A parametric linear or nonlinear regression model requires setting a specific functional form prior to estimation in order to describe the true but unknown relationship between the dependent and the independent variables. Consequently, potential specification errors are likely to occur, leading to an inconsistent estimation.
Although some economic models do explicitly suggest relationships among economic variables, most implications of economic theory are nonparametric. Therefore, if one has reservations about a particular parametric form, then a nonparametric function can be an alternative candidate. Nonparametric regression models permit the functional relationship to be unknown and nevertheless fit the data quite well without imposing restrictions beyond some degree of smoothness. They deliver estimators and inference procedures that are less reliant on the imposition of specific functional forms.
Inclusion of the nonparametric element may circumvent an inconsistent estimation arising from invalid parameterization. However, the inherent critical element of the
“curse of dimensionality＂limits the unknown function of a nonparametric model to contain a small number of variables to lessen the approximation error to the unknown function.
A researcher in some cases may be confident about a particular parametric form for one portion of the regression function, but less sure about the shape of another portion. Such prior beliefs justify the necessity for linking parametric with nonparametric techniques to formulate semiparametric regression models. The added value of semiparametric techniques consists in their competence to largely mitigate the curse of dimensionality distress, and the respective estimators of the parametric
and nonparametric components have their conventional rates of convergence.1 See, for example, Härdle (1990), Wand and Jone (1995), Fan et al. (1996), and Yatchew (1998, 2003).
Fan et al. (1996) first extended the traditional stochastic production frontier model, dated back to Aigner et al. (1977) and Meeusen and Van Den Broeck (1977), to a semiparametric frontier model in the context of cross section. They proposed pseudo-likelihood estimators and proved by Monte Carlo experiments that the finite-sample performance of their estimators is satisfactory. Deng and Huang (2008) further generalized it to a panel data setting and allowed for time-variant technical efficiency (TE) in the form of Battese and Coellli (1992). Nevertheless, almost all of the related works that use a semiparametric frontier model focus on the study of technical efficiency (TE). Kumbhakar and Wang (2006a) found that the assumption of fully allocative efficiency (AE) in a cost function tends to bias parameter estimates of the cost function and subsequent measures using these estimates.
To obtain both TE and AE measures, one is suggested to estimate the shadow cost system, consisting of an expenditure (cost) equation and the corresponding share equations, simultaneously using the maximum likelihood. Unfortunately, the highly nonlinear nature of the simultaneous equations makes the estimation almost untractable. Kumbhakar and Lovell (2000) proposed a two-step procedure with an eye to simplify somewhat the estimation problem of a pure parametric shadow cost system. The share equations are estimated in the first step by the method of nonlinear iterative seemingly unrelated regression (NISUR) to acquire the shadow price
1 Robinson (1988) showed that the parametric estimators are consistent at the parametric rate of
N1/ 2, while the nonparametric estimators converge at a slower rate than N1/ 2.
parameter estimates of interest. These estimates are treated as given in the second step, where the maximum likelihood technique is exploited to estimate the stochastic cost frontier alone after appropriately transforming the original expenditure equation using the first step estimates. This procedure is less efficient but computationally simpler.
However, Kumbhakar and Lovell (2000) did not address the properties of their proposed estimators. In addition, they repeat estimating the parameters in the second step and do not specify in which step the estimates should be used to calculate technical and allocative efficiencies respectively. Thereby, the main problem is: which estimates should we choose？Do the estimates in the second step behave more efficient than those in the first step？And is it possible to take all of the fist step estimates as given and then estimate the remaining parameters only？In this paper we propose three models in order to solve the aforementioned questions.
The purpose of the current work is four-fold. First, we relax the parametric restriction on a cost function representing technology in order to at least diminish the possible specification error. Second, the semiparametric stochastic shadow cost frontier offered by this paper differs from the standard semiparametric regression model and from the stochastic production frontier of Fan et al. (1996). Specifically, our model accommodates both TE and AE to avoid biased estimates of the technology parameters. To the best of our knowledge, no work has been done to introduce both efficiency measures into a semiparametric stochastic shadow cost frontier under the framework of panel data. It is hoped that this research will bridge the existing gap and to better characterize a firm’s optimization behavior. Third, a distinct five-step procedure from the one suggested by Kumbhakar and Lovell (2000) is proposed to facilitate the estimation. We argue for the new procedure due to the fact that its estimators of interest are shown to converge to the true values as the sample size
increases by applying Monte Carlo simulations. Finally, an empirical study using unbalanced panel data of commercial banks from 14 East European countries spanning 1993-2004 is carried out to illustrate the superiority of our semiparametric stochastic shadow cost frontier model.
The rest of this paper is organized as follows. Section 2 briefly reviews the relevant literature. Section 3 first presents the semiparametric stochastic cost frontier with shadow input prices and then proposes the estimation procedure. Section 4 introduces the design of Monte Carlo experiments to be conducted in the next section.
Section 5 provides and discusses the results of the experiments, which are intended to detect a suitable estimation procedure leading to consistent estimators. Section 6 illustrates the recommended estimation procedure with an empirical study, while the last section concludes the paper.