Consistent inference in fixed-effects stochastic frontier models

TLDR: In this paper, Chen et al. proposed two alternative estimators that achieve consistency for n → ∞ with fixed T and extended Chen et.al. (2014) results providing a feasible estimator when the inefficiency is heteroskedastic and follows a firstorder autoregressive process.

About: This article is published in Journal of Econometrics.The article was published on 2017-10-01 and is currently open access. It has received 142 citations till now. The article focuses on the topics: Estimator & Heteroscedasticity.

Figures

Table 9: Estimated inefficiency effects. Comparison between PDE and “Pooled” PDE. σ̄ = 1 N ∑n i=1 ∑Ti t=1 σ̂it. Significance levels: “*”:p < 10%;“**”:p < 5%, “***”:p < 1%.

Table 3: Simulation results for PDE and MLDVE with n = 250, T = 5. The Spearman rank correlation coefficient is in parentheses.

Table 8: Scale and output distance elasticities evaluated at the average values. Comparison between PDE and Pooled PDE. Y,X denotes return to scale estimated by ∑4 i=1 Y,Xi .

Table 6: Simulation results for the dynamic PDE. The Spearman rank correlation coefficient is in parentheses.

Table 1: Simulation results for PDE and MLDVE with n = 100, T = 5. The Spearman rank correlation coefficient is in parentheses.

Table 5: Simulation results for MMSLE and MMLE. The Spearman rank correlation coefficient is in parentheses.

Frequently asked questions

Q1. What are the contributions in this paper?

The classical stochastic frontier panel data models provide no mechanism for disentangling individual time-invariant unobserved heterogeneity from inefficiency. The authors propose two alternative estimation procedures that, by relying on a first-difference data transformation, achieve consistency when n goes to infinity with fixed T. Furthermore, they extend the approach of Chen et al. ( 2014 ) by providing a computationally feasible solution for estimating models in which inefficiency can be heteroskedastic and may follow a first-order autoregressive process. The authors investigate the finite sample behavior of the proposed estimators through a set of Monte Carlo experiments. 

Q2. How can the authors obtain an efficientestimator for i?

An efficientestimator for αi can be obtained by maximizing the log-likelihood function of the untransformed model where the other parameters are substituted by a consistent estimates. 

Q3. What is the first assumption needed to derive a valid draw?

The first assumption is needed to derive a valid draw to be used for the estimation and rulesout the possibility of using two-parameters distributions (e.g., truncated normal or gamma). 

Q4. How many sequencespers are needed to estimate the MMSLE?

Notice that allowing for heteroskedastic inefficiency makes the MMSLE far more computationally intensive (the rule of thumb is to use at least 50 sequencesper observation). 

Q5. What is the way to estimate the location/shape parameter?

While two-parameters flexible distributions have been widely and successfully applied in manystudies, the estimation of the location/shape parameter can be hard in small samples. 

Q6. How do the authors estimate the parameter vector?

in order to lower the computational burden, the authors propose to estimate the parameter vector θ = (β′,γ ′, δ′, ρ)′ by applying the pairwise estimation strategy to the marginal likelihoodfunction of a dynamic two-periods normal-half normal model. 

Q7. Why is the downward bias of due to the incidental parameter problem?

First of all, it is due to the incidentalparameters problem since the number of replications with non-zero ψ̂ increases with larger T ’sbut not with the cross-sectional dimension. 

Q8. What is the consistency property of i in the fixed-effects linear model?

This estimator is equivalent to the mean-adjusted estimator of αi in the fixed-effects linear model and, therefore, is consistent as T →∞. 

Q9. What are the types of hospitals that are assimilated to public structures?

Public hospitals include hospitals directly managed by Local Health Authorities, independent public hospitals (D.L. 502/92) and assimilated to public structures (L. 833/78).36 

Q10. What is the marginal likelihood function for a two-period panel?

The marginal likelihood function (17) implies the existence of H = ( T 2 ) consistent MMLEs, one for each “subsample” extracted considering two waves of the panel. 

Q11. What is the simplest way to evaluate a normal integral?

When T > 2, the evaluation of T -dimensional normal integrals, which makes problematic theextension of the Chen et al. (2014) approach to the heteroskedastic case, can be replaced by the approximation of ( T 2 ) 2-dimensional normal integrals whose evaluation has been shown to be accurate and computationally efficient (Genz, 2004). 

Q12. What is the resulting marginal likelihood function for a two-period panel?

by allowing the scale parameter of the inefficiency distribution to depend on a set of exogenous explanatory variables, σit = exp(zitγ), the resulting marginal likelihood function for a two-period panel isL∗(θ) = n∏ i=1 f(∆yit|θ,∆xit)= n∏ i=1 {∫ R f(∆yit|β, ψ,∆xit,∆uit)f(∆u|ς1, ς2)d∆uit }= n∏ i=1 {∫ R1(4πψ2)T/2 exp[ −12 ∆εit −∆uit 2ψ2] d∆uit }= n∏ i=1{ 1(4πψ2)T/2 [∫