Model averaging estimators for the stochastic frontier model

In this paper, we propose the model averaging estimation method for multiplicative error model and construct the corresponding weight choosing criterion based on the Kullback–Leibler divergence with a hyperparameter to avoid the problem of overfitting. The resulting model average estimator is proved to be asymptotically optimal. It is shown that the Kullback–Leibler model averaging (KLMA) estimator asymptotically minimizes the in‐sample Kullback–Leibler divergence and improves the forecast accuracy of out‐of‐sample even under different loss functions. In simulations, we show that the KLMA estimator compares favorably with smooth‐AIC estimator (SAIC), smooth‐BIC estimator (SBIC), and Mallows model averaging estimator (MMA), especially when some nonlinear noise is added to the data generation process. The empirical applications in the daily range of S&P500 and price duration of IBM show that the out‐of‐sample forecasting capacity of the KLMA estimator is better than that of other methods.

Bayesian Model Averaging and Weighted Average Least Squares: Equivariance, Stability, and Numerical Issues

Analysis of the behavior of technical inefficiency with respect to parameters and variables of a stochastic frontier model is a neglected area of research in frontier literature. An attempt in this direction, however, has recently been made. It has been shown that in a “standard” stochastic frontier model that both the firm level technical inefficiency and the production uncertainty are monotonically decreasing with observational error. In this paper we show, considering a stochastic frontier model whose error components are jointly distributed as truncated bivariate normal, that this property holds if and only if the distribution of observational error is negatively skewed. We also derive a necessary and sufficient condition under which both firm level technical inefficiency and production uncertainty are monotonically increasing with noise-inefficiency correlation. We next propose a new measure of the industry level production uncertainty and establish the necessary and sufficient condition for firm level technical inefficiency and production uncertainty to be monotonically increasing with industry level production uncertainty. We also study the limiting probabilistic behavior of these conditions under different parametric configuration of our model. Finally we carry out Monte Carlo simulations to study the sample behavior of the population monotonic property of the firm level technical inefficiency and production uncertainty in our model.

In statistical exercises where there are several candidate models, the traditional approach is to select one model using some data driven criterion and use that model for estimation, testing and other purposes, ignoring the variability of the model selection process. We discuss some problems associated with this approach. An alternative scheme is to use a model-averaged estimator, that is, a weighted average of estimators obtained under different models, as an estimator of a parameter. We show that the risk associated with a Bayesian model-averaged estimator is bounded as a function of the sample size, when parameter values are fixed. We establish conditions which ensure that a model-averaged estimator's distribution can be consistently approximated using the bootstrap. A new, data-adaptive, model averaging scheme is proposed that balances efficiency of estimation without compromising applicability of the bootstrap. This paper illustrates that certain desirable risk and resampling properties of model-averaged estimators are obtainable when parameters are fixed but unknown; this complements several studies on minimaxity and other properties of post-model-selected and model-averaged estimators, where parameters are allowed to vary.

The traditional Stochastic Frontier Model (SFM) suffers from a very restrictive assumption of independence of its error components and also limited ability to address heterogeneity (inefficiency effects) satisfactorily, thereby leading to potential biases in the estimation of model parameters, identification of inefficiency effect variables influencing efficiency and, ultimately, efficiency scores. This paper aims to investigate the consequences of ignoring any dependency in error components and heterogeneity in the stochastic frontier model, and proposes a copula-based SFM with heterogeneity to resolve such weaknesses based on a simulation study to prove its superiority over the traditional SFM, followed by an empirical application on a sample of rice producers from northern Thailand. We demonstrate that the proposed model, i.e., copula-based SFM with dependent error components and heterogeneity, is unbiased and robust. The simulation experiments show that the traditional SFM can cause biases in parameter estimation and severe overestimation of technical efficiency. The traditional SFM with heterogeneity also has similar consequences. However, just ignoring heterogeneity does not have a great impact on parameter estimation and technical efficiency compared to the consequence of ignoring dependency in error components. The empirical application of the proposed model results show that land, labor and material inputs are all significant drivers of rice production in our copula-based SFM with dependent error components and heterogeneity, whereas in the traditional SFM model only the land variable seems to be a significant driver of rice production. The mean technical efficiency (MTE) score was also overestimated by two points in the traditional SFM, i.e., MTE = 0.88 versus 0.86. Finally, results of the copula-based SFM with dependent error components and heterogeneity reveals that both subsistence pressure and the use of hired labor are significantly associated with technical inefficiency, whereas the traditional SFM could identify the effect of hired labor use only. Therefore, caution is necessary when interpreting results from the conventional SFM as the results may be biased, incomplete and/or inadequate.

Heterogeneity among firms has been an important issue in studying firms’ technical efficiencies. If firms do not randomly fall into different groups with different technologies but by self-selection, statistically it implies the data are subject to the sample selection bias. In this paper, we generalize the stochastic frontier (SF) model to accommodate heterogeneous technologies among firms by considering the threshold SF model with an endogenous threshold variable. We discuss the econometric techniques appropriate for the threshold SF model with panel data. To determine the optimal number of regimes, we use modified the model selection criteria of Gonzalo and Pitarakis (J Econom 110(2):319–352, 2002) and investigate their finite sample performance by some Monte Carlo experiments. Finally, we also demonstrate our approach by an empirical example.

Group-specific stochastic production frontier models with parametric specifications

In this paper, we focused on the new stochastic frontier model with the nonlinear autoregressive structure in panel data. In classic finance models, uncorrelated errors are considered, which are frequently not met in empirical situations. In our stochastic frontier model, the composite error consists of two components, the statistical error and technical inefficiency, whereas the technical inefficiency is assumed to be autocorrelated. A semiparametric method is suggested to estimate nonlinear autoregressive function by two steps procedure as parametric Taylor series expansion and nonparametric adjustment factor. For the model parameters estimation, the Expectation-Maximization approach is applied and the performance of the estimation is checked by Monte Carlo simulation. Finally, we investigate the applicability of new model using a real data set.

Specification tests for normal/gamma and stable/gamma stochastic frontier models based on empirical transforms

Heckman’s (Ann Econ Soc Meas 4(5), 475–492, 1976; Econometrica 47, 153–161, 1979) sample selection model has been employed in three decades of applications of linear regression studies. This paper builds on this framework to obtain a sample selection correction for the stochastic frontier model. We first show a surprisingly simple way to estimate the familiar normal-half normal stochastic frontier model using maximum simulated likelihood. We then extend the technique to a stochastic frontier model with sample selection. In an application that seems superficially obvious, the method is used to revisit the World Health Organization data (WHO in The World Health Report, WHO, Geneva 2000; Tandon et al. in Measuring the overall health system performance for 191 countries, World Health Organization, 2000) where the sample partitioning is based on OECD membership. The original study pooled all 191 countries. The OECD members appear to be discretely different from the rest of the sample. We examine the difference in a sample selection framework.

Observed and Unobserved Heterogeneity in Stochastic Frontier Models: An Application to the Electricity Distribution Industry

This letter considers the following two questions: How much are the estimates of the parameters of stochastic cost frontier and cost inefficiency of Japanese hospitals affected (1) by the utilization of the panel structure of data in estimation; and (2) by considering a hospital-specific intercept in the cost frontier? Three models are estimated: a stochastic cost frontier model with a hospital-specific intercept treated as a random variable for panel data; a stochastic cost frontier model without a hospital-specific intercept for panel data; and the conventional stochastic frontier model that treats the data as if they were cross-sectional. The main findings are: (1) the introduction of panel structure into the frontier estimation changes the cost inefficiency estimate significantly compared with the estimate based on cross-sectionally treated data; (2) the intercept of the cost frontier is significantly different among hospitals.

Recently, a skew-normal based stochastic frontier model has emerged as a promising tool for efficiency analysis. This paper introduces a Bayesian framework for statistical inference, integrating both informative and non-informative prior knowledge to estimate parameters of skew-normal distributions in stochastic frontier models. Through comprehensive evaluation using both simulation data and real data from a manufacturing productivity study, we demonstrate that the Bayesian approach provides more stable and accurate parameter estimates compared to the conventional maximum likelihood method. The results from both simulated and empirical analyses clearly highlight the superior performance of the Bayesian methodology, offering enhanced robustness and precision in estimating efficiency scores, thus contributing significantly to the advancement of stochastic frontier modeling.

The first chapter studies a varying coefficient panel data model with different smoothing variables and fixed effects using a two-step estimation approach. The first-step pilot estimator is constructed by ap- approximating the varying coefficients with B-spline functions. The pilot estimator is then used to perform a one-step backfitting to obtain the second-step efficient estimator with kernel local linear estimation. The second-step estimator is efficient in the sense of being equivalent to a procedure knowing the other components of the varying coefficient. The asymptotic properties of both the pilot and efficient estimators are obtained. A Monte Carlo simulation indicates that the proposed estimator performs well with finite sample size. The second chapter investigates the production efficiency of the U.S. coal-burning power plants that are covered under the Acid Rain Program from 2001 to 2005. To account for the influence of the Acid Rain Program on technical efficiency and the impact of environment variables on the marginal product of inputs, I introduce a panel data stochastic production frontier model with fixed effects and varying coefficients. The varying coefficients capture the relationship between power plant characteristics and the marginal return on inputs. Besides, the proposed model allows the inefficiency to depend on the environment variables, such as environmental policies. This set up enables the identification of the existence and magnitude of inefficiency. The proposed model relaxes the functional form assumption on coefficient functions and distribution assumption on the error terms. The empirical results reveal that the Acid Rain Program improves production efficiency for some power plants based on their operation and maintenance costs on sulfur removal equipment per unit of generating capacity. In addition, the marginal return on capital is affected by the years of operation of a power plant. The third chapter studies the performance of water utilities in China. Water shortage is of great concern in China. Improving water delivery efficiency is the direct and efficient method to alleviate the water crisis. This paper performs an efficiency analysis of 56 water utilities during the period from 2009 to 2013. A semiparametric stochastic production frontier model with smooth coefficient is introduced to evaluate the magnitude of technical inefficiency and to examine the impact of institutional and operational condition, such as customer density, non-household user rate, non-revenue water ratio, average water pressure in pipes, and the percentage of internal staff among all the employees. The discussion focuses on the impact of the internal

In this article, we describe the estimation of linear regression models with uncertainty about the choice of the explanatory variables. We introduce the Stata commands bma and wals, which implement, respectively, the exact Bayesian model-averaging estimator and the weighted-average least-squares estimator developed by Magnus, Powell, and Pr¨ufer (2010, Journal of Econometrics 154: 139–153). Unlike standard pretest estimators that are based on some preliminary diagnostic test, these model-averaging estimators provide a coherent way of making inference on the regression parameters of interest by taking into account the uncertainty due to both the estimation and the model selection steps. Special emphasis is given to several practical issues that users are likely to face in applied work: equivariance to certain transformations of the explanatory variables, stability, accuracy, computing speed, and out-of-memory problems. Performances of our bma and wals commands are illustrated using simulated data and empirical applications from the literature on model-averaging estimation.