Semiparametric Stochastic Frontier Estimation via Profile Likelihood

Abstract

We consider the estimation of a nonparametric stochastic frontier model with composite error density which is known up to a finite parameter vector. Our primary interest is on the estimation of the parameter vector, as it provides the basis for estimation of firm specific (in)efficiency. Our frontier model is similar to that of Fan et al. (1996), but here we extend their work in that: a) we establish the asymptotic properties of their estimation procedure, and b) propose and establish the asymptotic properties of an alternative estimator based on the maximization of a conditional profile likelihood function. The estimator proposed in Fan et al. (1996) is asymptotically normally distributed but has bias which does not vanish as the sample size n → ∞. In contrast, our proposed estimator is asymptotically normally distributed and correctly centered at the true value of the parameter vector. In addition, our estimator is shown to be efficient in a broad class of semiparametric estimators. Our estimation procedure provides a fast converging alternative to the recently proposed estimator in Kumbhakar et al. (2007). A Monte Carlo study is performed to shed light on the finite sample properties of these competing estimators.

Keywords:

• Nonparametric frontiers
• Profile likelihood estimation
• Stochastic frontier models

JEL Classification:

• C14
• C22

ACKNOWLEDGMENTS

We thank Daniel Henderson, Peter C. B. Phillips, and participants in the XX New Zealand Econometrics Study Group and Midwest Econometrics Group Meetings for helpful comments. We also thank Essie Maasoumi, an Associate Editor, and two referees for comments that improved the paper substantially. Any remaining errors are the authors’ responsibility.

Notes

¹See also van der Vaart (1999).

²It should be noted that the case of multiple outputs can be accommodated in our framework by adopting the polar coordinate representation of given in Simar and Zelenyuk (2011). See their Eqs. (2.5) and (2.6).

³See Fan (1993) and Li and Racine (2007).

⁴The proofs for Theorems 1 and 2 as well as the proof for Lemma 1 in Section 3.2 can be found in Martins-Filho and Yao (2011).

⁵Naturally, the asymptotic bias associated with can be eliminated by choosing a non optimal bandwidth decay rate (undersmoothing) for the estimator .

⁶We follow the usual practice of defining, for any two squared matrices A and B, A ≤ B if, and only if, B − A is positive semidefinite.

⁷See Martins-Filho and Yao (2011).