Addressing endogeneity when estimating stochastic ray production frontiers: a Bayesian approach

For a presentation of multiple-output production and duality theory, see Färe and Primont (1995, 1996).

Our approach invokes the same assumptions as dual approaches, i.e., availability of input (and output) prices and the assumption of profit maximization (or at least cost minimization). However, in certain empirical applications, our approach has the following advantages: (a) Dual approaches require notable variation in input (and output) prices between observations. However, if transaction costs are low and markets are working well, the “law of one price” approximately holds, which leaves too little variation between observations for estimating a dual approach. By our alternative approach, there is no requirement for variation in input (and output) prices. (b) To obtain estimates of output-oriented technical efficiency, a dual approach is (at best) very complicated to implement. (c) Although, in theory, primal and dual approaches should give the same result, in practice this is not assured. Hence, the preference is for a primal approach, as presented in this paper.

We refer the reader to Löthgren (1997), Gerdtham et al. (1999) and Löthgren (2000) for more details on the SRPF. The traditional specification of a Translog Shephard output distance function that corresponds to the stochastic ray output distance function specified in Eq. (1) is \(\ln D(x,y) = \ln y_m + F( \ln x, \ln ( y / y_m ) )\), where \(F(\cdot )\) denotes a quadratic functional form and m with \(1 \le m \le M\) indicates an arbitrarily chosen output that is used as numéraire to impose linear homogeneity in output quantities. Hence, the traditional Translog specification fulfills linear homogeneity through an arbitrarily chosen output and implicitly assumes that \(\ln (D(x,y))\) is a quadratic function of the logarithms of the input quantities and the logarithms of the normalized output quantities, while the Translog SRPF functional form fulfills linear homogeneity through the length of the vector of output quantities and implicitly assumes that \(\ln (D(x,y))\) is a quadratic function of the logarithms of the input quantities and the angles of the vector of output quantities.

An advantage of not taking logarithms of the angles (unlike the specification in Löthgren 1997) is that this specification can handle zero output quantities (Henningsen et al. 2017).

Note that \(\partial \vartheta _{m'}/\partial \ln y_{m}=\left( \partial \vartheta _{m'}/\partial y_{m}\right) y_{m}\), \(\partial \vartheta _{m'}/\partial \ln y_{m}=0\,\forall \,m<m'\) and \(\partial \mathrm {arccos}(z)/\partial z=-1/\sqrt{1-z^{2}}\).

Similar stochastic specifications can be found in Malikov et al. (2015) and Tsionas et al. (2015).

While this model specification does not assume that error terms \({\mathbf {v}}_{it}\) are independent of the input and output quantities, it assumes that the inefficiency term \(u_{it}\) is independent of the input and output quantities. This assumption is typical in the literature.

Our model specification assumes that the input prices w are exogenous, which is a typical assumption in many empirical analyses, e.g., in analyses based on cost minimization. However, under certain circumstances, the input prices w could be endogenous, e.g., if differences in input prices between banks reflect heterogeneous inputs rather than “true” differences in input prices (see, e.g., Quiggin and Bui-Lan 1984).

The use of parametric assumptions to approximate unavailable output prices may introduce approximation errors. The absence of output prices (or even input prices) is common in the literature. In deciding to proceed, we demonstrate that our method is applicable even if output prices are unavailable.

To measure observation-specific technical efficiency, we use \(\frac{1}{R}\sum _{j=1}^{R} \exp \left( -u_{it}^{\left( j\right) }\right) \) \(\forall \) \(i=1,\ldots ,n;t=1,\ldots ,T\), where \(\left\{ u_{it}^{\left( j\right) } \right\} _{j=1}^{R}\) is a MCMC sample drawn from the posterior.

More information on the sensitivity of our results to the ordering of the outputs is given in the Online Supplement of this article, where we present detailed statistics of the technological metrics for all 60 different orderings of the outputs.