Inference in dynamic stochastic frontier models

An important issue in models of technical efficiency measurement concerns the temporal behaviour of inefficiency. Consideration of dynamic models is necessary but inference in such models is complicated. In this paper we propose a stochastic frontier model that allows for technical inefficiency effects and dynamic technical inefficiency, and use Bayesian inference procedures organized around data augmentation techniques to provide inferences. Also provided are firm‐specific efficiency measures. The new methods are applied to a panel of large US commercial banks over the period 1989–2000. Copyright © 2006 John Wiley & Sons, Ltd.     

Bayesian estimation of inefficiency heterogeneity in stochastic frontier models

Estimation of the one sided error component in stochastic frontier models may erroneously attribute firm characteristics to inefficiency if heterogeneity is unaccounted for. However, unobserved inefficiency heterogeneity has been little explored. In this work, we propose to capture it through a random parameter which may affect the location, scale, or both parameters of a truncated normal inefficiency distribution using a Bayesian approach. Our findings using two real data sets, suggest that the inclusion of a random parameter in the inefficiency distribution is able to capture latent heterogeneity and can be used to validate the suitability of observed covariates to distinguish heterogeneity from inefficiency. Relevant effects are also found on separating and shrinking individual posterior efficiency distributions when heterogeneity affects the location and scale parameters of the one-sided error distribution, and consequently affecting the estimated mean efficiency scores and rankings. In particular, including heterogeneity simultaneously in both parameters of the inefficiency distribution in models that satisfy the scaling property leads to a decrease in the uncertainty around the mean scores and less overlapping of the posterior efficiency distributions, which provides both more reliable efficiency scores and rankings.     

Spatial stochastic frontier models: accounting for unobserved local determinants of inefficiency

This paper analyzes the productivity of farms across 370 municipalities in the Center-West region of Brazil. A stochastic frontier model with a latent spatial structure is proposed to account for possible unknown geographical variation of the outputs. The paper compares versions of the model that include the latent spatial effect in the mean of output or as a variable that conditions the distribution of inefficiency, include or not observed municipal variables, and specify independent normal or conditional autoregressive priors for the spatial effects. The Bayesian paradigm is used to estimate the proposed models. As the resultant posterior distributions do not have a closed form, stochastic simulation techniques are used to obtain samples from them. Two model comparison criteria provide support for including the latent spatial effects, even after considering covariates at the municipal level. Models that ignore the latent spatial effects produce significantly different rankings of inefficiencies across agents.

Semiparametric Bayesian inference for dynamic Tobit panel data models with unobserved heterogeneity

This paper develops semiparametric Bayesian methods for inference of dynamic Tobit panel data models. Our approach requires that the conditional mean dependence of the unobserved heterogeneity on the initial conditions and the strictly exogenous variables be specified. Important quantities of economic interest such as the average partial effect and average transition probabilities can be readily obtained as a by‐product of the Markov chain Monte Carlo run. We apply our method to study female labor supply using a panel data set from the National Longitudinal Survey of Youth 1979. Copyright © 2008 John Wiley & Sons, Ltd.     

Testing for unobserved heterogeneity in exponential and Weibull duration models

We examine the use of the likelihood ratio (LR) statistic to test for unobserved heterogeneity in duration models, based on mixtures of exponential or Weibull distributions. We consider both the uncensored and censored duration cases. The asymptotic null distribution of the LR test statistic is not the standard chi-square, as the standard regularity conditions do not hold. Instead, there is a nuisance parameter identified only under the alternative, and a null parameter value on the boundary of the parameter space, as in Cho and White (2007a). We accommodate these and provide methods delivering consistent asymptotic critical values. We conduct a number of Monte Carlo simulations, comparing the level and power of the LR test statistic to an information matrix (IM) test due to Chesher (1984) and Lagrange multiplier (LM) tests of Kiefer (1985) and Sharma (1987). Our simulations show that the LR test statistic generally outperforms the IM and LM tests. We also revisit the work of van den Berg and Ridder (1998) on unemployment durations and of Ghysels et al. (2004) on interarrival times between stock trades, and, as it turns out, affirm their original informal inferences.     

Modeling dependence in two-tier stochastic frontier models

The two-tier stochastic frontier model has seen widespread application across a range of social science domains. It is particularly useful in examining bilateral exchanges where unobserved side-specific information exists on both sides of the transaction. These buyer and seller specific informational aspects offer opportunities to extract surplus from the other side of the market, in combination also with uneven relative bargaining power. Currently, this model is hindered by the fact that identification and estimation relies on the potentially restrictive assumption that these factors are statistically independent. We present three different models for empirical application that allow for varying degrees of dependence across these latent informational/bargaining factors.

     

Consistent inference in fixed-effects stochastic frontier models

The classical stochastic frontier panel data models provide no mechanism to disentangle individual time invariant unobserved heterogeneity from inefficiency. Greene (2005a, b) proposed the so-called “true” fixed-effects specification that distinguishes these two latent components. However, due to the incidental parameters problem, his maximum likelihood estimator may lead to biased variance estimates. We propose two alternative estimators that achieve consistency for $n\to \infty$ with fixed $T$. Furthermore, we extend the Chen et al. (2014) results providing a feasible estimator when the inefficiency is heteroskedastic and follows a first-order autoregressive process. We investigate the behavior of the proposed estimators through Monte Carlo simulations showing good finite sample properties, especially in small samples. An application to hospitals’ technical efficiency illustrates the usefulness of the new approach.     

Technical efficiency and firm heterogeneity in stochastic frontier models: application to smallholder maize farms in Ethiopia

This study estimates the technical efficiency measures of maize producing farm households in Ethiopia using stochastic frontier (SF) panel models that take different approaches to model firm heterogeneity. The efficiency measures are found to vary depending on how the estimation model treats both unobserved and observed firm heterogeneity. Estimates from the ‘true’ random effects (TRE) models that treat firm effects as heterogeneity are found to be identical to those from pooled SF models. Those results differ from the ones generated from the basic random effects (RE) models that treat firm effects as part of overall technical inefficiency. The more flexible generalised ‘true’ random effects (GTRE) model that splits the error term into firm effects, persistent inefficiency, transient inefficiency, and a random noise component indicates the presence of higher levels of persistent inefficiency than transient inefficiency. The basic truncated-normal RE model and heteroscedastic RE model yields similar efficiency estimates. The GTRE model predict persistent efficiency measures similar to those from the basic RE and flexible RE model with environmental variables incorporated in the variance function as well as in the deterministic production frontier. These results imply that the RE and GTRE panel models provide reliable efficiency estimates for our data compared to the TRE models. All the estimated SF models generate comparable production function parameters in terms of magnitude and sign. Overall, the results underscore the importance of scrutinising stochastic frontier models for their reliability of analytical results before drawing policy inferences.

     

Censored regression models with unobserved,stochastic censoring thresholds

The ‘Tobit’ model is a useful tool for estimation of regression models with truncated or limited dependent variables, but it requires a threshold which is either a known constant or an observable and independent variable. The model presented here extends the Tobit model to the censored case where the threshold is an unobserved and not necessarily independent random variable. Maximum likelihood procedures can be employed for joint estimation of both the primary regression equation and the parameters of the distribution of that random threshold.     

Simple solutions to the initial conditions problem in dynamic,nonlinear panel data models with unobserved heterogeneity

I study a simple, widely applicable approach to handling the initial conditions problem in dynamic, nonlinear unobserved effects models. Rather than attempting to obtain the joint distribution of all outcomes of the endogenous variables, I propose finding the distribution conditional on the initial value (and the observed history of strictly exogenous explanatory variables). The approach is flexible, and results in simple estimation strategies for at least three leading dynamic, nonlinear models: probit, Tobit and Poisson regression. I treat the general problem of estimating average partial effects, and show that simple estimators exist for important special cases. Copyright © 2005 John Wiley & Sons, Ltd.     

Goodness of fit tests in stochastic frontier models

In this paper we discuss goodness of fit tests for the distribution of technical inefficiency in stochastic frontier models. If we maintain the hypothesis that the assumed normal distribution for statistical noise is correct, the assumed distribution for technical inefficiency is testable. We show that a goodness of fit test can be based on the distribution of estimated technical efficiency, or equivalently on the distribution of the composed error term. We consider both the Pearson χ ² test and the Kolmogorov–Smirnov test. We provide simulation results to show the extent to which the tests are reliable in finite samples.     

Identification in nonparametric limited dependent variable models with simultaneity and unobserved heterogeneity

We extend the identification results for nonparametric simultaneous equations models in Matzkin (2008) to situations where the observations on the vector of dependent variables might be limited, and where the number of exogenous unobservable variables is larger than the number of dependent variables.     

Testing for time-invariant unobserved heterogeneity in generalized linear models for panel data

Recent literature on panel data emphasizes the importance of accounting for time-varying unobservable individual effects, which may stem from either omitted individual characteristics or macro-level shocks that affect each individual unit differently. In this paper, we propose a simple specification test of the null hypothesis that the individual effects are time-invariant against the alternative that they are time-varying. Our test is an application of Hausman (1978) testing procedure and can be used for any generalized linear model for panel data that admits a sufficient statistic for the individual effect. This is a wide class of models which includes the Gaussian linear model and a variety of nonlinear models typically employed for discrete or categorical outcomes. The basic idea of the test is to compare two alternative estimators of the model parameters based on two different formulations of the conditional maximum likelihood method. Our approach does not require assumptions on the distribution of unobserved heterogeneity, nor it requires the latter to be independent of the regressors in the model. We investigate the finite sample properties of the test through a set of Monte Carlo experiments. Our results show that the test performs well, with small size distortions and good power properties. We use a health economics example based on data from the Health and Retirement Study to illustrate the proposed test.     

Unknown latent structure and inefficiency in panel stochastic frontier models

Journal of Productivity Analysis - This paper extends the fixed effect panel stochastic frontier models to allow group heterogeneity in the slope coefficients. We propose the first-difference...     

Simple tests of alternative specifications in stochastic frontier models

Previous work on stochastic production frontiers has generated a family of models, of varying degrees of complexity. Since this family is nested (in the sense that the more general models contain the less general), we can test the restrictions that distinguish the model. In this paper we provide tests of these restrictions, based on the results of estimating the simpler (restricted) models. Some of our tests are LM tests. However, in other cases the LM test fails, so we provide alternative simple tests.     

Formulation and estimation of stochastic frontier production function models

Previous studies of the so-called frontier production function have not utilized an adequate characterization of the disturbance term for such a model. In this paper we provide an appropriate specification, by defining the disturbance term as the sum of symmetric normal and (negative) half-normal random variables. Various aspects of maximum-likelihood estimation for the coefficients of a production function with an additive disturbance term of this sort are then considered.     

Nonparametric least squares methods for stochastic frontier models

When analyzing productivity and efficiency of firms, stochastic frontier models are very attractive because they allow, as in typical regression models, to introduce some noise in the Data Generating Process . Most of the approaches so far have been using very restrictive fully parametric specified models, both for the frontier function and for the components of the stochastic terms. Recently, local MLE approaches were introduced to relax these parametric hypotheses. In this work we show that most of the benefits of the local MLE approach can be obtained with less assumptions and involving much easier, faster and numerically more robust computations, by using nonparametric least-squares methods. Our approach can also be viewed as a semi-parametric generalization of the so-called “modified OLS” that was introduced in the parametric setup. If the final evaluation of individual efficiencies requires, as in the local MLE approach, the local specification of the distributions of noise and inefficiencies, it is shown that a lot can be learned on the production process without such specifications. Even elasticities of the mean inefficiency can be analyzed with unspecified noise distribution and a general class of local one-parameter scale family for inefficiencies. This allows to discuss the variation in inefficiency levels with respect to explanatory variables with minimal assumptions on the Data Generating Process.     

A note on the assumed distributions in stochastic frontier models

Stochastic frontier models all need an assumption on the distributional form of the (in)efficiency component. Generally this efficiency component is assumed to be half normally, truncated normally, or exponentially distributed. This paper shows that the exponential distribution is, just like the half normal distribution, a special case of the truncated normal distribution. Moreover, this paper discusses the implications that this finding has on estimation.     

Estimation and testing of stochastic frontier models using variational Bayes

We show how a wide range of stochastic frontier models can be estimated relatively easily using variational Bayes. We derive approximate posterior distributions and point estimates for parameters and inefficiency effects for (a) time invariant models with several alternative inefficiency distributions, (b) models with time varying effects, (c) models incorporating environmental effects, and (d) models with more flexible forms for the regression function and error terms. Despite the abundance of stochastic frontier models, there have been few attempts to test the various models against each other, probably due to the difficulty of performing such tests. One advantage of the variational Bayes approximation is that it facilitates the computation of marginal likelihoods that can be used to compare models. We apply this idea to test stochastic frontier models with different inefficiency distributions. Estimation and testing is illustrated using three examples.