Time-varying Technical Inefficiency Models

Most of the primal approaches used to handle time-varying inefficiency have specified it as a product of deterministic functions of time and the random effects, 𝑢𝑘, now reflecting the time-varying part of inefficiency. Note that Kumbhakar (1990) gives in the inefficiency term the following specification: 𝑢_𝑘𝑡 = [1 + 𝑒𝑥𝑝(𝑏𝑡 + 𝑐𝑡2)]−1_|𝑈

𝑘|, while the Battese and Coelli (1992) formulation of inefficiency is 𝑢_𝑘𝑡 = exp[−𝜂(𝑡 − 𝑇)] |𝑈𝑘|. Moving towards a time-decaying inefficiency framework, inefficiency will be a function of time and as such Battesse and Coelli (1992) suggest 𝑢_𝑘𝑡 =

104 _{Stevenson (1980) adopted a truncated normal distribution for 𝑢}

78 | P a g e

exp[−𝜂(𝑡 − 𝑇) + (−𝜂)(𝑡 − 𝑇)2] |𝑈𝑘|. The decay parameter determines whether inefficiency increases or decreases over time and remains constant across all units. These approaches require distributional assumptions for the inefficiency term and the most common candidate is the truncated normal 𝑢𝑘~𝑖𝑖𝑑𝑁+(𝜇, 𝜎𝑢2).

A more general case of the two is the version given by Lee and Schmidt (1993), in quite a flexible version without assuming any parametric function for the inefficiency term. Here, the behaviour of inefficiency is given by 𝑢_𝑘𝑡 = 𝑢_𝑘 𝜆_𝑡 𝑤𝑖𝑡ℎ 𝑡 = 1, … … … , 𝑇, as time-specific effects to be estimated. Both components 𝑢𝑘 and 𝜆𝑡 are deterministic, but in the estimation process 𝑢_𝑘 is considered to be random. It should be noted, that the temporal pattern of inefficiency is exactly the same for all units, which might be quite restrictive compared to other specifications. Placing all these models in a unified framework, a generic formula92 can be used where 𝑢_𝑘𝑡 = 𝐺(𝑡)𝑢_𝑘, 𝐺(𝑡) > 0 is a function of time (𝑡) representing a non-stochastic component common across units, while 𝑢_𝑘 is a unit-specific stochastic one. This type of time-dependent inefficiency varies over time and across individuals (Parmeter and Kumbhakar, 2014).

What follows in the analysis is the identification of cavities that need to be treated and a discussion of the models developed in literature to bridge such gaps. First, making reference to the distinction between any unobserved time-invariant individual specific heterogeneity and inefficiency is essential since the two are undistinguished so far. Hence, latent heterogeneity is confounded with inefficiency, so 𝑢̂𝑘 indicates heterogeneity in addition to, or even instead of, inefficiency (Greene 2005b). Second, the time-invariant nature of inefficiency is somehow misleading in long panel data since units in the long term should identify and treat any signs of inefficiency, otherwise a viable position in the market cannot be assured. Third, an undisputed question pertains to whether the time-invariant component should be considered persistent inefficiency or individual heterogeneity that captures the effects of unobserved time-invariant covariates having nothing to do with inefficiency.

Greene (2004b) argues that 𝑢_𝑘 would be absorbing a large amount of cross country heterogeneity that would inappropriately be measured as inefficiency. Hence, the ‘true’ fixed effects model in which inefficiency is time-varying irrespective of whether the time-invariant component is treated as inefficiency (persistent) or as an individual- specific effect (heterogeneity) has been developed, providing information on the transient part. The generic formula by Greene (2005a) is:

𝑞_𝑘𝑡 = 𝑎_𝑘+ 𝑥′_𝑘𝑡𝛽 + 𝑣_𝑘𝑡− 𝑢_𝑘𝑡 𝑣_𝑘𝑡~𝑁[0, 𝜎_𝑣2_]

𝑢𝑘𝑡 = |𝑈𝑘𝑡|

79 | P a g e

In the TFE model, 𝑎_𝑘 is a random variable that might be correlated with 𝑥_𝑘𝑡 and represents time-invariant heterogeneity. This is a simply pooled SFA model with unit- specific dummies capturing firm effects. TFE models have been reviewed extensively by Chen et al. (2014), who fitted the model by MLE estimator. Note that the inefficiency component here is only time-varying lacking of any measure of persistent inefficiency. The JLMS estimator is used directly for𝑢_𝑘𝑡.

The technical difficulty with TFE models is what is known in the literature as the incidental parameters problem (Neyman and Scott, 1948); Lancaster, (2000). This technical burden derives when the number of parameters to be estimated increases with the number of cross-sectional units in the data, since when 𝐾 → ∞ (number of cross sections increases), the number of 𝛼_𝑘 increases with 𝐾. In the ML framework the number of regressors is fixed, but in a fixed effects case, it increases with 𝐾 so that the desirable asymptotic properties of the MLE are violated with biased and poorly estimated parameters when 𝑇 (time periods) is small. This leads to a persistent bias in the MLE of the parameters in many fixed effects models estimated by ML (Greene, 2007). So, actually the problem with fixed effects is that the number of parameters grows with the number of observations and, therefore, the parameter estimates can never converge to their true value as the sample size increases (Hahn and Newey, 1994). Thus, the parameter estimates are severely unreliable. However, there have been recent advanced econometric developments105 _{using transformed or first-difference versions}

of the fixed effects framework to avoid entirely the incidental parameter problem described by Greene, (2005b).

In the TRE106 case 𝑎_𝑘is treated as uncorrelated with 𝑥_𝑘𝑡. Contrary to the simple RE case by Pitt and Lee, the inefficiency term does not contain any other time-invariant unmeasured sources of heterogeneity since these effects in the TRE models appear in a separate term labelled 𝑤_𝑘 and 𝑢_𝑘𝑡 picks up the inefficiency. The TRE model is:

𝑞_𝑘𝑡 = 𝑎_𝑘+ 𝑥′

𝑘𝑡𝛽 + 𝑣𝑘𝑡− 𝑢𝑘𝑡

𝑣𝑘𝑡~𝑁[0, 𝜎𝑣2 ]

𝑈_𝑘𝑡~𝑁[0, 𝜎_𝑢2], 𝑢_𝑘𝑡 = |𝑈_𝑘𝑡| 𝑎_𝑘= 𝛼 + 𝑤_𝑘, 𝑤_𝑘~𝑁[0, 𝜎_𝑤2_]

We can handle the model as a form of the random parameters (RP) model in which the only random parameter in the model is the constant term. The estimation technique of

105 _{See Chen et al. (2014) and Wang and Ho (2010) for the likelihood function of the within transformed and the first-difference}

model and their closed form expressions.

106 _{Here due to the fact that the time-invariant and unobserved heterogeneity appears in 𝑤}

𝑘 the estimated inefficiencies would be

80 | P a g e

the parameters for TRE models107 _{is the maximum simulated likelihood (MSL) method.}

To obtain an efficiency estimate the JLMS estimator is utilized indirectly by integrating

𝑤_𝑘 out of 𝐸(𝑢_𝑘𝑡|𝐸𝑘𝑡(𝑤𝑘)); in other words, 𝐸𝑘𝑡 is a function of 𝑤𝑘 and then 𝑤𝑘 is integrated out of 𝑢𝑘𝑡. For a Bayesian framework, the applied methods estimating an SF model have been analytically developed by Koop et al. (1997), Kim and Schmidt (2000) and Tsionas (2002).

A significant inexpediency commonly shared among the RE and TRE frameworks is the omitted variables bias, since unobserved variables may be correlated with the regressors. Mundlak (1978) suggests an auxiliary equation to treat this econometric issue of unobserved heterogeneity bias stemming from the questionable orthogonality assumptions of the random effects model. The idea of using an auxiliary equation dependent on a vector of the units’ means of all the time varying explanatory variables can be found in the literature under the term ‘correlated random effects (CRE) approach’. The formulation is as follows:

𝑞_𝑘𝑡 = 𝛽′𝑥_𝑘𝑡 + 𝛼_𝑘+ 𝐸_𝑘𝑡 𝛼_𝑘 = 𝛼 + 𝛾′𝑥̅̅̅ + 𝑤_{𝑘 𝑘}

With the assumption that 𝐸[𝑤𝑘𝑥𝑘𝑡] = 0. The auxiliary equation can be interpreted as a conditional mean function or as a projection (Greene, 2007). The method has been used extensively for various premises, i.e. robust tests108 _{controlling for correlation between}

heterogeneity and covariates on nonlinear models, aim to treat the incidental parameters problem, and average partial effects can be identified through CRE, etc. (Wooldridge, 2005).

It is vital to make a meaningful distinction among models with time-varying inefficiency components. Therefore, apart from the division between unobserved heterogeneity and time-varying inefficiency, it is crucial to discern the persistent part of inefficiency that might inaccurately distort our estimates. There are effects from unobserved inputs or inputs such as management that vary across units but not over time (Mundlak, 1961). Hence, estimating the magnitude of persistent inefficiency is vital, especially in short panels. Persistent inefficiency can change only occasionally since it entails structural or/and operational decisions to be made, while, time-varying efficiency can change over time due to a better reallocation of the resources in the short run. Let us consider the model by (Parmeter and Kumbhakar, 2014).

𝑞_𝑘𝑡 = 𝛽₀+ 𝑥′_𝑘𝑡𝛽 + 𝐸_𝑘𝑡 = 𝛽₀+ 𝑥′_𝑘𝑡𝛽 + 𝑣_𝑘𝑡 − 𝑢_𝑘𝑡 = 𝛽₀+ 𝑥′_𝑘𝑡𝛽 + 𝑣_𝑘𝑡− (𝑢_𝑘+ 𝜏_𝑘𝑡)

107 _{In an application of the Swiss nursing homes Farsi et al. (2005) expressed a preference for the TRE specification.} 108 _{Hausman test comparing random effects (RE) and fixed effects in a linear model.}

81 | P a g e

Here the error component 𝐸_𝑘𝑡 is decomposed to 𝑣_𝑘𝑡 − 𝑢_𝑘𝑡 and further the technical inefficiency part into 𝑢𝑘+ 𝜏𝑘𝑡 where 𝑢𝑘 denotes persistent inefficiency (time-invariant part) and 𝜏_𝑘𝑡 denotes the time-varying part of inefficiency (residual or transient). Both components are non-negative however the former is only unit specific, while the latter is both unit- and time-specific. Both components are quite informative in terms of the policy implications, since high values of 𝑢_𝑘 are of more concern from a long-term point of view, because of its persistent nature, than high values of 𝜏_𝑘𝑡. Regarding the estimation process, the model can be written as:

𝑞𝑘𝑡 = 𝛼𝑘+ 𝑥′𝑘𝑡𝛽 + 𝜔𝑘𝑡 = 𝛽0 − 𝑢𝑘− 𝐸(𝜏𝑘𝑡) + 𝑥′𝑘𝑡𝛽 + 𝑣𝑘𝑡 − [𝜏𝑘𝑡− 𝐸(𝜏𝑘𝑡)]

Therefore, it can perfectly fit a standard panel data model with unit-specific effects. The estimation technique here is twofold, either by the LSDV approach under the FE framework or by GLS under the RE framework. This model treats all time constant effects as persistent inefficiency even if some have time-invariant, unit-specific heterogeneity. If this is the case, then the model is likely to produce an upward bias in inefficiency since unobserved heterogeneity is treated as persistent inefficiency. These