Nonlinear production frontiers
The previous models both assumed that the production frontier was log-linear. However, many common production functions are inherently nonlinear in the parameters (e. g. the constant elasticity of substitution or CES or the asymptotically ideal model or AIM, see Koop et al., 1994). However, the techniques outlined above can be extended to allow for an arbitrary production function. Here we assume a model identical to the stochastic frontier model with common efficiency distribution (i. e. m = 1) except that the production frontier is of the form:12
Уі = /X; P) + Vi _ Zi.
The posterior simulator for everything except в is almost identical to the one given above. Equation (24.10) is completely unaffected, and (24.9) and (24.11) are slightly altered by replacing xP by f (x, в) = (f (x1; в)… f(xN; в))’.
However, the conditional posterior for в is more complicated, having the form:
Equation (24.17) does not take the form of any well known density and the computational algorithm selected will depend on the exact form of f(x; в). For the sake of brevity, here we will only point the reader in the direction of possible algorithms that may be used for drawing from (24.17). Two major cases are worth mentioning. First, in many cases, it might be possible to find a convenient density which approximates (24.17) well. For instance, in the case of the AIM model a multivariate-f density worked well (see Koop et al., 1994). In this case, importance sampling (Geweke, 1989) or an independence chain Metropolis – Hastings algorithm (Chib and Greenberg, 1995) should work well. On the other hand, if no convenient approximating density can be found, a random walk chain Metropolis-Hastings algorithm might prove a good choice (see Chib and Greenberg, 1995). The precise choice of algorithm will be case-specific and, hence, we do not discuss this issue in any more detail here.