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A Monte Carlo study of ranked efficiency estimates from frontier models

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Parametric stochastic frontier models yield firm-level conditional distributions of inefficiency that are truncated normal. Given these distributions, how should one assess and rank firm-level efficiency? This study compares the techniques of estimating (a) the conditional mean of inefficiency and (b) probabilities that firms are most or least efficient. Monte Carlo experiments suggest that the efficiency probabilities are easier to estimate (less noisy) in terms of mean absolute percent error when inefficiency has large variation across firms. Along the way we tackle some interesting problems associated with simulating and assessing estimator performance in the stochastic frontier model.

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  1. This has been accomplished in the semi-parametric, fixed-effect specification of the stochastic frontier, using the theory of multiple comparisons. See Horrace and Schmidt (2000).

  2. Since large n, small T is typical in panel datasets, perhaps time-invariant technical inefficiency is the empirically relevant case. In what follows we only consider the time-invariant case.

  3. The question of “how precisely \(\widehat{\theta }_{j}\) estimates u j ?” is interesting, but it not addressed here.

  4. For example, Feng and Horrace (forthcoming) consider the effects of the skewness of the technical inefficiency distribution on various technical efficiency estimates.

  5. There is a price one pays when selecting a subsample based on some external rule. That is, the firms with a similar characteristic (e.g. large size) may have a different technology from those firms that do not have the characteristic. Empiricists may select or group the firms from the sample based on some rule, but different groups may have different technologies.

  6. This is particularly difficult to predict for the efficiency probabilities.

  7. We omitted n = 500, T = 20 to save computing time for the entire exercise.

  8. This also allowed us to indirectly examine the validity of the subset efficiency probabilities introduced in Eqs. 7 and 8.

  9. We could have allowed the x jtm to be correlated within firms but did not.

  10. When CGLS fails due to \(\hat{\sigma}_{u}^{2}<0, \) we set \(\hat{\sigma}_{u}^{2}=0, \) per Waldman (1982).

  11. We also calculated mean absolute error for each measure, but the results were similar to those for MSE and are not reported.

  12. The imprecision may be worsen by the fact that the fixed effects estimator cannot exploit correlations between x and u, as they have not been built into the DGP.

  13. Of course there is no way to disentangle this phenomenon from the effect of the random effects estimator approaching the fixed effects estimator, but it is interesting to note.


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Correspondence to William C. Horrace.

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Horrace, W.C., Richards-Shubik, S. A Monte Carlo study of ranked efficiency estimates from frontier models. J Prod Anal 38, 155–165 (2012).

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