Time-invariant Technical Inefficiency Models

The stochastic panel model with time-invariant inefficiency can be estimated under either the fixed effects or random effects framework (Wooldridge, 2010). The selection of the framework is dependent on the level of relationship permitted between inefficiency and the explanatory variables of the model (Parmeter and Kumbhakar, 2014). The fixed effects approach permits correlation between 𝑥_𝑘𝑡 and 𝑢_𝑘 , whereas the random effects approach does not.

The received literature on the fixed effects model in the frontier modelling framework is based on Schmidt and Sickles’s (1984) modification on the linear regression model which incorporates a unit-specific intercept in the basic linear model framework. They propose a model that estimates the persistent part of the inefficiency without specifying an explicit distribution of the inefficiency, labelled the distribution free approach.

96 _{This is not the case with OLS (COLS, MOLS).}

97 _{A key question related to latent heterogeneity (the time-invariant individual effects) is whether the individual effects represent}

(persistent) inefficiency, or whether the effects are independent of the inefficiency and reflect (persistent) unobserved heterogeneity.

98 _{Some of the strong distributional assumptions used to disentangle the separate effects of inefficiency and noise can be relaxed}

(Coelli et al., 2005).

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Classic fixed effects models take advantage of the panel structure to increase the explanatory power of the model by incorporating a unit-specific, time-invariant effect. So, the linear fixed effects model can be reinterpreted as:

ln(qkt) = 𝛽0+ ∑ βixkti+ 𝑣𝑘𝑡 N i=1 − 𝑢𝑘 ln(qkt) = 𝛽0− 𝑢𝑘+ ∑ βixkti+ 𝑣𝑘𝑡 N i=1 ln(qkt) = 𝛼𝑘+ ∑ βixkti+ 𝑣𝑘𝑡 N i=1

Which can be estimated consistently and efficiently by OLS after including individual dummies as regressors100 for 𝛼_𝑘≡ 𝛽₀− 𝑢_𝑘. The model is reinterpreted by treating 𝛼_𝑘

as the firm-specific inefficiency effect. The purpose of the effect is to capture the impact of all the factors that are specific to the unit and constant over time, and which have not been included already in the model. This means that the time-invariant units’ heterogeneity, e.g. location characteristics (urban or rural), a person’s ability (when modelling income), prevailing environmental conditions, etc. is reflected in 𝛼𝑘. To retain the flavor of the frontier model once 𝛼̂𝑘 is available, the DMUs are compared on the basis of the following transformation to obtain an estimated value of 𝑢_𝑘(Schmidt and Sickles, 1984):

For production functions: 𝑢̂ = max_𝑘

𝑘 {𝛼̂ } − 𝛼𝑘 ̂𝑘, 𝑘 = 1, … … … … , 𝐾, where 𝛼̂𝑘 is the

𝑘 −th fixed effects estimate in the within-groups fixed effects linear regression model. This formulation implicitly assumes that the most efficient unit in the sample is 100 percent efficient. Therefore, the estimated inefficiency in the fixed-effects model is relative to the best unit in the sample. The unit-specific TE estimate equals 𝑇𝐸_𝑘= exp (−𝑢̂)𝑘 . For cost functions: 𝑢̂ = 𝛼𝑘 ̂ − min𝑘

𝑘 {𝛼̂ }𝑘 , 𝑘 = 1, … … … … , 𝐾 and the cost efficiency estimate equals 𝐶𝐸_𝑘 = exp (−𝑢̂)_𝑘 .

The fixed effects approach empoweres the model with an important implication to allow for correlation101 _{between 𝑥}

𝑘𝑡 and 𝑢𝑘.This may be a desirable property for empirical applications in which inefficiency is believed to be correlated with the inputs used (Mundlak, 1961). However, the model does not allow for separate identification of

100 _{This technique is often referred to as the least square dummy variable (LSDV) method. The coefficients of the dummies are the}

estimates of 𝛼𝑘.

101 _{An important limitation of the FE is that no other time-invariant variables, such as gender, race, region, etc., can be included in}

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inefficiency and individual heterogeneity, and this constitutes an important limitation that the recent literature aimed to capture, i.e. the true fixed effects (TFE) and true random effects (TRE) models (Greene, 2002).

The model proposed by Schmidt and Sickles (1984) is extended by Cornwell et al. (1990) in order to include a time-varying effect, without specifying an explicit distribution of the inefficiency. Early work on the model suggested direct manipulation of the fixed effects term; in other words, the time-varying part of the inefficiency term is defined as:

𝛼𝑘𝑡 = 𝜃𝑘0+ 𝜃𝑘1𝑡 + 𝜃𝑘2𝑡2

Despite the desirable decomposition between the time-invariant component 𝜃_𝑘0 and the time-varying component 𝜃_𝑘1𝑡 + 𝜃_𝑘2𝑡2 some further caveats need to be mentioned. First, it does not leave space for time-invariant heterogeneity that is not inefficiency; second, it assumes a unit-specific quadratic function of time102 _{to explain the time-}

varying part, which might be quite restrictive. Recent panel data literature has tried to relax the assumption of a time-invariant inefficiency in two components (Cornwell and Schmidt, 1996).

Turning to the random effects model developed by Pitt and Lee (1981) the inefficiency term 𝑢𝑘 is assumed to be constant through time and randomly distributed since it must be uncorrelated with independent variables. If so, time-invariant regressors such as gender, race, etc., can be included in the model without leading to collinearity with

𝛼𝑘. Also, the RE framework facilitates cases in which independent variables show very little variation between time periods; in such cases, the FE may fail to identify the statistical significance of those variables since all variation is captured by the effects. However, it should be stressed that the RE model is restrictive in the sense that is does not allow for correlation between the RE unobserved time-invariant inefficiency and the independent variables (i.e. the regressors) and noise. When the assumption of no correlation between the covariates and university efficiency is indeed correct, then estimation of the stochastic frontier panel data model by using the RE framework provides more efficient estimates than estimation under the FE framework.

According to Pitt and Lee (1981), distributional assumptions on the random components of the model can be imposed and then estimation of the parameters of the model is feasible through ML.103 _{The relevant log-likelihood for a random effects model with a}

half-normal distribution is derived by Lee and Tyler (1978) and discussed further by Battese and Coelli (1988). Inefficiency is not directly estimated via MLE, so once the parameters are estimated, JLMS-type conditional mean estimators can be used to receive an estimate for the unit-specific inefficiency (Kumbhakar, 1987).

102 _{Han et al. (2005) propose factor analytic forms for modelling 𝑎}

𝑘𝑡

103 _{An alternative to MLE is the use of generalised least squares (GLS) estimator; see Baltagi (2013) for an analytical review of the}

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The likelihood function for the 𝑘 − th observation is Pitt and Lee (1981):

𝑙𝑛𝐿𝑘 = 𝑐𝑜𝑛𝑡𝑎𝑛𝑡 + 𝑙𝑛Φ ( 𝜇_𝑘∗ 𝜎∗ ) +1 2ln(𝜎∗ 2_{) −}1 2{ 𝛴_𝑡𝐸_𝑘𝑡2 𝜎𝑣2 + (𝜇 𝜎𝑢 ) 2 − (𝜇𝑘∗ 𝜎∗ ) 2 } − 𝑇𝑙𝑛(𝜎𝑣) − ln(𝜎_𝑢) − 𝑙𝑛Φ (𝜇 𝜎_𝑢) With𝜇𝑘∗ = 𝜇𝜎𝑣2−𝜎𝑢2𝛴𝑡𝐸𝑘𝑡 𝜎𝑣2+𝑇𝜎𝑢2 and𝜎∗ 2 ₌ 𝜎𝑣2𝜎𝑢2

𝜎𝑣2+𝑇𝜎𝑢2, Τ sample size. For RE no assumptions on the

PDF of the inefficiency are made other than that inefficiency is an independently distributed random variable with non-negative values, i.e. 𝑢𝑘 ∼ 𝑖𝑖𝑑 ≥ 0 i.e. a half normal distribution. The stochastic error component is distributed as 𝑣_𝑘𝑡 ∼ 𝑖𝑖𝑑 𝑁(0, 𝜎𝑣2). The estimated parameters from the MLE estimation process are utilised in the next step, where the extended JLMS estimator of inefficiency is used to obtain an estimate of inefficiency. 𝐸(𝑢𝑘|𝐸𝑘) = 𝜇𝑘∗+ 𝜎∗[ 𝜑 (−𝜇_𝜎𝑘∗ ∗ ) 1 − Φ (−𝜇𝑘∗ 𝜎_∗) ]

If 𝑢_𝑘 ∼ |𝑈_𝑘|, 𝑈_𝑘 ∼ 𝑁[0, 𝜎_𝑢2_{] is distributed half-normally,}104 _{then 𝜇 = 0. Also, in some}

cases, 𝜇 may be a function of covariates of exogenous determinants of inefficiency, i.e.

𝜇 = (𝑧′ 𝑘𝛿).

As previously mentioned, the inefficiency term here has a time-invariant interpretation since inefficiency levels may vary for different individuals, but they do not change over time therefore, time variation is an issue to be accommodated in the literature. The array of models introduced in the next section aim to give a time-varying dimension to inefficiency since the implications of the time-invariant hypothesis are too restrictive. This implies that an inefficient unit (e.g., a university) does learn over time; therefore, if the latent goal is productivity and efficiency improvements, a time-varying inefficiency framework should be structured.