A James-Stein-type adjustment to bias correction in fixed effects panel models

Abstract

This paper proposes a James-Stein-type (JS) adjustment to analytical bias correction in fixed effects panel models that suffer from the incidental parameters problem. We provide high-level conditions under which the infeasible JS adjustment leads to a higher-order MSE improvement over the bias-corrected estimator, and the former is asymptotically equivalent to the latter. To obtain a feasible JS adjustment, we propose a nonparametric bootstrap procedure to estimate the JS weighting matrix and provide conditions for its consistency. We apply the JS adjustment to two models: (1) the linear autoregressive model with fixed effects, (2) the nonlinear static fixed effects model. For each application, we employ Monte Carlo simulations which confirm the theoretical results and illustrate the finite-sample improvements due to the JS adjustment. Finally, the extension of the JS procedure to a more general class of models and other policy parameters are illustrated.

Keywords:

• incidental parameters
• maximum likelihood
• panel data
• repeated measures
• shrinkage

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1 Even though we focus on the maximum likelihood estimator, the high-level conditions are general and apply to more general classes of estimators, such as M-estimators which are examined in Hahn and Kuersteiner (2011).

2 This is an important point that Hahn, Kuersteiner, and Newey (2002) elaborate on and that helps shed light on their results regarding the third-order equivalence between the bootstrap and jackknife bias corrections they proposed and those proposed by Pfanzagl and Wefelmeyer (1978).

3 Note that (9) $$\sqrt{nT}(\overline{B}(\widehat{\theta })-B)=\sqrt{nT}(B(\widehat{\theta })-B({\theta }_0))+\sqrt{nT}(\overline{B}({\theta }_0)-B({\theta }_0))+\sqrt{nT}(\overline{B}(\widehat{\theta })-B(\widehat{\theta })-(\overline{B}({\theta }_0)-B({\theta }_0)))\le \sqrt{nT}(B(\widehat{\theta })-B({\theta }_0))+\sqrt{nT}(\overline{B}({\theta }_0)-B({\theta }_0))+\underset{\theta \in \Theta }{\sup }\left|\frac{\partial \overline{B}(\theta )}{\partial \theta \prime }-\frac{\partial B(\theta )}{\partial \theta \prime }\right|O_p(1)+o_p(1).$$(9)

As a result, a more primitive condition for Condition 3.1.2 is for the derivative of the bias estimator to admit a uniform law of large numbers. Lemma D.3 provides the relevant primitive conditions for the case of the bias estimator in HN2004.

4 Here we define the MSE as the expectation of the outerproduct of ${\mathcal{O}}_{BC}-{\theta }_0,$ which yields a $k\times k$ matrix. To define a scalar objective function for Λ, we use the expectation of the inner product instead, which equals the trace of the matrix version of the MSE. For details, see Section B of the online appendix.

5 To ensure that we have a scalar objective function, we use the inner-product version of the MSE. For the derivation of the MSE-minimizing JS weight, see Section B.1.

6 This result relates to the higher-order effiency literature of the MLE. In the classical setting where the MLE is asymptotically efficient, Hahn, Kuersteiner, and Newey (2002) show that the estimation of the bias has no impact on the third-order MSE of the resulting estimator.

7 We provide primitive conditions for the consistency of the bootstrap JS weighting matrix for the nonlinear fixed effects models in HN2004.

8 For a review and discussion of the results in Mammen (1992), see Horowitz (2019)

9 To compute the infeasible JS weight, we numerically evaluate $B({\theta }_0),{\mathcal{V}}_{\mathcal{O}}$ and ${\mathcal{V}}_{\mathit{OB}}$ defined in Condition 3.1 using $n={10}^4$ and $T={10}^4.$ Then, we plug them into Λ_nT after a mean-variance decomposition of the numerator and denominator.

10 We do not report the rejection probabilities nor the SE/SD for the infeasible JS-BCE, since we cannot perform inference on it in practice.

11 These conditions include mixing conditions on the time series dependence in addition to conditions on higher-order moments and identification as in HN2004.