*3.2. Model 2: The True Fixed-E*ff*ects Model*

Model 1 is a standard panel data model where α<sup>i</sup> is an unobservable individual effect. The model can be estimated using the standard panel data fixed and random-effects estimators to estimate the model's parameters to obtain the estimated value of *ui*. The highest estimated value of αˆ*i*, namely *u*ˆ*i*, is used as a reference for the frontier.

However, there is a notable drawback in Model 1 s approach as it does not allow individual heterogeneity to be distinguished from inefficiency. In other words, all time-invariant heterogeneity such as enterprise infrastructure that is not necessarily inefficient is included as inefficiency [9,29]. Also, the time-invariant assumption of inefficiency is a potential issue with Model 1. If T is large, it seems implausible that the inefficiency in energy use will stay constant for an extended period of time, since the technological progress will eventually replace less efficient technologies. So, should one view the time-invariant component as persistent inefficiency or as individual heterogeneity? The optimal choice lies somewhere in between, that is, a part of the inefficiency might be persistent, while another part may be transitory.

To solve the problem that the two parts cannot be separated from time-invariant individual heterogeneity effects, we have to choose either a model wherein α<sup>i</sup> represents persistent inefficiency, or a model wherein α<sup>i</sup> represents an individual-specific heterogeneity effect.

Following Kumbhakar and Heshmati [29] we consider both specifications in this paper. Thus, the models we examine can be written as:

$$\begin{aligned} \mathit{ENE}\_{\mathrm{it}} &= \alpha\_{\mathrm{i}} + \mathfrak{x}\_{\mathrm{it}}' \beta + \mathfrak{e}\_{\mathrm{it}\prime} \in \boldsymbol{v}\_{\mathrm{it}} + \boldsymbol{u}\_{\mathrm{it}\prime} \\ \boldsymbol{v}\_{\mathrm{it}} &\sim \mathsf{N}(\mathbf{0}, \sigma\_{\mathrm{v}}^{2}), \ \boldsymbol{u}\_{\mathrm{it}} = \boldsymbol{h}\_{\mathrm{i}\prime} \boldsymbol{u}\_{\mathrm{i}\prime} \ h\_{\mathrm{it}} = \boldsymbol{f}(\boldsymbol{z}\_{\mathrm{i}\prime}' \boldsymbol{\delta}), \ \boldsymbol{u}\_{\mathrm{i}} &\sim \mathsf{N}^{+}(\boldsymbol{\mu}\_{\prime} \sigma\_{\mathrm{u}}^{2}). \end{aligned}$$

The key feature that allows for the model's transformation is the multiplicative form of inefficiency effects, *uit*, in which individual-specific effects, *ui*, appear in multiplicative forms with individual and time-specific effects, *hit*. As *u*<sup>∗</sup> *<sup>i</sup>* does not change with time, the within and first-difference transformations leave this stochastic term intact. Thus, the difference between Model 1 and Model 2 is that inefficiency in Model 2 is explained by its observable determinants (z), while in the former, the time patterns of inefficiency are explained by a trend, but inefficiency is not explained by any determinants. Thus, cost efficiency is obtained based on the separated *uit* components of the residual.
