Frequentist and Bayesian stochastic frontier models in Stata Federico - - PowerPoint PPT Presentation

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Frequentist and Bayesian stochastic frontier models in Stata

Federico Belotti† Silvio Daidone† Giuseppe Ilardi‡

†Università di Roma Tor Vergata, ‡Bank of Italy

Florence, November 19th, 2009

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Summary

1

Introduction

2

Frequentist Estimation

3

Bayesian inference

4

STATA commands

5

Empirical application

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Objectives of the paper

This paper focuses on stochastic frontier models for both cross-section and longitudinal data with a parametric approach to estimation Novel features: the newly available STATA command will be the first bayesian estimator of frontier parameters be comprehensive of most used and state-of-art frequentist estimators make extensive use of MATA functions

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

General framework -1-

• Starting from seminal study by Aigner, Lovell and Schmidt (1977),

theoretical literature on stochastic frontier has grown vastly.

• The range of applications of the techniques described is huge.
• The economic meaning of a frontier is to represent the best-practice

technology in a production process or in a particular economic sector.

• Cost frontiers describe the minimum level of cost given a certain output

level and certain input prices.

• Production frontiers represent the maximum amount of output that can

be obtained from a given level of inputs.

• The gap between the actual and the maximum output is a measure of

inefficiency and an important issue in many application fields, such as production studies.

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

General framework -2-

• A general stochastic frontier model may be written as

yi = x

′

iβ +ui +vi

(1) where yi is the performance of firm i (output, profits, costs), β is the vector of technology parameters, vi is the classical symmetric disturb, while ui is the inefficiency.

• As well as the functional assumption on the form of the frontier, we must

make some assumptions on the distribution and on the relations between the two errors in order to complete the statistical model.

• The typical assumptions in this model are

1

The independence between v e u.

2

vi ∼ N(0,σ2).

3

ui ∼ F, where F(x) is a generic family of distributions with x ∈ R+

• Objectives: in the first step we estimate the vector of technology

parameters β and in the second the efficiency of each producer.

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Cross-section -1-

In a cross-sectional setting, we present two different models: the normal-truncated normal and the normal-gamma. The former one is based

• n the following set of assumptions

vi ∼ N (0,σ2

vi )

ui ∼ N +(µit,σ2

ui )

µi = qitφ σ2

vi = exp(wiδi)

σ2

ui = exp(tiγi)

The log-likelihood function for i = 1,...,N firms is lnL = −1 2 ∑

i

ln[exp(wiδi)+exp(tiγi)]−N lnΦ

• − µi

σu

• + ∑

i

lnΦ µi σiλ − εiλi σi

• − 1

2 ∑

i

εi + µi σi 2 (2)

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Cross-section -2-

In the normal-gamma model ui ∼ iidΓ(m). This formulation introduced and developed by Greene generalizes the one-parameter exponential distribution. The corresponding log-likelihood function can be written as the likelihood for the normal-exponential model plus a term which has complicated the analysis to date lnL = N

• σ2

v

2σ2

u

• +∑

i

εi σu +∑

i

lnΦ

• −(εi +σ2

v /σu)

σv

• +

N [(m +1)lnσu −lnΓ(m +1)]+∑

i

lnh(m,εi) = lnLEXP +N [(m +1)lnσu −lnΓ(m +1)]+∑

i

lnh(m,εi) (3) where ∑i lnh(m,εi) = E[zr|z ≥ 0] and z ∼ N [µi,σ2

v ]

We estimate h(m,εi) by using the mean of a sample of draws from a normal distribution with underlying mean µi and variance σ2

v truncated at zero.

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Cross-section -3-

After technology parameters, the second step is to obtain an estimate of

• efficiency. For the truncated normal model we get both Jondrow, Lovell,

Materov and Schmidt (1982) and Battese and Coelli (1988) estimators of technical efficiency, respectively TEi = exp(−E{ui|εi}) (4) TEi = E(exp{−ui}|εi) (5) Bera and Sharma (1996) provide the formulas to get confidence intervals for these point estimators. While for the gamma model we numerically approximate the following expression E (ui|εi) = h(m +1,εi) h(m,εi) (6) where m is the shape parameter of the gamma distribution

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Panel -1-

Panel data estimation has received great coverage in the literature. Access to panel data enables one to avoid either strong distributional assumptions or the equally strong independence assumption. Latest developments in research community try to disentangle pure inefficiency from what is to be considered unobserved heterogeneity. Here we show the Greene (2005) “true” random effect model, the newest random effects formulations.

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Panel -2-

In its “true” random effects formulation Greene (2005) extends the conventional maximum likelihood estimation of random effects models yit = α +β ′xit +wi +vit ±uit (7) where wi is the random firm specific effect and vit and uit are the symmetric and one sided components. It is necessary to integrate the common term out of the likelihood function in order to estimate this random effects model by maximum likelihood. Since there is no closed form for the density of the compound disturbance in this model, we integrate and simulate the log-likelihood lnLS(β,λ,σ,ϑ) =

N

∑

i=1

ln 1 R

R

∑

r=1

• T

∏

t=1

2 σ φ εit|wir σ

• Φ

λεit|wir σ

• (8)

where ϑi are the parameters in the distribution of wi and wir is the r-th simulated draw for observation i.

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Historical notes on Bayesian estimation

The Bayesian inference in this context was proposed by van den Broeck et al. (1994). In this work, the authors computed Bayes factors between a series of parametric models. Koop et al. (1997) developed Bayesian inferential procedures to be applied to panel data, distinguishing between fixed and random effects models. There is only one existing work (Griffin and Steel (2004, JoE)) which adopts the semiparametric Bayesian inference. In this work, we consider two distributions: (i) an exponential and (ii) a flexible gamma (not just an Erlang) for the vector of inefficiencies u

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Priors -1-

In order to build a Bayesian regression model, we have to define a set of priors on the unknown vector of parameters η η η = (β β β,σ2,ν,λ). We assume the following prior structure π(η η η) = π(β β β,σ2,ν,λ) = π(β β β|σ2)π(σ2)π(ν)π(λ) where all distributions on the right-hand side will be proper, ensuring us to have a proper posterior distribution. In the exponential case π(ν) = 1.

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Priors -2-

Prior on β β β π(β β β|σ2) ∼ Nk(β0,σ2W) where β0 = 0 and W = d0Ik. The tuning of the hyperparameter d0 does not represent a critical point and, as reference value, we set d0 = 104. Moreover the choice of a different reasonable large value for the d0 should not produce a significative effect on the posterior inference. Prior on σ2 Analogously to the previous case, we elicit the variance with the most common informative solution: an Inverse Gamma prior π(σ2) ∼ IG(a0/2,b0/2). In panel data model (Fernandez et al., JoE 1997), we can relax this choice and use a non informative priors on (β β β,σ2).

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Priors -3-

Prior on ν and λ In these two cases we choose a Gamma distribution as a prior. In particular for λ −1, if we define efficiency as ri = exp(−ui), and adopt the prior distribution π(λ −1|φ) = Ga(φ,−ln(r∗)), then r∗ is the implied prior median efficiency. We can fix φ = 1 or we shall complete the prior for the general gamma inefficiency distribution by φ ∼ Ga(1,1) which is centered through the prior mean over the value leading to the exponential distribution, and has a reasonable prior variance for φ of unity.

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Likelihood

Since the joint density of y = (yi,...,yn) and u = (u1,...,unn) is given by f(y,u) =

n

∏

1=1

1 √ 2πσ2 exp

• (yi −x

′

i β −ui)2

2σ2

• ×

× λ −ν Γ(ν) ·uν−1

i

exp

• −ui

λ

• (9)

After marginalizing over u the relation (9) the likelihood function can be expressed as L(η η η|y) ∝

n

∏

1=1

λ −ν Γ(ν) exp

• σ2

2λ 2 +λ −1(yi −x

′

i β))

• ×

×

+∞

uν−1

i

1 √ 2πσ2 exp

• (yi −mi)2

2σ2

• dui,

(10) where mi = yi −x

′

i β −λ −1 ·σ2.

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Posterior distribution

The posterior distribution is proportional to the product of the priors π(η η η) and the likelihood π(y|η η η), i.e. π(η η η|y) ∝

n

∏

1=1

λ −ν Γ(ν) exp

• σ2

2λ 2 +λ −1(yi −h(β,xi))

• ×

×

+∞

uν−1

i

1 √ 2πσ2 exp

• (yi −mi)2

2σ2

• dui

× π(β β β|σ2)π(σ2)π(ν)π(λ). The posterior is analytically intractable We construct a Markov chain defined by conditional distributions of parameters. In this Markov chain, a Gibbs sampler, the random draws are made from each full-conditional posterior distribution. we apply a data augmentation scheme (Tanner and Wong 1987) to our model treating the latent random vector u as an unknown parameter vector to be estimated.

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

We provide four new Stata commands: s❢❝r♦ss and s❢♣❛♥❡❧ fit frequentist cross-sectional and panel stochastic frontier models, improving already existing commands ❢r♦♥t✐❡r and ①t❢r♦♥t✐❡r. ❜s❢❝r♦ss and ❜s❢♣❛♥❡❧ fit bayesian cross-sectional and panel stochastic frontier models. They are the first bayesian estimators within Stata which do not make use of WinBugs interface and the first general purpose bayesian estimators of stochastic frontier models.

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

The general syntax of these commands is as follows s❢❝r♦ss ❞❡♣✈❛r [✐♥❞❡♣✈❛rs] [✐❢] [✐♥] [,♦♣t✐♦♥s] s❢♣❛♥❡❧ ❞❡♣✈❛r [✐♥❞❡♣✈❛rs] [✐❢] [✐♥] [,♦♣t✐♦♥s] ❜s❢❝r♦ss ❞❡♣✈❛r [✐♥❞❡♣✈❛rs] [✐❢] [✐♥] [,♦♣t✐♦♥s] ❜s❢♣❛♥❡❧ ❞❡♣✈❛r [✐♥❞❡♣✈❛rs] [✐❢] [✐♥] [,♦♣t✐♦♥s]

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

We use Italian hospitals’ data coming from Lazio region. From the Lazio Public Health Agency we got Hospital Discharge Records that were used to build output measures. From the Italian ministry of Health we received input variables such as number of beds, physicians, etc. We limit our analysis to Acute care hospitals, since rehabilitation care and long-term care serve very different production functions Public and not-for-profit hospitals. While for private hospitals we study

• nly their activity which is publicly financed.

Years between 2000 and 2005, which represents an interesting period to assess the effect of DRG system. Overall we have a weakly balanced panel of 625 observations

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

✳ ❞ ❵▼❛✐♥❱❛r✐❛❜❧❡s✬ st♦r❛❣❡ ❞✐s♣❧❛② ✈❛❧✉❡ ✈❛r✐❛❜❧❡ ♥❛♠❡ t②♣❡ ❢♦r♠❛t ❧❛❜❡❧ ✈❛r✐❛❜❧❡ ❧❛❜❡❧ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❧♥♦r♠❴✇❡✐❣❤t❛ ❢❧♦❛t ✪✾✳✵❣ ❙✉♠ ♦❢ ❉❘● ✇❡✐❣❤ts ✐♥ ❛❝✉t❡ ❝❛r❡ ❛❧♣❤❛✶ ❢❧♦❛t ✪✾✳✵❣ ★ ❜❡❞s ❛❧♣❤❛✷ ❢❧♦❛t ✪✾✳✵❣ ★ ♣❤②s✐❝✐❛♥s ❛❧♣❤❛✸ ❢❧♦❛t ✪✾✳✵❣ ★ ♥✉rs❡s ❛❧♣❤❛✹ ❢❧♦❛t ✪✾✳✵❣ ★ ♦t❤❡r ✇♦r❦❡rs ❛❧♣❤❛✶✶ ❢❧♦❛t ✪✾✳✵❣ ❙q✉❛r❡❞ ★ ❜❡❞s ❛❧♣❤❛✷✷ ❢❧♦❛t ✪✾✳✵❣ ❙q✉❛r❡❞ ★ ♣❤②s✐❝✐❛♥s ❛❧♣❤❛✸✸ ❢❧♦❛t ✪✾✳✵❣ ❙q✉❛r❡❞ ★ ♥✉rs❡s ❛❧♣❤❛✹✹ ❢❧♦❛t ✪✾✳✵❣ ❙q✉❛r❡❞ ★ ♦t❤❡r ✇♦r❦❡rs ❛❧♣❤❛✶✷ ❢❧♦❛t ✪✾✳✵❣ ■♥t❡r❛❝t✐♦♥ ★ ❜❡❞s ✲ ★ ♣❤②s✐❝✐❛♥s ❛❧♣❤❛✶✸ ❢❧♦❛t ✪✾✳✵❣ ■♥t❡r❛❝t✐♦♥ ★ ❜❡❞s ✲ ★ ♥✉rs❡s ❛❧♣❤❛✶✹ ❢❧♦❛t ✪✾✳✵❣ ■♥t❡r❛❝t✐♦♥ ★ ❜❡❞s ✲ ★ ♦t❤❡r ✇♦r❦❡rs ❛❧♣❤❛✷✸ ❢❧♦❛t ✪✾✳✵❣ ■♥t❡r❛❝t✐♦♥ ★ ♣❤②s✐❝✐❛♥s ✲ ★ ♥✉rs❡s ❛❧♣❤❛✷✹ ❢❧♦❛t ✪✾✳✵❣ ■♥t❡r❛❝t✐♦♥ ★ ♣❤②s✐❝✐❛♥s ✲ ★ ♦t❤❡r ✇♦r❦❡rs ❛❧♣❤❛✸✹ ❢❧♦❛t ✪✾✳✵❣ ■♥t❡r❛❝t✐♦♥ ★ ♥✉rs❡s ✲ ★ ♦t❤❡r ✇♦r❦❡rs ❞②❡❛r✷✵✵✶ ❜②t❡ ✪✾✳✵❣ ❚✐♠❡ ❞✉♠♠②✿ ✷✵✵✶ ❞②❡❛r✷✵✵✷ ❜②t❡ ✪✾✳✵❣ ❚✐♠❡ ❞✉♠♠②✿ ✷✵✵✷ ❞②❡❛r✷✵✵✸ ❜②t❡ ✪✾✳✵❣ ❚✐♠❡ ❞✉♠♠②✿ ✷✵✵✸ ❞②❡❛r✷✵✵✹ ❜②t❡ ✪✾✳✵❣ ❚✐♠❡ ❞✉♠♠②✿ ✷✵✵✹ ❞②❡❛r✷✵✵✺ ❜②t❡ ✪✾✳✵❣ ❚✐♠❡ ❞✉♠♠②✿ ✷✵✵✺

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Frequentist paradigm: Cross-section truncated-normal model

Conditional mean model with explanatory variables for idiosyncratic error variance function ★❞❡❧✐♠✐t❀ ✳ s❢❝r♦ss ✩❨ ✩❳❧✐♥ ✩❳sq ✩❳✐♥t ✩❞❨✱ ❞✭t♥✮ ♠✉✭♣r✐✈❛t❡✶ ❡q✉✐♣✱ ♥♦❝♦♥s✮ ✈✭❧♥♦r♠❴❜❡❞s✮ t❡❝❤♥✐q✉❡✭♥r✮ ♥♦❧♦❣❀ ★❞❡❧✐♠✐t ❝r ❚r✉♥❝❛t❡❞✲♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ ✉ ◆✉♠❜❡r ♦❢ ♦❜s ❂ ✻✷✺ ❲❛❧❞ ❝❤✐✷✭✶✾✮ ❂ ✹✼✶✹✳✹✺ ▲♦❣✲❧✐❦❡❧✐❤♦♦❞ ❂ ✲✷✾✸✳✸✻✶✻ Pr♦❜ ❃ ❝❤✐✷ ❂ ✵✳✵✵✵✵ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❧♥♦r♠❴✇❡✐⑦t❛ ⑤ ❈♦❡❢✳ ❙t❞✳ ❊rr✳ ③ P❃⑤③⑤ ❬✾✺✪ ❈♦♥❢✳ ■♥t❡r✈❛❧❪ ✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❋r♦♥t✐❡r ⑤ ❛❧♣❤❛✶ ⑤ ✳✼✺✽✾✼✶✽ ✳✵✹✻✷✺✸✽ ✶✻✳✹✶ ✵✳✵✵✵ ✳✻✻✽✸✶✻✷ ✳✽✹✾✻✷✼✺ ❛❧♣❤❛✷ ⑤ ✳✷✹✻✹✷✶✼ ✳✵✹✷✾✾✺✺ ✺✳✼✸ ✵✳✵✵✵ ✳✶✻✷✶✺✷✶ ✳✸✸✵✻✾✶✹ ❛❧♣❤❛✸ ⑤ ✳✵✶✸✶✽✻✻ ✳✵✺✵✾✹✺✷ ✵✳✷✻ ✵✳✼✾✻ ✲✳✵✽✻✻✻✹✷ ✳✶✶✸✵✸✼✹ ❛❧♣❤❛✹ ⑤ ✲✳✵✷✽✶✹✹✹ ✳✵✹✶✶✺✽✶ ✲✵✳✻✽ ✵✳✹✾✹ ✲✳✶✵✽✽✶✷✽ ✳✵✺✷✺✷✹✶ ❛❧♣❤❛✶✶ ⑤ ✳✷✽✺✺✹✸✸ ✳✵✻✽✾✹✺✸ ✹✳✶✹ ✵✳✵✵✵ ✳✶✺✵✹✶✸ ✳✹✷✵✻✼✸✺ ❛❧♣❤❛✷✷ ⑤ ✳✶✶✹✺✶✸✼ ✳✵✸✽✺✾✹✹ ✷✳✾✼ ✵✳✵✵✸ ✳✵✸✽✽✼✵✶ ✳✶✾✵✶✺✼✹ ❛❧♣❤❛✸✸ ⑤ ✳✵✻✻✶✵✷ ✳✵✺✼✷✺✷✾ ✶✳✶✺ ✵✳✷✹✽ ✲✳✵✹✻✶✶✶✻ ✳✶✼✽✸✶✺✺ ❛❧♣❤❛✹✹ ⑤ ✲✳✵✸✶✼✺✺✾ ✳✵✷✽✽✶✽✾ ✲✶✳✶✵ ✵✳✷✼✵ ✲✳✵✽✽✷✸✾✽ ✳✵✷✹✼✷✽ ❛❧♣❤❛✶✷ ⑤ ✲✳✶✸✽✼✹✽✺ ✳✵✻✺✺✽✶✺ ✲✷✳✶✷ ✵✳✵✸✹ ✲✳✷✻✼✷✽✺✽ ✲✳✵✶✵✷✶✶✷

(Continued on next page)

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Frequentist paradigm: Cross-section truncated-normal model

Conditional mean model with explanatory variables for idiosyncratic error variance function

❛❧♣❤❛✶✸ ⑤ ✳✵✼✶✽✺✷✽ ✳✵✻✵✺✵✵✷ ✶✳✶✾ ✵✳✷✸✺ ✲✳✵✹✻✼✷✺✹ ✳✶✾✵✹✸✶✶ ❛❧♣❤❛✶✹ ⑤ ✲✳✶✵✵✼✻✶✶ ✳✵✹✷✶✷✺✺ ✲✷✳✸✾ ✵✳✵✶✼ ✲✳✶✽✸✸✷✺✺ ✲✳✵✶✽✶✾✻✻ ❛❧♣❤❛✷✸ ⑤ ✲✳✵✹✾✶✵✻✾ ✳✵✸✺✷✵✾✸ ✲✶✳✸✾ ✵✳✶✻✸ ✲✳✶✶✽✶✶✺✾ ✳✵✶✾✾✵✷✶ ❛❧♣❤❛✷✹ ⑤ ✳✵✽✵✽✷✽ ✳✵✹✾✸✼✸✾ ✶✳✻✹ ✵✳✶✵✷ ✲✳✵✶✺✾✹✸✶ ✳✶✼✼✺✾✾✷ ❛❧♣❤❛✸✹ ⑤ ✲✳✵✻✸✾✶✺ ✳✵✹✷✶✽✹✻ ✲✶✳✺✷ ✵✳✶✸✵ ✲✳✶✹✻✺✾✺✹ ✳✵✶✽✼✻✺✸ ❞②❡❛r✷✵✵✶ ⑤ ✳✵✹✺✺✽✷✸ ✳✵✹✼✺✷✸✼ ✵✳✾✻ ✵✳✸✸✼ ✲✳✵✹✼✺✻✷✺ ✳✶✸✽✼✷✼ ❞②❡❛r✷✵✵✷ ⑤ ✳✶✵✺✸✶✼✼ ✳✵✹✼✷✶✷✾ ✷✳✷✸ ✵✳✵✷✻ ✳✵✶✷✼✽✷✷ ✳✶✾✼✽✺✸✷ ❞②❡❛r✷✵✵✸ ⑤ ✳✶✽✸✽✽✾✽ ✳✵✹✼✹✶✺✽ ✸✳✽✽ ✵✳✵✵✵ ✳✵✾✵✾✺✻✺ ✳✷✼✻✽✷✸ ❞②❡❛r✷✵✵✹ ⑤ ✳✷✸✸✾✵✺✽ ✳✵✹✽✹✶✹✼ ✹✳✽✸ ✵✳✵✵✵ ✳✶✸✾✵✶✹✼ ✳✸✷✽✼✾✼ ❞②❡❛r✷✵✵✺ ⑤ ✳✸✸✺✹✶✹✸ ✳✵✺✵✸✽✼✸ ✻✳✻✻ ✵✳✵✵✵ ✳✷✸✻✻✺✼✶ ✳✹✸✹✶✼✶✺ ❴❝♦♥s ⑤ ✳✺✵✼✶✼✻✷ ✳✵✸✽✻✹✸✽ ✶✸✳✶✷ ✵✳✵✵✵ ✳✹✸✶✹✸✺✼ ✳✺✽✷✾✶✻✻ ✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ▼❯ ⑤ ♣r✐✈❛t❡✶ ⑤ ✳✵✷✹✵✻✼✻ ✳✶✸✵✸✻✹✺ ✵✳✶✽ ✵✳✽✺✹ ✲✳✷✸✶✹✹✷✶ ✳✷✼✾✺✼✼✸ ❡q✉✐♣ ⑤ ✲✷✳✸✺✺✶✻✸ ✳✻✺✹✽✾✻✷ ✲✸✳✻✵ ✵✳✵✵✵ ✲✸✳✻✸✽✼✸✻ ✲✶✳✵✼✶✺✾ ✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❯s✐❣♠❛ ⑤ ❴❝♦♥s ⑤ ✲✶✳✵✺✸✺✸✸ ✳✶✶✼✵✸✺✸ ✲✾✳✵✵ ✵✳✵✵✵ ✲✶✳✷✽✷✾✶✽ ✲✳✽✷✹✶✹✼✾ ✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❱s✐❣♠❛ ⑤ ❧♥♦r♠❴❜❡❞s ⑤ ✲✳✺✶✻✽✹✹✼ ✳✵✾✻✺✼✶✾ ✲✺✳✸✺ ✵✳✵✵✵ ✲✳✼✵✻✶✷✷✷ ✲✳✸✷✼✺✻✼✶ ❴❝♦♥s ⑤ ✲✷✳✽✶✸✽✸✸ ✳✶✷✷✶✵✼✼ ✲✷✸✳✵✹ ✵✳✵✵✵ ✲✸✳✵✺✸✶✻ ✲✷✳✺✼✹✺✵✻ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❍✵✿ ◆♦ ✐♥❡❢❢✐❝✐❡♥❝② ❝♦♠♣♦♥❡♥t✿ ③ ❂ ✲✸✻✳✺✾✸ Pr♦❜❁❂③ ❂ ✵✳✵✵✵ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Frequentist paradigm: Cross-section gamma model

s❢❝r♦ss ✩❨ ✩❳❧✐♥ ✩❳sq ✩❳✐♥t ✩❞❨✱ ❞✭❣✮ ♥s✐♠✭✶✵✵✮ s✐♠t②♣❡✭✸✮ ❜❛s❡✭✼✮ t❡❝❤♥✐q✉❡✭❜❤❤❤✮

• ❛♠♠❛ ❞✐str✐❜✉t✐♦♥ ♦❢ ✉

◆✉♠❜❡r ♦❢ ♦❜s ❂ ✻✷✺ ❲❛❧❞ ❝❤✐✷✭✶✾✮ ❂ ✻✽✾✸✳✼✽ ❙✐♠✉❧❛t❡❞ ▲♦❣✲❧✐❦❡❧✐❤♦♦❞ ❂ ✲✷✸✺✳✸✸✸✻ Pr♦❜ ❃ ❝❤✐✷ ❂ ✵✳✵✵✵✵ ◆✉♠❜❡r ♦❢ ❘❛♥❞♦♠✐③❡❞ ❍❛❧t♦♥ ❙❡q✉❡♥❝❡s ❂ ✶✵✵ ❇❛s❡ ❢♦r ❘❛♥❞♦♠✐③❡❞ ❍❛❧t♦♥ ❙❡q✉❡♥❝❡s ❂ ✼ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❧♥♦r♠❴✇❡✐⑦t❛ ⑤ ❈♦❡❢✳ ❙t❞✳ ❊rr✳ ③ P❃⑤③⑤ ❬✾✺✪ ❈♦♥❢✳ ■♥t❡r✈❛❧❪ ✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❋r♦♥t✐❡r ⑤ ❛❧♣❤❛✶ ⑤ ✳✽✸✺✷✼✽✻ ✳✵✸✾✸✶✺✷ ✷✶✳✷✺ ✵✳✵✵✵ ✳✼✺✽✷✷✷✷ ✳✾✶✷✸✸✺ ❛❧♣❤❛✷ ⑤ ✳✷✷✾✸✹✵✸ ✳✵✸✺✷✾✵✶ ✻✳✺✵ ✵✳✵✵✵ ✳✶✻✵✶✼✸ ✳✷✾✽✺✵✼✻ ❛❧♣❤❛✸ ⑤ ✲✳✵✺✵✶✼✽✾ ✳✵✸✹✷✷✽ ✲✶✳✹✼ ✵✳✶✹✸ ✲✳✶✶✼✷✻✹✹ ✳✵✶✻✾✵✻✼ ❛❧♣❤❛✹ ⑤ ✳✵✶✶✽✹✷ ✳✵✸✷✵✷✻✽ ✵✳✸✼ ✵✳✼✶✷ ✲✳✵✺✵✾✷✾✹ ✳✵✼✹✻✶✸✹ ❛❧♣❤❛✶✶ ⑤ ✳✸✶✼✺✽✶✷ ✳✵✺✶✻✾✸✼ ✻✳✶✹ ✵✳✵✵✵ ✳✷✶✻✷✻✸✺ ✳✹✶✽✽✾✽✾ ❛❧♣❤❛✷✷ ⑤ ✳✶✸✸✻✻✾ ✳✵✷✽✺✹✻✶ ✹✳✻✽ ✵✳✵✵✵ ✳✵✼✼✼✶✾✼ ✳✶✽✾✻✶✽✸ ❛❧♣❤❛✸✸ ⑤ ✲✳✵✷✸✶✸✹✾ ✳✵✸✶✺✽✽✼ ✲✵✳✼✸ ✵✳✹✻✹ ✲✳✵✽✺✵✹✼✻ ✳✵✸✽✼✼✼✽ ❛❧♣❤❛✹✹ ⑤ ✲✳✵✷✼✷✷✷✺ ✳✵✶✾✺✵✻✸ ✲✶✳✹✵ ✵✳✶✻✸ ✲✳✵✻✺✹✺✹✷ ✳✵✶✶✵✵✾✸

(Continued on next page)

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Frequentist paradigm: Cross-section gamma model

❛❧♣❤❛✶✷ ⑤ ✲✳✶✾✹✸✺✸✺ ✳✵✹✺✷✷✽✶ ✲✹✳✸✵ ✵✳✵✵✵ ✲✳✷✽✷✾✾✽✽ ✲✳✶✵✺✼✵✽✶ ❛❧♣❤❛✶✸ ⑤ ✳✶✸✺✶✽✽✶ ✳✵✸✼✽✶✼✽ ✸✳✺✼ ✵✳✵✵✵ ✳✵✻✶✵✻✻✺ ✳✷✵✾✸✵✾✻ ❛❧♣❤❛✶✹ ⑤ ✲✳✶✶✷✶✾✻✺ ✳✵✸✵✽✼✹ ✲✸✳✻✸ ✵✳✵✵✵ ✲✳✶✼✷✼✵✽✺ ✲✳✵✺✶✻✽✹✻ ❛❧♣❤❛✷✸ ⑤ ✲✳✵✶✻✶✷✾ ✳✵✷✶✹✽✺✺ ✲✵✳✼✺ ✵✳✹✺✸ ✲✳✵✺✽✷✸✾✽ ✳✵✷✺✾✽✶✾ ❛❧♣❤❛✷✹ ⑤ ✳✵✼✽✺✵✷✾ ✳✵✸✷✺✹✸✷ ✷✳✹✶ ✵✳✵✶✻ ✳✵✶✹✼✶✾✹ ✳✶✹✷✷✽✻✹ ❛❧♣❤❛✸✹ ⑤ ✲✳✵✺✾✽✶✻✸ ✳✵✷✻✷✹✽✽ ✲✷✳✷✽ ✵✳✵✷✸ ✲✳✶✶✶✷✻✸ ✲✳✵✵✽✸✻✾✼ ❞②❡❛r✷✵✵✶ ⑤ ✳✵✹✽✷✹✻✻ ✳✵✹✹✶✸✹ ✶✳✵✾ ✵✳✷✼✹ ✲✳✵✸✽✷✺✹✹ ✳✶✸✹✼✹✼✼ ❞②❡❛r✷✵✵✷ ⑤ ✳✶✷✹✹✷✻✷ ✳✵✹✹✸✵✼✶ ✷✳✽✶ ✵✳✵✵✺ ✳✵✸✼✺✽✺✽ ✳✷✶✶✷✻✻✺ ❞②❡❛r✷✵✵✸ ⑤ ✳✶✽✶✹✼✸✹ ✳✵✹✸✽✾✶✻ ✹✳✶✸ ✵✳✵✵✵ ✳✵✾✺✹✹✼✹ ✳✷✻✼✹✾✾✸ ❞②❡❛r✷✵✵✹ ⑤ ✳✷✺✶✼✾✵✾ ✳✵✹✹✷✺✼ ✺✳✻✾ ✵✳✵✵✵ ✳✶✻✺✵✹✽✾ ✳✸✸✽✺✸✸ ❞②❡❛r✷✵✵✺ ⑤ ✳✸✹✷✸✾✷✽ ✳✵✹✺✺✽✵✹ ✼✳✺✶ ✵✳✵✵✵ ✳✷✺✸✵✺✻✽ ✳✹✸✶✼✷✽✽ ❴❝♦♥s ⑤ ✳✸✵✻✸✽✺✻ ✳✵✸✸✾✵✺✸ ✾✳✵✹ ✵✳✵✵✵ ✳✷✸✾✾✸✷✸ ✳✸✼✷✽✸✽✽ ✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❚❤❡t❛ ⑤ t❤❡t❛ ⑤ ✷✳✹✷✺✽✻✸ ✳✷✷✶✸✸✹✾ ✶✵✳✾✻ ✵✳✵✵✵ ✶✳✾✾✷✵✺✺ ✷✳✽✺✾✻✼✷ ✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❱s✐❣♠❛✷ ⑤ s✐❣♠❛✷✈ ⑤ ✳✵✻✶✹✾✽✹ ✳✵✵✺✵✺✹✸ ✶✷✳✶✼ ✵✳✵✵✵ ✳✵✺✶✺✾✷✶ ✳✵✼✶✹✵✹✼ ✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❙❤❛♣❡ ⑤ ♠ ⑤ ✳✺✵✷✽✺✶✺ ✳✵✹✷✼✶✵✷ ✶✶✳✼✼ ✵✳✵✵✵ ✳✹✶✾✶✹✶✶ ✳✺✽✻✺✻✶✾ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❍✵✿ ◆♦ ✐♥❡❢❢✐❝✐❡♥❝② ❝♦♠♣♦♥❡♥t✿ ③ ❂ ✲✸✻✳✺✾✸ Pr♦❜❁❂③ ❂ ✵✳✵✵✵ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Gamma vs Truncated-normal JLMS technical efficiency estimates

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Truncated-normal technical efficiency estimates: JLMS vs BC estimator

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Bayesian paradigm: Cross-section exponential model

✳ ❜s❢❝r♦ss ✩❨ ✩❳❧✐♥ ✩❳sq ✩❳✐♥t ✩❞❨✱ ❞✭❡①♣✮ ✐t❡r❛t✐♦♥✭✷✵✵✵✮ ❜✉r♥✐♥✭✷✵✵✮ t❤✐♥✭✷✮ ♣r❡❞✭✹✮ ❇❛②❡s✐❛♥ ❙t♦❝❤❛st✐❝ ❢r♦♥t✐❡r ✲ ❊①♣♦♥❡♥t✐❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ ✉ Pr✐♦r ❤②♣❡r♣❛r❛♠❡t❡rs✿ ❙✐❣♠❛✷✲✲❃ ❛✿ ✶ ❜✿ ✶ ▲❛♠❜❞❛✲✲❃ ❛✿ ✶ ❜✿ ✳✷✷✸✶✹✸✻ ❙❡tt✐♥❣s✿ ■t❡r❛t✐♦♥s✿ ✷✵✵✵ ❇✉r♥✐♥✿ ✷✵✵ ❚❤✐♥♥✐♥❣✿ ✷ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❧♥♦r♠❴✇❡✐❣❤t❛ ⑤ ▼❡❛♥ ❙t❞✳❉❡✈✳ ⑤ ♣✷✺ ▼❡❞✐❛♥ ♣✼✺ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✰ ❛❧♣❤❛✶ ⑤ ✳✸✽✻✵✹✾✸ ✳✵✸✻✽✸✽✹ ⑤ ✳✸✻✶✹✸✹✶ ✳✸✽✺✹✹✺✼ ✳✹✶✵✸✼✼✼ ❛❧♣❤❛✷ ⑤ ✳✽✸✷✷✷✶ ✳✵✸✾✾✵✸✶ ⑤ ✳✽✵✹✼✻✾✾ ✳✽✸✶✾✶✾✷ ✳✽✺✾✻✹✽ ❛❧♣❤❛✸ ⑤ ✳✷✸✷✼✸✻ ✳✵✸✼✸✼✼✻ ⑤ ✳✷✵✻✷✷✼ ✳✷✸✸✵✻✺✶ ✳✷✺✽✺✼✷✸ ❛❧♣❤❛✹ ⑤ ✲✳✵✹✷✵✺✷✼ ✳✵✸✾✽✷✹✽ ⑤ ✲✳✵✻✽✽✽✶✺ ✲✳✵✹✷✵✺✹✹ ✲✳✵✶✹✺✻✹✺

(Continued on next page)

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Bayesian paradigm: Cross-section exponential model

❛❧♣❤❛✶✶ ⑤ ✳✵✵✹✼✸✵✼ ✳✵✸✻✽✷✵✼ ⑤ ✲✳✵✷✵✺✸✽✹ ✳✵✵✹✺✹✻✽ ✳✵✷✾✺✸✺✹ ❛❧♣❤❛✷✷ ⑤ ✳✸✷✼✻✹ ✳✵✺✼✻✶✽✹ ⑤ ✳✷✽✽✼✻✻✺ ✳✸✷✽✶✸✾✺ ✳✸✻✹✶✽✸✶ ❛❧♣❤❛✸✸ ⑤ ✳✶✸✻✼✵✷✸ ✳✵✸✹✽✻✺✼ ⑤ ✳✶✶✸✶✽✻✾ ✳✶✸✻✸✶✻✻ ✳✶✻✵✺✶✾✹ ❛❧♣❤❛✹✹ ⑤ ✲✳✵✶✵✻✾✽✹ ✳✵✹✻✼✷✼ ⑤ ✲✳✵✹✵✼✼✽✼ ✲✳✵✶✶✶✷✹✷ ✳✵✶✾✶✾✹✺ ❛❧♣❤❛✶✷ ⑤ ✲✳✵✷✶✺✾✷✶ ✳✵✷✺✼✺✺✼ ⑤ ✲✳✵✸✾✶✻✶✼ ✲✳✵✷✸✷✷✽✻ ✲✳✵✵✺✼✶✹✺ ❛❧♣❤❛✶✸ ⑤ ✲✳✶✾✻✺✽✺✹ ✳✵✺✹✷✵✵✹ ⑤ ✲✳✷✸✸✻✺✼✽ ✲✳✶✾✻✸✽✹✺ ✲✳✶✺✽✽✽ ❛❧♣❤❛✶✹ ⑤ ✳✶✸✹✻✵✺✶ ✳✵✺✵✽✼✻✾ ⑤ ✳✶✵✶✶✶✻✸ ✳✶✸✺✵✶✶✷ ✳✶✻✽✾✾✹ ❛❧♣❤❛✷✸ ⑤ ✲✳✶✶✹✾✺✸ ✳✵✸✽✼✺✽✽ ⑤ ✲✳✶✹✵✷✹✻✺ ✲✳✶✶✹✹✷✷✺ ✲✳✵✽✽✶✶✵✽ ❛❧♣❤❛✷✹ ⑤ ✲✳✵✷✵✼✵✻✶ ✳✵✸✸✵✻✷✼ ⑤ ✲✳✵✹✷✽✻✶ ✲✳✵✷✶✵✹✵✶ ✳✵✵✶✷✵✻✺ ❛❧♣❤❛✸✹ ⑤ ✳✵✽✵✷✼✼✼ ✳✵✹✹✶✷✾✻ ⑤ ✳✵✺✵✷✸✼✹ ✳✵✼✾✷✹✶✶ ✳✶✶✵✼✶✶✻ ❞②❡❛r✷✵✵✶ ⑤ ✲✳✵✻✻✵✼✶✾ ✳✵✸✺✾✽✾✷ ⑤ ✲✳✵✽✾✾✹✶✼ ✲✳✵✻✺✽✵✽✽ ✲✳✵✹✶✷✷✻✸ ❞②❡❛r✷✵✵✷ ⑤ ✳✵✺✵✵✻✷ ✳✵✹✼✺✷✻✽ ⑤ ✳✵✶✼✾✶✶✼ ✳✵✺✷✶✵✺✹ ✳✵✽✶✻✺✸✺ ❞②❡❛r✷✵✵✸ ⑤ ✳✶✷✷✽✻✹✾ ✳✵✹✻✾✹✵✼ ⑤ ✳✵✾✶✾✸✸✺ ✳✶✷✸✹✺✹✸ ✳✶✺✸✾✻✵✽ ❞②❡❛r✷✵✵✹ ⑤ ✳✶✽✺✵✸✾✷ ✳✵✹✼✸✼✼✾ ⑤ ✳✶✺✷✽✻✹✼ ✳✶✽✺✻✷✽✺ ✳✷✶✼✺✸✸✼ ❞②❡❛r✷✵✵✺ ⑤ ✳✷✺✻✶✸✵✸ ✳✵✹✻✼✵✷✷ ⑤ ✳✷✷✹✵✶✶✺ ✳✷✺✻✷✻✶✸ ✳✷✽✽✺✽✺✻ ❴❝♦♥s ⑤ ✳✸✺✷✾✹✼✸ ✳✵✹✼✽✽✷✻ ⑤ ✳✸✷✶✽✶✺✷ ✳✸✺✶✾✼✹✶ ✳✸✽✹✻✺✹✽ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ s✐❣♠❛✷ ⑤ ✳✵✻✸✽✽✺✺ ✳✵✵✺✽✸✻✶ ⑤ ✳✵✺✾✼✼✾✶ ✳✵✻✸✽✵✹ ✳✵✻✼✼✺✶✷ ❧❛♠❜❞❛ ⑤ ✸✳✹✸✷✶✹✸ ✳✷✷✹✼✼✽✺ ⑤ ✸✳✷✽✼✻✵✶ ✸✳✹✷✺✾✸✺ ✸✳✺✼✻✻✼✹ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Parameters’ simulation

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Bayesian cross-section estimate of technical efficiency: “mean” JLMS

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Frequentist paradigm: “True” RE model

★❞❡❧✐♠✐t❀ s❢♣❛♥❡❧ ✩❨ ✩❳❧✐♥ ✩❳sq ✩❳✐♥t ✩❞❨✱ ♠♦❞❡❧✭tr❡✮ ✐❞✭✐r❝❴✐❞✮ t✐♠❡✭②❡❛r✮ s✐♠t②♣❡✭✸✮ ❜❛s❡✭✸✼✮ ♥s✐♠✭✶✵✮ t❡❝❤♥✐q✉❡✭❞❢♣✮ ♥♦❧♦❣❀ ★❞❡❧✐♠✐t ❝r ❚r✉❡ ❘❛♥❞♦♠ ❊❢❢❡❝ts ♠♦❞❡❧ ✭❍❛❧❢✲◆♦r♠❛❧✮ ◆✉♠❜❡r ♦❢ ♦❜s ❂ ✻✷✺

• r♦✉♣ ✈❛r✐❛❜❧❡✿ ✐r❝❴✐❞

◆✉♠❜❡r ♦❢ ❣r♦✉♣s ❂ ✶✶✸ ❖❜s ♣❡r ❣r♦✉♣✿ ♠✐♥ ❂ ✶ ❛✈❣ ❂ ✺✳✺ ♠❛① ❂ ✻ ❙✐♠✉❧❛t❡❞ ▲♦❣✲❧✐❦❡❧✐❤♦♦❞ ❂ ✲✸✷✼✳✼✸✼✵ ◆✉♠❜❡r ♦❢ ❘❛♥❞♦♠✐③❡❞ ❍❛❧t♦♥ ❙❡q✉❡♥❝❡s ❂ ✶✵ ❇❛s❡ ❢♦r ❘❛♥❞♦♠✐③❡❞ ❍❛❧t♦♥ ❙❡q✉❡♥❝❡s ❂ ✸✼ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ⑤ ❙t❛♥❞❛r❞ ❧♥♦r♠❴✇❡✐⑦t❛ ⑤ ❈♦❡❢✳ ❙t❞✳ ❊rr✳ t P❃⑤t⑤ ❬✾✺✪ ❈♦♥❢✳ ■♥t❡r✈❛❧❪ ✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❋r♦♥t✐❡r ⑤ ❛❧♣❤❛✶ ⑤ ✳✼✼✵✹✸✺ ✳✵✹✾✺✽✶✷ ✶✺✳✺✹ ✵✳✵✵✵ ✳✻✼✸✵✶✽✹ ✳✽✻✼✽✺✶✺ ❛❧♣❤❛✷ ⑤ ✳✷✸✼✷✼✵✼ ✳✵✹✺✺✶✶✹ ✺✳✷✶ ✵✳✵✵✵ ✳✶✹✼✽✺✵✹ ✳✸✷✻✻✾✶ ❛❧♣❤❛✸ ⑤ ✲✳✵✷✼✻✽✹✻ ✳✵✹✺✻✼✼ ✲✵✳✻✶ ✵✳✺✹✺ ✲✳✶✶✼✹✸✵✶ ✳✵✻✷✵✻✵✾ ❛❧♣❤❛✹ ⑤ ✳✵✸✾✸✷✷✼ ✳✵✹✹✽✾✻✽ ✵✳✽✽ ✵✳✸✽✷ ✲✳✵✹✽✽✾ ✳✶✷✼✺✸✺✺ ❛❧♣❤❛✶✶ ⑤ ✳✸✸✼✵✽✻✼ ✳✵✻✶✶✷✻✶ ✺✳✺✶ ✵✳✵✵✵ ✳✷✶✻✾✽✻✽ ✳✹✺✼✶✽✻✺

(Continued on next page)

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Frequentist paradigm: “True” RE model

❛❧♣❤❛✷✷ ⑤ ✳✶✷✺✷✶✼✻ ✳✵✸✾✹✵✽ ✸✳✶✽ ✵✳✵✵✷ ✳✵✹✼✼✽✾✸ ✳✷✵✷✻✹✺✾ ❛❧♣❤❛✸✸ ⑤ ✳✵✺✷✵✸✸✾ ✳✵✺✹✸✾✶✽ ✵✳✾✻ ✵✳✸✸✾ ✲✳✵✺✹✽✸✹✹ ✳✶✺✽✾✵✷✶ ❛❧♣❤❛✹✹ ⑤ ✲✳✵✶✾✻✵✸✺ ✳✵✸✶✸✷✹ ✲✵✳✻✸ ✵✳✺✸✷ ✲✳✵✽✶✶✹✽✻ ✳✵✹✶✾✹✶✺ ❛❧♣❤❛✶✷ ⑤ ✲✳✶✽✾✷✼✾✺ ✳✵✻✵✶✽✸✻ ✲✸✳✶✺ ✵✳✵✵✷ ✲✳✸✵✼✺✷✼✺ ✲✳✵✼✶✵✸✶✺ ❛❧♣❤❛✶✸ ⑤ ✳✶✷✽✸✸✶✼ ✳✵✺✷✼✶✼✸ ✷✳✹✸ ✵✳✵✶✺ ✳✵✷✹✼✺✸✹ ✳✷✸✶✾✵✾✾ ❛❧♣❤❛✶✹ ⑤ ✲✳✶✷✷✾✶✸✶ ✳✵✹✶✻✹✹✹ ✲✷✳✾✺ ✵✳✵✵✸ ✲✳✷✵✹✼✸✺✺ ✲✳✵✹✶✵✾✵✻ ❛❧♣❤❛✷✸ ⑤ ✲✳✵✺✻✵✺✺✾ ✳✵✸✹✻✾✾✹ ✲✶✳✻✷ ✵✳✶✵✼ ✲✳✶✷✹✷✸✷✾ ✳✵✶✷✶✷✶✶ ❛❧♣❤❛✷✹ ⑤ ✳✶✶✷✹✷✺ ✳✵✹✺✾✽✹✾ ✷✳✹✹ ✵✳✵✶✺ ✳✵✷✷✵✼✹✹ ✳✷✵✷✼✼✺✻ ❛❧♣❤❛✸✹ ⑤ ✲✳✵✽✺✺✾✶✷ ✳✵✸✾✹✾✺✺ ✲✷✳✶✼ ✵✳✵✸✶ ✲✳✶✻✸✶✾✶✺ ✲✳✵✵✼✾✾✵✽ ❞②❡❛r✷✵✵✶ ⑤ ✳✵✺✺✻✷✽✸ ✳✵✺✷✵✻✻✻ ✶✳✵✼ ✵✳✷✽✻ ✲✳✵✹✻✻✼✶✺ ✳✶✺✼✾✷✽ ❞②❡❛r✷✵✵✷ ⑤ ✳✶✷✸✺✻✻✸ ✳✵✺✶✺✵✻✼ ✷✳✹✵ ✵✳✵✶✼ ✳✵✷✷✸✻✻✺ ✳✷✷✹✼✻✻✶ ❞②❡❛r✷✵✵✸ ⑤ ✳✶✽✺✺✻✷✹ ✳✵✺✶✼✺✻✶ ✸✳✺✾ ✵✳✵✵✵ ✳✵✽✸✽✼✷✼ ✳✷✽✼✷✺✷✷ ❞②❡❛r✷✵✵✹ ⑤ ✳✷✼✵✷✸✾✾ ✳✵✺✸✵✼✶✹ ✺✳✵✾ ✵✳✵✵✵ ✳✶✻✺✾✻✺✾ ✳✸✼✹✺✶✸✽ ❞②❡❛r✷✵✵✺ ⑤ ✳✸✽✺✽✹✺ ✳✵✺✻✸✺✽✸ ✻✳✽✺ ✵✳✵✵✵ ✳✷✼✺✶✶✸ ✳✹✾✻✺✼✼ ❴❝♦♥s ⑤ ✳✺✸✹✼✻✶✸ ✳✵✹✵✹✻✽✶ ✶✸✳✷✶ ✵✳✵✵✵ ✳✹✺✺✷✺ ✳✻✶✹✷✼✷✻ ✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ▲❛♠❜❞❛ ⑤ ❧❛♠❜❞❛ ⑤ ✸✳✵✼✵✽✾✺ ✳✷✾✺✻✸✺ ✶✵✳✸✾ ✵✳✵✵✵ ✷✳✹✾✵✵✸✺ ✸✳✻✺✶✼✺✺ ✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❙✐❣♠❛ ⑤ s✐❣♠❛ ⑤ ✳✻✹✻✾✹✸✸ ✳✵✷✸✷✺✻✶ ✷✼✳✽✷ ✵✳✵✵✵ ✳✻✵✶✷✹✾✾ ✳✻✾✷✻✸✻✼ ✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❚❤❡t❛ ⑤ t❤❡t❛ ⑤ ✲✳✶✷✵✸✺✵✽ ✳✵✶✽✼✻✵✻ ✲✻✳✹✷ ✵✳✵✵✵ ✲✳✶✺✼✷✶✶✸ ✲✳✵✽✸✹✾✵✷ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

True RE technical efficiency estimates: JLMS

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Bayesian paradigm: Panel exponential model

★❞❡❧✐♠✐t❀ ✳ ❜s❢♣❛♥❡❧ ✩❨ ✩❳❧✐♥ ✩❳sq ✩❳✐♥t ✩❞❨✱ ✐❞✭✐r❝❴✐❞✮ t✐♠❡✭②❡❛r✮ ❞✭❡①♣✮ ✐t❡r❛t✐♦♥✭✷✵✵✵✮ t❤✐♥✭✷✮ ✈✐❞ ♣r❡❞✭✺✮❀ ★❞❡❧✐♠✐t ❝r ❇❛②❡s✐❛♥ ❙t♦❝❤❛st✐❝ ❢r♦♥t✐❡r ✲ ❊①♣♦♥❡♥t✐❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ ✉ Pr✐♦r ❤②♣❡r♣❛r❛♠❡t❡rs✿ ❙✐❣♠❛✷✲✲❃ ❛✿ ✶ ❜✿ ✶ ▲❛♠❜❞❛✲✲❃ ❛✿ ✶ ❜✿ ✳✷✷✸✶✹✸✻ ❙❡tt✐♥❣s✿ ■t❡r❛t✐♦♥s✿ ✷✵✵✵ ❇✉r♥✐♥✿ ✷✵✵ ❚❤✐♥♥✐♥❣✿ ✷ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❧♥♦r♠❴✇❡✐❣❤t❛ ⑤ ▼❡❛♥ ❙t❞✳❉❡✈✳ ⑤ ♣✷✺ ▼❡❞✐❛♥ ♣✼✺ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ❛❧♣❤❛✶ ⑤ ✳✻✶✻✵✶✾✹ ✳✵✺✵✺✷✸✷ ⑤ ✳✺✽✶✼✺✵✾ ✳✻✶✺✸✶✽✷ ✳✻✹✾✽✼✶✽ ❛❧♣❤❛✷ ⑤ ✳✺✾✷✻✵✹ ✳✵✺✺✼✹✷✸ ⑤ ✳✺✺✺✺✺✷✺ ✳✺✾✸✺✻✽✶ ✳✻✷✽✸✻✷✸ ❛❧♣❤❛✸ ⑤ ✳✶✺✹✻✶✸✼ ✳✵✹✺✷✽✹✼ ⑤ ✳✶✷✷✻✷✷✻ ✳✶✺✹✻✽✽✻ ✳✶✽✺✵✼✻✼ ❛❧♣❤❛✹ ⑤ ✳✵✽✶✹✵✵✹ ✳✵✹✺✶✻✺✸ ⑤ ✳✵✺✵✾✵✾✼ ✳✵✽✵✹✷✸✹ ✳✶✶✶✹✼✽✹

(Continued on next page)

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Bayesian paradigm: Panel exponential model

❛❧♣❤❛✶✶ ⑤ ✳✵✺✷✺✵✷✹ ✳✵✸✷✸✵✽✶ ⑤ ✳✵✸✵✹✵✷ ✳✵✺✸✸✾✶✹ ✳✵✼✹✽✶✸✺ ❛❧♣❤❛✷✷ ⑤ ✳✹✷✼✷✾✾✹ ✳✵✻✶✸✻✽✾ ⑤ ✳✸✽✻✼✵✵✸ ✳✹✷✾✻✼✻✺ ✳✹✻✾✶✽✷✺ ❛❧♣❤❛✸✸ ⑤ ✳✶✺✽✻✶✽✸ ✳✵✸✻✷✶✶✸ ⑤ ✳✶✸✹✽✹✺✺ ✳✶✺✽✷✼✻✸ ✳✶✽✷✵✻✼✷ ❛❧♣❤❛✹✹ ⑤ ✳✵✽✼✼✶✶✺ ✳✵✸✼✼✵✼✼ ⑤ ✳✵✻✶✵✽✷ ✳✵✽✻✼✼✶✹ ✳✶✶✸✻✻✸✸ ❛❧♣❤❛✶✷ ⑤ ✳✵✶✶✺✹✺✾ ✳✵✶✽✶✺✸✷ ⑤ ✲✳✵✵✶✵✸✺✸ ✳✵✶✶✸✻✼✾ ✳✵✷✸✻✸✹✼ ❛❧♣❤❛✶✸ ⑤ ✲✳✶✾✽✹✼✺✻ ✳✵✺✺✼✸✹✼ ⑤ ✲✳✷✸✼✵✶✶✾ ✲✳✶✾✼✾✹✾ ✲✳✶✻✶✻✼✹✼ ❛❧♣❤❛✶✹ ⑤ ✳✵✹✼✷✸✽✶ ✳✵✹✾✻✽✾✾ ⑤ ✳✵✶✹✸✹✶✶ ✳✵✹✼✷✵✼✼ ✳✵✽✵✺✺✺✸ ❛❧♣❤❛✷✸ ⑤ ✲✳✵✺✺✺✵✼✷ ✳✵✸✶✻✼✶✸ ⑤ ✲✳✵✼✻✶✶✼✶ ✲✳✵✺✺✸✺✷✸ ✲✳✵✸✹✻✺✵✹ ❛❧♣❤❛✷✹ ⑤ ✲✳✵✽✵✺✷✷✶ ✳✵✷✹✺✹✼✼ ⑤ ✲✳✵✾✻✽✻✶ ✲✳✵✽✵✶✺✷✹ ✲✳✵✻✹✺✹✷✷ ❛❧♣❤❛✸✹ ⑤ ✳✵✺✷✺✸✵✹ ✳✵✸✺✵✼✸✾ ⑤ ✳✵✷✾✻✾✻✻ ✳✵✺✶✺✻✽✾ ✳✵✼✺✻✸✽✹ ❞②❡❛r✷✵✵✶ ⑤ ✲✳✵✹✹✸✸✺✶ ✳✵✷✾✷✾✻✺ ⑤ ✲✳✵✻✸✺✼✺✶ ✲✳✵✹✸✼✶✻✾ ✲✳✵✷✺✶✶✺✸ ❞②❡❛r✷✵✵✷ ⑤ ✳✵✻✻✶✺✽✼ ✳✵✸✵✻✵✼✺ ⑤ ✳✵✹✹✾✸✽✽ ✳✵✻✻✻✶✻✾ ✳✵✽✼✷✶✷✼ ❞②❡❛r✷✵✵✸ ⑤ ✳✶✵✻✻✵✵✹ ✳✵✸✵✺✷✽✾ ⑤ ✳✵✽✺✸✼✸✼ ✳✶✵✻✸✵✷ ✳✶✷✼✼✻✵✽ ❞②❡❛r✷✵✵✹ ⑤ ✳✶✼✾✷✼✽✾ ✳✵✸✶✾✾✾✷ ⑤ ✳✶✺✻✹✸✾✷ ✳✶✼✽✽✽✸✹ ✳✷✵✶✸✵✾✾ ❞②❡❛r✷✵✵✺ ⑤ ✳✷✹✵✷✼✷✼ ✳✵✸✶✻✸✹✽ ⑤ ✳✷✶✽✽✻✷ ✳✷✹✵✹✼✸✾ ✳✷✻✶✽✶✻✶ ❴❝♦♥s ⑤ ✳✷✽✽✹✹✷✽ ✳✵✸✷✹✽✷✶ ⑤ ✳✷✻✼✶✾✽✺ ✳✷✽✼✾✹✽✶ ✳✸✶✵✸✶✾✷ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✰✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ s✐❣♠❛✷ ⑤ ✳✵✹✺✵✶✵✸ ✳✵✵✸✵✼✹✽ ⑤ ✳✵✹✷✽✻✶✷ ✳✵✹✹✽✻✸✷ ✳✵✹✻✾✾✶✷ ❧❛♠❜❞❛ ⑤ ✶✳✽✻✾✼✶✸ ✳✷✸✹✻✾✾✾ ⑤ ✶✳✼✶✷✺✽✸ ✶✳✽✺✹✼✶✷ ✷✳✵✷✸✶✸✶ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata

Introduction Frequentist Estimation Bayesian inference STATA commands Empirical application

Bayesian panel estimate of technical efficiency: “mean” JLMS

Federico Belotti†, Silvio Daidone†, Giuseppe Ilardi‡ Frequentist and Bayesian stochastic frontier models in Stata