A Subset-Continuous-Updating Transformation on GMM Estimators for Dynamic Panel Data Models

Article A Subset-Continuous-Updating Transformation on GMM Estimators for Dynamic Panel Data Models Richard A. Ashley 1, and Xiaojin Sun 2,, 1 Department of Economics, Virginia Tech, Blacksburg, VA 24060; ashleyr@vt.edu 2 Department of Economics and Finance, University of Texas at El Paso, El Paso, TX 79968; xsun3@utep.edu * Correspondence: xsun3@utep.edu; Tel.: +1-915-747-7783 These authors contributed equally to this work. Academic Editor: name Version May 25, 2016 submitted to Econometrics; Typeset by LATEX using class file mdpi.cls 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Abstract: The two-step GMM estimators of Arellano and Bond 1] and Blundell and Bond 2] for dynamic panel data models have been widely used in empirical work; however, neither of them performs well in small samples with weak instruments. The continuous-updating GMM estimator proposed by Hansen, Heaton, and Yaron 3] is in principle able to reduce the small-sample bias but it involves high-dimensional optimizations when the number of regressors is large. This paper proposes a computationally feasible variation on these standard two-step GMM estimators by applying the idea of continuous-updating to the autoregressive parameter only, given the fact that the absolute value of the autoregressive parameter is less than unity for a dynamic panel data model to be stationary. We show that our subset-continuous-updating transformation does not alter the asymptotic distribution of the two-step GMM estimators and it therefore retains consistency. Our simulation results indicate that the transformed GMM estimators significantly outperform their standard two-step counterparts in small samples. Keywords: Dynamic Panel Data Models; Arellano-Bond GMM Estimator; Blundell-Bond GMM Estimator; Subset-Continuous-Updating Transformation JEL: C18, C23 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 1. Introduction In recent decades, dynamic panel data models with unobserved individual-specific heterogeneity have been widely used to investigate the dynamics of economic activities. Several estimators have been suggested for estimating the parameters in such a dynamic panel data model. A standard estimation procedure is to first-difference the model, so as to eliminate the unobserved heterogeneity, and then base GMM estimation on the moment conditions implied where endogenous differences of the variables are instrumented by their lagged levels. This is the well known Arellano-Bond estimator, or first-difference ( GMM estimator (see Arellano and Bond 1]. The GMM estimator was found to be inefficient since it does not make use of all available moment conditions (see Ahn and Schmidt 4]; it also has very poor finite sample properties in dynamic panel data models with highly autoregressive series and a small number of time series observations (see Alonso-Borrego and Arellano 5] and Blundell and Bond 2] since the instruments in those cases become less informative. To improve the performance of the GMM estimator, Blundell and Bond 2] proposed taking into consideration extra moment conditions from the level equation that rely on certain restrictions on the initial observations, as suggested by Arellano and Bover 6]. The resulting system (SYS GMM Submitted to Econometrics, pages 1 11 www.mdpi.com/journal/econometrics

Version May 25, 2016 submitted to Econometrics 2 of 11 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 estimator has been shown to perform much better than the GMM estimator in terms of finite sample bias and mean squared error, as well as with regard to coefficient estimator standard errors since the instruments used for the level equation are still informative as the autoregressive coefficient approaches unity (see Blundell and Bond 2] and Blundell et al. 7]. As a result, the SYS GMM estimator has been widely used for estimation of production functions, demand for addictive goods, empirical growth models, etc. However, it was pointed out later on, see Hayakawa 8] and Bun and Windmeijer 9], that the weak instruments problem still remains in the SYS GMM estimator. The work by Hansen, Heaton, and Yaron 3] suggests that the continuous-updating GMM estimator has smaller bias than the standard two-step GMM estimator. However, it involves high-dimensional optimizations when the number of regressors is large. Given the fact that the absolute value of the autoregressive parameter must be less than unity in order for a dynamic panel data to be stationary, we propose a computationally feasible variation on the two-step and SYS GMM estimators for dynamic panel data models, in which the idea of continuous-updating is applied solely to the autoregressive parameter. Following the jackknife interpretation of the continuous-updating estimator in the work of Donald and ewey 10], we show that the subset-continuous-updating transformation that we propose in this paper does not alter the asymptotic distribution of the two-step GMM estimators; and it hence retains consistency. It is computationally advantageous relative to the continuous-updating estimator in that it replaces a relatively high-dimensional optimization over unbounded intervals by a one-dimensional optimization limited to the stationary domain ( 1, 1 of the autoregressive parameter. We conduct Monte Carlo experiments and show that the proposed transformed versions of the and SYS GMM estimators significantly outperform their standard two-step counterparts in small samples. The layout of the paper is as follows. Section 2 describes the model specification and our proposed transformation method. Section 3 describes the Monte Carlo experiments and presents the results. Section 4 concludes the paper. 2. A Subset-Continuous-Updating Transformation 58 2.1. GMM Estimators for Dynamic Panel Data Models Consider a linear model with one dynamic dependent variable y it, additional explanatory variables X it = (x 1 it,..., xk it, individual-specific fixed effects µ i, and idiosyncratic shocks ν it : y it = θy i,t 1 + X it β + u it, where u it = µ i + ν it, (1 59 60 61 for i = 1,..., and t = 2,..., T where is large and T is small. Here, θ is the autoregressive parameter and we assume that it satisfies θ < 1; β is a column vector of remaining coefficients. We follow Hayakawa 8] and impose the following assumptions: 65 62 Assumption 1. {ν it } (t = 2,..., T; i = 1,..., are i.i.d. across time and individuals and independent of 63 µ i and y i1 with E(ν it = 0 and var(ν it = σν. 2 64 Assumption 2. {µ i } (i = 1,..., are i.i.d. across individuals with E(µ i = 0 and var(µ i = σµ. 2 Assumption 3. {xit k } (t = 2,..., T; i = 1,..., ; k = 1,.., K are i.i.d. across time and individuals and 66 independent of µ i with E(xit k it = (σk x 2. 1 The explanatory variable x k is 1 The independency of µ i seems to be an unusual assumption at first glance. This assumption does not matter for the first-differenced equation, in which the individual effects are differenced out. It does matter, however, for the level equation. In practice, economic variables, including the explanatory variables xit k, are usually persistent. In the common case where outside instruments are not available, researchers use the lagged difference ( xi,t 1 k as the instrument for the endogenous variable in the level equation when the endogenous variable is correlated with the individual effects and they use the lagged level (xi,t 1 k as the instrument when the endogenous variable is not correlated with the individual effects. The instrument is, of course, stronger in the latter case. In our model setup, we generally denote the instrument zit k and

Version May 25, 2016 submitted to Econometrics 3 of 11 67 68 69 endogenous in the sense that it is correlated with ν, with a correlation coefficient of ρ k xν. There exists a valid instrument z k for each endogenous x k. The strength of the instrument is measured by the correlation coefficient ρ k xz between x k and z k. Assumption 4. The initial observations satisfy: y i1 = µ i 1 θ + X i1β + ν i1, (2 70 71 where var(xi1 k = (σk x 2 /(1 θ 2 and var(ν i1 = σν 2 /(1 θ 2. Both xi1 k and ν i1 are independent of µ i. Under these assumptions, we consider both the GMM estimator of Arellano and Bond 1] and the SYS GMM estimator of Blundell and Bond 2]. These GMM estimators are derived from the following moment conditions: E(Z di u i = 0, E(Z siũi = 0, (3 where and y i1 0 0 0 0 Z i3 0 y Z di = i1 y i2 0 0 Z i4............, u i = 0 0 0 y i1 y i,t 2 Z it Z si = Z di 0 0 0 0 0 y i2 0 0 Z i3 0 0 y i3 0 Z i4........ 0 0 0 y i,t 1 Z it (, ũ i = u i u i u i3 u i4. u it, u i =, u i3 u i4. u it. Let y i = ( y i3, y i4,..., y it, y i, 1 = ( y i2, y i3,..., y it 1, y i = (y i3, y i4,..., y it, y i, 1 = (y i2, y i3,..., y it 1, ỹ i = ( y i, y i, ỹ i, 1 = ( y i, 1, y i, 1, X i = ( X i3, X i4,..., X it, X i = (X i3, X i4,..., X it, X i = ( X i, X i, and y, y 1, y, y 1, ỹ, ỹ 1, X, X, X, u, u, and ũ are stacked across individuals. The two-step GMM estimators are obtained from ( θ, β ( 1 = argmin Z di u i ( θ SYS, β ( 1 SYS = argmin Z siũi Ω Ω ( θ, β ] ( 1 1 Z di i u, (4 ( θ SYS, β ] ( 1 1 SYS Z siũi, (5 72 with Ω ( θ, β = 1 Ω ( θ SYS, β SYS = 1 Z di u i ( θ, β ] u i ( θ, β ] Zdi, (6 Z si where ( θ, β and ( θ SYS, β SYS are preliminary estimators. ũ i ( θ SYS, β SYS ] ũ i ( θ SYS, β SYS ] Zsi, (7 control the strength of the instrument with the correlation coefficient ρ k xz; in our Monte Carlo experiments, we vary the value of ρ k xz to account for both weaker and stronger instrument cases.

Version May 25, 2016 submitted to Econometrics 4 of 11 73 2.2. A Subset-Continuous-Updating Transformation Let g(z, θ, β be a vector of functions of a data observation Z, as well as a scalar parameter θ and a vector of parameters β satisfying the moment condition: Eg(Z, θ 0, β 0 ] = 0, (8 Let z i (i = 1,..., denote the observations and g i (θ, β = g(z i, θ, β. The sample first and second moments of the g are given by: ĝ(θ, β = 1 Ω(θ, β = 1 g i (θ, β, (9 g i (θ, βg i (θ, β. (10 Two-Step GMM Estimator: The two-step GMM estimator is the solution to the following minimization problem: ( θ, β = argmin ĝ(θ, β Ω( θ, β ĝ(θ, β, (11 where ( θ, β is a preliminary estimator, e.g., the first-step estimator. The first-order conditions are: ĝ θ Ĝ β Ω( θ, β ĝ = 0, (12 Ω( θ, β ĝ = 0, (13 74 where ĝ = ĝ( θ, β, ĝ θ = ĝ( θ, β/ θ and Ĝ β = ĝ( θ, β/ β. Subset-Continuous-Updating GMM Estimator: The subset-continuous-updating GMM estimator is obtained as the solution to a minimization problem as follows: ( θ, β = argmin ĝ(θ, β Ω(θ, β ĝ(θ, β, (14 where β is a preliminary estimator. The first-order conditions are: ĝ θ Ω( θ, β ĝ ĝ Ω( θ, β Λ Ω( θ, β ĝ = 0, (15 Ĝ β Ω( θ, β ĝ = 0, (16 75 76 77 78 79 80 where ĝ = ĝ( θ, β, ĝ θ = ĝ( θ, β/ θ, Ĝ β = ĝ( θ, β/ β, and Λ = ĝiĝ θ,i /. Here, ĝ i = g(z i, θ, β and ĝ θ,i = g(z i, θ, β/ θ. We call the solution to the above minimization problem a subset-continuous-updating estimator to reflect the fact that we are applying the idea of continuous-updating of Hansen, Heaton, and Yaron 3] on a subset of model parameters. This estimator is consistent according to the following jackknife interpretation in the work of Donald and ewey 10]:

Version May 25, 2016 submitted to Econometrics 5 of 11 Let B = Ω( θ, β Λ denote the matrix of coefficients from the multivariate regression of ĝ θ,i on ĝ i, and η i = ĝ θ,i B ĝ i the vector of associated residuals. By the usual orthogonality property of least squares residuals, n η iĝ i / = 0. We can rewrite the first-order conditions as: ( where  θ,i = j =i η j 0 = (ĝ θ B ĝ Ω( θ, β ĝ, ( 1 β = η i Ω( θ, ĝ, = 1 2 ( = 1 j=1 1 η j j =i Ω( θ, β ĝi, η j Ω( θ, β ĝ i, = 1  θ,i g(z i, θ, β, (17 β] 1 Ω( θ, / and 81 82 83 84 85 86 87 88 89 ( where  β = j=1 Ĝβ,j 0 = 1  β g(z i, θ, β, (18 β] 1 Ω( θ, / and Ĝ β,j = g(z i, θ, β/ β. For β, the usual interpretation of a GMM estimator, where a linear combination of the moments is set equal to zero, applies; see Equation (18. For θ, Equation (17 shows the jackknife interpretation of the continuous updating estimator. The usual interpretation is modified to allow a linear combination coefficient for each observation, that excludes the own observation from the Jacobian of the moments. This does not affect the asymptotic distribution of the estimator, because ĝ = o p (1 means that each  θ,i will have the same limit as. It does, however, affect the small sample distribution of the estimator. 2.3. Transformed GMM Estimators for Dynamic Panel Data Models To apply the subset-continuous-updating transformation on the GMM estimators for dynamic panel data models, we simply let g = Z d u for the GMM estimator and g = Z sũ for the SYS GMM estimator. The subset-continuous-updating version of and SYS GMM estimators (denoted T and TSYS are derived from ( θ T, β ( 1 T = argmin Z di u i ( θ TSYS, β ( 1 TSYS = argmin Z siũi ( Ω θ T, β ] ( 1 1 T Z di u i, (19 ( Ω θ TSYS, β ] ( 1 1 TSYS Z siũi, (20 with Ω (θ T, β T = 1 Ω (θ TSYS, β TSYS = 1 Z di u i (θ T, β ] ( T u i θ T, β ] T Zdi, (21 Z si ũ i (θ TSYS, β TSYS ] ũ i ( θ TSYS, β TSYS ] Zsi, (22

Version May 25, 2016 submitted to Econometrics 6 of 11 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 where β T and β TSYS are preliminary estimators. Applying the result obtained in Section 2.2, the T and TSYS GMM estimators have exactly the same asymptotic distributions as their standard two-step counterparts. They are computationally advantageous relative to the continuous-updating estimator in that they replace a K + 1 dimensional numerical optimization over the unbounded domain of (θ, β by the sequence of a one-dimensional optimization over θ ( 1, 1 followed by an (ordinarily analytic optimization over β. 3. Monte Carlo Experiments We consider the model and assumptions specified in Section 2. Without loss of generality, we consider one additional explanatory variable beyond the lagged dependent variable, i.e., we restrict K = 1; and we let β = 1, σx 2 = 1, and σν 2 = 1. This restriction is made for expositional clarity here, as our proposed transformed estimation method applies to models with multiple explanatory variables (K > 1 without any extra computational burden because the first stage optimization is imposed solely on the bounded scalar parameter θ. For each Monte Carlo replication, we draw a multivariate normal distribution for µ i, ν it, x it, and z it. We then construct the dependent variable y it according to the model setup in Section 2. The parameters that are varied in the Monte Carlo experiments are the autoregressive parameter θ, the variance ratio σµ/σ 2 ν, 2 the correlation coefficient between x and ν, denoted ρ xν, and the correlation coefficient between x and z, denoted ρ xz. The parameters ρ xν and ρ xz measure the endogeneity of the regressor x and the strength of the instrumental variable z, respectively. For each of the two sample sizes, = 100 and T = {6, 11}, we consider 36 schemes, with σµ/σ 2 ν 2 = {1/4, 1, 4}, θ = {,, }, ρ xν = {, }, and ρ xz = {, }. The two values of ρ xv correspond to a mildly endogenous regressor and a moderately endogenous regressor and the two ρ xz values specify a weak instrument and a relatively strong instrument. All results are obtained from 10,000 Monte Carlo replications. Tables 1 and 2 compare the transformed estimators to the standard two-step estimators from the perspective of estimation accuracy, quantified by mean squared errors (MSE. The upper panel of each table presents statistics on the β coefficient estimators and the lower panel focuses on the θ coefficient estimators. For both sample sizes that we consider, the standard two-step GMM estimators have small mean squared errors when the regressor x is mildly endogenous (ρ xv =, and our proposed transformation does not improve matters much. When the regressor becomes moderately endogenous (ρ xv =, however, the standard two-step GMM estimators yield sizeable β mean squared errors, no matter how strong the instrument is. By applying our proposed transformation we are able to reduce the β mean squared errors to a large extent, especially when the instrumental variable is relatively strong (ρ xz =. Even when the instrument is not highly correlated with the endogenous regressor (ρ xz =, our transformation successfully lowers mean squared errors of the standard two-step estimators of β by about 60%. We do not see much MSE difference in the θ estimation between standard two-step estimators and their transformed counterparts. Thus, our proposed T and TSYS GMM estimators provide significant MSE improvements in estimating β without worsening the estimation precision with respect to θ.

Version May 25, 2016 submitted to Econometrics 7 of 11 Table 1. Monte Carlo Comparison of Mean Squared Errors, = 100 and T = 6 Panel I: Mean Squared Error of β Estimates θ (ρ xv, ρ xz T SYS TSYS T SYS TSYS T SYS TSYS (, 0.0402 0.0678 0.0388 0.0518 0.0403 0.0671 0.0399 0.0579 0.0407 0.0662 0.0420 0.0684 (, 0.0401 0.0144 0.0390 0.0116 0.0409 0.0148 0.0404 0.0135 0.0402 0.0151 0.0417 0.0154 (, 466 0.1005 451 0.0844 450 0.0994 451 0.0847 463 0.1000 507 0.0860 (, 459 0.0153 445 0.0126 455 0.0160 460 0.0141 449 0.0157 496 0.0152 (, 0.0398 0.0693 0.0390 0.0536 0.0393 0.0684 0.0408 0.0582 0.0383 0.0678 0.0448 0.0674 (, 0.0398 0.0149 0.0390 0.0117 0.0391 0.0154 0.0406 0.0136 0.0380 0.0160 0.0447 0.0161 (, 447 0.1005 444 0.0834 436 0.1032 483 0.0857 410 0.1007 593 0.0860 (, 438 0.0163 444 0.0130 433 0.0167 489 0.0144 411 0.0174 584 0.0160 (, 0.0370 0.0710 0.0390 0.0532 0.0341 0.0685 0.0418 0.0590 0.0322 0.0726 0.0503 0.0825 (, 0.0366 0.0172 0.0388 0.0122 0.0342 0.0209 0.0423 0.0146 0.0316 0.0271 0.0507 0.0189 (, 371 0.1026 444 0.0834 255 0.1030 531 0.0809 103 0.1069 786 0.0906 (, 368 0.0184 442 0.0129 256 0.0205 531 0.0144 128 0.0257 797 0.0174 Panel II: Mean Squared Error of θ Estimates θ (ρ xv, ρ xz T SYS TSYS T SYS TSYS T SYS TSYS (, 0.0024 0.0028 0.0018 0.0021 0.0029 0.0036 0.0022 0.0026 0.0040 0.0051 0.0035 0.0036 (, 0.0023 0.0028 0.0018 0.0022 0.0029 0.0036 0.0022 0.0026 0.0040 0.0050 0.0035 0.0034 (, 0.0014 0.0019 0.0011 0.0015 0.0017 0.0024 0.0014 0.0018 0.0023 0.0034 0.0023 0.0025 (, 0.0014 0.0021 0.0011 0.0016 0.0017 0.0026 0.0014 0.0019 0.0023 0.0037 0.0022 0.0024 (, 0.0037 0.0046 0.0022 0.0027 0.0055 0.0073 0.0032 0.0038 0.0091 0.0135 0.0073 0.0071 (, 0.0036 0.0044 0.0021 0.0027 0.0055 0.0072 0.0031 0.0038 0.0090 0.0130 0.0072 0.0063 (, 0.0021 0.0030 0.0013 0.0018 0.0031 0.0046 0.0021 0.0027 0.0050 0.0084 0.0050 0.0049 (, 0.0021 0.0033 0.0013 0.0020 0.0032 0.0051 0.0021 0.0028 0.0051 0.0093 0.0050 0.0044 (, 0.0121 0.0166 0.0025 0.0035 0.0279 0.0417 0.0045 0.0064 0.0522 0.0787 0.0127 0.0149 (, 0.0119 0.0154 0.0026 0.0039 0.0276 0.0384 0.0046 0.0062 0.0534 0.0718 0.0125 0.0138 (, 0.0064 0.0108 0.0016 0.0025 0.0134 0.0265 0.0032 0.0044 0.0278 0.0563 0.0101 0.0104 (, 0.0065 0.0109 0.0016 0.0028 0.0136 0.0233 0.0032 0.0043 0.0279 0.0455 0.0100 0.0089 ote: The dynamic model we estimate is y it = θy i,t 1 + βx it + µ i + ν it. The variance ratio is defined as σ 2 µ/σ 2 ν, where σ 2 µ is the variance of fixed effects and σ 2 ν is the variance of idiosyncratic shocks. The parameters ρ xv and ρ xz are the correlation coefficients between x it and ν it and between x it and z it. They characterize the endogeneity of the regressor x and the strength of the instrumental variable z, respectively. and SYS stand for the standard two-step Arellano-Bond and Blundell-Bond GMM estimators. Their transformed counterparts are denoted T and TSYS.

Version May 25, 2016 submitted to Econometrics 8 of 11 Table 2. Monte Carlo Comparison of Mean Squared Errors, = 100 and T = 11 Panel I: Mean Squared Error of β Estimates θ (ρ xv, ρ xz T SYS TSYS T SYS TSYS T SYS TSYS (, 0.0355 0.0342 0.0357 0.0259 0.0353 0.0338 0.0359 0.0265 0.0353 0.0343 0.0369 0.0315 (, 0.0353 0.0076 0.0355 0.0056 0.0354 0.0075 0.0359 0.0064 0.0352 0.0076 0.0370 0.0081 (, 386 0.1181 392 0.0806 381 0.1189 396 0.0805 378 0.1193 419 0.0818 (, 380 0.0159 387 0.0089 379 0.0163 392 0.0099 378 0.0166 412 0.0117 (, 0.0350 0.0340 0.0357 0.0257 0.0345 0.0336 0.0362 0.0264 0.0343 0.0340 0.0380 0.0312 (, 0.0347 0.0076 0.0355 0.0056 0.0346 0.0075 0.0362 0.0064 0.0342 0.0076 0.0381 0.0080 (, 375 0.1178 393 0.0804 365 0.1185 403 0.0802 355 0.1186 449 0.0822 (, 370 0.0159 388 0.0089 363 0.0162 399 0.0099 355 0.0165 441 0.0118 (, 0.0324 0.0330 0.0359 0.0255 0.0302 0.0323 0.0377 0.0263 0.0286 0.0320 0.0431 0.0291 (, 0.0321 0.0077 0.0356 0.0057 0.0303 0.0079 0.0377 0.0065 0.0286 0.0085 0.0432 0.0075 (, 323 0.1161 399 0.0794 275 0.1152 448 0.0794 234 0.1142 614 0.0812 (, 317 0.0157 395 0.0089 272 0.0160 444 0.0098 221 0.0164 600 0.0112 Panel II: Mean Squared Error of θ Estimates θ (ρ xv, ρ xz T SYS TSYS T SYS TSYS T SYS TSYS (, 0.0009 0.0019 0.0007 0.0021 0.0010 0.0023 0.0008 0.0023 0.0012 0.0027 0.0013 0.0025 (, 0.0009 0.0019 0.0007 0.0019 0.0010 0.0023 0.0008 0.0020 0.0012 0.0028 0.0013 0.0025 (, 0.0005 0.0013 0.0005 0.0017 0.0006 0.0015 0.0005 0.0018 0.0007 0.0018 0.0008 0.0021 (, 0.0005 0.0015 0.0004 0.0015 0.0006 0.0017 0.0005 0.0015 0.0007 0.0020 0.0008 0.0019 (, 0.0011 0.0025 0.0008 0.0022 0.0015 0.0034 0.0010 0.0026 0.0019 0.0046 0.0024 0.0037 (, 0.0011 0.0025 0.0008 0.0020 0.0014 0.0035 0.0010 0.0025 0.0018 0.0047 0.0024 0.0034 (, 0.0006 0.0016 0.0005 0.0017 0.0008 0.0022 0.0007 0.0021 0.0011 0.0030 0.0016 0.0027 (, 0.0006 0.0018 0.0005 0.0015 0.0008 0.0024 0.0007 0.0018 0.0011 0.0033 0.0016 0.0023 (, 0.0024 0.0074 0.0009 0.0026 0.0045 0.0187 0.0018 0.0041 0.0065 0.0311 0.0074 0.0090 (, 0.0025 0.0075 0.0009 0.0024 0.0044 0.0165 0.0018 0.0037 0.0065 0.0275 0.0075 0.0071 (, 0.0013 0.0047 0.0006 0.0022 0.0022 0.0116 0.0013 0.0031 0.0033 0.0238 0.0057 0.0078 (, 0.0013 0.0046 0.0006 0.0017 0.0022 0.0111 0.0012 0.0027 0.0033 0.0194 0.0057 0.0051 ote: The dynamic model we estimate is y it = θy i,t 1 + βx it + µ i + ν it. The variance ratio is defined as σ 2 µ/σ 2 ν, where σ 2 µ is the variance of fixed effects and σ 2 ν is the variance of idiosyncratic shocks. The parameters ρ xv and ρ xz are the correlation coefficients between x it and ν it and between x it and z it. They characterize the endogeneity of the regressor x and the strength of the instrumental variable z, respectively. and SYS stand for the standard two-step Arellano-Bond and Blundell-Bond GMM estimators. Their transformed counterparts are denoted T and TSYS. 128 129 130 131 132 As Tables 3 and 4 show, the improvement we observe in the previous two tables is primarily in the bias instead of in the sampling variance. In case of moderately endogenous regressors (ρ xv = and weak instruments (ρ xz =, the biases in β can be halved by applying the transformation on the standard two-step estimators. When the instrument becomes relatively strong, our transformation removes 90% of the biases.

Version May 25, 2016 submitted to Econometrics 9 of 11 Table 3. Monte Carlo Comparison of Biases, = 100 and T = 6 Panel I: Bias in β Estimates θ (ρ xv, ρ xz T SYS TSYS T SYS TSYS T SYS TSYS (, 0.1908 0.0811 0.1892 0.0749 0.1908 0.0848 0.1909 0.0647 0.1910 0.0866 0.1946 0.0334 (, 0.1906 0.0147 0.1896 0.0102 0.1916 0.0168 0.1918 0.0082 0.1898 0.0149 0.1940-0.0036 (, 0.4936 227 0.4927 136 0.4918 163 0.4922 0.1942 0.4929 201 0.4973 0.1674 (, 0.4929 0.0461 0.4921 0.0437 0.4924 0.0472 0.4932 0.0391 0.4915 0.0481 0.4962 0.0308 (, 0.1888 0.0925 0.1898 0.0823 0.1860 0.0870 0.1927 0.0624 0.1811 0.0843 007 0.0239 (, 0.1887 0.0167 0.1896 0.0107 0.1856 0.0140 0.1921 0.0037 0.1805 0.0148 004-0.0088 (, 0.4913 210 0.4919 100 0.4898 236 0.4953 0.1971 0.4864 218 053 0.1594 (, 0.4904 0.0485 0.4919 0.0445 0.4894 0.0492 0.4959 0.0391 0.4863 0.0495 044 0.0272 (, 0.1739 0.0853 0.1892 0.0737 0.1535 0.0696 0.1952 0.0415 0.1275 0.0554 146-0.0013 (, 0.1733 0.0152 0.1889 0.0070 0.1542 0.0076 0.1967-0.0063 0.1262-0.0079 155-0.0248 (, 0.4814 212 0.4918 069 0.4660 164 000 0.1797 0.4435 059 243 0.1397 (, 0.4810 0.0502 0.4916 0.0408 0.4660 0.0494 0.4999 0.0310 0.4457 0.0468 254 0.0155 Panel II: Bias in θ Estimates θ (ρ xv, ρ xz T SYS TSYS T SYS TSYS T SYS TSYS (, -0.0050 0.0024 0.0018 0.0029-0.0057 0.0031 0.0069 0.0073-0.0075 0.0046 0.0178 0.0117 (, -0.0049 0.0027 0.0021 0.0029-0.0056 0.0038 0.0075 0.0067-0.0085 0.0043 0.0172 0.0080 (, -0.0027 0.0039 0.0022 0.0035-0.0037 0.0034 0.0053 0.0057-0.0047 0.0065 0.0137 0.0101 (, -0.0031 0.0035 0.0021 0.0029-0.0031 0.0051 0.0064 0.0063-0.0058 0.0062 0.0124 0.0078 (, -0.0095 0.0032 0.0034 0.0046-0.0149 0.0042 0.0127 0.0116-0.0243 0.0066 0.0386 0.0203 (, -0.0098 0.0039 0.0028 0.0034-0.0152 0.0050 0.0117 0.0079-0.0247 0.0083 0.0393 0.0144 (, -0.0053 0.0062 0.0032 0.0044-0.0072 0.0083 0.0114 0.0111-0.0143 0.0111 0.0306 0.0186 (, -0.0068 0.0053 0.0022 0.0034-0.0087 0.0087 0.0109 0.0090-0.0128 0.0154 0.0314 0.0145 (, -0.0344 0.0029 0.0054 0.0049-0.0712-0.0197 0.0274 0.0124-0.1173-0.0517 0.0884 0.0215 (, -0.0352 0.0068 0.0053 0.0023-0.0720-0.0103 0.0286 0.0064-0.1221-0.0404 0.0868 0.0060 (, -0.0181 0.0136 0.0065 0.0056-0.0404 0.0085 0.0254 0.0154-0.0739-0.0130 0.0766 0.0240 (, -0.0191 0.0189 0.0055 0.0030-0.0410 0.0186 0.0242 0.0083-0.0712 0.0108 0.0772 0.0169 ote: The dynamic model we estimate is y it = θy i,t 1 + βx it + µ i + ν it. The variance ratio is defined as σ 2 µ/σ 2 ν, where σ 2 µ is the variance of fixed effects and σ 2 ν is the variance of idiosyncratic shocks. The parameters ρ xv and ρ xz are the correlation coefficients between x it and ν it and between x it and z it. They characterize the endogeneity of the regressor x and the strength of the instrumental variable z, respectively. and SYS stand for the standard two-step Arellano-Bond and Blundell-Bond GMM estimators. Their transformed counterparts are denoted T and TSYS.

Version May 25, 2016 submitted to Econometrics 10 of 11 Table 4. Monte Carlo Comparison of Biases, = 100 and T = 11 Panel I: Bias in β Estimates θ (ρ xv, ρ xz T SYS TSYS T SYS TSYS T SYS TSYS (, 0.1837 0.1274 0.1851 0.0993 0.1829 0.1265 0.1850 0.0916 0.1829 0.1255 0.1863 0.0828 (, 0.1829 0.0324 0.1844 0.0170 0.1831 0.0327 0.1850 0.0154 0.1824 0.0326 0.1864 0.0144 (, 0.4870 0.3210 0.4879 588 0.4865 0.3222 0.4881 552 0.4862 0.3227 0.4899 450 (, 0.4864 0.1009 0.4873 0.0644 0.4863 0.1032 0.4877 0.0655 0.4862 0.1037 0.4892 0.0643 (, 0.1821 0.1266 0.1851 0.0985 0.1805 0.1253 0.1857 0.0905 0.1797 0.1238 0.1891 0.0831 (, 0.1813 0.0315 0.1844 0.0167 0.1808 0.0317 0.1857 0.0150 0.1792 0.0311 0.1892 0.0149 (, 0.4859 0.3205 0.4880 584 0.4848 0.3215 0.4888 543 0.4837 0.3215 0.4929 463 (, 0.4853 0.1005 0.4874 0.0641 0.4845 0.1024 0.4884 0.0653 0.4837 0.1029 0.4921 0.0651 (, 0.1740 0.1214 0.1853 0.0962 0.1667 0.1160 0.1897 0.0871 0.1610 0.1093 029 0.0773 (, 0.1731 0.0271 0.1846 0.0148 0.1670 0.0242 0.1896 0.0128 0.1607 0.0196 030 0.0120 (, 0.4802 0.3172 0.4886 563 0.4749 0.3152 0.4933 521 0.4703 0.3125 095 483 (, 0.4795 0.0983 0.4882 0.0633 0.4746 0.0984 0.4929 0.0632 0.4689 0.0963 082 0.0627 Panel II: Bias in θ Estimates θ (ρ xv, ρ xz T SYS TSYS T SYS TSYS T SYS TSYS (, -0.0055 0.0006 0.0000 0.0036-0.0067-0.0005 0.0032 0.0051-0.0070 0.0010 0.0151 0.0074 (, -0.0055 0.0018 0.0001 0.0032-0.0055 0.0030 0.0040 0.0062-0.0073 0.0019 0.0148 0.0063 (, -0.0034 0.0024 0.0003 0.0032-0.0036 0.0032 0.0032 0.0056-0.0045 0.0035 0.0120 0.0066 (, -0.0036 0.0033 0.0001 0.0022-0.0036 0.0047 0.0031 0.0047-0.0048 0.0049 0.0117 0.0070 (, -0.0090 0.0002 0.0000 0.0046-0.0120-0.0013 0.0065 0.0077-0.0140 0.0006 0.0297 0.0105 (, -0.0092 0.0013 0.0001 0.0040-0.0108 0.0027 0.0072 0.0083-0.0142 0.0022 0.0298 0.0095 (, -0.0055 0.0023 0.0006 0.0033-0.0067 0.0032 0.0059 0.0070-0.0088 0.0043 0.0239 0.0097 (, -0.0057 0.0040 0.0005 0.0029-0.0067 0.0062 0.0058 0.0067-0.0092 0.0072 0.0235 0.0100 (, -0.0239-0.0047 0.0018 0.0059-0.0378-0.0153 0.0224 0.0137-0.0491-0.0251 0.0758 0.0172 (, -0.0242-0.0017 0.0016 0.0047-0.0369-0.0045 0.0226 0.0113-0.0492-0.0137 0.0765 0.0157 (, -0.0139 0.0021 0.0029 0.0053-0.0215-0.0001 0.0191 0.0141-0.0302-0.0084 0.0656 0.0199 (, -0.0143 0.0070 0.0028 0.0038-0.0217 0.0111 0.0190 0.0116-0.0303 0.0124 0.0658 0.0215 ote: The dynamic model we estimate is y it = θy i,t 1 + βx it + µ i + ν it. The variance ratio is defined as σ 2 µ/σ 2 ν, where σ 2 µ is the variance of fixed effects and σ 2 ν is the variance of idiosyncratic shocks. The parameters ρ xv and ρ xz are the correlation coefficients between x it and ν it and between x it and z it. They characterize the endogeneity of the regressor x and the strength of the instrumental variable z, respectively. and SYS stand for the standard two-step Arellano-Bond and Blundell-Bond GMM estimators. Their transformed counterparts are denoted T and TSYS. 133 134 135 136 137 138 139 140 141 The value of the autoregressive parameter θ and the size of the variance ratio σµ/σ 2 ν 2 mainly affect the accuracy of the θ estimation. In line with the evidence in the literature, the general pattern is that estimation mean square error and bias increase with both θ and σµ/σ 2 ν 2 when employing the standard two-step GMM estimators. This pattern does not change when our transformation is applied. In conclusion, the subset-continuous-updating transformation that we propose in this paper is shown to improve the estimation accuracy on the endogenous explanatory variable coefficient without sacrificing the accuracy on the autoregressive parameter in a dynamic panel data model. 2 o extra computational burden is incurred when we apply this transformation to models with multiple explanatory variables. 2 While the results shown in paper are associated with the scenario of zero correlation between the endogenous variable x it and the individual effects µ i, we also conduct Monte Carlo experiments to take into account the possibility of a non-zero correlation coefficient, say. Our main results (not reported here remain unchanged. This result is consistent with our argument in Section 2.1 that the assumption of zero correlation only matters for the level equation and we can vary the value of ρ xz to account for both cases of zero and non-zero correlation.

Version May 25, 2016 submitted to Econometrics 11 of 11 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 4. Conclusion The two-step GMM estimators of Arellano and Bond 1] and Blundell and Bond 2] for dynamic panel data models have been widely used in empirical work. However, neither of them performs well in small samples with weak instruments. The continuous-updating GMM estimator proposed by Hansen, Heaton, and Yaron 3] is in principle able to reduce the small-sample bias but it involves high-dimensional optimizations when the number of regressors is large. Given the fact that the absolute value of the autoregressive parameter is less than unity for a dynamic panel data model to be stationary, we propose a computationally feasible variation on the standard two-step GMM estimators by applying the idea of continuous-updating to the autoregressive parameter only. We show that our subset-continuous-updating transformation does not alter the asymptotic distribution of the two-step GMM estimators and hence retains consistency. According to our Monte Carlo results, the transformed GMM estimators significantly outperform their standard two-step counterparts in small samples. Author Contributions: Overall, both authors contributed equally to this project. Conflicts of Interest: The authors declare no conflict of interest. Bibliography 1. Arellano, M.; Bond, S. Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. Review of Economic Studies 1991, 58, 277 297. 2. Blundell, R.; Bond, S. Initial conditions and moment restrictions in dynamic panel data models. Journal of Econometrics 1998, 87, 115 143. 3. Hansen, L.P.; Heaton, J.; Yaron, A. Finite-sample properties of some alternative GMM estimators. Journal of Business & Economic Statistics 1996, 14, 262 280. 4. Ahn, S.C.; Schmidt, P. Efficient estimation of models for dynamic panel data. Journal of Econometrics 1995, 68, 5 27. 5. Alonso-Borrego, C.; Arellano, M. Symmetrically normalized instrumental-variable estimation using panel data. Journal of Business & Economic Statistics 1999, 17, 36 49. 6. Arellano, M.; Bover, O. Another look at the instrumental variable estimation of error-component models. Journal of Econometrics 1995, 68, 29 51. 7. Blundell, R.; Bond, S.; Windmeijer, F. Estimation in dynamic panel data models: improving on the performance of the standard GMM estimator. In Advances in Econometrics, Badi H. Baltagi, Thomas B. Fomby and R. Carter Hill, Emerald Group Publishing Limited 2001, pp. 53 91. 8. Hayakawa, K. Small sample bias properties of the system GMM estimator in dynamic panel data models. Economics Letters 2007, 95, 32 38. 9. Bun, M.J.G.; Windmeijer, F. The weak instrument problem of the system GMM estimator in dynamic panel data models. Econometrics Journal 2010, 13, 95 126. 10. Donald, S.G.; ewey, W.K. A Jackknife Interpretation of the Continuous Updating Estimator. Economics Letters 2000, 67, 239 243. c 2016 by the authors. Submitted to Econometrics for possible open access publication under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/.