The Stochastic Frontier Model with Cross-Sectional Data
The model given in equation (24.2) implicitly assumes that all deviations from the frontier are due to inefficiency. This assumption is also typically made in the DEA approach. However, following standard econometric practice, we add a random error to the model, Z, to capture measurement (or specification) error,4 resulting in:
The addition of measurement error makes the frontier stochastic, hence the term "stochastic frontier models". We assume that data for i = 1 … N firms is available and that the production frontier,/(•), is log-linear (e. g. Cobb-Douglas or translog). We define Xi as a 1 x (k + 1) vector (e. g. Xi = (1 Li K) in the case of a Cobb – Douglas frontier with two inputs, L and K) and, hence, (24.3) can be written as:
Vi = X-P + vi – zu (24.4)
where P = (Po… pk)’, Vi = ln (Y), vi = ln (Z,), zi = – ln (t) and xi is the counterpart of Xi with the inputs transformed to logarithms. zi is referred to as inefficiency and, since 0 < t < 1, it is a nonnegative random variable. We assume that the model contains an intercept with coefficient P 0. Equation (24.4) looks like the standard linear regression model, except that the "error" is composed of two parts. This gives rise to another name for these models, viz. "composed error models".
For future reference, we define y = (y… yN)’, v = (v1… vN)’, z = (z1… zN)’ and the N x (k + 1) matrix x = (x 1… x’N)’. Also, let fG(a | b, c) denote the density function of a Gamma distribution with shape parameter b and scale c so that a has mean b/c and variance b/c2. p(d) = fN(d|g, F) indicates that d is r-variate normal with mean g and covariance matrix F. We will use I(-) to denote the indicator function; i. e. I(G) = 1 if event G occurs and is otherwise 0. Furthermore, IN will indicate the N x N identity matrix and iN and N x 1 vector of ones. Sample means will be indicated with a bar, e. g. у = N^y.