A Transformed System GMM Estimator for Dynamic Panel Data Models. February 26, 2014

Transcription

1 A Transformed System GMM Estimator for Dynamic Panel Data Models Xiaojin Sun Richard A. Ashley February 26, 20 Abstract The system GMM estimator developed by Blundell and Bond (998) for dynamic panel data models has been widely used in empirical work; however, it does not perform well with weak instruments. This paper proposes a variation on the system GMM estimator, based on a simple transformation of the dependent variable. Simulation results indicate that, in finite samples, this transformed system GMM estimator greatly outperforms its conventional counterpart in estimating the coefficient of the lagged dependent variable, especially when the variation in the fixed effects is large relative to that in the idiosyncratic shocks and when the dependent variable is highly persistent. Applying this transformation also substantially strengthens the reliability of inferences on the overall model specification based upon the Sargan/Hansen test. As illustrations, the transformed system GMM estimator is applied to two empirical examples from the literature: a production function and an employment equation. Keywords: Dynamic Panel Data Models, Transformed System GMM Estimator, Sargan/Hansen Test. JEL Classification: C8, C23, D2. We thank Maurice Bun, Suqin Ge, Kazuhiko Hayakawa, Kwok Ping Tsang, and Wen You for useful comments. Correspondence: Xiaojin Sun: Department of Economics, Virginia Tech, Blacksburg, VA, 206 ( aaronsun@ vt.edu); Richard A. Ashley: Department of Economics, Virginia Tech, Blacksburg, VA, 206 ( ashleyr@vt. edu). Electronic copy available at:

2 Introduction In recent decades, dynamic panel data models with unobserved individual-specific heterogeneity have been widely used to investigate the dynamics of economic activity. Several estimators have been suggested in order to estimate the parameters in a dynamic panel data model. A standard estimation procedure is to first-difference the model in order to wipe out the unobserved heterogeneity and to utilize the moment conditions where endogenous differences of the variables are instrumented by their lagged levels. This is the well known Arellano-Bond estimator or first-difference (DIF) GMM estimator (see Arellano and Bond (99)). The DIF GMM estimator was found to be inefficient since it does not make use of all available moment conditions (see Ahn and Schmidt (995)); 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 (999) and Blundell and Bond (998)), since the instruments in those cases become less informative. To improve the performance of the DIF GMM estimator, Blundell and Bond (998) 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 (995). The resulting system (SYS) GMM estimator has been shown to perform much better than the DIF 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 (998) and Blundell, Bond and Windmeijer (200)). 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 (2007) and Bun and Windmeijer (200), that the weak instruments problem still remains in the SYS GMM estimator. The concentration parameter (see Staiger and Stock (997) and Stock, Wright and Yogo (2002)) of the level equation, which is utilized by the SYS GMM estimator in addition to the first-difference equation, is even more sensitive to the variance ratio than that of the first-difference equation, which yields sizeable bias in the SYS GMM estimator when the variance ratio is high, where the variance ratio is defined as the variation in the fixed effects divided by the variation in the idiosyncratic shocks. In this paper, we propose a new estimator for dynamic panel data models based on a simple 2 Electronic copy available at:

3 transformation of the dependent variable. In particular, we consider a panel model including an exogenous and an endogenous variable - both moderately persistent - in addition to the lagged dependent variable. Our innovation is to transform this model by moving the lagged dependent variable to the left hand side and applying the SYS GMM estimator to the transformed model. We then estimate the autoregressive coefficient (on the lagged dependent variable) using a grid search to minimize the GMM objective function, i.e., the Sargan/Hansen statistic; an estimated standard error for this coefficient is obtained via a distribution-free residual bootstrap. We show that this transformed estimator is equivalent to the optimal SYS GMM estimator, which estimates the parameters and updates the weighting matrix simultaneously, and hence consistent for large number of cross-sections and finite number of time periods, in the special case of a simple autoregressive model. In the general case that additional explanatory variables are included, Monte Carlo experiments show that this transformation yields a significant improvement relative to the performance of the SYS GMM estimator in terms of finite sample bias, mean squared error and coverage probability of the 95% confidence interval for the autoregressive coefficient. The main contributions of this paper are: to exhibit the weak instruments problem for SYS GMM estimator; to propose a transformed system GMM estimator and evaluate the performance of this new estimator relative to its conventional counterpart using Monte Carlo simulations; and to verify our findings by providing applications to both a production function and an employment equation. We also show that the Sargan/Hansen instrument validity test is very poorly sized in SYS GMM estimations at reasonable sample lengths and that our proposed method distinctly improves matters in this regard. 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 applies the transformed system GMM estimator to an illustrative production function example and an employment equation example in the literature. Section 5 concludes the paper. The Sargan/Hansen test actually only addresses overall model misspecification - instrument validity itself is not directly testable - see Ashley and Parmeter (203) for a sensitivity analysis solution to this non-testability problem. 3

4 2 A Transformed System GMM Estimator Consider a linear model with one dynamic dependent variable, additional explanatory variables, and individual-specific fixed effects: y it = θy i,t + X itβ + u it, u it = µ i + ν it, (2.) for i =,..., N and t = 2,..., T where T is small and N is large. Here, θ is the autoregressive coefficient and β is a vector of the remaining coefficients. 2. A Simple AR() Model and Previous GMM Estimators We begin by considering the simplest dynamic panel data model, a simple AR() model; the case with additional explanatory variables X is discussed at the end of this section. Thus, consider an AR() model of the form: y it = θy i,t + u it, u it = µ i + ν it, (2.2) and specify the following assumptions on the error components structure: E(µ i ) = 0, E(ν it ) = 0, E(ν it µ i ) = 0 for i =,..., N and t =,..., T, (2.3) E(ν it ν is ) = 0 for i =,..., N and t s, (2.) as well as additional restrictions concerning the initial conditions (see Ahn and Schmidt (995)): E(y i ν it ) = 0 for i =,..., N and t = 2,..., T. (2.5) For the estimation of such an autoregressive model described by (2.2) with assumptions (2.3)- (2.5), Arellano and Bond (99) proposed transforming the model into first-differences and utilizing the following m d = 0.5(T )(T 2) moment conditions: E(Z di u i) = 0, (2.6)

5 where Z di is a (T 2) m d instrument matrix and u i is a (T 2) column vector of residuals for the first-difference equation of individual i, Z di = y i y i y i ; u i = u i3 u i y i y i,t 2 u it Consider one additional assumption: E(µ i y i2 ) = 0, (2.7) a restriction on the initial conditions (see Arellano and Bover (995)), which holds when the process is mean stationary. This condition, together with (2.3)-(2.5), yields the following m l = 0.5(T )(T 2) moment conditions, based on which the level (LEV) GMM estimator is constructed, for the level equation: E(Z li u i) = 0, (2.8) where Z li is a (T 2) m l instrument matrix and u i is a (T 2) column vector of residuals for the level equation of individual i, Z li = y i y i2 y i ; u i = u i3 u i y i2 y i,t u it To guarantee that y i,t is not correlated with µ i, we require the initial conditions restriction [( E y i µ ) ] i µ i = 0. (2.9) θ Blundell and Bond (998) and Blundell, Bond and Windmeijer (200) pointed out that the instruments used by the DIF GMM estimator become less informative when the value of the au- 5

6 toregressive coefficient θ increases towards unity and when the variance of the individual effects, σµ, 2 increases relative to the variance of the idiosyncratic errors, σν. 2 Because of the important role it plays in Blundell and Bond (998) and Blundell, Bond and Windmeijer (200), the quotient σµ/σ 2 ν 2 is standardly called the variance ratio ; this designation is used below, and frequently abbreviated vr. They proposed using what is called the SYS GMM estimator, which utilizes all the m d moment conditions for the DIF GMM estimator and a non-redundant subset of the moment conditions for the level equation, a total of m s = m d + (T 2) as follows: E(Z siũ i ) = 0, (2.0) where ũ i is a vector of error terms in the first-difference equation followed by those in the level equation of individual i, i.e., ũ i = ( u i, u i ), and Z si is a 2(T ) m s instrument matrix for both equations of individual i: Z si = Z di 0 0 Z p li = Z di y i y i y i,t. The one-step SYS GMM estimator takes the form: θ s = (ỹ Z s W s Z sỹ ) ỹ Z s W s Z sỹ, (2.) where ỹ i = ( y i, y i ) and W s = (Z s(i N H s )Z s ) with H s = H d 0 0 I T 2, where H d is a (T 2) square matrix which has twos in the main diagonal, minus ones in the first subdiagonals, and zeros otherwise. 6

7 The two-step SYS GMM estimator is given by replacing the weighting matrix W s with ( ) N W s = Z N si ũ i ũ iz si, (2.2) i= where ũ i are first-step residuals. The validity of the moment conditions (2.0) is tested by the Sargan test of overidentifying restrictions (or J test, see Sargan (958) and Hansen (982).) For the SYS GMM estimator, the Sargan/Hansen s J test statistic is given by: J s = N ũ 2 Zs W s Z s ũ 2, (2.3) where W s is the weighting matrix in (2.2) and ũ 2 are the second-step residuals. Under the null hypothesis that the underlying model is well-specified and that the moment conditions used in the system GMM estimation are all valid, J s is asymptotically chi-squared distributed with m s k degrees of freedom where m s stands for the number of moment conditions in system GMM estimation and k is the number of estimated parameters. In case that a vector of additional regressors X, which are correlated with the individual effects µ, are included in the model, additional valid moment conditions can be utilized depending on three possible correlation structures between regressor x it and the idiosyncratic error term ν it.. x it is strictly exogenous, i.e., E(x is ν it ) = 0 for s =,..., T and t = 2,..., T, E(x is u it ) = 0 for t = 3,..., T and s T, (2.) E(u it x it ) = 0 for t = 2,..., T. (2.5) 2. x it is weakly exogenous (or predetermined), i.e., E(x is ν it ) = 0 for s =,..., t and t = 2,..., T and E(x is ν it ) 0 for s = t +,..., T and t = 2,..., T, E(x i,t s u it ) = 0 for t = 3,..., T and s t, (2.6) E(u it x it ) = 0 for t = 2,..., T. (2.7) 3. x it is endogenous, i.e., E(x is ν it ) = 0 for s =,..., t and t = 2,..., T and E(x is ν it ) 7

8 0 for s = t,..., T and t = 2,..., T, E(x i,t s u it ) = 0 for t = 3,..., T and 2 s t, (2.8) E(u it x i,t ) = 0 for t = 3,..., T. (2.9) 2.2 TSYS GMM A Transformed System GMM Estimator Here we consider a dynamic panel data model with just two additional explanatory variables beyond the lagged dependent variable, so that one of these two additional variables can be constructed so as to be correlated with both the individual effect µ i and the idiosyncratic error ν it, whereas the other is constructed so as to be correlated with µ i only. The model then can be represented by: y it = θy i,t + β x x it + β z z it + u it, u it = µ i + ν it, (2.20) where both x and z follow AR() processes: x it = ρ x x i,t + τ x µ i + λν it + ε it, (2.2) z it = ρ z z i,t + τ z µ i + ξ it, (2.22) with µ i N (0, σ 2 µ), ν it N (0, σ 2 ν), ε it N (0, σ 2 ε), ξ it N (0, σ 2 ξ ), and the three autoregressive coefficients θ, ρ x, and ρ z all lie in the interval (, ). For the SYS GMM estimator, in order to utilize the lagged differences as instruments for the level equation, we first consider under what conditions y it, x it, and z it are all uncorrelated wih µ i. Notice that t 3 x it = ρx t 2 x i2 + ρ s x(λ ν i,t s + ε i,t s ), (2.23) s=0 t 3 z it = ρz t 2 z i2 + ρ s z ξ i,t s, (2.2) s=0 8

9 so that x it and z it will be correlated with µ i if and only if x i2 and z i2 are correlated with µ i. To guarantee E[ x i2 µ i ] = E[ z i2 µ i ] = 0, we require the initial conditions: [( E x i τ ) ] xµ i µ i = 0, (2.25) ρ x [( E z i τ ) ] zµ i µ i = 0. (2.26) ρ z Given the initial conditions on x it and z it, according to Equation (2.20), writing y it as y it = θ y i,t + β x x it + β z z it + v it t 3 t 3 t 3 = θ t 2 y i2 + β x θ s x i,t s + β z θ s z i,t s + θ s ν i,t s, (2.27) s=0 s=0 shows that y it will be correlated with µ i if and only if y i2 is correlated with µ i. To guarantee E[ y i2 µ i ] = 0, we then require the similar initial conditions: [( E y i µ ( i βx τ x + β )) ] zτ z + µ i = 0. (2.28) θ ρ x ρ z s=0 Given (2.25), (2.26) and (2.28), the following moment conditions are valid: E(Z siũ i ) = 0, (2.29) where Z si = Z di 0 0 Z p li, (2.30) with Z di = y i x i z i3 0 y i y i x i x i2 0 0 z i y i y i,t x i x i,t 2 z it, 9

10 and Z p li = y i2 0 0 x i2 0 0 z i3 0 y i3 0 0 x i3 0 z i y i,t 0 0 x i,t z it. Z di and Z p li are matrices of size (T 2) [(T )(T 2)+] and (T 2) [2(T 2)+] respectively. Even though the two-step SYS GMM estimator is appropriate, i.e., consistent for large N and finite T, and it is widely used for the estimation of model (2.20)-(2.22), it is known to have sizeable finite sample bias when the fixed effects have large variance relative to the idiosyncratic errors, i.e., when the variance ratio, σµ/σ 2 ν, 2 is large. More specifically, the SYS GMM estimator is a weighted average of the DIF GMM and the LEV GMM estimators, where the weight on the LEV GMM estimator increases with the autoregressive coefficient and with the variance ratio. However, the concentration parameter for the level model (LEV GMM) is much more sensitive to the value of σµ/σ 2 ν 2 than the concentration parameter of the difference model (DIF GMM). 2 This induces very poor performance in the SYS GMM estimator when σµ/σ 2 ν 2 is large (see Hayakawa (2007) and Bun and Windmeijer (200)). 3 See also Doran and Schmidt (2006) on the poor finite sample properties of GMM estimators in highly overidentified models. In order to address this problem, we transform the model into the following form, by moving the lagged dependent variable to the left-hand side of Equation (2.20) and constructing a new dependent variable y (θ), y it(θ) = y it θy i,t = β x x it + β z z it + µ i + ν it. (2.3) This transformation complicates the estimation of θ (and especially the sampling variance of this parameter estimate), but conditional on θ it yields a non-dynamic model, GMM estimation of which can have, as will be demonstrated below, desirable features. 2 The concentration parameter is a unitless measure of strength of the instruments, see Stock, Wright and Yogo (2002). In our framework, it quantifies the instrumental strength of y i,..., y i,t 2, x i,..., x i,t 2, and z it in the first-difference equation, and the strength of y i,t, x i,t, and z it in the level equation. 3 Hayakawa (2007) conducted Monte Carlo experiments, showing that the bias of the system GMM estimator is sizeable for any true value of autoregressive coefficient between 0 and when the variance ratio, σ 2 µ/σ 2 ν, is equal to in a sample of N = 50 and T =. According to Bun and Windmeijer (200), the absolute bias of the system GMM estimator is an increasing function of variance ratio, σ 2 µ/σ 2 ν. 0

11 Equation (2.3) is then estimated in two steps,. Using a grid search, specify a sequence of different values of θ in the domain (, ) and perform the two-step SYS GMM on the regression of yit ( θ) = y it θy i,t on x it and z it for each θ, using the same instrument set as used by the SYS GMM estimator, Z s = (Z s, Z s2,, Z sn ) where Z si is defined in (2.30). For each posited θ value, collect the Sargan/Hansen statistic. The θ which corresponds to the least value of Sargan/Hansen statistic is chosen to be the estimate θ. 2. Substitute θ back into Equation (2.3) and obtain the two-step SYS GMM estimates of β x and β z from the regression of y it ( θ) on x it and z it as β x and β z using Z s as instruments. In particular, for any posited θ, y it ( θ) = y it θy i,t, the transformed model takes the form y it( θ) = β x x it + β z z it + µ i + ν it. (2.32) The one-step SYS GMM estimator over the transformed regression (2.3) can be written as β x ( θ) β z ( θ) = ([ x z] Z s W s Z s[ x z]) [ x z] Z s W s Z sỹ ( θ), (2.33) where x, z and ỹ are stacked across individuals with x i = ( x i, x i ), z i = ( z i, z i ), ỹ i = ( y i, y i ). Z s and W s are the same instrument set and weighting matrix as used by the SYS GMM estimator on the untransformed model. replacing the weighting matrix with i= The two-step SYS GMM estimator is given by ( ) N W s = Z N si ũ i ũ iz si, (2.3) where ũ i are first-step residuals, i.e., ũ i = ỹ i ( θ) x i βx ( θ) z i βz ( θ). The Sargan/Hansen statistic is then a function of θ: J( θ) = N ũ 2 Zs W s Z s ũ 2, (2.35)

12 where ũ 2 are second-step residuals and W s takes the form of (2.3), both of which vary with θ. The estimate of θ is obtained by minimizing the Sargan/Hansen statistic in the domain (, ), θ = argminj( θ) = θ (,) N ũ 2 Zs W s Z s ũ 2, (2.36) and the corresponding estimates of β s are given by: β x = β x ( θ), βz = β z ( θ). (2.37) It is worth noting that, in a simple AR() model without any regressors beyond the lagged dependent variable, our transformed system (denoted TSYS hereafter) GMM estimator collapses to the optimal (or continuously updating) SYS GMM estimator, which estimates θ simultaneously with updating the weighting matrix W s. The optimal SYS GMM estimator is asymptotically equivalent to the traditional two-step SYS GMM estimator, since the extra term in the minimization problem with continuously updating weight matrix does not distort the limiting distribution (see Pakes and Pollard (989)). 5 By initially moving the lagged dependent variable to the left-hand side, the procedure specified above does not provide an estimated asymptotic standard error for θ, however. Consequently, we obtain the standard error of θ through a distribution-free bootstrap. 3 Monte Carlo Experiments The SYS GMM estimator is consistent and asymptotically more efficient than the DIF GMM estimator, but is known to perform poorly in finite samples, especially when the variance ratio is high and when the dependent variable is highly persistent. This section provides finite sample results of the TSYS GMM estimator and comparisons to the SYS GMM estimator via Monte Carlo experiments. In practice, the sign of the persistence in the data is easy to determine. One might search over (0, ) if the dependent variable exhibits positive persistence, as in most of the cases. 5 Hansen, Heaton and Yaron (996) compared the finite sample properties of some alternative GMM estimators and found that the optimal GMM estimator demonstrates a better performance than the traditional two-step GMM estimator in Monte Carlo experiments. 2

13 3. Monte Carlo Setup We consider the model specified in Equations (2.20)-(2.22) with the initial observations drawn from the following mean-stationary distributions: x i, T0 = τ xµ i + λν i, T0 + ε i, T0, (3.) ρ x z i, T0 = τ zµ i + ξ i, T0, (3.2) ρ z y i, T0 = µ ( i βx τ x + β ) zτ z + + ν i, T0, (3.3) θ ρ x ρ z with T 0 = 30, namely a 30-period pre-sample is drawn to ensure covariance stationarity in x it, z it, and y it. 6 These initial observations in Equations (3.)-(3.3) render x it, z it and y it mean stationary in the sense that their conditional mean given µ i is time-invariant, i.e., E(x it µ i ) = τ xµ i for t =,..., T and i =,..., N, (3.) ρ x E(z it µ i ) = τ zµ i for t =,..., T and i =,..., N, (3.5) ρ z E(y it µ i ) = µ ( i βx τ x + β ) zτ z + for t =,..., T and i =,..., N. (3.6) θ ρ x ρ z The parameters that are varied in the simulations are the three autoregressive coefficients θ, ρ x, ρ z and the variance ratio, vr = σ 2 µ/σ 2 ν, which measures the variance of fixed effects relative to that of idiosyncratic errors. We consider six schemes, with vr taking values of, and for two different sample sizes, N = 500, T = 6 and N = 500, T = respectively. For each scheme, there are twelve designs with θ = {0.2, 0.5, 0.8} and ρ x, ρ z = {0.5, 0.8}. These three population values of θ correspond to low time-series persistence, medium persistence, and high persistence. The parameter values adopted (and held fixed) in the various Monte Carlo results are: β x =, β z =, λ = 0.2, σ 2 ν =, σ 2 ε = 0.6, σ 2 ξ = 0.6. These choices for β x and β z ensure that their estimates are statistically different from zero in the results presented below; The parameter λ, which characterizes the endogeneity of variable x, is 6 Qualitatively similar simulation results are also obtained without utilizing this pre-sample. 3

14 specified so as to ensure a correlation between x and v of around 0.3 when ρ x = 0.8 and around 0. when ρ x = 0.5. Since we fix the variance of the idiosyncratic errors, ν it, at, the variance of the fixed effects, µ i, is denoted by the variance ratio, i.e., vr = σµ. 2 Notice that the simulation results depend only on the relative value vr, not on the total variance σµ 2 + σν. 2 We choose τ x and τ z to be τ x = 0.2/ vr, τ z = 0.2/ vr. We evaluate the performance of the TSYS GMM estimator in estimating θ, the coefficient of lagged dependent variable, as the variance ratio takes on various values in different sample sizes. By specifying τ x and τ z to be inversely related to the square root of variance ratio, we are able to keep both the endogeneity in x it and z it and the strength of corresponding instruments as stable as possible, in both the first-difference equation and the level equation. More specifically, in our setup, the endogeneity in x it and z it does not vary with the variance ratio in the first-difference equation and varies only slightly in the level equation; the strength of lagged levels of x it and first-difference of z it as instruments for the first-difference equation do not vary with the variance ratio, nor do the strength of lagged difference of x it and first-difference of z it as instruments for the level equation. Thus, the variance ratio mainly affects the endogeneity and instrument strength involving the lagged dependent variable, so that we are able to focus particularly on this autoregressive coefficient. All results are based on,000 Monte Carlo replications, with new values for initial observations drawn in each replication. To bootstrap the standard error of θ, we first estimate the residuals, ν it with t = 3,..., T, and obtain a bootstrap sample for the dependent variable by resampling the residuals and conditioning on the initial values y i and y i2. The bootstrapped standard error is obtained by calculating the sample standard deviation of the SYS GMM estimator using 50 draws from the bootstrapped sampling distribution. 7 For a fair comparison, we also bootstrap the standard error for the SYS GMM estimator, though the asymptotic standard error is available. 7 To bootstrap the standard error, we employ the SYS GMM estimator, instead of our TSYS GMM estimator, in order to economize on these bootstrap calculations, since these two estimators have similar precision, as shown in our simulation results below. On the number of replications, Efron and Tibshirani (993) stated that, for standarderror estimation, 50 bootstrap replications are often enough to give a good estimate and very seldom are more than 200 replications needed. In the present case, 50 replications suffice in order for the bootstrapped standard errors to converge. Note that these bootstrap replications provide an estimated standard error for θ T SY S and hence must be repeated for Monte Carlo simulation of the estimators. In practice, of course, this bootstrap calculation would only be done once, for the θ T SY S obtained using the actual sample data.

15 3.2 Monte Carlo Results From this point on, we apply a superscript TSYS to an estimated parameter to denote the transformed system GMM parameter estimate and SYS for a conventional system GMM estimate. In particular, θ T SY S and θ SY S represent the autoregressive coefficient estimates provided by these two estimators, respectively; β T SY S x, β T SY S z and β SY S x, β SY S x differences between these two estimators in the following four subsections. are defined similarly. We discuss the 3.2. Parameter Estimates The estimates of θ, β x, and β z given by both the TSYS GMM estimator and the SYS GMM estimator are presented in Table for a short panel (T = 6) and in Table 2 for a panel of moderate length (T = ). In each of these two tables, Panels I through IV correspond to the four possible combinations of ρ x and ρ z, the posited first-order autocorrelation coefficients in x it and z it, each takes on a value of 0.5 and 0.8. Within each panel, there are nine schemes, in which θ takes on three different values θ = {0.2, 0.5, 0.8} and vr takes on three different values vr = {,, }. The finite sample bias ( bias ) and mean squared error ( mse ) over,000 Monte Carlo simulations are displayed in the first two columns for each scheme. We highlight three important findings below. First, the evidence that the SYS GMM estimates are sensitive to the variance ratio is consistent with the results of Hayakawa (2007) and Bun and Windmeijer (200). In particular, for any given value of θ, the SYS GMM estimate of the autoregressive coefficient becomes more biased as the variance ratio increases for each combination of ρ x and ρ z. For example, with (θ, ρ x, ρ z ) = (0.8, 0.8, 0.8), the bias of θ SY S becomes 8 times larger as the variance ratio increases from to in Table with T = 6 and 9 times larger in Table 2 with T =. Concomitantly, the mean squared error, a more informative indicator of estimator imprecision, increases 7 times and 0 times, respectively. In contrast, the TSYS GMM estimates are much less sensitive to the variance ratio, increasing by a factor of only 2 in this instance. Second, compared to the SYS GMM estimator, the TSYS GMM estimator is less biased in estimating θ, the coefficient on the lagged dependent variable especially in the case of high variance ratio and high time-series persistence in y it, where the weak instrument problem becomes more prominent. For example, with vr = and θ = 0.8, the mean squared error of θ T SY S is about 5

16 one half that of θ SY S with T = 6 and only one fifth that of θ SY S with T = across the different combinations of ρ x and ρ z. Third, the TSYS and SYS GMM estimators do not differ much from each other in estimating β x and β z, in either bias or mean squared error, even the former estimator gives rise to larger but insignificant bias in the estimate of β x in some cases. Thus, one need not sacrifice β x or β z estimation accuracy in using the TSYS GMM estimator to obtain more accurate θ estimation, although the TSYS GMM estimation does require additional computations to obtain the bootstrapped standard error estimates for θ T SY S. Where, as is common in applied work, an analysis of the economic dynamics is a primary focus, an accurate estimation of the lagged dependent variable coefficient, θ, is essential. The conventional SYS GMM estimator tends to substantially over-estimate this lagged dependent variable coefficient when the variance ratio and persistence in the dependent variable are both sizeable. In particular, our results imply that, if the true value of θ is high enough, θ SY S can exceed one and the estimated 95% confidence interval will likely include values beyond the parameter space within which an autoregressive model is stationary. Thus, in such cases the SYS GMM estimation results imply a model which is dynamically unstable, and one might wrongly proceed to choose a unit root model instead. In contrast, the TSYS GMM estimator more accurately estimates the lagged dependent variable coefficient and, in addition, dramatically reduces the incidence of truly poor inference with regard to the model dynamics Standard Error Estimates The standard deviation ( std ) over the,000 Monte Carlo replications is displayed in the third column of each scheme in Tables and 2, followed by the standard error ( se ) in the fourth column. The std (third) column displays the observed dispersion of each parameter estimator around its mean across the Monte Carlo simulations of the equation system arguably, this is quantifying the square root of the actual sampling variance of this estimator. The se (fourth) column displays the average value across the Monte Carlo simulations of the corresponding parameter estimator standard error estimates obtained via the bootstrap replications. The bootstrapped standard error estimates are in all cases good approximations to these actual standard deviations across the Monte Carlo simulations. 6

17 3.2.3 Coverage Probabilities Note that, with high variance ratio and high persistence in y it, the bias of θ SY S is usually several times larger than the corresponding simulated standard deviation and asymptotic standard error. Hence, we next report on the fraction of each set of,000 Monte Carlo simulations for which the true parameter value lies within the estimated 95% confidence interval. These 95% confidence interval coverage estimates are presented in Table 3 for the short panel (T = 6) and in Table for the panel of moderate length (T = ). Two measures of coverage probability are reported. One of these is based on the confidence intervals using in each case the dispersion of the standard deviation across the Monte Carlo simulations; this is denoted C std. The other coverage probability is for the actually implementable confidence intervals, using the bootstrapped standard error estimates; this is denoted C se. With a low variance ratio, i.e., vr =, the coverage probabilities, either C std or C se, of the 95% confidence intervals for both estimators are all around 95%. However, as the variance ratio rises to, C std for θ SY S drops dramatically to around 60% with T = 6 and around 0% with T =, across different combinations of ρ x, ρ z, and θ; C se becomes even lower. In contrast, the 95% confidence interval for θ T SY S, using either the simulated standard deviation or bootstrapped standard error, maintains a probability of around 95%. These results strongly suggest that θ T SY S (and its bootstrapped standard error) yields much more reliable inference result with regard to θ, as well as a more accurate estimate of θ Size of the Sargan/Hansen Test Given the fact that the endogenous regressor x it, exogenous regressor z it and the dependent variable y it are drawn from mean stationary processes as described in Equations (3.)-(3.6), lagged levels and lagged differences of x it and y it are valid instruments for the first-difference equation and level equation, respectively, and lagged z it are valid instruments for both equations. Thus, presuming that the model is in all other respects well-specified, the null hypothesis of the Sargan/Hansen test is satisfied. Table 5 shows the rejection frequencies of the Sargan/Hansen test for both estimators, using their asymptotic critical values at a nominal 5% level, in the Monte Carlo experiments of Tables and 2. 7

18 With either a short panel, T = 6, or a moderate panel, T =, the rejection frequencies of the Sargan/Hansen test for the TSYS GMM estimator lie around 5% whereas those for the SYS GMM estimator are much higher, even exceeding 5% when the dependent variable y it is highly persistent (θ = 0.8) and the variance ratio is high (vr = ). These results indicate that the small sample bias found for the conventional SYS GMM estimator in the presence of weak instruments tends to give rise to serious size distortions in the behavior of the associated Sargan/Hansen test. 8 Of course, since the Sargan/Hansen test is only asymptotically justified, the size of the test is expected to improve as the number of cross-sections, N, increases. For example, with θ = 0.8, ρ x = 0.8, ρ z = 0.8, vr = and T = 6, the rejection frequency is 9.% when N = 500; For simulations with N = 3, 000 (not reported here) the rejection frequency does drop to 8%, however. These results indicate that a huge sample size is required in order for the Sargan/Hansen test associated with the conventional SYS GMM estimator to be well-sized. In practice, such huge datasets are rarely available. The Monte Carlo simulation results in Tables and 2 indicate that the TSYS GMM estimator yields reliable inference on the overall model specification based on the Sargan/Hansen test even in reasonably modest samples. The Sargan/Hansen statistic can also be used to test the validity of a subset of instruments, via a difference-in-sargan/hansen test, also known as a C statistic. 9 Table 6 reports the probability of wrongly rejecting the validity of the level moment conditions. By comparing those probabilities to 5%, we find the same phenomenon as shown in Table 5: The rejection frequencies of differencein-sargan/hansen test are beyond the acceptable region using the SYS GMM estimator whereas those for the TSYS GMM estimator match the pre-sample criterion of 5%. These large size distortions in both the Sargan/Hansen and the difference-in-sargan/hansen test for the conventional SYS GMM estimator indicate that these tests are likely to be quite misleading unless the sample size is quite huge. In contrast, the Sargan/Hansen test sizes are far closer to the nominal level using the TSYS GMM estimator. Thus, the TSYS GMM estimator not only improves the accuracy with which the unknown parameters in model can be estimated and tested but also strengthens the reliability of inferences on the instrument validity and/or overall model specification 8 Roodman (2009) suggested not taking comfort in a Sargan/Hansen test p-value below 0. but viewing higher values as potential signs of trouble. 9 Of course, the validity of these tests rests on an assumption that the remaining instruments and the rest of the model specification are all valid. 8

19 using the Sargan/Hansen test. Empirical Examples In this section, we apply both the SYS GMM estimator and the TSYS GMM estimator to two empirical examples, the production function and the employment equation, both have been studied in the literature.. The Production Function Using the conventional SYS GMM estimator, Blundell and Bond (2000) estimated a Cobb- Douglas production function which has a dynamic representation as follows: y it = θy i,t + β k it + β 2 k i,t + β 3 n it + β n i,t + µ i + γ t + ν it, (.) subject to two non-linear common factor restrictions β 2 = θβ and β = θβ 3. Here, y it is log real output of firm i in year t, k it is log real capital stock, and n it is log employment. Of the error components, µ i is an unobserved firm-specific effect, γ t is a year-specific intercept (reflecting a common technology shock), and ν it is an idiosyncratic shock. Using data for a panel of 509 R&D-performing US manufacturing companies observed for the 8 years, comprising 982 to 989, they found that the SYS GMM estimator yields much more reasonable parameter estimates by exploiting additional instruments (these additional instruments are based on reasonable stationarity restrictions imposed on the initial conditions process, beyond those utilized by the DIF GMM estimator). In this section, we apply both the SYS GMM estimator and the TSYS GMM estimator to the estimation of the production function in Equation (.). The data we use, similar to that used in Mairesse and Hall (996) and that in Blundell and Bond (2000), is a balanced panel of 586 R&D-performing US manufacturing companies over the period. 0 0 Our sample is larger than the sample of 2 firms in Mairesse and Hall (996), where they more or less arbitrarily discarded observations with jumps in the stock variables of absolute value greater than 200 percent and in the flow variables of absolute value greater than 300 percent. Our sample is also larger than the 509 firms used in Blundell and Bond (2000), where they chose 200 percent and 900 percent as the thresholds for the stock variables and the flow variables, respectively. While we are using only three variables out of more than ten in the original dataset, instead of imposing an arbitrary threshold, we keep all the observations and end up with a sample of 586 firms. Diagnostic checks indicate that our results are not driven by our retention of these somewhat less typical firms in the data set. 9

20 The production series, y it, is defined as the log of nominal sales (in millions), deflated by the manufacturing sector output deflator; k it is log of the nominal plant & equipment (in millions) at the beginning of year t, deflated by the manufacturing sector investment deflator; and n it is log of fulltime employment (in thousands). Both deflators are taken from NBER-CES Manufacturing Industry Database see Bartelsman and Gray (996) and use 987 as their base year. All three series, y it, k it, and n it, are highly persistent. So as to make the estimation comparable to our simulation setup, we eliminate the year-specific effects at the outset by cross-sectionally demeaning the data. This approach is equivalent to including time dummies in the regression models. To be consistent with Blundell and Bond (2000), we maintain the assumption that both capital stock (k it ) and employment (n it ) are potentially correlated with the firm-specific effects (µ i ) and the idiosyncratic shocks (ν it ). In this sense, k it and n it are endogenous and their first lag k i,t and n i,t are predetermined. Values of y it, k it, and n it lagged two and more periods can consequently be used as instruments for the first-difference equation and their lagged first-differences can be used for the level equation. The estimation results for this model are presented in Table 7. In the first two columns of Table 7, we estimate Equation (.) using the entire sample of 586 firms over 3 years. Results using the SYS GMM estimator and using our TSYS GMM estimator are given in columns () and (2), respectively. Note that the Sargan/Hansen test rejects the overall model specification in both columns. This is consistent with what was found by Blundell and Bond (2000), to the effect that the validity of lagged levels dated t 2 and earlier as instruments in the first-difference equations is clearly rejected by the Sargan/Hansen test of overidentifying restrictions. Their explanation of this phenomenon focuses on the presence of measurement errors. They showed that, in the case of measurement errors, the idiosyncratic shocks ν it violate the requisite IID assumption and become a moving average process of order one instead. Blundell and Bond (2000) consequently instead utilized lagged levels dated t 3 and earlier as instruments in the firstdifference equation, combined with lagged first differences dated t 2 as instruments in the level equation. Instead of discarding part of the instruments, we retain the entire set of instruments but confine the estimation to just the first ten years, ; these estimation results are given in columns (3) and () of Table 7. The Sargan/Hansen test still rejects the overall model specification but the corresponding p-value is close to the threshold. Then, we further confine the estimation to the 20

21 subperiod and present the results in columns (5) and (6). In each column, fitting the model in Equation (.) to the data over this period is supported by the Sargan/Hansen test of a p-value higher than 5%. With a short sample of just 7 years and a low variance ratio which is estimated to be 2.7 in the present case, the TSYS GMM estimate of the coefficient on lagged production ( θ T SY S ), 0.578, is similar to the sample realization of the SYS GMM estimator ( θ SY S ), This is consistent with our simulation results that the SYS GMM and TSYS GMM estimators have similar performance in estimating the coefficient of the lagged dependent variable when the fixed effects exhibit low variation relative to the idiosyncratic shocks..2 The Employment Equation We consider a dynamic employment equation of the form: n it = θn i,t + β w it + β 2 w i,t + β 3 k i,t + β k i,t + µ i + γ t + ν it, (.2) where n it is the log of U.K. employment in company i at the end of year t, w it is the log of the real product wage, and k it is the log of gross capital. Of the error components, µ i is an unobserved firm-specific effect, γ t is a year-specific intercept, and ν it is an idiosyncratic shocks. The data we use is from the published accounts of 0 U.K. manufacturing companies with seven or more continuous observations during the period 976-8; see Arellano and Bond (99) for a detailed description. We utilize a balanced sub-sample of 38 companies between 977 and 982, over which period we have most observations and eliminate the year-specific effects at the outset by cross-sectionally demeaning the data. We maintain the assumption that both wage (w it ) and capital (k it ) are endogenous and their first lag w i,t and k i,t are predetermined. Values of n it, w it, and k it lagged two and more periods can consequently be used as instruments for the first-difference equation and their lagged first-differences can be used for the level equation. The estimation results for this model are presented in Table 8. Results using the SYS GMM estimator and using the TSYS GMM estimator are given in columns () and (2), respectively. The Sargan/Hansen test associated with the SYS GMM estimation rejects Blundell and Bond (998) estimated Equation (.2) using the unbalanced panel of all 0 firms and obtained similar results; see their Table. 2

22 the validity of instruments (and the overall model specification), but note that the Sargan/Hansen test based on the TSYS GMM estimation does not reject. This results is consistent with the simulation results discussed in Section 3 above, indicating that the Sargan/Hansen test is well-sized when our TSYS GMM estimator is applied whereas it tends to over-reject a true null hypothesis after a conventional SYS GMM estimation. The SYS GMM estimate of the lagged dependent variable coefficient ( θ SY S ) of 0.80 is substantially larger than that of the TSYS GMM estimator ( θ T SY S ), whose sample realization here equals just This is a difference of 0.29 on a coefficient estimate with an estimated standard error of something like to 0.099, so is clearly significant. 2 Thus, Thus, θ T SY S is providing a significantly different estimate of θ than is θ SY S in the case of high variance ratio, which is estimated to be 5.56 here; the estimated standard error of θ T SY S (of versus 0.099) is also smaller than that of θ SY S, suggesting that θ T SY S is also providing a more precise estimate of θ. This example clearly illustrates the simulation-based results of Section 3, indicating that the conventional SYS GMM estimator tends to over-estimate the lagged dependent variable coefficient in the case of high variance ratio, whereas the TSYS GMM estimator does not. 5 Conclusion In this paper, we propose a new estimator (denoted TSYS GMM above) which is implemented as a transformation of the conventional system GMM estimator (denoted SYS GMM above) for dynamic panel data models with additional regressors beyond the lagged dependent variable. The finite sample properties of both estimators are compared using Monte Carlo simulation methods. The transformed system GMM estimator dramatically improves on the performance of the conventional system GMM estimator, in terms of finite sample bias, mean squared error, and the 95% confidence interval coverage, especially when the fixed effects have large variation relative to the idiosyncratic shocks and when the dependent variable is highly persistent. In addition, Monte Carlo experiments indicate that the finite sample bias of the conventional system GMM estimator in the presence of weak instruments gives rise to quite serious size distortions in the behavior of the associated Sargan/Hansen test. The transformed system GMM estimator, how- 2 The exact significance, of course, depends on the correlation between the sampling error in θ T SY S and θ SY S. 22

23 ever, renders the Sargan/Hansen test well-sized. This result strongly suggests that this transformed system GMM estimator is substantially more reliable than its conventional counterpart in terms of providing Sargan/Hansen test based inferences concerning the validity of instruments and/or the overall model specification. 23

24 References Ahn, Seung C. and Peter Schmidt, Efficient estimation of models for dynamic panel data, Journal of Econometrics, 995, 68 (), Alonso-Borrego, Cesar and Manuel Arellano, Symmetrically normalized instrumental-variable estimation using panel data, Journal of Business & Economic Statistics, 999, 7 (), Arellano, Manuel and Olympia Bover, Another look at the instrumental variable estimation of errorcomponent models, Journal of Econometrics, 995, 68 (), and Stephen Bond, Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations, Review of Economic Studies, 99, 58 (2), Ashley, Richard A. and Christopher F. Parmeter, Sensitivity analysis of inference in GMM estimation with possibly-flawed moment conditions, Working Paper, 203, pp. 3. Bartelsman, Eric J. and Wayne Gray, The NBER manufacturing productivity database, Technical Working Paper 205, National Bureau of Economic Research, 996, pp. 3. Blundell, Richard and Stephen Bond, Initial conditions and moment restrictions in dynamic panel data models, Journal of Econometrics, 998, 87 (), 5 3. and, GMM estimation with persistent panel data: an application to production functions, Econometric Reviews, 2000, 9 (3), ,, and Frank Windmeijer, Estimation in dynamic panel data models: improving on the performance of the standard GMM estimator, Vol. 5, Emerald Group Publishing Limited, Bun, Maurice J. G. and Frank Windmeijer, The weak instrument problem of the system GMM estimator in dynamic panel data models, Econometrics Journal, 200, 3 (), Doran, Howard E. and Peter Schmidt, GMM estimators with improved finite sample properties using principal components of the weighting matrix, with an application to the dynamic panel data model, Journal of Econometrics, 2006, 33 (), Efron, Bradley and Robert J. Tibshirani, An introduction to the bootstrap, New York: Chapman & Hall, 993. Hansen, Lars Peter, Large sample properties of generalized method of moments estimators, Econometrica, 982, 50 (),

25 , John Heaton, and Amir Yaron, Finite-sample properties of some alternative GMM estimators, Journal of Business & Economic Statistics, 996, (3), Hayakawa, Kazuhiko, Small sample bias properties of the system GMM estimator in dynamic panel data models, Economics Letters, 2007, 95 (), Mairesse, Jacques and Bronwyn H. Hall, Estimating the productivity of research and development: an exploration of GMM methods using data on French and United States manufacturing firms, NBER Working Papers 550, National Bureau of Economic Research, Inc., 996, pp. 33. Pakes, Ariel and David Pollard, Simulation and the asymptotics of optimization estimators, Econometrica, 989, 57 (5), Roodman, David, How to do xtabond2: an introduction to difference and system GMM in Stata, The Stata Journal, 2009, 9 (), Sargan, J. D., The estimation of economic relationships using instrumental variables, Econometrica, 958, 26 (3), Staiger, Douglas and James H. Stock, Instrumental variables regression with weak instruments, Econometrica, 997, 65 (3), Stock, James H., Jonathan H. Wright, and Motohiro Yogo, A survey of weak instruments and weak identification in generalized method of moments, Journal of Business & Economic Statistics, 2002, 20 (),

26 Table : Monte Carlo Results, N=500, T=6, β x = β z = Panel I: ρ x = 0.5, ρ z = 0.5 θ = 0.2 θ = 0.5 θ = 0.8 vr bias mse std se bias mse std se bias mse std se θ T SY S θ SY S βx T SY S βx SY S βz T SY S βz SY S θ T SY S θ SY S βx T SY S βx SY S βz T SY S βz SY S θ T SY S θ SY S βx T SY S βx SY S βz T SY S βz SY S Panel II: ρ x = 0.5, ρ z = 0.8 θ = 0.2 θ = 0.5 θ = 0.8 vr bias mse std se bias mse std se bias mse std se θ T SY S θ SY S βx T SY S βx SY S βz T SY S βz SY S θ T SY S θ SY S βx T SY S βx SY S βz T SY S βz SY S θ T SY S θ SY S βx T SY S βx SY S βz T SY S βz SY S