Grouped coefficients to reduce bias in heterogeneous dynamic panel models with small T

Abstract

We propose the grouped coefficients estimator to reduce bias in dynamic panels with small T that have a multilevel structure to the coefficient and factor loading heterogeneity. If groups are chosen such that the within-group heterogeneity is small, then the grouped coefficients estimator can lead to substantial bias reduction compared to pooled GMM dynamic panel estimators. We also propose using a Wald test that can be used to assess whether pooled estimators suffer from heterogeneity bias. We illustrate the usefulness of grouped coefficients with an application to labour demand in which the coefficients are grouped by sub-sector. Our results suggest that the standard pooled estimates are substantially biased.

Keywords:

• dynamic panel
• varying coefficient
• cross-sectional dependence
• multilevel
• hierarchical

JEL Classification:

• C23
• C52

Notes

¹ In many cases, the average effect is of the greatest interest since it corresponds to the aggregate effect. The distribution of coefficients may also be of interest. Our estimator gives group-specific estimates of the coefficients – which is useful – but does not describe the full distribution of individual coefficients, so we focus attention on the average coefficient. Estimating the distribution of individual coefficients requires different methods and may require more stringent assumptions about the model (e.g., Hsiao et al., 1999). Individual coefficients cannot be consistently estimated with small T.

² The relevant asymptotic theory here is one in which group size goes to infinity and T is fixed.

³ Pesaran and Smith (1995) call their estimator the mean group estimator. This name could also describe our estimator, but we use the term grouped coefficients instead to avoid confusion with Pesaran and Smith’s mean group estimator. In addition, for the remainder of this paper, we refer to the mean group estimator as individual coefficients OLS to distinguish from grouped coefficients estimators that estimate separate models for groups of individuals.

⁴ The Arellano–Bond estimator is also commonly referred to as difference GMM and the Blundell–Bond estimator as system GMM. The Blundell–Bond estimator is also sometimes called the Arellano-Bover estimator since Arellano and Bover (1995) also proposed using moment conditions based on the level of the composite error.

⁵ Alternatively, if all coefficients are expected to be homogeneous except for the factor loadings of common factors, then bias can be reduced by estimating a single model with group-specific coefficients on time fixed effects.

⁶ Pesaran and Smith call this the mean group estimator.

⁷ The incidental parameters problem underlies the bias of fixed effects in linear dynamic panels as well (see Lancaster, 2000).

⁸ That is, we report the probability of rejecting the null of no second-order autocorrelation of the first differences.

⁹ We obtain the result that grouped coefficients OLS is unbiased because we assume that there is no heterogeneity of the intercepts when $\delta =0$. In practice, there may be substantial heterogeneity of the intercepts within groups in which case the grouped coefficients Blundell–Bond estimator would be preferred.

¹⁰ We could instead have implemented the test in the more conventional Hausman form as $H={\left({\hat{\theta }}_{GC}-{\hat{\theta }}_{PL}\right)}^{{ }^{\prime }}{\left(N^{-1}{\hat{\mathbf{V}}}_H\right)}^{-1}\left({\hat{\theta }}_{GC}-{\hat{\theta }}_{PL}\right),$ where ${\hat{\theta }}_{GC}$ denotes the grouped coefficients estimate, ${\hat{\theta }}_{PL}$ denotes the analogous pooled estimate and ${\hat{\mathbf{V}}}_H$ denotes an estimate of the asymptotic variance of the limiting distribution of the root-N difference of the two estimates. Hsiao and Pesaran (2008) suggest a similar approach based on individual coefficients OLS when T is large. Standard asymptotic theory implies that the H is asymptotically distributed ${\chi }^2\left(k\right)$ under the null hypothesis.

¹¹ A table with the number of firms in each year is in the supplemental appendix.

¹² We dropped data from sub-sectors labelled ‘3’ and ‘6’ in the data. Sub-sector 6 was dropped since it only contained data from five firms. Sub-sector 3 was dropped because Arellano–Bond and Blundell–Bond estimates of the coefficient on the lagged dependent variable were −1.22 and −0.91, respectively. Including sub-sector 3 in our sample would further strengthen our result that grouped coefficients estimates give a smaller coefficient on the lagged dependent variable.