, (5) u it = j= KX δ si θ st + it = δiθ 0 t + it, (6) s= where the idiosyncratic term, it, is assumed to be 0,σ 2 i and σ 2 i < for all i. Also, θ st and it are assumed to be independent of each other. he common factor has E kθ st k 4 <, plim P t= θ2 st = σ 2 sθ with factor loadings of kδ sik D<, and plim kδ 0 δ/ M δ k = 0 with δ =(δ,...,δ ) 0 for some M δ > 0, where δ i =(δ i,..., δ Ki ) 0. his latent model has the observationally equivalent factor representation y it = μ i + δ 0 if t + z it, (7) 0

12 where F t =(F t,...,f kt ) 0, F st = P j=0 ρj θ st j and z it = P j=0 ρj it j. Phillips and Sul (2007) show that the asymptotic bias of the WG estimator becomes random under the cross-sectional dependence. o derive the asymptotic random bias formula under cross-sectional dependence of y it, we first introduce the following notation. Let ˆρ F,WG = ρ + P Fst F s θst θ s P Fst F s 2 (8) be the WG estimator of ρ from the regression of F st = m s + ρf st + θ st where F s = ( ) P t= F st for s =,...,K. According to Phillips and Sul (2007), the WG estimator ˆρ WG CSD in (4)-(6) can be decomposed into ˆρ WG CSD =( η) ˆρ WG CSI + η ˆρ F,WG + o p where ˆρ WG CSI represents the WG estimator under cross-sectional independence, η = m 2 δ σ2 θ σ 2 + m 2 δ σ2 θ, m 2 P δ =(K) P K i= s= δ2 si, P σ2 θ =(K) K P s= t= θ2 st, and σ 2 = P P i= t= 2 it. Here η can be interpreted as the degree of cross-sectional dependence, such that η =for perfect crosssectional dependence and η =0for cross-sectional independence. Since ˆρ F,WG does not depend on the dimension of cross-section (), the asymptotic bias for ˆρ WG CSD now depends on the inconsistency of ˆρ F,WG. hat is, as shown by Phillips and Sul (2007), plim (ˆρ WG CSD ρ) = ( η) +ρ + η (ˆρ F,WG ρ)+o p, as. Similarly, we can directly apply this approach to the RMA estimator by letting ˆρ F,RMA = ρ + P Fst F st θst ( ρ) F st P Fst F st 2 (9) be the RMA estimator from F st F st = ρ F st F st + εst where ε st = θ st ( ρ) F st and F st =(t ) P t j= F sj. And let ˆρ RMA CSD be the RMA estimator under cross-sectional dependence. hen the asymptotic bias of the RMA estimator can be shown as follows. Proposition 2: ( asymptotic bias under cross-sectional dependence). Under Assumption 2, for first and then, the probability limit of ˆρ RMA CSD is given by plim ˆρRMA CSD ρ =( η) B (ρ, )+η (ˆρ F,RMA ρ)+o p, (20) where B(ρ, ) is shown in Proposition. he proof is presented in Appendix D. ote that the asymptotic bias of ˆρ RMA CSD now depends on

13 three components: (i) the degree of cross-sectional dependence, η; (ii) the inconsistency of ˆρ RMA CSI, the RMA estimator under cross-sectional independence; and (iii) the bias of the time-series estimator ˆρ F,RMA (K =case). Obviously, Proposition applies when η =0, while the inconsistency becomes purely random when η =. AsK, however, the asymptotic bias expression is identical to that in Proposition regardless of the value of η. But the direction of the inconsistency is not clear because ˆρ F,RMA is downward biased for a small while B (ρ, ) is always positive. Unfortunately the exact finite sample bias formula for ˆρ F,RMA is available yet. Figure 4 demonstrates the randomness of the asymptotic bias by plotting the empirical distributions of ˆρ WG CSD and ˆρ RMA CSD for = 2000 and =20with various values of K = {, 0, 50}. When K =, the empirical mean of ˆρ RMA CSD is very close to the true value of ρ =0.5. his is the case when the asymptotic positive bias from pooling is offset by the negative univariate bias of ˆρ F,RMA. As K increases, the empirical distribution of ˆρ RMA CSD becomes tighter but the mean value also increases slightly. By contrast, ˆρ WG CSD has a noticeable downward asymptotic bias for all K although it tends to decrease as K grows. Overall, ˆρ RMA CSD dominates ˆρ WG CSD and the dominance stands out when the factor number is relatively small WG with K 50 RMA with K WG with K 0 RMA with K WG with K RMA with K Figure 4: Random Asymptotic Bias under Cross-sectional Dependence, = 20, = 2000, ρ = 0.5 δ is (0, ), θ st (0, ), it (0, ) 2

14 4 Monte Carlo Experiments In this section, we report the results of Monte Carlo experiments designed to examine the precision and the effectiveness of bias reduction achieved by the RMA estimators in small and moderate samples for ρ [0, ). We vary the environments by the autoregressive order and by the degree of cross-sectional dependence. o economize on space, we have been selective in terms of which results to report especially when log (/) is relatively large because in this case B (ρ, ) remains in the distribution of the RMA estimator. An extensive set of simulation results are available at the author s web site. 8 We consider four cases here. Case : AR() with cross-sectionally independent observations. For this widely studied environment, several bias reduction methods have been proposed. We compare two of these to RMA and WG estimators. he first is the GMM estimator studied by Arellano and Bover (995, hereafter AB) and the other is the estimator proposed by Hahn and Kuersteiner (2002, hereafter HK). he data generating process is, y it = μ i + z it, z it = ρz it + it, where it (0, ), μ i,σμ 2. As in Assumption, the initial observation obeys yi = ³ μ i + z i,σμ 2 +, which produces weakly stationary sequences of y ρ 2 it. We consider sample sizes of {50, 00, 200} and {6,, 2} so that the time series observations used in the regression are 0 = {5, 0, 20}. he asymptotic variance of the AB estimator depends on the nuisance parameter ψ = σ /σ μ = /σ μ, whereas the variances of RMA, WG, and HK do not. o explore the potential small-sample dependence of the AB estimator on the variability of the individual-specific effect, we consider alternative values of relative variance of the individual-specific component of the error term, σ μ {, 5, 0}, or ψ {, 0.2, 0.}. able reports the bias and mean-square error of the four estimators under comparison. he RMA and AB estimators are seen to be upward biased while WG and HK are biased downwards. RMA compares well to HK for small. Although the relative performance of HK improves as grows, when 0 =5for example, the HK estimator bears substantial downward bias ( 0.22 for ρ =0.9) evenwhen is as large as 200. For relatively large, the performance of HK is comparable to that of RMA particularly when ρ is relatively small. he GMM estimator due to AB performs well for ψ =, but its performance deteriorates substantively for ψ =0.2 and ψ =0.. Even for ψ =, it is dominated by RMA for moderate values of ρ. In sum, RMA dominates HK both in terms of attenuating bias and in precision for small and it is typically more precise than AB, whose 8 Full reports (MS excel format) are available at 3

15 performance is quite sensitive to ψ. he dominance of RMA over HK is particularly noticeable for small or for highly persistent ρ when is relatively large. Case 2: AR(2) with cross-sectionally independent observations. Since it is not straightforward to correct for bias with HK or AB in the AR(2) case, we only report the performance results for RMA in comparison with WG. For simplicity but without loss of generality, the DGP for this case is y it = μ i + z it, z it = ρ z it + ρ 2 z it 2 + it, where it (0, ), μ i 0,σμ 2. We consider the lag coefficients (ρ, ρ2 ) {(0.9, 0.2), (0.5, 0.2), (0.9, 0.3), (0.5, 0.3), (0.9, 0.4), (0.5, 0.4)} where ρ = ρ + ρ 2. Here we report the results for (ρ, ρ 2 ) {(0.9, 0.2), (0.5, 0.2)} only because the results of the other cases are largely similar. able 2 reports the bias, variance and mean square error (MSE) of RMA and WG. We note that the bias of WG in the AR(2) case is much more serious than that in AR() environment. For example, when ρ =0.9 and =50, the WG bias for ρ is -0.62, -0.32, -0.6 for 0 =5, 0, and 20 respectively, whereas the biases were -0.47, -0.25, and -0.2 for the corresponding values of 0 in AR() case. As in AR() case, the bias hinges upon rather than. Although the variance of RMA is slightly larger than that of WG, the MSE of RMA is consistently much smaller than that of WG estimator. Case 3: AR() and AR(2) with exogenous regressor. Since Phillips and Sul (2007) show that ˆβ WG is asymptotically unbiased when the exogenous variable enters with a level effect, we concentrate on the following DGP that allows an exogenous variable to enter in difference form, y it = μ i + z it, z it = ρ z it + ρ 2 z it 2 + γq it + it, where it (0, ), μ i (0, ) and q it (0, ). Wesetγ =and report the size of the t-test under the null hypothesis of γ =. Among the various values for the parameters considered, we report in able 3 only the results for ρ =0.9 forthear()caseandρ =0.9 and ρ 2 =0.2 for the AR(2) case (again ρ = ρ + ρ 2 ). Several features of able 3 are noteworthy. First, the WG estimator for γ is biased downwards in both the AR() and AR(2) cases and the bias directly distorts the size of the t-test. Whereas the size distortion for RMA is relatively small and remains fairly constant, the distortion of the WG based t-test increases with. At the nominal size of 0.05, the size of t-test based on WG estimator is as large as 0.94 when = 200 and 0 =5in AR() case, while the corresponding size of t-test based on RMA is merely Second, RMA reduces the bias and variance significantly for all cases 4

16 considered. However, in the AR(2) case for very small, there is some size distortion in the RMA based t-test when is relatively large, mainly due to the large second order bias of RMA. he size distortion, however, diminishes quickly as increases. Case 4: AR() with cross-sectionally dependent observations. he DGP for this case is y it = a i + ρy it + u it, u it = δ i F t + it, where it (0, ),δ i (, ) and F t (0, ). We consider exactly the same simulation environment with that of Case except that the error term is now cross-sectionally correlated as reflected by the common factor F t. able 4 reports the bias and MSE of the four estimators under comparison. As in able, we present the AB estimator for three different values of relative variance of the individual-specific component of the error term. We first note from able 4 that the bias of RMA estimator is smaller for moderate ρ and/or large than when the observations are cross-sectionally independent. his contrasts to the competing estimators whose biases are larger under cross-sectional dependence. his result is consistent with the predictions of Proposition 2 that the bias of RMA (WG) is smaller (larger) under cross-sectional dependence. Consequently, the performance of RMA estimator stands out even when the observations are cross-sectionally dependent. he dominance of the RMA estimator is particularly noticeable when and are moderate. Unlike the case of cross-sectional independence, the RMA estimator continues to dominate the alternative estimators even when and are large. ake = 200 and =20for instance, the HK estimator had a comparable performance to the RMA estimator under cross-sectional independence, but the bias of RMA estimator is now much smaller when observations are cross-sectionally correlated. he story remains much the same in terms of MSE. Although the MSE of RMA estimator is larger than when the observations are cross-sectionally independent mainly due to the increased variance, it decreases more rapidly than the alternative estimators as and grow. As a result, the RMA estimator has smaller MSE than the other estimators particularly when the underlying processes are highly persistent. o summarize, our simulation results suggest that the finite sample performance of the RMA estimator is appealing especially when the observations are cross-sectionally correlated. 5 Conclusion In this paper, we extend the idea of recursive mean adjustment as a bias reduction strategy to estimating the dominant root in dynamic panel data regressions. Specifically we develop the RMA estimators under general AR(p) process under both cross-sectional independence and dependence. 5

17 We show that the RMA estimator delivers effective bias reduction when the observations are independent across individuals. When the observations are correlated across individuals and when this dependence arises from an underlying factor structure, we find that effective bias reduction still can be achieved by using the RMA estimator. Our simulation results based on small and larger suggest that the RMA estimator dominates comparable estimators in terms of bias, variance and MSE reduction both when error terms are cross-sectionally independent and when they are crosssectionally correlated. his finding still holds in the presence of exogenous regressors especially in terms of t-test performance. Overall our method is efficient and effective in reducing bias and more importantly is straightforward to implement. In light of the fact that mean and median unbiased estimators are generally unavailable for higher ordered panel autoregression models, the recursive mean adjustment procedure advocated in this study is believed to fill an important gap in the dynamic panel literature. 6

18 References [] Alvarez J. and Arellano, M. (2003), he ime Series and Cross-Section Asymptotics of Dynamic Data Estimators, Econometrica, 7, [2] Andrews, D.W.K. (993), Exactly Median-unbiased Estimation of First Order Autoregressive/UnitrootModels, Econometrica, 6, [3] Andrews, D.W.K. and Chen, H., (994). Approximately median-unbiased estimation of autoregressive models, Journal of Business and Economic Statistics, 2, [4] Arellano, M. A. and Bover, O. (995), Another Look at the Instrumental Variable Estimation of Error-Components Models, Journal of Econometrics, 68, pp [5] Bai, J. (2003), Inference on Factor Models of Large Dimensions, Econometrica, 7, [6] Bai, J., and g, S. (2002), Determining the umber of Factors in Approximate Factor Models, Econometrica 70, [7] Balestra, P. and M. erlove (966), Pooling cross section and time series data in the estimation of a dynamic model: he demand for natural gas. Econometrica, [8] Blundell, R. and S. Bond (998), Initial Conditions and Moment Restrictions in Dynamic Panel Data Models, Journal of Econometrics, 87, [9] Blundell, R., S. Bond, and F. Windmeijer (2000), Estimation in Dynamic Panel Data Models: Improving on the Performance of the Standard GMM Estimators, Working Paper 00-2, he Institute for Fiscal Studies. [0] Bun, M. and Carree, M. (2005), Bias-corrected Estimation in Dynamic Panel Data Models, Journal of Business and Economic Statistics, 23, [] Campbell, J. and Lapham, B. (2004), Real Exchange Rate Fluctuations And the Dynamics of Retail rade Industries on the US-Canada Border, American Economic Review, 94, [2] Elliott, G.,.J. Rothenberg, and J.H. Stock (996), Efficient ests for an Autoregressive Unit Root, Econometrica, 64(4), [3] Giraitis, L. and P.C.B. Phillips (2006), Uniform Limit heory for Stationary Autoregression, Journal of ime Series Analysis, 27(), [4] Hahn, J., and Kuersteiner, G. (2002), Asymptotically Unbiased Inference for a Dynamic Panel Model with Fixed Effects When Both n and are Large, Econometrica 70(4), [5] Hahn, J., J. Hausman, and G. Kuersteiner (2007). Long Difference Instrumental Variables Estimation for Dynamic Panel Models with Fixed Effects. Journal of Econometrics, 40(2),

19 [6] Kendall, M.G., (954), ote on Bias in the Estimation of Autocorrelation, Biometrika 4, [7] Kiviet, J. F. (995). On Bias, Inconsistency, and Efficiency of Various Estimators in Dynamic Panel Data Models. Journal of Econometrics, 68, [8] Marriott, F. H. C., and Pope, J. A.. (954), Bias in the Estimation of Autocorrelations, Biometrika 4, [9] ickell, S. (98), Biases in Dynamic Models with Fixed Effects, Econometrica 49, [20] Park, J. Y. (2003): Weak Unit Roots, mimeo, Rice University. [2] Pesaran, H. (2006), Estimation and Inference in Large Heterogeneous Panels with Multifactor Error Structure, Econometrica, 74 (4) [22] Pesaran, H., Shin, Y and Smith, R. (999), Pooled Mean Group Estimation of Dynamic Heterogeneous Panels, Journal of the American Statistical Association, Vol. 94, [23] Pesaran, H. and Smith, R. (995), Estimating Long-Run Relationships from Dynamic Heterogeneous Panels, Journal of Econometrics, Vol. 68, [24] Pesaran, H. and Yamagata,. (2008), esting Slope Homogeneity in Large Panels, Journal of Econometrics, Vol 42, [25] Phillips, P.C.B., J. Park, and Y. Chang (2004), onlinear Instrumental Variable Estimation of an Autoregression, Journal of Econometrics 8, [26] Phillips, P.C.B. and Sul, D. (2007), Bias in Dynamic Panel Estimation with Fixed Effects, Incidental rends and Cross Section Dependence, Journal of Econometrics, Vol. 37 (), [27] So, B.S. and Shin, D.W. (999a), Recursive Mean Adjustment in ime-series Inferences, Statistics & Probability Letters, 43, pp [28] So, B.S. and Shin, D.W. (999b), Cauchy Estimators for Autoregressive Processes with Application to Unit-Root ests And Confidence Intervals, Econometric heory, Vol. 5, pp [29] So, B.S. and Shin, D.W. (2002), Recursive Mean Adjustment and ests for onstationarities, Economics Letters, Vol 75, pp [30] Sul, D. (2008), Panel Unit Root ests under Cross Section Dependence with Recursive Mean Adjustment, mimeo, University of Auckland. [3] Sul, D., P.C.B. Phillips, and C.Y. Choi, (2005), Prewhitening bias in HAC estimation. Oxford Bulletin of Economics and Statistics 67,

20 [32] aylor, R.A.M. (2002), Regression-Based Unit Root ests with Recursive Mean Adjustment for Seasonal and onseasonal ime Series, Journal of Business and Economic Statistics, 20,

21 echnical Appendix Define z it = t P t s= z is and ȳ it = t Appendix A: Proof of Proposition P t s= y is where t = t for the notational convenience. First we consider AR() case, and then use the result to establish the proof of Proposition. ote that for general AR(p) case, we have E X i= t= = γ (z) 0 + = γ (z) 0 (z it z it ) 2 t= t= ( γ (z) 0 +2 t ( t γ (z) 0 2 t tx h= t X h= γ (z) h hγ (z) h + 2 t ). tx h= hγ (z) h ) 2 t= ( tx ) γ (z) t h h=0 Hence for AR() case, we have E X i= t= µ (z it z it ) 2 = σ2 log ρ 2 + O σ2 as, (A-) ρ2 where σ 2 = P i= σ2 i and σ 2 i = P t= it < for all i. From a standard central limit theorem for panel autore- AppendixA: ProofofRemark gressive processes, we have X i= d z it it µ0, σ 4 ρ 2. From (A-), we have Ã!" X # X z it ε it (z it z it ) 2 i= i= d 0, ρ 2. ow note that ³ P P i= (ˆρRMA ρ) = z P it it P P P i= (z it z it ) 2 ( ρ) i= (z it z it ) z it P P i= (z it z it ) 2. (A-2) 20

22 From direct calculation, we have where P P B (ρ, )= ( ρ) plim i= (z it z it ) z it P (z it z it ) 2 B = P i= = ρ log + B ( ρ) B 2 log + ρ 2 B = ρ log B 2 t= For calculating its variance, first consider à X E z it z it X t ρt = O (), B 2 =2ρ t= 2 z it! 2 = Hence the variance of the second term in (A-2) is given by herefore, à Var ( ρ) t= + O 2 ρ t t 2 ( ρ) 2 = O (). t σ2 z + O () = O (log ). P! (z µ it z it ) z it log P (z it z it ) 2 = O 2. à r! Ãr! ³ P P i= z it it (ˆρRMA ρ) =O p log + O p 3 + P P i= (z it z it ) 2 q so that as, but log q If / 3 0 but log 0, we have (ˆρRMA ρ) d 0, ρ 2. ζ where ζ is a constant, then we have (ˆρRMA ρ B (ρ, )) d 0, ρ 2 or (ˆρRMA ρ) d ζb (ρ, ), ρ 2. 2

23 Appendix A2: Proof of (6) in Proposition For ease of reference, we restate (4) here as y + it ȳ it = ρ (y it ȳ it )+e it, where Xp e it = ( ρ) z it + it + ³φ j j,wg ˆφ z it j. j= oting that E P t=p z it j z it = O () and plim ³ ˆφj,WG φ j = O, we have P P p plim ˆρ RMA ρ i= = ( ρ) plim [(y it ȳ it ) z it ] P P i= (y it ȳ it ) 2 P P i= h(y it ȳ it ) P p j= ³φ j ˆφ i j,wg z it j + plim P P i= (y it ȳ it ) 2 which establishes (6) in the text. = B (ρ, )+ O D (ρ, ) = B (ρ, )+O 2, Appendix A3: Proof of (7) in Proposition For the simplicity of analysis, we consider an AR(2) case and then show how the logic can be generalized to an AR(p) case. First, consider the regression error of it = it ˆρ p RMA ρ y it. Using the fact that (y it μ i ) (y it 2 μ i )=(y it y it 2 ) where μ i = E (y it ), the inconsistency of the pooled estimator ˆφ p RMA can be written as ˆφ p RMA φ = ˆρ p RMA ρ P i= P (y it y it 2 )(y it y i ) P i= P t= (y it y it 2 ) 2 {z } P P i= + (y it y it 2 )( it i ) P P i= (y it y it 2 ) 2, A where y i and i are time-series averages. As, the term labeled A above has the limiting value of P P P i= plim (y it y it 2 )(y it y i ) ³γ (z) 0 γ (z) P P i= (y it y it 2 ) 2 = 2 P ³ = γ (z) 0 γ (z) 2. 22

24 It follows that plim ³ˆφp φ RMA = 2 plim p ˆρ RMA ρ + O plim X (y it y it 2 )( it i ). i= oting that the AR(2) model has the representation of y it = c λ L c 2 it + λ 2 L it, where λ and λ 2 are the roots of ρ z ρ 2 z 2, and because à E X! à X c s λ j su it j it E X! X c s λ j su it j it j=0 t= j=0 t= µ = c s ( λ ) c s ( λ )+O 2 = O for s =, 2, where c = λ (λ λ 2 ) and c 2 = λ 2 / (λ λ 2 ). It follows that plim or equivalently X i= (y it y it 2 )( it i ) = plim X i= plim ³ˆφp φ RMA = 2 plim p ˆρ RMA ρ + O 2. (y it i y it 2 i ) =O, It is apparent that this logic goes through in the AR(p) case. We can therefore say ³ φ à plim ˆφp RMA = P 2 plim p ˆρ RMA ρ i= + plim = 2 plim p ˆρ RMA ρ + O 2. P! t= z itũ it P P i= t= z2 it Appendix A4: Proof of (8) in Proposition Let ˆρ p RMA ρ = C, D + C 2, D, 23

25 where C, = ( ρ) C, D X = B (ρ, )+O p i= [(y it ȳ it ) z it ]+ µ, à X i=! z it it note that plim B (ρ, )=B (ρ, ), and C 2, = X i= Xp (y it ȳ it ) Since ³φ j ˆφ j,wg and Finally we have X i= j= ˆρ p RMA ρ = C, D q Hence as, but, log Ãr! ³φ j j,wg ˆφ z it j = O p 3. = O p µ + O µ, µ (y it ȳ it ) z it j = O p. ζ, we have à + O p log r! µ + O p. ˆρ p RMA ρ B (ρ, ) d 0, ρ 2 or ˆρ p RMA ρ d ζb (ρ, ), ρ 2. AppendixB:ProofofRemark3(LinearrendCase) Appendix B: Proof of Iterative Recursive Detrending in (2) We address ˆρ IRD by considering the AR() case with incidental trends and then discuss how it can be generalized to the AR(p) case. We work with the latent model and whether z it is observable does not matter for this analysis. he point estimate ˆb i = β i,wg /( ˆρ WG ) becomes equivalent to the point estimate from the regression for moderately large, y it ˆρ WG z it = μ i + b i t + it +(ρ ˆρ WG ) z it. 24

26 It follows that ˆbi b i = P t= ³ t P t= t h³ it P it ³ t P t= ³ +(ρ ˆρ WG ) and by direct calculation, E ³ˆbi b i 2 =2 3 σ2 + O 2. ow let b i ³ˆbi = b i, then ˆρ IRD can be obtained either from y it = y it ˆb i t = μ i + ³b i i ˆb t + z it = μ i b i t + z it z it P z it i P t= t 2, or from y it = μ i ( ρ) b i ρ b i ( ρ) t + ρy it + it. Using the fact that ȳ it = t it follows that y it ȳ it ³y = ρ it ȳ it t X s= y is = μ i + z it 2 b i t, ½ + ( ρ) z it + it b i ρ ¾ 2 b i ( ρ) t, ote that Using these results, we have i= y it ȳ it = 2 b i t + b i +(z it z it ). E t= ˆρ IRD ρ Ã! X ³ = y 2 it ȳ it X i= ³ y it ȳ it ³ y it ȳ it 2 = D (ρ, )+O (). ½ ( ρ) z it + it + b i ρ + ¾ 2 b i ( ρ) t. 25

27 From the direct calculation, we have E (z it z it ) b i t = O, Eb i (t 2) z it = O X, E (b i ) 2 (t 2) = O, but since E (b i ) 2 X Eb i (t 2) t =4σ 2 + O, (t 2) it = σ 2 + O, it follows that ( ρ) 4 E X X (b i ) 2 (t 2) t =( ρ) σ 2 + O, i= 2 Eb i à P! (t 2) it = (t t)( it i ) X P (t t) 2 (t 2) it = 2 σ2 + O. he sum of the O () terms is ( ρ) σ 2 2 σ2 = 2 ( 2ρ) σ2, while the denominator is D (ρ, )= ρ 2 σ2 + O (). It follows that the second stage inconsistency of the IRD estimator is plim (ˆρ IRD ρ) =B (ρ, )+ 2 ( 2ρ) D (ρ, ) + O 2. Appendix B2: Proof of Double Recursive Detrending in () obtain the inconsistency of ˆρ τ RMA as From direct calculation, we plim (ˆρ τ RMA ρ) =G (ρ, )= C F / A B / ρ where A = +4 µ 2 ρ 2 µ t ( ρ) 2 t 2ρ ρt ρ 2 4 µ µ ρ t ρ t, 26

28 µ B = + 2ρ ρ 2 X 4 4 k= P 2 k= ( ρk ) k ρ j j= s=k+j 2 +4 X s j= s=j k=2 X 2 k X 4 s k= +8 j= k ρ k j= ρ k j s=j s s=j, s s=j+(k ) s à X µ 2 ρ 2 µ C = ρ +4 t ( ρ) 2 t 2ρ! ρt ρ 2 à X µ µ ρ t µ µ ρ t! 2 + ρ, ρ ρ k= t ³ F = +2 P P j j=2 t= ρt + ρ 2 +4 X k +2 ρ k X s s s j= s=j k=2 j= s=j s=j+(k ) 2 k X 2(+ρ) + ρ j X 2 k X + ρ k j s s 2 ρ X 3 2 ρ j j=0 i=j+2 s=i j= s=k+j Appendix C: Proof of Remark 5 (GLS-Demeaning) Assume the true DGP is given by s. t k= j= y it = a i + z it, z it = ρ z it + it, ρ = c. s=j Define and y g it = ( y it c ( yit if t> y i if t =, Z t = P u it = y it yg it Z t. P Z2 t c if t> if t =, ote that Zt 2 =+ t= ( a) 2 = =

29 From direct calculation, we have P u it = y it yg it Z t P Z2 t z i 2 +7(7 c) = y it a i = z it z i + i ³ P ³ it z +7 2 P it where ³ 49 7(7 c) P ³ it z +7 2 P it i = 2 z i Hence as, we may say that à i = 49 z 7(7 c) i! à z it 7! it + O p 2. Let ˆρ GLS = P P u it u it P P u 2 it = C D. ow evaluate at c =7. hen we have à i = 49 z i 7! it + O p 2 and " plim C = E " E X i= X i= = I + II + III + IV # " (z it z i )(z it z i ) + E # " i (z it z i ) E X i= X 2 i i= # # i (z it z i ) where " X II = E 2 i i= # = E X i= à à 49 z i 7 it!! 2 = 492 σ 2 z +49 σ2 + O 2 = O, 28

30 " X III = E i= =49σ 2 z 49 σ2 z " X IV = E Ã 49 z i 7 µ ρ i= ρ =49σ 2 z +7σ 2 + O, Ã! it + O p 2! # (z it z i ) + O =49σ 2 z + O, Ã Ã 49 z i 7! it + O p 2! # (z it z i ) and " plim D = E " 2 E X i= X i= = I 2 + II +2 III. # " (z it z i ) 2 + E # i (z it z i ) X 2 i i= # ote that the two terms, I and I 2 willbegivenin(a-3)and(a-4),respectively. ow consider C plim = I +98σz 2 +7σ 2 + O D I 2 +98σz 2 + O (. ) ote that we can define the long-differencing regression as so that we have z it z i = ρ (z it z i ) ( ρ) z i + it, ˆρ long = P P i= (z it z i )(z it z i ) P (z it z i ) 2 = ρ ( ρ) P i= P P i= (z it z i ) z i P (z it z i ) 2, P i= and plim ˆρ long = ρ + ρ 2, since with a moderate large " # X E (z it z i ) z i = σz ( 2 ) i= ρ, (A-3) ρ 29

31 " X E i= herefore, for a large we have Plugging ρ = 7/ yields # (z it z i ) 2 =2σz ( 2 ) C plim = 2σ2 zρ+( ρ) σz 2 +98σz 2 +7σ 2 D 2σz 2 +98σz 2 + O 2 = ρ +( ρ) ( + ρ) + + O plim (ˆρ GLS ρ) =( ρ) ( + ρ) + + O = 35 + O 2. Appendix D: Proof of Proposition 2 ρ. (A-4) ρ According to Phillips and Sul (2007), the WG estimator ˆρ WG CSD of the model (4)-(6) can be decomposed into ˆρ WG CSD =( η) ˆρ WG CSI + ηˆρ F,WG + o p, where ˆρ WG CSI is the WG estimator when the observations are cross-sectionally independent, η = m 2 δ σ2 θ σ 2 + m 2 δ θ σ2, m 2 δ = (K) P P K i= s= δ2 si, σ2 θ = P (K) K P s= t= θ2 st, and σ 2 = P P i= t= 2 it. η has the interpretation of the degree of cross-sectional dependence. When η =, the observations are maximally dependent and when η =0they are independent. Since ˆρ F,WG does not depend on, the asymptotic bias for ˆρ WG CSD is seen to depend on the inconsistency of ˆρ F,WG. As shown by Phillips and Sul (2007), plim (ˆρ WG CSD ρ) = ( η) +ρ + η (ˆρ F,WG ρ)+o p, as. he strategy of the proof follows the strategy taken by Phillips and Sul (2007). For the inconsistency for WG estimator, see Proposition 3 in Phillips and Sul (2007). Here we take K =case and then provide inconsistency for RMA estimator under cross-sectional dependence. For RMA, define C C plim ˆρ RMA CSD = ( ρ) plim D C, where C C ( ρ) P P i= (y it ȳ it ) z it + P P i= (y it ȳ it ) u it, D C P i=p t= (y it ȳ it ) 2. 30

32 Further decompose B (ρ, ) into with B (ρ, )=plim ˆρRMA CSI ρ = B C = ( ρ) plim P P i= t= (z it z it ) z it B P D plim P i= t= (z it z it ) 2. In the single factor(k =) case, the latent model representation is Since y it ȳ it = z it z it, we have plim CC P = ( ρ) plim i= = ( ρ) plim y it = μ i + z it,z it = ρz it + u it, u it = δ i θ t + it, X X z it = δ i ρ j θ t j + ρ j it j = δ i F t + x it. j=0 j=0 P (y it ȳ it ) z it P h P i= t= (δ if t + x it ) P δ i Ft + x it i 2 t= δi Ft + x it P nh = ( ρ) plim P i= t= x it x it P i h t= x2 it + δi 2 P t= F t F t P F io t= t 2 h = B C + m 2 P δ t= F t F t P F i t= t 2 because all probability limits of the cross product terms are zero. hat is, Eδ i x it =Eδ i x it =0. Similarly, we have P plim DC = plim P i= (z it z it ) 2 P = plim P i= t= (x it x it ) 2 P + plim P i= δ2 i t= Ft F 2 t = B D + m 2 δp t= Ft F 2 t. Combining the two results gives plim CC plim DC = h B C ( ρ) m 2 P δp t= t= F t F t P F i t= t 2 h P B D + m 2 δ t= Ft F i 2. t Let to have P t= Ft F 2 σ 2 t = θ ρ 2 + O p ³ /2. 3