Abstract
The productive efficiency of a firm can be seen as composed of two parts, one persistent and one transient. The received empirical literature on the measurement of productive efficiency has paid relatively little attention to the difference between these two components. Ahn and Sickles (Econ Rev 19(4):461–492, 2000) suggested some approaches that pointed in this direction. The possibility was also raised in Greene (Health Econ 13(10):959–980, 2004. doi:10.1002/hec.938), who expressed some pessimism over the possibility of distinguishing the two empirically. Recently, Colombi (A skew normal stochastic frontier model for panel data, 2010) and Kumbhakar and Tsionas (J Appl Econ 29(1):110–132, 2012), in a milestone extension of the stochastic frontier methodology have proposed a tractable model based on panel data that promises to provide separate estimates of the two components of efficiency. The approach developed in the original presentation proved very cumbersome actually to implement in practice. Colombi (2010) notes that FIML estimation of the model is ‘complex and time consuming.’ In the sequence of papers, Colombi (2010), Colombi et al. (A stochastic frontier model with short-run and long-run inefficiency random effects, 2011, J Prod Anal, 2014), Kumbhakar et al. (J Prod Anal 41(2):321–337, 2012) and Kumbhakar and Tsionas (2012) have suggested other strategies, including a four step least squares method. The main point of this paper is that full maximum likelihood estimation of the model is neither complex nor time consuming. The extreme complexity of the log likelihood noted in Colombi (2010), Colombi et al. (2011, 2014) is reduced by using simulation and exploiting the Butler and Moffitt (Econometrica 50:761–764, 1982) formulation. In this paper, we develop a practical full information maximum simulated likelihood estimator for the model. The approach is very effective and strikingly simple to apply, and uses all of the sample distributional information to obtain the estimates. We also implement the panel data counterpart of the Jondrow et al. (J Econ 19(2–3):233–238, 1982) estimator for technical or cost inefficiency. The technique is applied in a study of the cost efficiency of Swiss railways.
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Notes
Books and surveys on the measurement of the level of productive efficiency do make the distinction between models that estimate time-varying inefficiency indicators and models that produce time invariant indicators. See, for example, Kumbhakar and Lovell (2003) or Greene (2008). Measurement of the distinct parts has proved challenging.
For a discussion of these models see Greene (2008).
Other extensions of the basic frontier model have also considered exponential and truncated normal distributions for the inefficiency term. See for instance Battese and Coelli (1992).
The received literature provides many alternative stochastic frontier models that allow the estimation of time-varying inefficiency. In addition to those noted above, we would include the model proposed by Lee and Schmidt (1993) that extends the Kumbhakar (1990) and Battese and Coelli (1992) models, and the model proposed by Kumbhakar and Wang (2005). There are others as well. See Greene (2008) or Parmeter and Kumbhakar (2015) for surveys.
The Mundlak auxiliary equation has been proposed for a random effects linear regression model. This approach, based on normality and the linear model, might not strictly apply to stochastic frontier models estimated by ML, as these models possess an asymmetric composite error term ε i . As the model captures the correlation between the individual effects and the explanatory variables at least partly, the resulting heterogeneity bias is expected to be minimal. The general approach has been used elsewhere in a variety of settings under the heading of ‘correlated random effects models.’ See, e.g., Wooldridge (2010).
There is no requirement that the number of observations, T i , be the same for each firm. T i is assumed here to be constant only for convenience to avoid another subscript in the presentation.
The estimation strategy in (11) could, in principle, be applied to estimation of the stochastic frontier model in (6) by integrating u it out of the conditionally normal linear regression model in (5). Maximization of (6) directly is extremely straightforward, however, and the MSLE would provide no improvement over direct MLE.
The authors allowed \(\sigma_{u}^{2}\) to vary by period. This aspect can be added to the estimating equations by changing σ 2 u I to Ψ = diag(σ 2 u,1 ,…,σ 2 u,T ) in the definition of V. In the formulations below, the parameterization in terms of σ and λ would have to be replaced with the original parameterization in terms of σ v and σ u,t . Their results on the presence of this type of heteroscedasticity are mixed.
In the original work, Colombi (2010) notes use of a self-developed R routine named SNF-maxlik.
The computation was done using a modified version of LIMDEP/NLOGIT that will be generally available in 2015. Adaptation of the method to other platforms, such as the mixed estimation packages in R or Stata should be straightforward given the availability of the TRE model to begin with. Running time for estimation of the cost model (20) was approximately 2 min and we did not experience convergence problems. The model is essentially a true random effects model, which has proved to be straightforward to estimate in a number of received applications since 2004. The computation of technical efficiency firm by firm based on (17), which does require a one time, post estimation application of GHK simulation, takes roughly the same amount of time as estimation of the model, itself.
The data set is available at http://people.stern.nyu.edu/wgreene/Text/Edition7/TableF19-1.txt.
The values of the coefficients of the MGTRE are very close to the coefficients obtained using a classical fixed effects model.
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Filippini, M., Greene, W. Persistent and transient productive inefficiency: a maximum simulated likelihood approach. J Prod Anal 45, 187–196 (2016). https://doi.org/10.1007/s11123-015-0446-y
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DOI: https://doi.org/10.1007/s11123-015-0446-y