Bootstrap Based Bias Correction for Homogeneous Dynamic Panels*

Transcription

1 FACULTEIT ECONOMIE EN BEDRIJFSKUNDE HOVENIERSBERG 24 B-9000 GENT Tel. : 32 - (0) Fax. : 32 - (0) WORKING PAPER Bootstrap Based Bias Correction for Homogeneous Dynamic Panels* Gerdie Everaert 1 Lorenzo Pozzi 2 October /263 *Thanks to Ferre DeGraeve for useful comments and suggestions. We wish to acknowledge support from the Interuniversity Attraction Poles Program - Belgian Science Policy, contract no. P5/21. 1 Gerdie.Everaert@ugent.be, SHERPPA, Ghent University 2 Lorenzo.Pozzi@ugent.be, Fund for Scientific Research (Flanders, Belgium) and SHERPPA, Ghent University. D/2004/7012/49

2 Abstract The within or least squares dummy variable estimator is severely biased in homogeneous dynamic panel models with moderate T. We present a bias correction for this estimator based on an iterative bootstrap procedure. Monte Carlo simulations show that this procedure is a good alternative for the analytical correction by Kiviet (1995, JE). The bootstrap (i) improves on the analytical correction when the variance of the individual effects increases, (ii) is straightforward to extend to less restrictive settings and (iii) allows for a correction of the longrun coefficient that is independent of the correction of the short-run coefficients. JEL Classification: C13, C23 Keywords: Bias correction, within estimator, dynamic panel, GMM estimator, Monte Carlo simulation, Bootstrap

3 1 Introduction It is a well-known result that the within estimator or least squares dummy variable (LSDV) estimator in homogeneous dynamic panels is unbiased only when the time dimension (T ) grows large. With moderate T, the bias of the LSDV estimator can be substantial. Nickell (1981) gives an analytical derivation of the bias and shows that the bias on the coefficient of the lagged dependent variable (i) is negative if the population value of this coefficient is positive (ii) is increasing in the value of the population parameter and (iii) does not disappear when the cross-section dimension (N) grows large. Alternative dynamic panel data estimators are the first-differenced Generalised Method of Moments (GMM) estimator of Arellano and Bond (1991) and the system GMM estimator of Arellano and Bover (1995). These estimators are conceived for typical microeconomic panels as they are only consistent for N. Therefore, they are of less interest in a typical macroeconomic panel, i.e. when T is large relative to N. Moreover, a large time dimension necessitates the use of a large number of instruments. This may lead to biases in panels with low or moderate N. 1 In practice researchers often (i) use the first-differenced GMM estimator and/or the system GMM estimator when N is large, i.e. in a typical microeconomic panel, and (ii) use the LSDV estimator when T is large, i.e. in a typical macroeconomic panel. With respect to this practice, two notes should be made. First, Judson and Owen (1999) use Monte Carlo simulations to show that the bias of the LSDV estimator can be sizeable, i.e. as high as 20% of the true value of the parameter, even for T =30. Therefore, they recommend against using the LSDV estimator 1 Tauchen (1986) demonstrates a bias/efficiency trade-off for GMM estimators in a time series context. When the sample size is small, an increase in the number of moment conditions increases efficiency but may also lead to biased estimates. Biases are caused because the optimal set of moment conditions may contain instruments dated far into the past that have low correlation with the instrumented variables (see Nelson and Startz (1990) ) or because there is a correlation between the sample moments and the sample weight matrix (see Altonji and Segal (1994)). The recommendation in these cases is to use a suboptimal instrument set (i.e. a subset of the total number of instruments available). 2

4 even in typical macroeconomic panels. Second, Monte Carlo simulations (see e.g. Arellano and Bond (1991) and Kiviet (1995)) show that the LSDV estimator, although severely biased, has a relatively small standard error in comparison with the various GMM estimators. Given this relatively small dispersion, Kiviet (1995) proposes an analytical bias correction for the LSDV estimator. This corrected estimator works remarkably well in terms of bias reduction while maintaining the high degree of efficiency of the uncorrected LSDV estimator. From their Monte Carlo simulations, Judson and Owen (1999) conclude that this corrected LSDV estimator consistently outperforms the other estimators. 2 It is less practical, though, as it has been devised under a number of restrictions. For instance, the implementation of the corrected LSDV estimator requires a balanced panel, at most one lag of the dependent variable, strongly exogenous explanatory variables, no heteroscedasticity over time or across countries and independence of errors across cross-sections. More recent papers somewhat relax these restrictions (see also Baltagi and Kao (2000)). For instance Kiviet (1999) allows for weak instead of strong exogeneity of explanatory variables. Bun (2003) considers an analytical bias correction for the LSDV estimator with a nonscalar disturbance covariance matrix. To obtain unbiased estimates under even more general assumptions however, researchers must derive a proper analytical correction. This may not be as straightforward for the applied researcher as it is for the econometrician. The purpose of this paper is therefore to introduce an iterative bootstrap algorithm as an alternative for the analytical bias correction proposed by Kiviet (1995). Starting from the biased LSDV estimates, the basic idea is to search over the parameter space until we find values for the set of unknown population parameters that give us the original biased LSDV estimates as average LSDV estimates when sampling using these population parameters. These coefficients can then be considered to be unbiased estimates for the true population parameters. The search over 2 Note that Judson and Owen (1999) do not include the system GMM estimator in their simulations. 3

5 the parameter space is computationally implemented through an iterative bootstrap method. A highly similar bootstrap procedure for bias correction in time series autoregressive models can be found in Tanizaki (2004, Chap. 5). A clear advantage of this bootstrap approach over the analytical correction initiated by Kiviet (1995) is that it is straightforwardly extendable to settings with less restrictive assumptions. As it is often of interest to estimate the long-run effect of the exogenous variables, we also focus on the bootstrap estimation of the long-run coefficients. Given its relevance it is somewhat surprising that little attention has been paid to this issue in existing Monte Carlo studies. Pesaran, Smith, and Im (1996) investigate the small sample properties of various estimators of the long-run coefficients in a dynamic heterogeneous panel data context. They discuss a number of cases where the mean group estimator of the long-run coefficient (that is, the mean of the long-run coefficients of all cross-sections in the panel) is biased. Pesaran and Zhao (1999) examine the effectiveness of alternative bias correction procedures in reducing the small sample bias of the mean group estimator for the long-run coefficient. Their Monte Carlo results show that using a bias corrected estimator of the short-run coefficients (for instance one obtained from an analytical correction) to calculate a bias corrected long-run coefficient, which is referred to as the "naive" approach, fails in all cases. This is due to the nonlinear dependence of the long-run coefficients on the short-run coefficients. A bootstrap approach for correcting the long-run coefficient, on the other hand, improves significantly on this "naive" approach. In the homogeneous panels that we consider in this paper, these nonlinear dependencies are also present. If the short-run coefficients are corrected through an analytical correction like the one by Kiviet (1995), only the "naive" approach can be used to correct the long-run. With our bootstrap approach, however, we can correct the bias in the long-run coefficient in a more direct manner along the lines suggested by Pesaran and Zhao (1999). 4

6 Monte Carlo simulations are used to examine the small sample properties of the bootstrap corrected LSDV estimator in terms of bias and efficiency in a homogeneous dynamic panel model with one lag of the dependent variable and one exogenous variable. The following alternative estimators are included as a benchmark: (i) the uncorrected LSDV estimator (ii) the analytically corrected LSDV estimator by Kiviet (1995) (iii) the first difference GMM estimator by Arellano and Bond (1991) and (iv) the system GMM estimator by Arellano and Bover (1995). With respect to the short-run coefficients, three important conclusions stand out. First, as already emphasized by Judson and Owen (1999), the LSDV estimator can be severly biased even for moderate T. Second, the performance of the iterative bootstrap approach and the analytical bias correction are very similar, i.e. both are able to correct (to some extent) the bias of the LSDV estimator while preserving its small dispersion. The main differences are that (i) the bootstrap correction performs better if the relative variance of the individual-specific componentsishigh and (ii) the analytical correction tends to perform better when N increases. Third, the bootstrap and analytical correction tend to outperform the various GMM estimators in cases where T is not small but moderate. In cases with moderate T and moderate N, the GMM estimators are typically more biased compared to the bootstrap and analytical correction. In cases with moderate T and large N, the system GMM estimators are less biased but are relatively inefficient so that the bootstrap and analytical correction perform better than the system GMM estimator in some of these cases as well. With respect to estimating the long-run coefficients, we find no significant difference between the performance of the bootstrap procedure versus any of the other ("naive") estimators. Further, as also noted by Pesaran and Zhao (1999), the long-run coefficient is often poorly estimated using any of the considered estimators when the coefficient of the lagged dependent variable is high, e.g. around 0.8. (even in cases where the coefficients on the lagged dependent and exogenous variable are well estimated). 5

7 The paper is organized as follows. In section 2 we present a bootstrap approach for correcting the biased LSDV estimator. In section 3 we outline the Monte Carlo set-up we use to investigate the performance of this bootstrap based bias corrected LSDV estimator. In section 4 we compare the performance of our corrected LSDV estimator in terms of bias reduction and efficiency with the performance of other homogeneous dynamic panel data estimators. Section 5 concludes. 2 Bootstrapbasedbiascorrectionforthewithinestimator 2.1 Model and assumptions We consider the following homogeneous dynamic panel model with individual effects, y it = γy it 1 + βx it + η i + ε it (1) for i =1,...,N (cross-sectional dimension) and t =2,...,T (time dimension) and where y it is the dependent variable, x it is an explanatory variable, η i is the unobserved heterogeneity or individual effect with η i i.i.d.(0,σ 2 η) (with σ 2 η > 0), and ε it is an unobserved disturbance term with ε it i.i.d. (0,σ 2 ε) (with σ 2 ε > 0). We assume there is a dynamically stable relationship between y it and x it so that γ < 1. We further assume that, E(ε it ε js )=0(for i 6= j ) E(η i η j )=0 (for i 6= j) E(η i ε jt )=0 ( i, j, t) E(x it ε js )=0( i, j, t, s) The first assumption states that the errors are mutually uncorrelated over time and across cross-sections. The second and third assumptions state that the individual effects are uncorrelated and exogenous. The last assumption states that x it is strongly exogenous. We do not 6

8 make an assumption about E(x it η j ) ( i, j, t) because our bootstrap corrected estimator and the other estimators considered in this paper are appropriate whatever its value (i.e. all can be applied to "fixed effects" models). Note that even though the assumptions underlying the model we use may appear quite restrictive, the bootstrap corrected within estimator discussed in this section could, conditional on investigating its properties, quite easily be extended to a much larger amount of cases, e.g. to cases where x it is not strongly but weakly exogenous (in the sense of being affected by lagged values but not by current or future values of y it ), to unbalanced panels, to cases where higher order lags of y it and x it are present, to cases where x it is a vector instead of a scalar, Bias correction Let δ be the vector of unknown parameters, δ =(γ,β) 0 in eq.(1), and b δ be the LSDV estimate of δ using the data (y it,y it 1,x it ) for i =1,...,N and t =2,...,T. We know that in dynamic panels, b δ is biased, i.e. ³ E bδ Z + zf bδ (z) dz 6= δ, (2) where f bδ ( ) is the probability distribution of b δ. Now suppose that, through repeated sampling from the population described in eq.(1), we n areabletogenerateasequence, bδ 1 (δ),..., b δ o J (δ), of J biased estimates of δ. From this sequence of estimates, the integration in eq.(2) can be written as, ³ E bδ = lim J 1 J JX j=1 b δ j (δ) (3) 7

9 From eq.(3) it is clear that δ is an unbiased estimator for δ if it satisfies the following equation: b δ = lim J 1 J JX j=1 b δ j δ (4) i.e. if we would sample repeatedly from a population with parameters δ and calculate the LSDV estimate b δ j δ in each sample, δ is an unbiased estimator for δ, given(yit,y it 1,x it ), if the average of b δ j δ over J iterations corresponds to the LSDV estimate b δ of δ basedontheoriginal data. 3 For practical purposes, a bias corrected estimate for δ can be obtained by searching over the parameter space until a set of parameters δ is found that satisfies eq.(4). In this paper, this search is implemented through an iterative bootstrap algorithm. The core of this algorithm consists of a bootstrap procedure which simulates the distribution of the LSDV estimator when sampling from eq.(1) with some vector of known parameters, say e δ. The purpose of the iteration around this bootstrap procedure is to find a set of parameters e δ that gives us the original biased LSDV estimates b δ as the average, denoted by e δ b, of the bootstrap LSDV distribution of e δ.the biased LSDV estimate b δ can be thought of as being our first guess for the vector of population parameters δ. Thus, we initialise the algorithm by setting e δ = b δ. After each iteration, we evaluate e δ as an estimate of δ. If e δ is to be an unbiased estimate of δ, e δ b should equal b δ,i.e. ω = b δ e δ b =0. If this condition is satisfied, e δ is taken to be the unbiased estimate δ for δ defined in eq.(4). If this condition is not satisfied, e δ is updated as e δ = b δ + ω and we iterate over the bootstrap procedure and update e δ until this condition is satisfied. More formally, the iterative bootstrap procedure is described by the following steps: 1. Obtain b δ by applying the LSDV estimator to the original data and initialise the bootstrap 3 See Tanizaki (2004, Chap. 5) for a similar expression of an unbiased estimator in time series AR models. 8

10 ³ procedure by setting e δ = b δ,with e δ = eγ, β e. 2. Calculate the individual effect eη i as: Ã eη i = 1 X T y it eγ T 1 t=2 TX y it 1 β e t=2! TX x it t=2 (5) 3. Calculate the residuals bε it as: bε it = y it eγy it 1 e βx it eη i (6) and rescale bε it to obtain eε it (described below). 4. Choose the number of bootstrap samples B (we set B = 2500 in all cases), and proceed as follows in bootstrap sample j, withj =1,...B: (a) For each cross-section i, draw (with replacement) a sample eε b it of size T from the rescaled residuals eε it. (b) Calculate a new sample y b it for the variable y it as follows: y b it = eγy b it 1 + e βx it + eη i + eε b it (7) with initialisation y b i1 = y i1. (c) Obtain e δ b j = ³ eγ b j, β e b j by applying the LSDV estimator to the data yit b,yb it 1,x it. 5. Calculate the averages of e δ b j over the B bootstrap samples as: e δ b = 1 B BX j=1 ³ b e δ eδ j (8) 9

11 6. Calculate ω as: ω = b δ e δ b (9) i.e. ω measures how far the average of the bootstrap LSDV distribution of e δ is from the original LSDV estimates b δ. 7. Update e δ by: e δ = b δ + ω (10) Repeat steps 2-7 until e δ is stable, i.e. ω = Take e δ as the bias corrected estimator δ for δ definedineq.(4). In step 3, the residuals are rescaled since the variance of the unadjusted residuals is too small (see e.g. Johnston and Dinardo (1997, Chap. 11)). Define a 1 2 matrix W it as W it =(y it 1, x it ). Then note that the LSDV residual has a variance given by: E bε 2 it =(1 hit ) σ 2 (11) with h it = f W it ³ fw 0f W 1 fw 0 it where f W it is a 1 (2 + N) matrix given by fw it =(W it,d it ) where D it =(δ i1,...,δ in ) with δ ij =1for i = j and δ ij =0for i 6= j and where the (N(T 1)) (2 + N) matrix f W is given by, 4 We set the convergence criterion to and the maximum number of iterations to 50. We encountered no convergence problems. 10

12 0 fw = W11 f... W1T f 1... WN1 f... WNT 1 f Therefore, the residuals are rescaled as: eε it = bε it p (1 hit ) The bootstrap procedure outlined above corrects the LSDV estimates of the short-run coefficients γ and β. It is however often of interest to estimate the long-run impact θ = β 1 γ of the exogeneous variable x on the dependent variable y as well. The Monte Carlo experiments of Pesaran and Zhao (1999) show that, due to nonlinear dependencies, a procedure which corrects the long-run coefficients by using a bias corrected estimator of the short-run coefficients (i.e. what they call the "naive" approach) does not lead to the largest possible bias reduction. They propose analytical and bootstrap procedures to calculate the bias of the estimates of the long-run coefficients which can then be used to construct a bias corrected estimator. It is straightforward to implement their bootstrap correction for the long-run parameter in the algorithm outlined above by conducting the following steps: Calculate the LSDV estimate of the long-run impact of the exogenous variable x as b θ = and move immediately to the last iteration (for which ω =0). b β 1 bγ At the beginning of the last iteration (which starts at step 2) calculate θ = From e δ b j = ³ eγ b j, β e b j obtained in step 4(c) calculate θ b = 1 P B B j=1 eβ b j 1 eγ b j. e β 1 eγ. Calculate the unbiased estimator for the long-run coefficient as θ c = b θ + θ θ b. Note that the bootstrap corrected estimator for the long-run coefficient by Pesaran and Zhao (1999) is not iterative and is thus obtained in one step so that θ = b θ and θ c =2 b θ θ b. Note 11

13 also that a "naive" estimator for the long-run coefficient can be calculated from our bootstrap as θ n = β 1 γ where β and γ are the bootstrap corrected short-run parameters obtained at the end of our iterative bootstrap procedure. 3 Monte Carlo design 3.1 Data generation To compare the performance of our bootstrap bias corrected within estimator with different existing panel data estimators in terms of bias and efficiency, we conduct a Monte Carlo simulation. This simulation is based on Kiviet (1995). As before we consider the dynamic panel data model given by eq.(1) and we maintain the same assumptions discussed in section 2.1. We now also assume that η i and ε it are normally distributed. Further, we assume that the variable x it is generated through, x it = ρx it 1 + ξ it (12) where ρ < 1 and ξ it i.i.d. N(0,σ 2 ξ ) (with σ2 ξ > 0). Data for x it and y it are generated through eqs.(12) and (1) under different parameter combinations (the number of Monte Carlo samples equals 1000 in all cases). The initial values for x it and y it are set to their unconditional means, thus x i1 =0and y i1 =(1 γ) 1 η i.thefirst 50 observations are discarded from the generated samples. Besides the normalizing restriction σ 2 ε =1, the following restrictions are also imposed on the data generating process, β =1 γ σ η = µσ ε (1 γ) (where µ>0) The first restriction implies that the long-run multiplier of x it with respect to y it is unity, 12

14 while the second restriction makes it possible to control the relative impact on y it of the disturbance versus the individual effect (through a value for µ). Finally, we calculate the signal-to-noise ratio in our model since Kiviet (1995) argues that varying this ratio significantly alters the relative biases of the estimators. Define v it y it (1 γ) 1 η i and substitute this into eq.(1) to obtain v it = γv it 1 + βx it + ε it. The signal-to-noise ratio, which measures the usefulness of x it and of past values y it to explain y it,isthengivenby, σ 2 s = var(v it ε it ) = β 2 σ 2 ξ 1+(γ + ρ) 2 (1 + γρ) 1 (γρ 1) γ 2 ρ 2 1 +(1 γ 2 ) 1 σ 2 εγ 2 When generating data we control σ 2 s by setting it at a number of values Estimation Given the generated data we then estimate eq.(1) using the bootstrap bias corrected within estimator (LSDVb) which we discussed in section 2.2. In this section we discuss the other panel data estimators that we use in our Monte Carlo simulation: the classical within estimator (LSDV) on which our bootstrap is based, the first difference GMM estimator (GMMd) by Arellano and Bond (1991) and the system GMM estimator (GMMs) by Arellano and Bover (1995). For a discussion of the analytical bias correction for the within estimator (LSDVa), we refer to Kiviet (1995). Suppose we collect the time series for the lagged dependent variable and the exogenous variable in a (T q) 2 matrix W i where q is minimally equal to 1 and further depends on the data transformations and the number of instruments used in the estimation (see below). As before, we define the parameter vector δ =(γ,β) 0. The LSDV, GMMd and GMMs estimators 5 Note that some parameter configurations are unfeasible if they imply σ 2 ξ < 0 13

15 all have the following form, where A N = b δ = "Ã N X i=1 W 0 i Z i! A N Ã N X i=1 Z 0 iw i!# 1 Ã X N! Ã N! X Wi 0 Z i A N Ziy 0 i i=1 i=1 (13) ³ N 1 P N i=1 Z0 i H iz i 1 and where W i and y i denote some transformation of W i and y i : deviations from individual means for LSDV, first differences for GMMd and combinations of first differences and levels for GMMs. In the first two cases W i and y i are (T q) 2 and (T q) 1 matrices respectively, in the latter case these are (2(T q)) 2 and (2(T q)) 1 matrices respectively. Z i is a matrix of instruments of dimension (T q) K for LSDV and GMMd and dimension (2(T q)) K for GMMs (where K is the number of moment conditions). For the LSDV estimator W i and y i contain deviations of individual means, K =2, q =1, H i = I T q (where I is an identity matrix) and Z i = Wi. For GMMd and GMMs we always report the second step estimates, i.e. estimates obtained using H i = v i v 0 i where v i are consistent estimates of the first step residuals. 6 These are obtained by using eq.(13) with a specific firststep weighting matrix H i = H (where we refer to Arellano and Bond (1991)and Arellano and Bover (1995) for the exact form of H). For GMMd both the first and second step estimators are asymptotically equivalent if ε it are i.i.d. For GMMs, on the other hand, the second step estimator is asymptotically more efficient (regardless whether ε it are i.i.d. or not). For both GMMd and GMMs we consider two instrument sets. GMMd1 is estimated with y it 2 and x it for each t = 3,...,T implying q = 2 (since we do not include periods where instruments are available only for x it )andk =2T 4. GMMs1 has the same instruments for the first difference part of the system and has y it 1 for each t =3,...,T for the levels part 6 First differenced residuals for GMMd and first differenced and level residuals for GMMs 14

16 of the system implying q =2and K =3T 6. 7 GMMd3 is estimated with y it 2, y it 3, y it 4 (if available) and x it, x it 1 and x it+1 for each t =3,...,T (where obviously only x it, x it 1 are included for t = T ) implying q =2and K =6T 16. GMMs3 has the same instruments as GMMd3 for the first difference part of the system and has y it 1 for each t =3,...,T for the levels part of the system implying q =2and K =7T 18. As noted before, the process of generating data and estimating eq.(1) on the generated data (using the various estimators) is repeated 1000 times. The obtained distributions of the estimators for γ and β are used to calculate quantities like bias and root mean squared error which summarize the performance of these estimators. The performance results are presented in section 4. 4 Performance results In this section we investigate and compare the performance of the different dynamic panel data estimators discussed in the previous sections. In tables 1 to 5 we present the performance results of these estimators when estimating the coefficient on the lagged dependent and on the exogenous variable. We discuss these results in sections 4.1 to 4.4. For first-difference GMM we only report the results for GMMd3. The results for GMMd1 are available upon request. In table 6 and in figures 1-5 we present a number of performance results of the estimators when estimating the long-run impact of the exogenous variable. These results are discussed in section 4.5. y i1 x i y i2 x i For instance, for T =5GMMs1 has Z i = y i3 x i y i y i y i4 15

17 4.1 Results known from the literature From tables 1-5 a number of conclusions known from the existing literature can immediately be drawn. First, the within estimator LSDV for γ is severely biased and this bias disappears as T grows larger but not as N increases (see Nickell (1981)). Second, the within estimator LSDV for β is much less biased (see Kiviet (1995)). Third, except for low T (T =5), the Kiviet correction LSDVa works remarkably well to reduce the bias of γ while maintaining the low dispersion associated with the uncorrected within estimator (see Judson and Owen (1999)). Fourth, the bias in LSDVa disappears as T increases but also diminishes as N grows large since the approximation error is O(N 1 T 3/2 ) (see Kiviet (1995)). Fifth, as noted by Blundell and Bond (1998), the first difference GMM estimator GMMd3 performs quite poorly in terms of bias and dispersion when the persistence in the series becomes higher (compare tables 1 and 3 where γ =0.8 with tables 2 and 4 where γ =0.6). The system GMM estimators GMMs1 and GMMs3 clearly deal with this problem and are in general to be preferred over the the GMMd estimator. 8 Sixth, if we compare the two system GMM estimators, GMMs1 and GMMs3, we can see that the higher efficiency of the latter (caused by the use of more information) is, at least in the cases in tables 1,2 and 5, compensated by the lower bias in the former (i.e. the bias-efficiency trade-off reported by Tauchen (1986) and Altonji and Segal (1994)) Performance of the corrected LSDV estimators Comparing the performance of the bootstrap correction LSDVb to the Kiviet correction LSDVa the following conclusions can be drawn. First, table 1 (with γ =0.8) and table 2 (with γ =0.6) 8 The initial conditions that must be fulfilled to use the system GMM estimator are satisfied under the parameter values chosen in our simulations (see Blundell, Bond, and Windmeijer (2000)). 9 In tables 3 and 4 where µ =5we find that the biases in the GMMs estimators for γ are not negative but positive. For most cases here we find that the GMMs3 estimator performs better than GMMs1 not only in terms of dispersion but also in terms of bias. 16

18 show that in cases where N =20the performance of the LSDVb estimator is very similar to that of LSDVa in terms of bias, standard deviation and thus root mean squared error (RMSE). Second, in contrast to LSDVa, LSDVb does not improve as N grows larger. Thus in the cases where N =100in tables 1 and 2, LSDVa performs better than LSDVb (especially in table 1whenT =5). Third, tables 3 and 4 (where µ =5instead of µ =1) show that LSDVb outperforms LSDVa when the relative variance of the individual-specific effects is higher (except in the case where N = 500). 4.3 Performance of the GMM estimators If we look at the performance of the GMMs estimators in terms of RMSE we find it quite surprising that they work reasonably well in moderate N samples. As far as we know this performance of the system GMM estimator in moderate samples has not been reported before in the literature. Existing studies investigating the performance of the system GMM estimator like Blundell and Bond (1998) and Blundell, Bond, and Windmeijer (2000) investigate cases where N is minimally equal to 100. Even though the system GMM estimator does not perform badly in typical macroeconomic samples (where N =20with T =10, T =20or T =40) the corrected LSDV estimators LSDVa and LSDVb perform unambiguously better in all these cases. As is to be expected the GMMs estimators are better alternatives when N increases. Note however that even for large N the corrected LSDV estimators, LSDVa and LSDVb, are preferred over the system GMM estimator when T is not small but moderate (e.g. T =10 or T =20). In tables 2 to 5 for instance, LSDVb outperforms the system GMM estimator in the case N = 100 and T =10. This is due to the relative inefficieny of instrumental variable estimators in comparison to least squares estimators. 17

19 4.4 Other results In small T,smallN samples (case N =20, T =5) all estimators perform badly when γ = 0.8 (table 1). Given the high persistence in the data, more data are needed to estimate the parameters precisely. Thus the time series dimension must be increased for LSDVa or LSDVb to be applicable or the cross-section dimension must be increased for GMMs1 or GMMs3 to be applicable. When γ =0.6 (table 2) the estimators all perform better when N =20and T =5 given the lower persistence in the data. In table 5 we investigate the effects of increasing the signal to noise ratio (σ 2 s =8instead of σ 2 s =2as in tables 1-4). The conclusion is that for all estimators in all cases this tends to reduce bias and standard error. The conclusions drawn are in general identical as for the case with σ 2 s =2. A final conclusion that can be derived from tables 1 to 5 is that for β the relative inefficiency and bias of the GMM estimators implies that the corrected LSDV estimators for β perform better in almost all cases. 4.5 Results for the long-run parameter β/(1 γ) We now turn to the discussion of the estimation of the long-run parameter β/(1 γ). We compare the performance of the bootstrap based estimator of the long-run coefficient discussed in section 2.2, namely θ c (LSDVb), with the performance of a number of "naive" estimators. The latter are based on short-run estimates of β and γ obtained through LSDV, the analytically corrected Kiviet (1995) estimator (LSDVa) and the system GMM estimator (GMMs3). We do not report simulation results for the "naive" estimator based on the bootstrap corrected shortrun coefficients, namely θ n (see section 2.2), because the results are quite similar to the ones 18

20 for LSDVa. Figures 1 to 5 show the simulated distributions for the estimates of the long-run parameter for all cases considered in tables 1-5 (except for the T =10, N =100case which is available upon request). In table 6 we present statistics for a selection of cases that are helpful to illustrate some problems with the estimation of β/(1 γ). The most important conclusion from these figures is that in a lot of cases the long-run parameter β/(1 γ) is very poorly estimated using any of the considered estimators, especially when few data are available and/or the persistence measured by γ is high. For instance from the figures 1 and 3 we can see that no estimator succeeds in obtaining an unbiased and efficient estimate for the long-run parameter when γ = 0.8 unless the signal to noise ratio is increased to 8 (see figure 5). Trivially, identifying the long-run from moderate amounts of data is difficult if deviations from it are persistent (i.e if values for γ are high) and if there is a lot of noise (i.e if values for σ 2 s are low). When γ =0.6 the long-run is more easy to estimate, with (i) the estimators LSDVa and LSDVb to be preferred when T is moderate and N is small and (ii) GMMs to be preferred when T is small and N is large. Concerning the former case, GMMs3 is even considerably upward biased for β/(1 γ) when µ = 5. Thisfinding should warn against the use of GMMs3 in panels with small N and large individual effects,asmightbethecasein a typical macroeconomic panel. Moreover, the comparison of different estimators using the RMSE criterion may be problematic for the long-run parameter. The reason is the possible occurrence of large positive outliers in the distribution of the long-run parameter. These occur when the estimated values for γ are close to 1. These outliers lead to a trade-off between the bias of an estimator for γ and the efficiency of the estimator for β/(1 γ). Estimators which are (downward) biased for γ, like the LSDV estimator, may have a relatively small dispersion in their estimate for β/(1 γ). Estimators which correct this bias, on the other hand, may be very inefficient in their estimate 19

21 for β/(1 γ). This trade-off will be sharper (i) the closer γ is to 1 and (ii) the lower the efficiency of the estimator for γ. As an example consider the case T =5,N = 100 in table 1. On the one hand, GMMs3 performs very well in terms of RMSE to estimate γ compared to the other estimators. The bias in γ is small and the dispersion is moderate. On the other hand, the RMSE of the GMMs3 estimator for β/(1 γ) (see table 6, case 3) is the highest among all estimators. This stems from the fact that the estimates for γ are on average close to 0.8 but are dispersed enough to reach values close to 1. Estimates for γ closeto1tendtoblowuptheestimate for β/(1 γ), resulting in a heavy right tail in the distribution of the GMMs3 estimator for β/(1 γ). The 95th percentile and the difference between the mean and median indeed suggest that the distribution is skewed to the right (see also figure 1 panel (e)). In contrast, the LSDV estimator is severely downward biased for γ. Given its small dispersion, the estimates do not reach values close to 1. This results in a low dispersion for β/(1 γ) and, despite the large bias, a RMSE which is considerably lower than the one obtained from using GMMs3. 20

22 5 Conclusions In dynamic panels with a small or moderate time dimension the bias of the within or least squares dummy variable estimator may be substantial. In this paper we present a new correction method for this estimator based on an iterative bootstrap approach. For applied researchers this method may be easier to implement than existing analytical bias corrections. We conduct Monte Carlo simulations for a panel data regression model with one lag of the dependent variable and one exogenous explanatory variable. Our results for the autoregressive parameter suggest that the performance (measured in terms of root mean squared error) of our bootstrap correction is very similar to that of the analytical bias correction by Kiviet (1995). The Kiviet correction performs slightly better when the cross-section dimension grows large, but the bootstrap approach seems more interesting when differences between cross-sections are more pronounced. When crosssections are countries this case could for instance imply the use of a sample that consists of both industrialized and low development countries. For panels with a moderate time dimension and a small to large cross-section dimension both types of corrections are preferred over the system GMM estimator by Arellano and Bover (1995). For panels with a small time dimension and a moderate to large cross-section dimension the system GMM estimator performs best. Further results are that (i) all dynamic panel data estimators considered perform badly when both panel dimensions are small, especially when the true value of the autoregressive parameter is close to one, (ii) the corrected within estimators considered (bootstrap and analytical) are generally better at estimating the coefficient on the exogenous variable than the GMM estimators considered, (iii) the long-run impact of the exogenous variable is often poorly estimated even in cases where good estimates for the coefficients on the lagged dependent and exogenous variables can be obtained. 21

23 References Altonji, J., and L. Segal (1994): Small sample bias in GMM estimation of covariance structures, NBER Technical Working Paper, 156. Arellano, M., and S. Bond (1991): Some tests of specification for Panel Data: Monte Carlo evidence and an application to Employment Equations, Review of Economic Studies, 58, Arellano, M., and O. Bover (1995): Another look at the Instrumental Variable estimation of Error-Components Models, Journal of Econometrics, 68, Baltagi, B., and C. Kao (2000): Nonstationary Panels, Cointegration in Panels and Dynamic Panels: A Survey, in Nonstationary Panels, Panel Cointegration, and Dynamic Panels, ed. by B. Baltagi, vol. 15 of Advances in Econometrics. Elsevier Science, Amsterdam. Blundell, R., and S. Bond (1998): Initial conditions and moment restrictions in dynamic panel data models, Journal of Econometrics, 87, Blundell, R., S. Bond, and F. Windmeijer (2000): Estimation in Dynamic Panel Data Models: Improving on the Performance of the Standard GMM Estimator, in Nonstationary Panels, Panel Cointegration, and Dynamic Panels, ed.byb.baltagi,vol.15ofadvances in Econometrics. Elsevier Science, Amsterdam. Bun, M. (2003): Bias correction in the dynamic panel data model with a nonscalar disturbance covariance matrix, Econometric Reviews, 22, Johnston, J., and J. Dinardo (1997): Econometric Methods. McGraw-Hill, NewYork, fourth edn. 22

24 Judson, A., and A. Owen (1999): Estimating Dynamic Panel Data Models: A Guide for Macroeconomists, Economics Letters, 65, Kiviet, J. (1995): On Bias, Inconsistency, and Efficiency of Various Estimators in Dynamic Panel Data Models, Journal of Econometrics, 68, Kiviet, J. (1999): Expectations of Expansions for Estimators in a Dynamic Panel Data Model: Some Results for Weakly Exogenous Regressors, in Analysis of Panels and Limited Dependent Variables Models, ed. by C. Hsiao, K. Lahiri, L.-F. Lee, and M. Pesaran, pp Cambridge University Press, Cambridge. Nelson, C., and R. Startz (1990): The distribution of the instrumental variables estimator and its t-ratio when the instrument is a poor one, Journal of Business, 63, Nickell, S. (1981): Biases in dynamic models with fixed effects, Econometrica, 49(6), Pesaran, M., R. Smith, and K. Im (1996): Dynamic linear models for heterogeneous panels, in The econometrics of panel data, ed. by L. Matyas, and P. Sevestre, pp Kluwer Academic Publishers. Pesaran, M., and Z. Zhao (1999): Bias reduction in estimating long-run relationships from dynamic heterogeneous panels, in Analysis of panels and limited dependent variable models, ed. by C. Hsiao, K. Lahiri, L. Lee, and M. Pesaran, chap. 12, pp Cambridge University Press. Tanizaki, H. (2004): Computational Methods in Statistics and Econometrics, vol. 173 of Statistics: Textbooks and Monographs. MarcelDekker,NewYork. 23

25 Tauchen, G. (1986): Statistical properties of GMM estimators of structural parameters obtained from financial market data, Journal of Business and Economic Statistics, 4,

26 Table 1: Monte Carlo results for γ =0.8,ρ=0.5,σ 2 s =2,µ=1 T N Bias γ Stv γ Rmse γ Bias β Stv β Rmse β 5 20 LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs

27 Table 2: Monte Carlo results for γ =0.6,ρ=0.5,σ 2 s =2,µ=1 T N Bias γ Stv γ Rmse γ Bias β Stv β Rmse β 5 20 LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs

28 Table 3: Monte Carlo results for γ =0.8,ρ=0.5,σ 2 s =2,µ=5 T N Bias γ Stv γ Rmse γ Bias β Stv β Rmse β 5 20 LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs

29 Table 4: Monte Carlo results for γ =0.6,ρ=0.5,σ 2 s =2,µ=5 T N Bias γ Stv γ Rmse γ Bias β Stv β Rmse β 5 20 LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs

30 Table 5: Monte Carlo results for γ =0.8,ρ=0.5,σ 2 s =8,µ=1 T N Bias γ Stv γ Rmse γ Bias β Stv β Rmse β 5 20 LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs LSDV LSDVa LSDVb GMMd GMMs GMMs

31 Table 6: Monte Carlo results for the long-run parameter β/(1 γ) T N Case Mean Median Rmse 5th 95th bias bias percentile percentile 5 20 γ =0.8,ρ =0.5, LSDV σ 2 s =2,µ=1 LSDVa LSDVb GMMs γ =0.8,ρ =0.5, LSDV σ 2 s =2,µ=1 LSDVa LSDVb GMMs γ =0.8,ρ =0.5, LSDV σ 2 s =2,µ=1 LSDVa LSDVb GMMs γ =0.8,ρ =0.5, LSDV σ 2 s =2,µ=1 LSDVa LSDVb GMMs γ =0.6,ρ =0.5, LSDV σ 2 s =2,µ=1 LSDVa LSDVb GMMs γ =0.6,ρ =0.5, LSDV σ 2 s =2,µ=1 LSDVa LSDVb GMMs γ =0.8,ρ =0.5, LSDV σ 2 s =2,µ=5 LSDVa LSDVb GMMs γ =0.8,ρ =0.5, LSDV σ 2 s =2,µ=5 LSDVa LSDVb GMMs γ =0.6,ρ =0.5, LSDV σ 2 s =2,µ=5 LSDVa LSDVb GMMs γ =0.6,ρ =0.5, LSDV σ 2 s =2,µ=5 LSDVa LSDVb GMMs