Panel Data Models

Panel data techniques are common in empirical economic analyses due to the possibility of incorporating both cross sectional and time series observations into one framework. Therefore, panel data permit the observation of a cross-section of firms’

efficiency over a period of interest. Clearly, this provides more information about firm performance compared to a snapshot of efficiency among firms in just a period of time.

For the traditional panel data models, it is not necessary to make certain strong distributional assumptions about the composed error term (especially the u) as is the case with the MLE estimation of cross-sectional SFA models. Amongst other important points or benefits, repeated observations on a set of firms can substitute for some strong assumptions. Moreover, repeated observations on firms yield additional information which cannot be derived from adding more firms to a cross-sectional sample (efficiency from larger sample). Hence, technical efficiency for each firm can be estimated consistently as → ∞. To this end, panel data frontier models are likely to produce more consistent estimates of an individual firm’s efficiency (Kumbhaka and Lovell, 2003).

Additionally, with panel data, it is possible to relax other standard model as-sumptions such as homoscedasticity (panel SFA models allow for the treatment of heteroscedasticity). Also, panel data allows for a richer range of models and yields greater testing power. Further, since the independence assumption is not necessary

for panel data models, they are able to incorporate time-invariant regressors into some of the models (Murillo-Zamorano, 2004).

In some panel data SFA models, technical efficiency varies across firms but is assumed to be time-invariant for each firm. This assumption would appear unrealistic in this thesis given the relatively long time frame of the rebound analysis undertaken in this study. Consequently, the focus of the following sections is on panel data mod-els that allow for time varying technical efficiency. I start with the traditional panel data techniques such as the fixed and random effects techniques.

2.6.2.3.1 Fixed-effects Model

The fixed effects (FE) model is generally more suitable when the symmetric noise component is independently distributed over time and across firms and uncorrelated with the regressors. In this case, there is no need to make distributional assumption about and it can be correlated with the regressors or with . is assumed to be iid (0, . Since varies across firms only, it can be treated as constant/fixed so it may be used as firm-specific intercept parameters in a fixed effects model, which can be estimated via OLS (or least squares with dummy variables (LSDV)):

ln ∑ ln (2.24)

where represents firm-specific intercept terms. The FE model can be estimated in three different ways: (i) by suppressing and estimating I firm-specific intercepts (i.e. adding a dummy variable for each of the I firms; (ii) by keeping the intercept term and estimating (I-1) firm-specific intercepts/dummies; or (iii) by employing the within transformation, in which all the data are expressed in terms of deviations from firm means (i.e. , etc.), such that the I intercepts are

then recovered as means of firm residuals. Under this FE framework, the intercepts are derived following

max

(2.25) so that the technical inefficiency component of the residuals is estimated as:

(2.26) which ensures that all 0 so that firm-specific technical efficiency can be derived as:

exp (2.27)

As is the case with COLS, under the FE estimator, at least one firm is assumed to be 100% efficient against which all the other firms’ relative efficiency is measured.

However, in addition to the FE estimator not requiring the distributional or inde-pendence assumptions, when compared to the cross-sectional MLE models, it provides consistent estimates of and technical efficiency as either → ∞ or

→ ∞, while consistent estimates of only requires that → ∞.

Notwithstanding the benefits of the FE estimator above, a number of major limitations exist. The fixed effects, which capture firm-specific is likely to pick-up other time invariant factors across firms such as location, regulatory environment and sunk costs. This is especially true for the within transformation. This challenge could potentially result in the over-estimation of technical inefficiency. Moreover, even when such time-invariant factors are included in the model, there is the additional problem that they would be correlated with other regressors. Thus, a random-effects model may be adopted where it is possible to make assumptions that the fixed effects

are uncorrelated with the other regressors, in addition to the random assumption about the distribution of the effects.

2.6.2.3.2 Random-effects model

Moreover, unlike the FE model, is taken to be randomly distributed with con-stant mean and variance, but is uncorrelated with the regressors. This makes it possible to incorporate time-invariant factors since the inefficiency measure/effects and the regressors are independent of each other. However, just like with the FE es-timator, the iid assumption about is retaind so that a random effects (RE) model can be written as:

ln ln

∗ ln ^∗

(2.28) Both components of the error term and ^∗ have zero means and the RE model can be estimated using either the two-step generalised least squares (GLS) technique or MLE (when distributional assumptions about the error components are tenable such as the normal-half-normal distribution (Pitt and Lee, 1981); normal-truncated normal distribution (Kumbhakar, 1987; Battese and Coelli, 1988)). When distributional assumptions are not required on the composed error terms, the GLS approach is appropriate for estimating the model and technical effi-ciency. First, OLS is used to estimate the model parameter estimates and then in the second stage, ^∗and are reestimated using feasible GLS.

As shown by Kumbhaka and Lovell (2003), ^∗ does not depend on i, since is a positive constant, such that only one intercept term is estimated. Once ^∗

and have been reestimated using feasible GLS, ^∗can be derived by the means of the residuals:

∗ 1

ln ^∗ ln

(2.29) Estimates of can be obtained through means of the normalization:

max ^{∗ ∗} (2.30)

Or alternatively can be estimated using the best linear unbiased predictor (BLUP).

The BLUP of ^∗ is given as:

∗ ∙ ln ^∗ ln

(2.31)

The RE model estimates above as consistent as both → ∞ and → ∞. The firm-specific technical efficiency estimates are derived by substituting into (2.27) above. As with the FE estimator, the RE model also requires that at least one of the firms is 100% technically efficient, relative to the other inefficient firms. In general also, both the FE and RE models do not necessarily require distributional assump-tions in the specification and estimation of a stochastic frontier (Murillo-Zamorano, 2004).

The RE estimator has the desirable possibility of permitting time invariant re-gressors in the model, which might yield lower technical inefficiency estimates.

Notwithstanding the relative strengths of the RE model, it requires the extra (trade-off) assumption that the inefficiency term is not correlated with the regressors.