Abstract
We consider method-of-moments fixed effects (FE) estimation of technical inefficiency. When dealing with a large number of cross-sectional observations, N, it is possible to obtain consistent moment estimators of the inefficiency distribution. It is well known that the classical FE estimator may be seriously upward biased when N is large and T, the number of time observations, is small. The method-of-moments FE estimators do not suffer from this type of bias in large-N settings. The proposed methodology bridges classical FE and maximum likelihood estimation, leading to a reduction in bias without making the random effects assumption.
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Notes
The conditional and unconditional averages of these estimators converge in mean to the counterparts of the inefficiency random variable as \(N\rightarrow \infty\).
The letter \(\nu\) without sub-script denotes the population random variable of the random error. Later on in the text, we also use \(\alpha\) and u to denote the corresponding population random variables for the fixed effects and inefficiency.
If \(E|\hat{u}_i-\hat{u}_i^a|\rightarrow 0\), then convergence in mean for the conditional expectations is proved as follows: \(E(|E(\hat{u}_i-\hat{u}_i^a|u_i)|)\le E(E(|(\hat{u}_i-\hat{u}_i^a||u_i))=E(|\hat{u}_i-\hat{u}_i^a|)\rightarrow 0\), where the inequality is given by the triangle inequality and the equality by the law of iterated expectations.
By the Cauchy–Swartz inequality: \(E|\bar{x}_i(\beta -\hat{\beta })|\le ( E(\bar{x}_i)^2) ^{1/2}( E(\beta -\hat{\beta })^2) ^{1/2}\rightarrow 0\) in quadratic mean, since the mean-square error of the least-squares estimator tends to zero as \(N\rightarrow \infty\).
Results for consistency and asymptotic normality are given in the Appendix provided in the supplementary materials of this publication. Several of the results are adopted from Wikström (2015) who consider similar method of moment estimation for the ‘true’ FE model.
Throughout this paper, we use the within estimator of \(\beta\) and within-transformations such as \(\ddot{z}_{it}=z_{it}-\bar{z}_{i}\) (e.g. Wooldridge 2010).
Corresponding to \(\theta\) in the general setting.
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Acknowledgments
We thank an anonymous Associate Editor for very helpful comments.
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Wikström, D. Modified fixed effects estimation of technical inefficiency. J Prod Anal 46, 83–86 (2016). https://doi.org/10.1007/s11123-016-0473-3
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DOI: https://doi.org/10.1007/s11123-016-0473-3