A SubsetContinuousUpdating Transformation on GMM Estimators for Dynamic Panel Data Models


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1 Article A SubsetContinuousUpdating 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; 2 Department of Economics and Finance, University of Texas at El Paso, El Paso, TX 79968; * Correspondence: Tel.: 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 Abstract: The twostep 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 continuousupdating GMM estimator proposed by Hansen, Heaton, and Yaron 3] is in principle able to reduce the smallsample bias but it involves highdimensional optimizations when the number of regressors is large. This paper proposes a computationally feasible variation on these standard twostep GMM estimators by applying the idea of continuousupdating 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 subsetcontinuousupdating transformation does not alter the asymptotic distribution of the twostep GMM estimators and it therefore retains consistency. Our simulation results indicate that the transformed GMM estimators significantly outperform their standard twostep counterparts in small samples. Keywords: Dynamic Panel Data Models; ArellanoBond GMM Estimator; BlundellBond GMM Estimator; SubsetContinuousUpdating Transformation JEL: C18, C Introduction In recent decades, dynamic panel data models with unobserved individualspecific 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 firstdifference 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 ArellanoBond estimator, or firstdifference ( 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 AlonsoBorrego 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
2 Version May 25, 2016 submitted to Econometrics 2 of 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 continuousupdating GMM estimator has smaller bias than the standard twostep GMM estimator. However, it involves highdimensional 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 twostep and SYS GMM estimators for dynamic panel data models, in which the idea of continuousupdating is applied solely to the autoregressive parameter. Following the jackknife interpretation of the continuousupdating estimator in the work of Donald and ewey 10], we show that the subsetcontinuousupdating transformation that we propose in this paper does not alter the asymptotic distribution of the twostep GMM estimators; and it hence retains consistency. It is computationally advantageous relative to the continuousupdating estimator in that it replaces a relatively highdimensional optimization over unbounded intervals by a onedimensional 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 twostep 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 SubsetContinuousUpdating Transformation 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, individualspecific 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, ( 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: 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 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 firstdifferenced 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
3 Version May 25, 2016 submitted to Econometrics 3 of 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, ( 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 i Z i3 0 y Z di = i1 y i2 0 0 Z i , u i = y i1 y i,t 2 Z it Z si = Z di y i2 0 0 Z i3 0 0 y i3 0 Z i 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 twostep 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.
4 Version May 25, 2016 submitted to Econometrics 4 of A SubsetContinuousUpdating 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 TwoStep GMM Estimator: The twostep 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 firststep estimator. The firstorder 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. SubsetContinuousUpdating GMM Estimator: The subsetcontinuousupdating 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 firstorder 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, ( 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 subsetcontinuousupdating estimator to reflect the fact that we are applying the idea of continuousupdating 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]:
5 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 firstorder conditions as: ( where Â 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 Â θ,i g(z i, θ, β, (17 β] 1 Ω( θ, / and ( where Â β = j=1 Ĝβ,j 0 = 1 Â 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 Â θ,i will have the same limit as. It does, however, affect the small sample distribution of the estimator Transformed GMM Estimators for Dynamic Panel Data Models To apply the subsetcontinuousupdating 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 subsetcontinuousupdating 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
6 Version May 25, 2016 submitted to Econometrics 6 of 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 twostep counterparts. They are computationally advantageous relative to the continuousupdating estimator in that they replace a K + 1 dimensional numerical optimization over the unbounded domain of (θ, β by the sequence of a onedimensional 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 twostep 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 twostep 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 twostep 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 twostep estimators of β by about 60%. We do not see much MSE difference in the θ estimation between standard twostep 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.
7 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 (, (, (, (, (, (, (, (, (, (, (, (, Panel II: Mean Squared Error of θ Estimates θ (ρ xv, ρ xz T SYS TSYS T SYS TSYS T SYS TSYS (, (, (, (, (, (, (, (, (, (, (, (, 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 twostep ArellanoBond and BlundellBond GMM estimators. Their transformed counterparts are denoted T and TSYS.
8 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 (, (, (, (, (, (, (, (, (, (, (, (, Panel II: Mean Squared Error of θ Estimates θ (ρ xv, ρ xz T SYS TSYS T SYS TSYS T SYS TSYS (, (, (, (, (, (, (, (, (, (, (, (, 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 twostep ArellanoBond and BlundellBond GMM estimators. Their transformed counterparts are denoted T and TSYS 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 twostep estimators. When the instrument becomes relatively strong, our transformation removes 90% of the biases.
9 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 (, (, (, (, (, (, (, (, (, (, (, (, Panel II: Bias in θ Estimates θ (ρ xv, ρ xz T SYS TSYS T SYS TSYS T SYS TSYS (, (, (, (, (, (, (, (, (, (, (, (, 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 twostep ArellanoBond and BlundellBond GMM estimators. Their transformed counterparts are denoted T and TSYS.
10 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 (, (, (, (, (, (, (, (, (, (, (, (, Panel II: Bias in θ Estimates θ (ρ xv, ρ xz T SYS TSYS T SYS TSYS T SYS TSYS (, (, (, (, (, (, (, (, (, (, (, (, 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 twostep ArellanoBond and BlundellBond GMM estimators. Their transformed counterparts are denoted T and TSYS 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 twostep GMM estimators. This pattern does not change when our transformation is applied. In conclusion, the subsetcontinuousupdating 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 nonzero 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 nonzero correlation.
11 Version May 25, 2016 submitted to Econometrics 11 of Conclusion The twostep 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 continuousupdating GMM estimator proposed by Hansen, Heaton, and Yaron 3] is in principle able to reduce the smallsample bias but it involves highdimensional 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 twostep GMM estimators by applying the idea of continuousupdating to the autoregressive parameter only. We show that our subsetcontinuousupdating transformation does not alter the asymptotic distribution of the twostep GMM estimators and hence retains consistency. According to our Monte Carlo results, the transformed GMM estimators significantly outperform their standard twostep 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, Blundell, R.; Bond, S. Initial conditions and moment restrictions in dynamic panel data models. Journal of Econometrics 1998, 87, Hansen, L.P.; Heaton, J.; Yaron, A. Finitesample properties of some alternative GMM estimators. Journal of Business & Economic Statistics 1996, 14, Ahn, S.C.; Schmidt, P. Efficient estimation of models for dynamic panel data. Journal of Econometrics 1995, 68, AlonsoBorrego, C.; Arellano, M. Symmetrically normalized instrumentalvariable estimation using panel data. Journal of Business & Economic Statistics 1999, 17, Arellano, M.; Bover, O. Another look at the instrumental variable estimation of errorcomponent models. Journal of Econometrics 1995, 68, 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 Hayakawa, K. Small sample bias properties of the system GMM estimator in dynamic panel data models. Economics Letters 2007, 95, Bun, M.J.G.; Windmeijer, F. The weak instrument problem of the system GMM estimator in dynamic panel data models. Econometrics Journal 2010, 13, Donald, S.G.; ewey, W.K. A Jackknife Interpretation of the Continuous Updating Estimator. Economics Letters 2000, 67, 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/.
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