The Stochastic Frontier Model with Panel Data
It is increasingly common to use panel data13 in the classical econometric analysis of the stochastic frontier model. Some of the statistical problems (e. g. inconsistency of point estimates of firm specific efficiency) of classical analysis are alleviated with panel data and the assumption of a particular distributional form for the inefficiency distribution can be dispensed with at the cost of assuming time-invariant efficiencies (i. e. treating them as "individual effects"). Schmidt and Sickles (1984) is an early influential paper which develops a relative efficiency measure based on a fixed effects specification and an absolute efficiency measure based on a random effects specification. In this paper, we describe a Bayesian alternative to this classical analysis and relate the random/fixed effects distinction to different prior structures for the efficiency distribution.
Accordingly, assume that data is available for i = 1… N firms for t = 1… T time periods. We will extend the notation of the previous section so that yi and v are now T x 1 vectors and xi a T x k matrix containing the T observations for firm
i. Note, however, that the assumption of constant efficiency over time implies that zi is still a scalar and z an N x 1 vector. For future reference, we now define y = (y 1… yN)’ and v = (v1… v’N)’ as NT x 1 vectors and x = (x1… x’N)’ as an NT x k matrix. In contrast to previous notation, xi does not contain an intercept. We assume that the stochastic frontier model can be written as:
Vi = Poll – + xi § + V – ZiiT, (24.18)
where P 0 is the intercept coefficient and vi is iid with pdf fN(vi |0, ЬЛІТ). As discussed in Fernandez et al. (1997), it is acceptable to use an improper noninformative prior for h when T > 1 and, hence, we assume p(h) We discuss different
choices of priors for p 0 and zi in the following material.
Bayesian fixed effects model
Equation (24.18) looks like a standard panel data model (see, e. g., Judge et al., 1985, ch. 13). The individual effect in the model can be written as:
a = p o – zi,
and the model rewritten as:
Vi = a iiT + xi§ + vi. (24.19)
Classical fixed effects estimation of (24.19) proceeds by making no distributional assumption for a i, but rather using firm-specific dummy variables. The Bayesian analog to this is to use flat, noninformative priors for the a i s.14 Formally, defining a = (a1… aN)’, we then adopt the prior:
p(a, 5, h) – Г>(5). (24.20)
The trouble with this specification is that we cannot make direct inference about zi (since P 0 is not separately identified) and, hence, the absolute efficiency of firm i: Ti = exp(-Zj). However, following Schmidt and Sickles (1984), we define relative inefficiency as:
zf = zi – min(z,) = max (a,) – ai. (24.21)
In other words, we are measuring inefficiency relative to the most efficient firm (i. e. the firm with the highest ai).15 Relative efficiency is defined as rf = exp (-zf) and we assume that the most efficient firm has rf = 1.
It is worth noting that this prior seems like an innocuous noninformative prior, but this initial impression is false since it implies a rather unusual prior for rf. In particular, as shown in Koop et al. (1997), p(rf) has a point mass of N-1 at full efficiency and is p(rf) — 1/rf for rf C (0, 1). The latter is an L-shaped improper prior density which, for an arbitrary small a C (0, 1) puts an infinite mass in (0, a) but only a finite mass in (a, 1). In other words, this "noninformative" prior strongly favors low efficiency.
Bayesian inference in the fixed effects model can be carried out in a straightforward manner, by noting that for uniform p(5), (24.19)-(24.20) is precisely a normal linear regression model with Jeffreys’ prior. The vector of regression coefficients (a’ 5′)’ in such a model has a (N + k)-variate student-t posterior with N(T – 1) – k degrees of freedom (where we have assumed that N(T – 1) > k, which
implies T > 1). For typical values of N, T, and k the degrees of freedom are enormous and the student-f will be virtually identical to the normal distribution. Hence, throughout this subsection we present results in terms of this normal approximation.
Using standard Bayesian results for the normal linear regression model (e. g. Judge ef al., 1985, ch. 4), it follows that the marginal posterior for 5 is given by (for general p(5)):
p(5|y, x) = fN (51X, hr1S-1)p(5),
X = S-1 X (Xi – It©)'(Vi – EilT),
S = X Si, © = т1’тхі
Si = (Xi – Itx)(Xi – It©).
Note that (24.23) is the standard "within estimator" from the panel data literature. Finally,
= NT – —k X(У – 7 1t – X-S),( Vi – 7 і It – X^S),
N(T 1) k i=1
where 7i is the posterior mean of ai defined below.
The marginal posterior of a is the N-variate normal with means
7 і = Ei – © X,
where A(i, j) = 1 if i = j and 0 otherwise. Thus, analytical formulae for posterior means and standard deviations are available and, if interest centers on these, posterior simulation methods are not required. However, typically interest centers on the relative efficiencies which are a complicated nonlinear function of a, viz.,
and, hence, posterior simulation methods are required. However, direct Monte Carlo integration is possible since the posterior for a is multivariate normal and can easily be simulated. These simulated draws of a can be transformed using (24.24) to yield posterior draws of rf. However, this procedure is complicated by the fact that we do not know which firm is most efficient (i. e. which firm has largest a) and, hence, is worth describing in detail.
We begin by calculating the probability that a given firm, i, is the most efficient:
P(rf = 1| y, x) = P(ai = max (a,) | y, x), (24.25)
which can be easily calculated using Monte Carlo integration. That is, (24.25) can simply be estimated by the proportion of the draws of a which have ai being the largest.
Now consider the posterior for rf over the interval (0, 1) (i. e. assuming it is not the most efficient):
p(rf | y, x) = ^ p(rf |y, x, rf = 1)P(rf = 1|y, x). (24.26)
Here P(rf = 1| y, x) can be calculated as discussed in the previous paragraph. In addition, p(rf | y, x, rf = 1) can be calculated using the same posterior simulator output. That is, assuming firm j is most efficient, then rf = exp (ai – a,) which can be evaluated from those draws of a that correspond to a, = maxl(a l). Hence, posterior analysis of the relative efficiencies in a Bayesian fixed effects framework can be calculated in a straightforward manner.16