Semiparametric Efficient Distribution Free Estimation of Panel Models

Abstract

This article generalizes results from Park et al. (1998) and Adams et al. (1999) on semiparametric efficient estimation of panel models. The form of semiparametric efficient estimators depends on the statistical assumptions imposed. Normality assumptions on the transitory error are sometimes inappropriate. We relax the normality assumption used in the articles above to derive more general semiparametric efficient estimators. These estimators are illustrated in a Monte Carlo simulation and an analysis of banking productivity.

Keywords:

• Banking efficiency
• Efficient estimation
• Information bound
• Panel models
• Semiparametric estimation

Mathematics Subject Classification:

• C14
• C23
• G21

Acknowledgments

This paper reflects the views of the authors and not those of the Federal Reserve Board or the Federal Reserve System. We wish to thank Jinyong Hahn and participants of the Winter Econometric Society Meetings (1998) for their comments. Ying Fang provided needed editorial assistance.

Notes

¹Park et al. (1998) show that within is efficient when all regressors are correlated with the effects.

²Horowitz and Markatou (1996) have considered alternative deconvolution methods for the random effects error components models and Horrace (1997) has modified their approach to consider stochastic frontier estimators for single cross-sections.

³It is assumed at every (θ₀, η₀) the mapping (θ, η) → P _{(θ, η)} is continuously Hellinger differentiable (see Ibragimov and Has'minskii, 1981).

⁴The proof can be found in Appendix A.

⁵See Newey (1990) and Bickel et al. (1993).

⁶See Bickel et al. (1993).

⁷Adams et al. (1997) analyzed the impact of kernel functions and of variance of optimal bandwidth selection criteria on the semiparametric efficient estimator. We use a normal kernel and higher-order normal kernel as opposed to the logistic kernel used by Park et al. (1998).

⁸For a discussion of the output distance function with multiple outputs, see Adams et al. (1999).

⁹The proof can be found in Appendix A.

¹⁰In all tables, RMSE has been multiplied by 100. Binwidth has been set to 0.5.

¹¹This result is expected. See Park et al. (1998).

¹²See Berger et al. (1995).

¹³We use the asset approach to output determination. See Berger (1993) and Adams et al. (1999) for a discussion of the different approaches to output determination in the banking industry. We also only consider traditional banking markets.

¹⁴This is especially true for parameter variance estimates, which can be sensitive to binning.

¹⁵This is a simplification and other methods could be applied.

¹⁶These averages are trimmed at the 0.1 level in keeping with Berger (1993).

¹⁷See Park et al. (1998) as well as Newey (1990).

¹⁸The efficient score, , can be calculated in a similar fashion.