2. OBJETIVOS
2.2. Objetivos específicos
2
1( ) ( )
1 t T t T
it = +η − +η −
η (2.22)
where η1 and η2 are unknown parameters. This model permits producer effects to be convex or concave, but the time-invariant model is the special case in which η1 = η2 = 0. In the Battese and Coelli (1992) model, the last period (t=Ti) for producer i contains the base level of inefficiency for that producer (Uit = Ui) and the efficiencies are measured relative to a frontier that may be regressing over time.
2.6.2. The Battese and Coelli (1995) specification
Battese and Coelli (1995) propose a model which is equivalent to the Kumbhakar, Ghosh and McGukin (1991) specification, with the exceptions that allocative efficiency is imposed, the first-order profit maximizing conditions removed, and panel data is permitted. The Battese and Coelli (1995) approach models both the stochastic and the technical inefficiency effects in the frontier, in terms of observable variables, and estimating all parameters by the method of maximum likelihood, in a one-step analysis38.
One may add a forth restriction of T = 1 to return to the original cross sectional, half - normal formulation of Aigner, Lovell and Schmidt (1977). Obviously, a large number of permutations exist.
For example, if all these restrictions excepting µ = 0 are imposed, the model suggested by Stevenson (1980) results. Furthermore, if the cost function option is selected, we can estimate the model specification in Schmidt and Lovell (1979) specification, which assumed allocative efficiency.
These latter two specifications are the cost function analogues of the production functions in Battese and Coelli (1988) and Aigner, Lovell and Schmidt (1977), respectively.
38 Battese and Coelli (1995) suggested that under the assumption of truncated normal one-sided error term, the mean of the truncated normal distribution could be expressed as a function of certain covariates, a closed form likelihood function can be derived, and the method of maximum likelihood may be used to obtain parameter estimates, and provide inefficiency measures.
According to Battese and Coelli (1995), the explanatory variables can include intercept terms or any variables in both the frontier and the model for the inefficiency effects, provided the inefficiency effects are stochastic. Battese and Coelli (1995) also suggested that under the assumption of truncated normal one-sided error term, the mean of the truncated normal distribution could be expressed as a function of certain covariates, a closed form likelihood function can be derived, and the method of maximum likelihood may be used to obtain parameter estimates, and provide inefficiency measures39. The Battese and Coelli (1995) model also overcomes the contradiction of the ‘two – step’ models and allows the simultaneous estimation of the parameters of the stochastic frontier and the inefficiency model (Puig-Junoy, 2001)40.
The original Battese and Coelli’s (1995) specification involved a production function with an error term incorporating two components, one to account for random effects (vi) and one to capture the unobservable inefficiency factor (ui).
The model consists of two equations, one to represent the production frontier and a second to measure technical inefficiency:
) exp( it ti it
it x V U
Y = β + − (2.23)
and
(2.24)
39 As in Movshuk (2004), while early stochastic frontier models were devised form cross – sectional data, Battese and Coelli (1995) model is formulated for panel data, which may also be unbalanced. The model not only estimates inefficiency levels of particular industries, but also explains their inefficiency in terms of potentially important explanatory variables, decomposing TFP growth into two components: technological growth: a shift of production possibility frontier set by best – practice industries, and inefficiency changes: deviations of actual output level form the production possibility frontier.
40 The two-stage analysis of explaining levels of technical efficiency (or inefficiency) was criticized by Battese and Coelli (1995) as being contradictory, in the assumptions made in the separate stages of the analysis.
where:
i =1, …., N, t = 1, …., T
Yit is (the logarithm of) the production of the ith producer in the tth time period.
xit is a k×1 vector of input quantities of the ith producer in the tth period β is a vector of unknown parameters
Vit are random variables which are assumed to be iid. N(0, σ V2
) and independent of the Ui which are non – negative random variables which are assumed to account for technical inefficiency in production, and assumed to be iid. as truncations at zero of the N(µ, σ U2
) distribution m =it zitδ where:
zit is a p×1 vector of variables which may influence the efficiency of a producer, and
δ is a 1×p vector of parameters to be estimated.
The parameterisation used in this model form is the one by Battese and Corra (1977) who replacedσU2 and σV2 with σ2 =σV2 +σU2 andγ =σU2 /
(
σV2 +σU2)
.In the first equation, Yit represents output of the i-th producer at time t. Xijt is a vector of productive inputs and indicator variables for the i-th producer at time t.
Following Battese and Coelli (1995), the Uits are assumed non-negative random variables that represent the stochastic shortfall of outputs from the most efficient production. It is assumed that Uit is defined by truncation of the normal distribution with mean:
+ ∑
=
= J
j j jit
it Z
1
0 δ
δ
µ (2.25)
and variance, σ2, where Zjit is value of the j-th explanatory variable associated with the technical inefficiency effect of country i in year t; and δ0and δj are unknown parameters to be estimated.
The output-based measure of technical efficiency may be estimated as41:
To obtain an observation – specific estimate of technical inefficiency (u), we use the Jondrow et al. (1982) result; that is, estimate u from uˆ=E
(
u|v−u)
in which(
v −u)
is replaced by the residuals of the production function. Because estimation procedures yield merely the residuals ε rather than the inefficiency term u, this term in the model must be observed indirectly (Greene, 1993, Cullinane and Song, 2003). Jondrow et al.
(1982) suggest the conditional expectation of uit, conditioned on the realized value of the error term εit = (vit-uit) as an estimator of uit and, in other words, E[uit|εit] is the conditional mean productive inefficiency for the ith industry at any time t. Measures of technical efficiency (TEi) for each producer can be calculated as42:
( )
Here, Zit is a vector of demographic and socioeconomic characteristics that might be correlated with inefficiency and which might vary over time. The inefficiency model's
41 Jondrow et al. (1982) provided an initial solution by deriving the conditional distribution of [-ui| (vi – ui)] which contains all the information (vi – ui) contains about ui. This enabled to derive the expected value of this conditional distribution, from which they proposed to estimate the technical efficiency of each producer: ˆ ( , )
{
exp{ [
ˆ |( ) ] } }
1 10 xi yi = E−ui vi −ui − ≥
E
T , which is a function of the MLE
parameter estimates. Later, Batesse and Coelli (1988) proposed to estimate the technical efficiency of each producer from: ˆ ( , )
{
[exp{ ˆ }|( )
]}
1 10 xi yi = E −ui vi −ui − ≥
E
T , which is slightly different
function of the same MLE parameter estimates.
42 The Batesse and Coelli model (1992, 1995) is modelling the time varying inefficiency in which time trend is specified to inefficiency term written as u(i,t)=exp(eta(t-T)| u(i)|.
random component, w, is not identically distributed nor is it required to be non-negative (Battese and Coelli, 1995) 43, 44.
A crucial issue concerning the model being estimated is what comprises the vector z.
The results obtained suggest that efficiency levels in different industries were not always the result of homogeneous influences. Factors that influence efficiency include scale effects, foreign – ownership, plant age, the proportion of workers in non – manual occupations, capital intensity and population density. The emphasis is on modelling inter-industry differences in (relative) efficiency. Typically, variables are included to reflect competitive factors in the industry such as market share and profitability (Caves, 1990, Hay and Liu, 1997), investment in new technology, industry dynamics and product differentiation, as well as the importance of scale economics (Harris, 1993). The distribution in efficiency across time is considered, as is the question of whether efficiency levels were converging over time.
Battese and Coelli (1995) model has become popular thanks to its computational simplicity as well as its ability to examine the effects of various producer-specific variables on technical efficiency in an econometrically consistent manner, as opposed to a traditional two-step procedure, which is inconsistent with the assumption of independently and identically distributed technical inefficiency effects in the stochastic frontier.
43 As referred to Coelli (1996), this model specification also encompasses a number of other model specifications as special cases. If we set T = 1 and zit contains the value one and no other variables (i.e.
only a constant term), then the model reduces to the truncated normal specification in Stevenson (1980), where δ0 (the only element in δ) will have the same interpretation as the µ parameter in Stevenson (1980). It should be noted, however, that the two above mentioned models are not special case one to each other. Thus these two model specifications are non – nested and hence no set of restrictions can be defined to permit a test of one specification versus the other.
44 This model specification also encompasses a number of other model specifications as special cases.
Particularly, the model of Stevenson (1980) is a particular case of the Battese and Coelli (1995) model that can be obtained for the cases in which T is equal to 1 (for cross-sectional data).
The main advantage of this technique over the two-stage technique is that it incorporates producer specific factors in the estimation of the production frontier because these factors may have a direct impact on efficiency.