Improving GMM Efficiency in Dynamic Models for Panel Data with Mean Stationarity

Abstract

Estimation of dynamic panel data models largely relies on the generalized method of moments (GMM), and adopted sets of moment conditions exploit information up to the second moment of the variables. However, in many microeconomic applications, the variables of interest are skewed (typical examples are individual wages, size of the firms, number of employees, etc.); therefore, third moments might provide useful information for the estimation process. In this paper, we propose a moment condition, to be added to the set of conditions customarily exploited in GMM estimation of dynamic panel data models, that exploits third moments. The moment condition we propose is based on the data generating process that, under mean stationarity, characterizes the initial observation $y_{i0}$ and the long-run mean of the dependent variable. In the literature on dynamic panel data models and in the way how Monte Carlo simulations are implemented therein for mean stationary processes, this condition is always fulfilled, but never explicitly exploited for estimation. Monte Carlo experiments show remarkable efficiency improvements when the distribution of individual effects, and thus of $y_{i0}$, is indeed skewed.

Appendices

Appendix A: Detailed Results of Monte Carlo Experiments

In this Appendix, detailed tables reporting the results of the Monte Carlo experiments are presented. Table 13.4 reports the full set of results for the Monte Carlo experiments related to the pure dynamic model (Sect. 13.3.1; Table 13.1), whereas Table 13.5 reports results for the experiments including a simultaneously determined regressor (Sect. 13.3.2; Table 13.2). The tables show the mean, variance, median and interquartile range (IQR) of Monte Carlo estimates.

Table 13.4 Detailed results of Monte Carlo experiments within a pure dynamic framework (Sect. 13.3.1; Table 13.1)

Table 13.5 Detailed results of Monte Carlo experiments with an endogenous regressor (Sect. 13.3.2; Table 13.2)

Appendix B: Variance Comparison

In this Appendix, we derive the variance of the estimator we propose, based on moment conditions (13.2), (13.4) and (13.6), and we compare it with the variance of the traditional system GMM estimator.

We denote the set of R moment conditions in the population as $E(\mathbf {f}(\beta ))=0$. In order to simplify computations, we take into account the case of model (13.1), so that $\beta$ (and the variance of its estimator) is a scalar.

From general GMM theory (Hansen 1982), the asymptotic variance of a GMM estimator is given by

$$ V(\hat{\beta }) = (B'\varLambda ^{-1} B)^{-1} $$

with $B= \partial \mathbf {f}(\beta ) / \partial \beta$ and $\varLambda = E(\mathbf {f}(\beta ) \mathbf {f}(\beta )')$ the efficient weighting matrix of the GMM procedure.

In this case, we can partition B into two components (and $\varLambda$ accordingly): (i) the elements corresponding to the moment conditions that characterize the traditional system GMM estimator, and (ii) the one additional moment condition we propose in this paper. Assuming that the moment condition we propose is the last moment condition in the full set composed of (13.2), (13.4) and (13.6), we can write

$$ B = \left( \begin{array}{c} B_1 \\ B_2 \end{array} \right) \; \; \text { and } \; \; \varLambda = \left( \begin{array}{cc} \varLambda _1 &{} \varLambda _2 \\ \varLambda _3 &{} \varLambda _4 \end{array} \right) $$

with $B_1$ related to the moment conditions in (13.2) and (13.4), and $B_2$ related to the moment condition in (13.6). $\varLambda$ is partitioned accordingly (with $\varLambda _3=\varLambda _2'$). Therefore, we have

$$\begin{aligned} V(\hat{\beta }) = (B'\varLambda ^{-1} B)^{-1} = \left\{ \left( \begin{array}{cc} B_1'&B_2' \end{array} \right) \left( \begin{array}{cc} \varLambda _1 &{} \varLambda _2 \\ \varLambda _3 &{} \varLambda _4 \end{array} \right) ^{-1} \left( \begin{array}{c} B_1 \\ B_2 \end{array} \right) \right\} ^{-1} \end{aligned}$$

(13.14)

By using the formula of the inverse of a partitioned matrix, we get

$$\begin{aligned} V(\hat{\beta }) = (B_1' \varLambda _1^{-1} B_1 + (\varLambda _3 \varLambda _1^{-1} B_1 - B_2)' A (\varLambda _3 \varLambda _1^{-1} B_1 - B_2))^{-1} \end{aligned}$$

(13.15)

with $A=(\varLambda _4 - \varLambda _3 \varLambda _1^{-1} \varLambda _2)^{-1} = (\varLambda _4 - \varLambda _2' \varLambda _1^{-1} \varLambda _2)^{-1}$ (a scalar). Note that the first element in (13.15) corresponds to the inverse of the variance of the system GMM estimator of $\beta$. Therefore, the variance of the GMM estimator based on (13.2), (13.4), and the additional moment condition (13.6) will be smaller than the variance of the system GMM estimator if

$$ (\varLambda _3 \varLambda _1^{-1} B_1 - B_2)' A (\varLambda _3 \varLambda _1^{-1} B_1 - B_2) >0 $$

As the scalar $A = (\det (\varLambda ) / \det (\varLambda _1))^{-1} > 0$,¹⁶ this condition will be always satisfied unless

$$\begin{aligned} \varLambda _3 \varLambda _1^{-1} B_1 - B_2 = 0 \end{aligned}$$

(13.16)

No efficiency gain will be obtained adding the moment condition we propose when condition (13.16) is satisfied. On the contrary, if condition (13.16) is not satisfied ($\varLambda _3 \varLambda _1^{-1} B_1 - B_2 \ne 0$), efficiency gains are obtained by adding the moment condition we propose to the moment conditions that characterize the system GMM estimator.

Fig. 13.1

Ratio of the asymptotic variance of the estimator we propose with respect to the asymptotic variance of the system GMM estimator, for different (i) levels of asymmetry in the distribution of $\alpha _i$, $e_{i0}$; (ii) time periods; (iii) ratios of unconditional variance of $\alpha$ with respect to e; (iv) values of $\beta$

By relying on further simulations (not reported in the main text of the paper) with $T+1=3$, it is possible to show that $\varLambda _3 \varLambda _1^{-1} B_1 - B_2 \ne 0$ if $\alpha _i$ or $e_{i0}$ have an asymmetric distribution (the third moment is not zero). Coherently, the Monte Carlo experiments reported in Sect. 13.3 show gains in performance in the cases in which the error components are assumed to have an asymmetric (log-normal or chi-squared) distribution.

Figure 13.1 shows the ratio of the asymptotic variance of the estimator we propose with respect to the asymptotic variance of the system GMM estimator. The baseline experiment consider $\beta =0.5$, $T+1=3$, a $\chi ^2_3$ distribution for $e_{i0}$ and $\alpha _i$, and equal unconditional variances of $\alpha _i$ and $e_{it}$ (both equal to 1). From the baseline experiment, we changed: (i) the level of asymmetry in the distribution of $\alpha _i$, $e_{i0}$, accomplished by changing the number of degrees of freedom of the $\chi ^2$ distribution; (ii) the number of time periods $T+1$ from 3 to 10; (iii) the ratio of unconditional variance of $\alpha$ with respect to e (from 1 to 9; we changed the value of the unconditional variance of $\alpha _i$, leaving the unconditional variance of e fixed at 1); (iv) the values of $\beta$ from 0.1 to 0.9. Coherently with the results in the Monte Carlo experiments in Sect. 13.3, the gains in performance led by the additional moment condition we propose are stronger for larger asymmetry in the distribution of $\alpha _i$ and $e_{i0}$ (thus, of $y_{i0}$), shorter observation in time, and values of $\beta$ that approaches 1. Also, the gains in performance decrease when the unconditional variance of $\alpha _i$ increases with respect to the unconditional variance of $e_{it}$.