*3.1. Model 1: The Time-Variant E*ffi*ciency Model*

Ref [8] considered a production model wherein technical inefficiency effects were modeled in a stochastic frontier function for panel data. In this paper, we specify a factor demand version of the model. The objective is to minimize the use of a factor in the production of a given output, factor price, and technology. This is similar to [27] who analyzed labor use efficiency in the banking industry. Here we use the same approach but in the context of energy use. Separability between energy and other inputs is assumed. The assumption is supported by the fact that we use aggregate output and aggregate individual inputs. A cost function is appropriate for the current case as energy use is cost for producing a given output, which is desirable to be minimized. Provided the inefficiency effects are stochastic, the model permits the estimation of both technical change or a shift in function over time and time-varying technical inefficiencies. The model is estimated using the maximum likelihood method which allows for estimating the effects of inefficiency's determinants. In this case inefficiency is a function of time.

In Model 1 we use the following generic formulation to discuss the various components in a unifying network:

$$\begin{aligned} \text{ENE}\_{it} &= f(\mathbf{x}\_{it}, \boldsymbol{\beta}) + \in\_{it} \; \epsilon\_{it} = v\_{it} + u\_{it}, \\ u\_{it} &= \mathbf{G}(t) u\_{i}, \; v\_{it} \sim \mathcal{N}(0, \sigma\_{v}^{2}), \; u\_{it} \sim \mathcal{N}^{+}(\mu, \sigma\_{u}^{2}), \; G(t) = \left[1 + \exp\left(\gamma\_{1}t + \gamma\_{2}t^{2}\right)\right]^{-1} \end{aligned}$$

where *ENE* is energy use and *G*(*t*) > 0 is a function of time (t); in this model, inefficiency (*uit*) is not fixed for a given individual, instead it both changes over time and across individuals. Inefficiency is composed of two distinct components: the nonstochastic time component, G(t) and a stochastic individual component, *ui*. The stochastic component, *uit*, uses the panel structure of the data in this model. The *ui* component is individual-specific and the *G(t)* component is time-varying and is common for all the individuals. We consider some specific forms of *G(t)* used in [28] model which assumes *G*(*t*) > 0, given that *ui* > 0, and thus *uit* ≥ 0 is ensured by having a non-negative *G(t). G(t)* can be monotonically increasing (decreasing) or concave (convex) depending on the signs and magnitude of γ<sup>1</sup> and γ2. Inefficiency changes in this model are time driven and a nonlinear exponential function of

time. However, the trend pattern is similar for all individuals; the differences in performance among individuals are due to the *ui* component. The random and nonlinear nature of the model requires iterative estimation by the maximum likelihood (ML) estimation method. Cost efficiency is estimated assuming truncated normal distribution using the product of the individual specific *ui* and the time variant *G(t).* The product of the two is in the interval between 0 and 100 where 100 represents a full cost-efficient unit.
