Some models for stochastic frontiers with endogeneity
Introduction
Studies of stochastic frontier models that allow for correlation between inefficiency effects and regressors are few and have been mainly done under a fixed effects framework in which a panel data model with a two-sided error term is estimated first, and the inefficiency effects are later estimated by subtracting the effects from their maximum (see e.g. Sickles, 2005 and references cited therein). Given that stochastic frontier models are more commonly estimated based on a one-sided random effects assumption, it is useful to investigate estimation within a framework where the one-sided random effects are correlated with the regressors. Also of interest are methods for accommodating correlation between the idiosyncratic error term and the regressors. The purpose of this paper is to propose a relatively general approach to modelling of stochastic frontiers with endogeneity, where one-sided efficiency effects, and idiosyncratic error terms, can be correlated with the regressors. We show that by transforming the inefficiency term into a normally distributed random term and modelling endogeneity through the mean or covariance of the normal errors, a range of stochastic frontier models with endogeneity can be handled.
We first consider a panel stochastic frontier model in which correlations between the effects and the regressors are based on a generalisation of the correlated random effects model proposed by Mundlak (1978), extended by Chamberlain (1984), and described further by Wooldridge (2010). Inefficiency effects are assumed to be correlated with the regressors through the mean of a transformation of the inefficiency errors. The main focus is on a log transformation implying the inefficiency errors have a lognormal distribution whose first argument depends on the regressors. Pursuing Bayesian estimation of the model, we derive conditional posterior densities for the parameters and the inefficiency errors for use in a Gibbs sampler. We then extend the model in two directions. Following Colombi et al. (2011), we add a time-varying inefficiency error leading to a model with both time invariant (permanent) and time-varying (transient) inefficiency errors; endogeneity is assumed to occur through correlation between the regressors and the time-invariant error. Necessary changes to the previously specified conditional posteriors are described. The second extension is to a more general model where endogeneity can exist because both the inefficiency errors and the idiosyncratic errors are correlated with the regressors. So that estimation can proceed, a “reduced form” type equation with instrumental variables is added to the earlier model. Details of how to estimate the model using both maximum simulated likelihood and Bayesian methods are provided.
The paper is organised as follows. The basic Mundlak-type model where the mean of the transformed error is a function of the regressors is considered in Section 2. In Section 3 we extend this model to include both permanent and transient inefficiency errors. Specification and estimation of the model that makes provision for instrumental variables and accommodates endogeneity more generally are considered in Section 4. An application using Philippine rice data and the models from Sections 2 Modelling correlation with a Chamberlain–Mundlak device, 3 Extension to a time-varying inefficiency model is provided in Section 5.
Section snippets
Modelling correlation with a Chamberlain–Mundlak device
In the first instance we consider the following random effects stochastic production frontier model with a time invariant inefficiency term In Eq. (2.1), indexes the firms and indexes time, is a row vector of functions of inputs (e.g., logs of inputs and squared logs of inputs), represents the logarithm of output, is the log of the frontier production function (e.g., translog), is a non-negative random error which accounts for
Extension to a time-varying inefficiency model
A deficiency of the model considered in the previous section, and one that is likely to be particularly critical if the number of time periods is large, is the time invariance of the firm inefficiencies. One way to remedy this deficiency is to specify the inefficiency error as , allowing it to vary freely over both firms and time. In this case we can specify and derive a corresponding set of conditional posterior densities. We do so in Section 4, but for a more general
A model with full endogeneity and instrumental variables
In the previous two sections endogeneity was modelled as correlation between the inefficiency errors and the inputs. However, in a number of studies (e.g. Kutlu, 2010, Karakaplan and Kutlu, 2013, Tran and Tsionas, 2013) allowance is made for correlations between idiosyncratic error terms and the inputs. In this section we consider a model that, in its most general form, allows for (i) time varying inefficiencies, (ii) correlation between the inputs and both the inefficiency error and the
An application to Philippines rice data
Since at least the 1970s some studies have reported an inverse relationship between farm size and productivity (efficiency) in developing countries (see e.g., Bardhan, 1973 or Sen, 1975), or they have argued that such a relationship exists because smaller farms use better-motivated or monitored family labour, whereas bigger farms use less-motivated hired labour. Imperfections in the labour or credit markets have also been put forward as possible reasons for the relationship. However, other
Conclusions
By transforming the inefficiency error to a normally distributed random term, we have been able to construct a relatively general model for introducing endogeneity into stochastic frontier analysis. Endogeneity can be introduced through either the mean of the transformed inefficiency error, or the covariance structure of the various errors, or both. The model can accommodate the introduction of instrumental variables, can be used with time-invariant and time-varying inefficiency terms, permits
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2020, European Journal of Operational ResearchCitation Excerpt :Recently, Griffiths and Hajargasht (2016) describe a Bayesian SFA approach to address endogeneity in a panel data setting, see also Liu, Sickles, and Tsionas (2017). Griffiths and Hajargasht use a Chamberlain-Mundlak device in a fixed effect structure. Specifically, the authors assume there is a transformation of firm specific fixed effect ui, H(ui), that is normally distributed with a mean that depends on the firm’s average of some of the input or functions of them.