Nonparametric least squares estimation in derivative families

Cost function estimation often involves data on a function and a family of its derivatives. Such data can substantially improve convergence rates of nonparametric estimators. We propose series-type estimators which incorporate the various derivative data into a single nonparametric least-squares procedure. Convergence rates are obtained and it is shown that for low-dimensional cases, much of the beneficial impact is realized even if only data on ordinary first-order partials are available. In instances where root-n$n$ consistency is attained, smoothing parameters can often be chosen very easily, without resort to cross-validation. Simulations and an illustration of cost function estimation are included.     

A smoothed least squares estimator for threshold regression models

We propose a smoothed least squares estimator of the parameters of a threshold regression model. Our model generalizes that considered in Hansen [2000. Sample splitting and threshold estimation. Econometrica 68, 575–603] to allow the thresholding to depend on a linear index of observed regressors, thus allowing discrete variables to enter. We also do not assume that the threshold effect is vanishingly small. Our estimator is shown to be consistent and asymptotically normal thus facilitating standard inference techniques based on estimated standard errors or standard bootstrap for the slope and threshold parameters.     

Confidence statements for efficiency estimates from stochastic frontier models

This paper is an empirical study of the uncertainty associated with technical efficiency estimates from stochastic frontier models. We show how to construct confidence intervals for estimates of technical efficiency levels under different sets of assumptions ranging from the very strong to the relatively weak. We demonstrate empirically how the degree of uncertainty associated with these estimates relates to the strength of the assumptions made and to various features of the data.     

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.

     

Nonparametric regression function estimation using interaction least squares splines and comlexity regularization

Let (X, Y) be a pair of random variables withsupp(X)⊆[0,1] ^l andEY ²<∞. Letm ^* be the best approximation of the regression function of (X, Y) by sums of functions of at mostd variables (1≤d≤l). Estimation ofm ^* from i.i.d. data is considered. For the estimation interaction least squares splines, which are defined as sums of polynomial tensor product splines of at mostd variables, are used. The knot sequences of the tensor product splines are chosen equidistant. Complexity regularization is used to choose the number of the knots and the degree of the splines automatically using only the given data. Without any additional condition on the distribution of (X, Y) the weak and strongL ₂-consistency of the estimate is shown. Furthermore, for everyp≥1 and every distribution of (X, Y) withsupp(X)⊆[0,1] ^l ,y bounded andm ^* p-smooth, the integrated squared error of the estimate achieves up to a logarithmic factor the (optimal) rate      

Likelihood ratio tests for model selection of stochastic frontier models

An important issue when conducting stochastic frontier analysis is how to choose a proper parametric model, which includes choices of the functional form of the frontier function, distributions of the composite errors, and also the exogenous variables. In this paper, we extend the likelihood ratio test of Vuong, Econometrica 57(2):307–333, (1989) and Takeuchi’s, Suri-Kagaku (Math Sci) 153:12–18, (1976) model selection criterion to the stochastic frontier models. The most attractive feature of this test is that it can not only be used for testing a non-nested model, but also still be applicable even when the general model is misspecified. Finally, we also demonstrate how to apply this test to the Indian farm data used by Battese and Coelli, J Prod Anal 3:153–169, (1992), Empir Econ 20(2):325–332, (1995) and Alvarez et al., J Prod Anal 25:201–212, (2006).     

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.     

Estimation and inference in unstable nonlinear least squares models

There is compelling evidence that many macroeconomic and financial variables are not generated by linear models. This evidence is based on testing linearity against either smooth nonlinearity or piece-wise linearity, but there is no framework that encompasses both. This paper provides an econometric framework that allows for both breaks and smooth nonlinearity in between breaks. We estimate the unknown break-dates simultaneously with other parameters via nonlinear least-squares. Using new central limit results for nonlinear processes, we provide inference methods on break-dates and parameter estimates and several instability tests. We illustrate our methods via simulated and empirical smooth transition models with breaks.     

Partially generalized least squares and two-stage least squares estimators

A class of partially generalized least squares estimators and a class of partially generalized two-stage least squares estimators in regression models with heteroscedastic errors are proposed. By using these estimators a researcher can attain higher efficiency than that attained by the least squares or the two-stage least squares estimators without explicitly estimating each component of the heteroscedastic variances. However, the efficiency is not as high as that of the generalized least squares or the generalized two-stage least squares estimator calculated using the knowledge of the true variances. Hence the use of the term partial.     

A Bayesian estimator for stochastic frontier models with errors in variables

A Bayesian estimator is proposed for a stochastic frontier model with errors in variables. The model assumes a truncated-normal distribution for the inefficiency and accommodates exogenous determinants of inefficiency. An empirical example of Tobin??s Q investment model is provided, in which the Q variable is known to suffer from measurement error. Results show that correcting for measurement error in the Q variable has an important effect on the estimation results.     

On maximum likelihood estimation of stochastic frontier production models

The iterative algorithm suggested by Greene (1982) for the estimation of stochastic frontier production models does not necessarily solve the likelihood equations. Corrected iterative algorithms which generalize Fair's method (1977) and solve the likelihood equations are derived. These algorithms are compared with the Newton method in an empirical case. The Newton method is more time saving than these algorithms.     

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.     

Estimation of stochastic frontier models based on multimodel inference

In most empirical studies, once the best model has been selected according to a certain criterion, subsequent analysis is conducted conditionally on the chosen model. In other words, the uncertainty of model selection is ignored once the best model has been chosen. However, the true data-generating process is in general unknown and may not be consistent with the chosen model. In the analysis of productivity and technical efficiencies in the stochastic frontier settings, if the estimated parameters or the predicted efficiencies differ across competing models, then it is risky to base the prediction on the selected model. Buckland et al. (Biometrics 53:603?C618, 1997) have shown that if model selection uncertainty is ignored, the precision of the estimate is likely to be overestimated, the estimated confidence intervals of the parameters are often below the nominal level, and consequently, the prediction may be less accurate than expected. In this paper, we suggest using the model-averaged estimator based on the multimodel inference to estimate stochastic frontier models. The potential advantages of the proposed approach are twofold: incorporating the model selection uncertainty into statistical inference; reducing the model selection bias and variance of the frontier and technical efficiency estimators. The approach is demonstrated empirically via the estimation of an Indian farm data set.     

Two-step two-stage least squares estimation in models with rational expectations

The paper develops a two-step estimator for use in rational-expectations models with autocorrelated residuals and predetermined, but not strictly exogenous, instruments. The estimator extends the applicability of McCallum's (1976) error-in-variablesapproach to estimating such models, and is asymptotically efficient in a class of intrumental-variables estimators. As an application we use instrumental-variables techniques to estimate Taylor's (1979) rational-expectations macroeconomic model of the United States.     

On the efficiency of least squares estimators in non-linear models

Summary The identity of least squares estimators å and maximum likelihood estimators â is studied in non-linear models of the type z=g(a), where z are observable quantities with a probability density function pr(z). This identity was proved for independent random variables z and for distributions pr(z), of which the arithmetic sample mean is an optimal estimate.     

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.

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.     

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.     

Understanding prediction intervals for firm specific inefficiency scores from parametric stochastic frontier models

This paper makes two important contributions to the literature on prediction intervals for firm specific inefficiency estimates in cross sectional SFA models. Firstly, the existing intervals in the literature do not correspond to the minimum width intervals and in this paper we discuss how to compute such intervals and how they either include or exclude zero as a lower bound depending on where the probability mass of the distribution of \( u_{i} |\varepsilon_{i} \) resides. This has useful implications for practitioners and policy makers, with greatest reductions in interval width for the most efficient firms. Secondly, we propose an ‘asymptotic’ approach to incorporating parameter uncertainty into prediction intervals for firm specific inefficiency (given that in practice model parameters have to be estimated) as an alternative to the ‘bagging’ procedure suggested in Simar and Wilson (Econom Rev 29(1):62–98, 2010). The approach is computationally much simpler than the bagging approach.