Bias-corrected method of moments estimators for dynamic panel data models
Introduction
Dynamic panel data models are now used in a wide area of empirical applications. Since the work of Anderson and Hsiao (1981), instrumental variables and generalized method of moments (GMM) estimators have been extensively applied in the estimation of linear dynamic panel data models. However, it is known that the GMM estimator by Holtz-Eakin et al. (1988) and Arellano and Bond (1991) suffers from the weak-instruments problem when the persistency of the data is strong, as demonstrated by Blundell and Bond (1998). They also showed that the GMM estimator for models in levels with first-differenced instruments mitigates that problem and they proposed the so-called system GMM estimator that combines moment conditions for the models in first differences and in levels. Nowadays, the system GMM estimator is most frequently used in practice, albeit Bun and Windmeijer (2010) showed that it still suffers from the weak-instruments problem when the variance of the individual-specific effects is larger than that of the idiosyncratic errors.
As alternatives to the GMM approach, maximum likelihood (ML) estimators and bias-corrected within-groups (WG) estimators were proposed. Hsiao et al. (2002) suggested a transformed ML estimator that adapts the ML approach to the differenced variables. Hayakawa and Pesaran (2015) extended this transformed ML estimator to allow for cross-sectional heteroskedasticity and proposed robust standard errors. With regard to bias-corrected WG estimators, Kiviet (1995) and Judson and Owen (1999) demonstrate that they are attractive alternatives to GMM estimators. Although the bias-corrected WG estimator of Kiviet (1995) is based on a higher-order expansion of the bias term, the analytical results are based on the unknown parameters that have to be estimated by some consistent initial estimator. Accordingly, the asymptotic distribution of this estimator is unknown. Bun and Carree (2005) proposed an alternative bias-corrected WG estimator which iteratively solves a nonlinear equation with regard to unknown parameters. Dhaene and Jochmans (2016) obtain an adjusted profile likelihood function by integrating a bias-corrected profile score. Kruiniger (2018) proposes a generalized version of the modified ML estimator that addresses identification problems when the autoregressive parameter equals unity or when .
In this paper, we demonstrate that a bias-corrected estimator can be obtained as a method of moments estimator. The adjusted profile score is transformed into nonlinear moment conditions that can be easily solved with standard numerical methods. Asymptotic results are readily available. The underlying estimating equations are equivalent to those of the Dhaene and Jochmans (2016) estimator when we adopt a fixed-effects assumption for the exogenous regressors. For the first-order autoregressive model, they are also equivalent to those of Bun and Carree (2005) and Kruiniger (2018). Furthermore, we re-emphasize an earlier finding by Bun et al. (2017) that the ML estimators of Hsiao et al. (2002) and Bai (2013) are based on a modified log-likelihood function that leads to an asymptotically equivalent bias correction of the first-order condition. Yet, these ML estimators rely on additional assumptions about the initial observations that are not required for our approach.
Within our method of moments framework, we can easily differentiate between regressors that are correlated with the individual-specific effects and those that are uncorrelated with them. All it requires is a slight modification of the respective moment conditions. This allows for the estimation of dynamic fixed-effects and dynamic random-effects models, as well as hybrid versions. Under appropriate orthogonality assumptions, time-invariant regressors can be incorporated as well.
Moreover, the model allows for individual-specific heteroskedasticity in the large-, fixed- framework. When both and are large, the estimator is also robust to time series heteroskedasticity. Furthermore, we propose cluster-robust/panel-corrected standard errors that account for cross-sectional dependence, and we extend our bias-corrected method of moments approach to higher-order autoregressive models. Monte Carlo experiments suggest that these estimators perform well in terms of efficiency and correctly sized tests, relative to uncorrected WG and GMM approaches.
Section snippets
Bias-corrected method of moments estimation
To motivate the bias correction approach, we initially consider the pure first-order autoregressive modelwhere . The log-likelihood function conditional on the initial observations is given byProfiling out the nuisance parameters yieldswhere and . For estimating the parameter in such an
Relationship to maximum likelihood estimation
In this section, we show that the ML estimation procedures proposed by Hsiao et al. (2002) and Bai (2013) can be seen in a similar light as our bias-corrected method of moments estimator. In particular, we demonstrate that their first-order conditions with respect to can be decomposed into two terms, and a bias correction term. Accordingly, the main differences between those approaches are the assumptions on the initial conditions that result in variations of the bias correction
Higher-order dynamics
Similar to Dhaene and Jochmans (2016), we can extend the bias-corrected method of moments estimator to an autoregressive model of order . To simplify the exposition, strictly exogenous regressors are initially neglected. The respective moment functions would remain the same as in the AR(1) model. Consider the AR() modelFor notational convenience, we suppress the subscript 0 in denoting the true coefficient values in this
Cross-sectional dependence
In many macroeconomic applications, it is reasonable to assume that the elements of the error vector are correlated: Assumption 3 (i) The errors are independent across but dependent across with and for all . The largest eigenvalue of the positive-definite matrices is bounded as . Assumption 1 (ii)–(iv) continue to hold.
For notational convenience, we focus on the AR(1) model. The results continue to hold for the AR() model.
Small-sample properties
To assess the small-sample properties of the bias-corrected method of moments estimator in comparison to alternative estimators that have been suggested in the literature, we perform some Monte Carlo experiments. In the baseline scenario, the data are generated from a simplified homoskedastic version of the dynamic panel data model considered by Kiviet et al. (2017) in their simulation exercise:The regressor is strictly exogenous with
Conclusion
We proposed an estimator that is based on a simple set of moment conditions that can be easily solved with standard numerical optimization procedures. It is straightforward to generalize the estimator to higher-order autoregressive models or dynamic random-effects models. An estimator of the asymptotic covariance matrix is readily available, as are robust standard errors that effectively adjust for cross-sectional dependence, which is a relevant feature in the analysis of macroeconomic panel
Declaration of Competing Interest
No conflict of interests
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