SUMMARY
 Top of page
 SUMMARY
 INTRODUCTION
 THE ECONOMETRIC MODEL
 BAYESIAN NUMERICAL INFERENCE PROCEDURES
 ARTIFICIAL EXAMPLES AND SAMPLING PERFORMANCE OF BAYES ESTIMATORS
 AN APPLICATION TO US BANKING
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
 Supporting Information
This paper considers a panel data stochastic frontier model that disentangles unobserved firm effects (firm heterogeneity) from persistent (timeinvariant/longterm) and transient (timevarying/shortterm) technical inefficiency. The model gives us a fourway error component model, viz., persistent and timevarying inefficiency, random firm effects and noise. We use Bayesian methods of inference to provide robust and efficient methods of estimating inefficiency components in this fourway error component model. Monte Carlo results are provided to validate its performance. We also present results from an empirical application that uses a large panel of US commercial banks. Copyright © 2012 John Wiley & Sons, Ltd.
INTRODUCTION
 Top of page
 SUMMARY
 INTRODUCTION
 THE ECONOMETRIC MODEL
 BAYESIAN NUMERICAL INFERENCE PROCEDURES
 ARTIFICIAL EXAMPLES AND SAMPLING PERFORMANCE OF BAYES ESTIMATORS
 AN APPLICATION TO US BANKING
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
 Supporting Information
In recent years the use of stochastic frontier models to estimate (in)efficiency in production and cost functions has been growing exponentially. Many of these applications use panel models that utilize the data in more efficient ways. The standard stochastic frontier panel data models have been extended in several directions. Estimation of some of these models can be conducted under less demanding assumptions and at the same time using more flexible modeling approaches. For example, heterogeneous technologies have been the focus of much research, including random coefficient stochastic frontier models. Other examples include latent class or mixture models and Markov switching models.1 More recently, an important line of research has been the formulation and estimation of panel models in which firm effects are separated from inefficiency.
In a standard panel data model, the focus is mostly on controlling firm heterogeneity due to unobserved timeinvariant covariates. The innovation in the timeinvariant stochastic frontier models (developed in the 1980s) was to make these firm effects onesided so as to give them an inefficiency interpretation. In some models these inefficiency effects were treated as fixed parameters (Schmidt and Sickles, 1984) while others treat them as random variables (Pitt and Lee, 1981; Kumbhakar, 1987). The models proposed by Kumbhakar (1991), Kumbhakar and Heshmati (1995) and Kumbhakar and Hjalmarsson (1993, 1995) treated firm effects as persistent inefficiency and included another random component to capture timevarying technical inefficiency. These formulations are in contrast to the ‘true’ random or fixedeffect models proposed by Greene (2005a,2005b) in which firm effects are not parts of inefficiency. The truth might be somewhere in between. That is, part of the firm effects in Greene (2005a,2005b) might be persistent inefficiency. Similarly, part of persistent inefficiency in the models proposed by Kumbhakar and coauthors might include unobserved firm effects. Since none of the assumptions used in the abovecited models are fully satisfactory, we consider a generalized true randomeffects (GTRE) model that decomposes the timeinvariant firm effects into a random firm effect (to capture unobserved heterogeneity à la Greene, 2005a,2005b) and a persistent technical inefficiency effect (as in Pitt and Lee, 1981; Schmidt and Sickles, 1984; Kumbhakar, 1987).
Most of the popular panel data models that are widely used in empirical applications do not control for unobserved firm effects. For example, the inefficiency specification used by Battese and Coelli (1992) (which is a variant of the model due to Kumbhakar, 1991) and its variants allow inefficiency to change over time but without firm effects.2 Thus these models confound firm effects with inefficiency. Two other panel data models, viz., the ‘truefixed’ and ‘truerandom’ effects (Greene, 2005a,2005b Kumbhakar and Wang, 2005) frontier models separate firm effects (fixed or random) from inefficiency, where inefficiency can be a function of exogenous variables.
Some of the models that are widely used in the literature can be summarized as in Table 1. Details on estimation of these models can be found in Kumbhakar et al. (2011).
Table 1. Main characteristics of some of the panel data models  Model 1  Model 2  Model 3  Model 4  Model 5  Model 6 

Firm effect:  No  No  Yes (fixed)  Yes (random)  No  Yes (random) 


Technical inefficiency 
Persistent  No  No  No  No  Yes  Yes 
Transient  No  No  No  No  Yes  Yes 
Overall technical inefficiency 
Mean  Timeinv.a  Timeinv.  Timeinv.  Zero trunc.b  Zero trunc.  Zero trunc. 
Variance  Homo.  Hetero.  Hetero.  Hetero.  Homo.  Homo. 
Symmetric error term 
Variance  Homo.  Hetero.c  Hetero.  Homo.  Homo.  Homo. 
Before proceeding further it might be worth asking questions like: Is there any economic rationale for all these components? Why do we need a model with all these components? To address these questions, we start with the following. Assume that α_{i} represents firm heterogeneity (effect of unobserved factors). The necessity to control α_{i} in estimating the regression function is well understood in the panel data literature. So it is not necessary to rationalize modeling α_{i} separate from v_{it}. The nest question is: Why do we want to decompose inefficiency into various components? To justify this consider the case where inefficiency is associated with (unobserved) management, and assume that management is timeinvariant. Consequently, inefficiency will also be timeinvariant (i.e. ). This gives us a threeway error component model, i.e., , in which one needs to separate α_{i} from , both of which are timeinvariant. Now assume that management changes over time (which is probably more realistic), although most of it might be timeinvariant. That is, management has a timeinvariant and a timevarying component. If, as before, inefficiency is associated with management, we have a situation in which one part of inefficiency is timeinvariant and the other part is timevarying (0) (to be consistent with the management story). This example illustrates the rationale for a fourway error component model, i.e. , which we consider in this paper. Colombi et al. (2011) give some other examples to justify the need for a fourway error component model. If the policy makers (regulators) are interested in eliminating persistent inefficiency that is often attributed to regulation, it is necessary to estimate them first. Estimating a model with only one inefficiency component (with or without controlling for firm effects) is likely to give incorrect estimates of inefficiency.
In view of the above discussion we consider a general model, which in a cost function framework is3
 (1)
where the dependent variable is (log) cost and the independent variables, represented by the vector x, are input prices and outputs (log).4 Subscripts i and t refer to firm and time (i = 1, 2, …, n and t = 1, 2, …, T), respectively. Note that the model in (1) has four error components. If we denote the composed error where the superscript (+) indicates nonnegative value of the corresponding error component, we can give a meaningful interpretation of each of the error components. First, the random noise component is v_{it}, which is similar to the noise component in a standard regression model. Second, the persistent (longrun) technical inefficiency component is . Third, shortrun or transient technical inefficiency is . Fourth, firmspecific random effects (firm heterogeneity) are captured by the α_{i} term.
The generality of the model in (1) can be viewed from both crosssectional and panel points of view. If only crosssectional data are available, the model in (1) will have only the crosssectional components in the error term, viz., , which is the original stochastic frontier model proposed by Aigner et al. (1977) and Meeusen and van den Broeck (1977). A standard pooled panel model will have two error components and the error term will be of the form . Greene's true randomeffects model will have the error specification . The composite error term in the Kumbhakar–Heshmati (1995) and Kumbhakar–Hjalmarsson (1995) model is of the form . However, the fully flexible error specification is . It is possible to identify each of these components using the standard distributional assumptions used in the stochastic frontier models. For example, can be viewed as the firmspecific component in a standard panel model, the predicted value of which can be obtained from estimating (1) using a standard oneway error component panel estimation. Decomposing from δ_{i} is standard in a crosssectional stochastic frontier model (the Jondrow et al., 1982, procedure) where δ_{i} can be viewed as the twosided noise term and is inefficiency. Similarly, estimation of the panel model also gives predicted values of , from which one can obtain estimates of using the exact same procedure.5 Colombi et al. (2011) considered a singlestep maximum likelihood method to estimate the technology parameters and technical efficiency components of this model.
In this paper we consider an alternative approach, viz., a Bayesian Markov chain Monte Carlo (MCMC) approach to estimate the GTRE model. There are some advantages to the Bayesian approach compared to the ML approach used in Colombi et al. (2011). First, it has good finitesample properties even in samples where n and T are relatively small. Second, in light of the recent advances in the treatment of the incidental parameters problem, there is reason to believe that average likelihood or a fully Bayesian approach can perform much better relative to samplingtheory treatments. We provide both simulation results and results from real data.
Note that there are no Bayesian frontier models that allow random firm effects along with timeinvariant inefficiency. Koop and Steel (2003) proposed a panel data model where technical inefficiency can be timevarying. This is based on a known parametrization of technical inefficiency through γ = Du, where u is technical inefficiency and D is a known matrix. However, nothing is mentioned about persistent inefficiency or separation of inefficiency from firm effects.6 Thus our model is not just another application of an ‘offtheshelf’ Bayesian MCMC approach. It contributes to the Bayesian stochastic frontier literature in terms of a new model that has not been applied before.
In summary, the contributions of this paper are twofold. We propose two parametrizations for the Gibbs sampler that effectively provide accurate inferences and less autocorrelation in the MCMC scheme to address the problem of the relationship between timeinvariant (persistent) inefficiency and firm effects. We also propose an efficient reparametrization of the MCMC scheme to account for the correlation of the three randomeffect components (and ). This reparametrization improves considerably the performance of MCMC. We show in artificial experiments that the ‘straightforward’ Gibbs sampler suffers from problems of slow convergence and extremely high autocorrelations and thus it cannot help in a full exploration of the posterior. In sampling experiments where our reparametrized Gibbs sampler is used, the Bayes posterior mean (or median) performs quite well for the sample sizes typically encountered in economic applications (n > 100 and T = 5 or 10).
The rest of the paper is organized as follows. The econometric model is discussed in Section 2, followed by the Bayesian inference procedure in Section 3. Performance of the Bayesian estimation procedure in the light of some artificial examples is discussed in Section 4. Section 5 reports results from an application using a panel data on US banks. Section 6 concludes the paper.
THE ECONOMETRIC MODEL
 Top of page
 SUMMARY
 INTRODUCTION
 THE ECONOMETRIC MODEL
 BAYESIAN NUMERICAL INFERENCE PROCEDURES
 ARTIFICIAL EXAMPLES AND SAMPLING PERFORMANCE OF BAYES ESTIMATORS
 AN APPLICATION TO US BANKING
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
 Supporting Information
We consider the model in (1), i.e.
and rewrite it as
 (2)
If we rewrite the model as
 (3)
then (3) resembles the model proposed by Kumbhakar and Heshmati (1995) and Kumbhakar and Hjalmarsson (1993, 1995), among others. The only difference is that in the papers by Kumbhakar and coauthors ξ_{it} is assumed to be i.i.d., whereas in (3) ξ_{it} is not i.i.d., although v_{it} and α_{i} are i.i.d. unless the variance of α_{i} is zero.7 The formulation used by Kumbhakar and Heshmati (1995) is thus opposite that of Greene (2005a,2005b), in the sense that the former ignores firm effects whereas the latter ignores persistent inefficiency. Thus both formulations are misspecified, although their impacts on estimated inefficiency are not the same. The Greene formulation is likely to produce downward bias in the estimate of overall inefficiency. Perhaps it would be better to say that the Greene formulation gives an estimate of transient inefficiency, and therefore it will give a downward bias of overall inefficiency, especially if persistent inefficiency exists. Given these shortcomings, it is important to consider estimation of the model in (1).
At this point it is useful, perhaps, to explain why we use a randomeffects formulation. The model with has received a lot of attention in the literature. It is known that the model is subject to the incidental parameters problem, since the number of α_{i}s increases with the sample size, leading to inconsistent inferences by the method of maximum likelihood. Recently, semiBayesian approaches have been proposed using the fact that artificial priors can be introduced for the incidental parameters. From that point of view it is natural to be explicit about the nature of the random effects and use a fully likelihoodbased procedure.8
The transformation in Chen et al. (2011) that used the multivariate CSN is one class of transformations, but there are many transformations that are possible because the true fixedeffects model does not have the property of information orthogonality (Lancaster, 2000). The best transformation, the one that is maximally bias reducing, cannot be taking deviations from the means because the information matrix is not block diagonal with respect to . Other transformations might be more effective.
AN APPLICATION TO US BANKING
 Top of page
 SUMMARY
 INTRODUCTION
 THE ECONOMETRIC MODEL
 BAYESIAN NUMERICAL INFERENCE PROCEDURES
 ARTIFICIAL EXAMPLES AND SAMPLING PERFORMANCE OF BAYES ESTIMATORS
 AN APPLICATION TO US BANKING
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
 Supporting Information
In this section we present results from the models we discussed earlier using the US banking data. The data used comes from the Reports of Income and Condition (Call Reports). We used a balanced panel of banks from 1998 to 2005 as in Feng and Serletis (2009). To address technological heterogeneity, we follow Feng and Serletis (2009), who classified the sample banks into three size categories (large, medium and small). Each of these categories is then subdivided into four categories, giving a total of 12 groups. Here we use three of their groups (group 1: the very large banks; group 6 is the lower end of medium banks; and group 10: in the middle of small banks) (see Table 3 of their paper for details).
Since efficiency is intimately related to the technology, it is important that the technology is specified to be as flexible as possible and the specified technology also satisfies theoretical properties. Since banking outputs are services which cannot be stored, the standard practice is to specify the technology in terms of a dual cost function, thereby meaning that banks minimize cost taking outputs as given. Flexible functional forms (mostly Translog) are used in the literature. Since the Translog function tends to violate theoretical properties of a cost function (viz., concave in input prices), Feng and Serletis (2009) used the Fourier cost function, which satisfies global regularity conditions. Here we use the popular Translog form and impose the regularity constraints using the procedure by Terrell (1996).
The other issue is specification of inefficiency. Even if the cost function is globally well behaved but inefficiency is not modeled properly, estimates of economic variables of interest (returns to scale, technical change, etc.) as well as inefficiency might be wrong. For example, in specifying the cost function and inefficiency, Feng and Serletis (2009) neither controlled for bank effects nor allowed persistent (bankspecific) inefficiency. In our model as well as in its closest cousin—the TRE model proposed by Greene (2005a,b)—bank effects are controlled in estimating inefficiency. In the application, we show that efficiency results (as well as estimated technical change) differ across models, and therefore one has to consider appropriateness of alternative models.
Specification of inputs and outputs for financial service providers like banks is not straightforward. This is because many of the services are jointly produced and prices are typically assigned to a bundle of financial services. The role of commercial banks is generally defined as collecting the savings of households and other agents to finance the investment needs of firms and consumption needs of individuals. In addition, they provide various financial services relating to fund transfer, trade, investments, etc. What banks do produce, using what, is a longstanding debate in the banking literature. Two approaches dominate: the production approach and the intermediation approach. Both approaches apply the traditional microeconomic theory of the firm to banking and differ only in the specification of banking activities. Under the production approach banks are primarily viewed as providers of services to customers. The inputs under this approach include physical variables (e.g. labor, material, space or information systems) and the outputs represent the services provided to customers. Under the intermediation approach banks produce intermediation services through the collection of deposits and other liabilities and apply them in interestearning assets, such as loans, securities and other investments. This approach includes both operating and interest expenses as inputs, whereas loans and other major assets count as outputs. The appropriateness of each approach varies according to the issues and problems addressed. It is apparent that banks perform many activities within the broad framework of both approaches. Here we use the balancesheet approach of Sealy and Lindley (1977), in which all liabilities (core deposits and purchased funds) and financial equity capital provide funds and are treated as inputs. Similarly, all assets (loans and securities) use bank funds and are treated as outputs. This approach is different from the intermediation approach, which is consistent with the valueadded definition of output (Das and Kumbhakar, 2012).
Following Feng and Serletis (2009), we use three traditional outputs consumer loans (y_{1}); nonconsumer loans (y_{2}) composed of industrial and commercial loans and real estate loans; and securities (y_{3}), which includes all nonloan financial assets, i.e. all financial and physical assets minus the sum of consumer loans, nonconsumer loans, securities and equity. All outputs are deflated by the consumer price index (CPI) to the base year 1998. In addition to these traditional outputs we also include the following two nontraditional outputs, viz., financial equity capital (y_{4}) and nontraditional banking activities (y_{5}). The time trend variable (t) is included in the cost function to capture technological change. The input variables used are: labor, borrowed funds and physical capital. Prices of these are: the wage rate for labor (w_{1}); the interest rate for borrowed funds (w_{2}); and the price of physical capital (w_{3}). Prices of these inputs are calculated by dividing total expenses on each input categories by their respective quantities. For example, the wage rate is computed by dividing total salaries and benefits by the number of fulltime employees. Similarly, the price of capital is total expenses on premises and equipment divided by premises and fixed assets, and the price of deposits and purchased funds equals total interest expense divided by total deposits and purchased funds. Total cost is simply the cost of these three inputs. This specification of outputs and inputs is similar to most of the previous banking studies (see, for example, Berger and Mester, 1997).
As mentioned earlier, Feng and Serletis (2009) emphasized that imposition of monotonocity and concavity constraints is quite important. However, instead of using the Fourier functional form, we used the Translog function and followed the approach in Terrell (1996) to impose these constraints. Specifically, we first impose the constraints at the means, and then we use rejection sampling to the whole set of observations until at least 99% of the observations satisfy the restrictions.
We analyze the group 1 bank (the largest banking group in the sample) results in greater detail and crosscheck some key results with some other groups. The main reason for not reporting results for all 12 groups is that the groups did not seem to give different results so far as technical inefficiency, returns to scale, etc., are concerned. For comparison (robustness check), we use (i) the traditional stochastic frontier model (SFM) which is used in Feng and Serletis (2009), (ii) the true random effects (TRE) model proposed by Greene (2005a,b), and (iii) the generalized TRE (GTRE) model introduced in this paper. Note that GTRE is the most general model, followed by TRE and the traditional SFM. Since GTRE nests the other two models, it is possible to test empirically which model is appropriate for the data at hand.
Table 4. Bayes factor for model comparison  Group 1  Group 6  Group 10 

GTRE vs. TRE  45.343  189.212  212.016 
TRE vs. SFM  212.376  512.991  565.127 
Instead of reporting returns to scale (RTS), we report output cost elasticity (), which is the reciprocal of RTS. Scale economies are said to exist if RTS exceed unity (or E_{cy} < 1). Note that for a Translog cost function E_{cy} is observationspecific (i.e. it varies with bank and over time). We also report technical change (TC = ∂ ln C/∂ t), a negative of which will indicate technical progress (cost diminution over time, ceteris paribus). TC is also observationspecific (i.e. it varies with bank and over time). Posterior distributions of E_{cy} and TC19 are reported in Figure 3. It can be seen that E_{cy} results are quite similar across three models. This is, however, not the case with TC (reported in the lower panel of Figure 3). Estimates of TC from the traditional SFM and GTRE are quite close and show large variations, in contrast to those from the TRE model. Given that banks in this group are quite heterogeneous in size, it is expected to observe large variations. Furthermore, since the specification test results favor the GTRE model, we rely more on the results from the GTRE model.
In Figure 4 we present posterior distributions of overall technical inefficiency from the abovementioned three models. The traditional SFM and the GTRE models show that inefficiency averages around 12% and 5%, respectively. On the contrary, results from TRE show average inefficiency of close to 2.6% (calculations not shown). Further, results from the TRE model show that in fact it is quite implausible to expect inefficiency values higher than 8%. For US banks this is quite hard to believe, and goes against almost all the efficiency studies (e.g. Berger and Humphrey, 1997). Since TRE does not allow persistent inefficiency whereas GTRE does, it is expected that inefficiency (persistent and transient combined) from GTRE will be higher. In other words, TRE is likely to give low estimates of inefficiency because persistent inefficiency will be treated as bank effects—not inefficiency. It is, however, not clear whether results from the traditional SFM will be lower or higher compared to GTRE because the former model fails to take bank effects and persistent technical inefficiency into account. If these two effects are negatively correlated (as we find in these data), one might cancel the other and inefficiency results might go up or down depending on whether these timeinvariant effects are picked up by the inefficiency term or not.
One advantage of the GTRE model is that it identifies and estimates two sources of inefficiency. Inefficiency in the other two models is captured by the transient component (). Consequently, in comparing inefficiency across different models we used the overall measure (sum of persistent and transient components in GTRE model) of inefficiency. In Figure 5 we plot the posterior distributions of persistent (η^{+}, solid line) and transient (u^{+}, broken line) inefficiency components. The persistent component does not seem to exceed 4% and the average of both components is close to 3%. Persistent inefficiency in excess of about 6% is practically impossible, although the transient component has a long right tail that allows inefficiency values as large as 20%. The broken line in Figure 4 combines (as opposed to a naïve algebraic approach that would simply add up the means of inefficiency) the posterior probabilistic evidence for both u^{+} and η^{+} shown in Figure 5.
Bankspecific posterior distributions of the bank effects, α, for both TRE and GTRE models are shown in Figure 6. It can be seen from the figure that bank effects from GTRE are much larger (in both tails) compared to those in the TRE model. Since the TRE model does not allow persistent technical inefficiency, the resulting ‘pseudo bank effects in the TRE model' will capture the joint effects of persistent inefficiency and bank effects. If bank effects and persistent technical inefficiency are negatively correlated, the pseudo bank effects the TRE model might be more concentrated (as in Figure 6) and likely to underestimate the magnitude of ‘true’ bank effects.
Now we turn our attention to other groups. For brevity we report results from group 6 and group 10 banks.20 These groups are representative of medium and small banks. Furthermore, to conserve space we focus our attention to the TRE and GTRE models. The results are summarized in Figures 7 and 8. The results indicate increasing returns to scale (E_{cy} < 1) and technical progress (TC < 0) for most of the banks. Average persistent technical inefficiency is around 1.5% but highly likely to be as large as 3%. Since TRE does not allow persistent inefficiency, for a fair comparison of inefficiency between these two models we have to compare the overall inefficiency in the GTRE with transient inefficiency (u^{+}) in the TRE model. We find some differences in the distribution of overall inefficiency between the TRE and GTRE models for group 6 banks. The difference is very small for group 10 banks. Since the TRE model considers only transient inefficiency, it is likely that the overall inefficiency from TRE overestimates the probability of full efficiency.
Note that in Figures 7 and 8 we are comparing distributions of overall inefficiency from two models. Even if this difference is small it is not clear whether estimated inefficiency for each bank under these two models are close to each other. For this, we report scatterplots of posterior estimates of overall inefficiency from the GTRE and TRE models in Figure 9 (for group 6; for group 10 we obtained similar results). Whether the efficiency scores are approximately the same can be examined by looking at scatterplots of estimates of inefficiencies from TRE and GTRE models. If the scores are similar (identical) we would expect all pairs to lie on the 45° line (or close to it). However, the scatterplots reveal that the correlations are quite low and therefore classification of banks according to GTRE inefficiency scores are likely to be quite different from the scores obtained from the TRE models.