4
Economics Letters 118 (2013) 151–154 Contents lists available at SciVerse ScienceDirect Economics Letters journal homepage: www.elsevier.com/locate/ecolet Misspecification in allocative inefficiency: A simulation study Levent Kutlu School of Economics, Georgia Institute of Technology, Atlanta, GA, 30332-0615, USA article info Article history: Received 18 August 2012 Received in revised form 26 September 2012 Accepted 2 October 2012 Available online 11 October 2012 JEL classification: C2 C3 Keywords: Technical inefficiency Allocative inefficiency Stochastic frontier Panel data abstract We make an extensive simulation analysis in order to investigate the consequences of ignoring the potentially complex and data dependent effects of allocative inefficiency on the estimation of stochastic frontier panel data models. Generally system estimators perform worse than single equation estimators. This result holds even when we approximate the allocative inefficiency. © 2012 Elsevier B.V. All rights reserved. 1. Introduction In the stochastic frontier framework, a cost–share system is rarely estimated as the inefficiency term in the cost equation and the deviations from the optimal shares from the observed shares are complicated functions of allocative inefficiency. Kumbhakar (1997) provides an exact solution but it is very difficult to estimate this model. Hence, Kumbhakar and Tsionas (2005) and Brissimis et al. (2009) provide first-order approximations to Kumbhakar’s (1997) model. In the cross sectional data and single equation framework, Kumbhakar and Wang (2006) show (by simulations) that ignoring the allocative inefficiency can lead to non-negligible biases in the parameter estimates. Hence, ignoring allocative inefficiency can be problematic even for the single equation models. In contrast to Kumbhakar and Wang (2006), who only use a single equation maximum likelihood estimator, we use a variety of estimators in our study. This enables us to compare the performances of different estimators. One promising candidate for this setting is Kumbhakar’s (1997) exact model. We make a Taylor series approximation to Kumbhakar’s (1997) model. 1 In addition to this estimator we include two regression based (fixed effects (FE) and Cornwell et al. (1990) within (CSSW)) and two maximum likelihood based (single equation and system) estimators. Tel.: +1 404 894 4453; fax: +1 404 894 1890. E-mail address: [email protected]. 1 We follow Kumbhakar and Tsionas (2005) and Brissimis et al. (2009). In Section 2, we explain the models that will be examined in our simulations. Section 3 gives more details about the simulations and the results. Finally, in Section 4, we make our conclusions. 2. Models We consider five different models for our experiments. The first and second models are FE and CSSW, respectively. Since these estimators are well known, we omit the details. The third and fourth models are single equation maximum likelihood estimator (MLSE) and the system maximum likelihood estimator (MLSYS), respectively. We assume that ln C it W M it = ln ˜ C it W M it + u it + v it = β 0 + k β Y k ln(Y k it ) + m β W m ln W m it W M it + 1 2 k,l β Y kl ln(Y k it ) ln(Y l it ) + 1 2 m,n β W mn ln W m it W M it ln W n it W M it + k,m β YW km ln(Y k it ) ln W m it W M it + u it + v it (1) S it = ˜ S it + e it , (2) 0165-1765/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2012.10.004

Misspecification in allocative inefficiency: A simulation study

Embed Size (px)

Citation preview

Page 1: Misspecification in allocative inefficiency: A simulation study

Economics Letters 118 (2013) 151–154

Contents lists available at SciVerse ScienceDirect

Economics Letters

journal homepage: www.elsevier.com/locate/ecolet

Misspecification in allocative inefficiency: A simulation studyLevent Kutlu ∗

School of Economics, Georgia Institute of Technology, Atlanta, GA, 30332-0615, USA

a r t i c l e i n f o

Article history:Received 18 August 2012Received in revised form26 September 2012Accepted 2 October 2012Available online 11 October 2012

JEL classification:C2C3

Keywords:Technical inefficiencyAllocative inefficiencyStochastic frontierPanel data

a b s t r a c t

We make an extensive simulation analysis in order to investigate the consequences of ignoring thepotentially complex and data dependent effects of allocative inefficiency on the estimation of stochasticfrontier panel data models. Generally system estimators perform worse than single equation estimators.This result holds even when we approximate the allocative inefficiency.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

In the stochastic frontier framework, a cost–share system israrely estimated as the inefficiency term in the cost equation andthe deviations from the optimal shares from the observed sharesare complicated functions of allocative inefficiency. Kumbhakar(1997) provides an exact solution but it is very difficult to estimatethis model. Hence, Kumbhakar and Tsionas (2005) and Brissimiset al. (2009) provide first-order approximations to Kumbhakar’s(1997) model. In the cross sectional data and single equationframework, Kumbhakar and Wang (2006) show (by simulations)that ignoring the allocative inefficiency can lead to non-negligiblebiases in the parameter estimates. Hence, ignoring allocativeinefficiency can be problematic even for the single equationmodels. In contrast to Kumbhakar and Wang (2006), who onlyuse a single equation maximum likelihood estimator, we use avariety of estimators in our study. This enables us to compare theperformances of different estimators. One promising candidate forthis setting is Kumbhakar’s (1997) exact model. We make a Taylorseries approximation to Kumbhakar’s (1997) model.1 In additionto this estimator we include two regression based (fixed effects(FE) and Cornwell et al. (1990) within (CSSW)) and two maximumlikelihood based (single equation and system) estimators.

∗ Tel.: +1 404 894 4453; fax: +1 404 894 1890.E-mail address: [email protected].

1 We follow Kumbhakar and Tsionas (2005) and Brissimis et al. (2009).

0165-1765/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.econlet.2012.10.004

In Section 2, we explain themodels thatwill be examined in oursimulations. Section 3 givesmore details about the simulations andthe results. Finally, in Section 4, we make our conclusions.

2. Models

We consider five differentmodels for our experiments. The firstand second models are FE and CSSW, respectively. Since theseestimators are well known, we omit the details. The third andfourth models are single equation maximum likelihood estimator(MLSE) and the system maximum likelihood estimator (MLSYS),respectively. We assume that

ln

Cit

WMit

= ln

Cit

WMit

+ uit + vit

= β0 +

k

βYk ln(Y kit)

+

m

βWm lnWm

it

WMit

+

12

k,l

βYkl ln(Y kit) ln(Y l

it)

+12

m,n

βWmn lnWm

it

WMit

ln

W nit

WMit

+

k,m

βYWkm ln(Y kit) ln

Wm

it

WMit

+ uit + vit (1)

Sit = Sit + eit , (2)

Page 2: Misspecification in allocative inefficiency: A simulation study

152 L. Kutlu / Economics Letters 118 (2013) 151–154

where Cit is the total cost for firm i, Y kit is the kth output for firm

i, Wmit is the price index for the mth input for firm i, uit ≥ 0

is a random variable representing the total cost of inefficiency,vit ∼ N(0, σ 2

v ) is the traditional error term, Sit is the vector ofobserved input shares,2 Sit is the optimal input share implied bythe Shephard’s lemma, and eit ∼ N(0, Σe) is the error term forthe share equations.3 We assume that uit , vit , and eit are mutuallyindependent. MLSE estimates Eq. (1) and MLSYS estimates thesystem. For MLSE and MLSYS, we assume that uit = ui ∼

N+(0, σ 2u ). MLSYS naively assumes that the error term for the share

equations is independent from the inefficiency term in the costequation. This assumption is inconsistent, because the error termin the share equations cannot be independent from the cost ofallocative inefficiency.

The final estimator, MLKT, is a system estimator that approx-imates Kumbhakar’s (1997) exact model.4 This is done by usinga first-order Taylor series approximation of the cost of allocativeinefficiency and the allocative inefficiency error term around zeroallocative inefficiency. Kumbhakar (1997) assumes a translog func-tional form for the cost function as given in Eq. (1). Shephard’slemma implies the following input share equations:

Smit =∂ ln(Cit)

∂ ln(Wmit )

= βWm +

n

βWmn ln

W nit

WMit

+

k

βYWkm ln(Y kit). (3)

The observed input share equations are given by

Smit = Smit + ξmit . (4)

Let uit = uTit + uA

it , where uTit , u

Ait ≥ 0, with uT

it = BTt u

Ti , where BT

tis a function of time. Here, uT

it and uAit represent the cost of technical

inefficiency and the cost of allocative inefficiency, respectively. Thecost of allocative inefficiency is modelled by utilizing the followingrelationship:

Pfm(xit; β)

PfM(xit; β)=

Wmit

WMit

exp(ηmit ) =

Wm∗

it

WMit

, m = 1, 2, . . . ,M − 1, (5)

where Pf is the production function frontier dual to C and ηmit =

Bηt η

mi , where Bη

t is a function of time. In what follows, we set BTt =

Bηt = 1. It can be shown that5

uAit = ln(C∗

it ) − ln(Cit) + ln(Git), (6)

where C∗

it = C(Yit ,W ∗

it ; β),W ∗

it = (W 1∗it ,W 2∗

it , . . . ,WM−1∗it ,

WMit ),Git =

m[

∂ ln C∗it

∂ lnWm∗it

] exp(−ηmit ) =

m Sm∗

it exp(−ηmit ), and

Sm∗

it = S(Yit ,W ∗

it ; β). The input share equations are given by

Smit =Sm∗

it

Git exp(ηm)+ ξm

it + emit = Smit + ξmit + emit , (7)

where ξmit =

Sm∗it

Git exp(ηmit )−Smit represents deviations from the optimal

input shares due to input-specific allocative inefficiency and emit isa term added to account for measurement errors. For the translogcost function,

2 One of the input shares is omitted for obvious reasons.3 The linear homogeneity restrictions for the input prices is imposed by

normalizing the cost and the prices by the price index for materials.4 KT stands for Kumbhakar and Tsionas (2005) as it uses the approximation

proposed by them. Indeed, MLKT is a variation of the estimator proposed byBrissimis et al. (2009).5 See Kumbhakar (1997).

Table 1Simulation results.

Base

PARAM FE CSSW MLSE MLSYS MLKT

β0 0.4852 0.4551 0.5206 0.5172 0.5173βw 0.4996 0.4996 0.4997 0.4989 0.4989βy 0.9003 0.9005 0.9002 0.9053 0.9051βww −0.1018 −0.1017 −0.1016 −0.0918 −0.0923βwy 0.1017 0.1018 0.1016 0.0920 0.0924βyy 0.0982 0.0980 0.0983 0.0979 0.0979

SIZE FE CSSW MLSE MLSYS MLKT

β0 – – 0.2510 0.2530 0.2545βw 0.0535 0.0640 0.0645 0.6670 0.6650βy 0.0590 0.0730 0.0665 0.1010 0.1010βww 0.0475 0.0540 0.0540 0.6495 0.6530βwy 0.0580 0.0590 0.0675 0.6570 0.6635βyy 0.0555 0.0635 0.0665 0.0695 0.0700

RMSE 0.1253 0.1339 0.1235 0.1483 0.1480MAE 0.0850 0.0909 0.0838 0.1019 0.1018RMSEeff 0.0392 0.0771 0.0373 0.0373 0.0488MAEeff 0.0312 0.0593 0.0298 0.0297 0.0368Mean u 0.2994 0.3294 0.2639 0.2650 0.2897

True u 0.2837 True Eff 0.7667

uAit = lnGit +

m

βWmηmit +

m,n

βYWmn lnWm

it

WMit

ηnit

+12

m,n

βWmnηmit η

nit +

k,m

βYWkm ln(Y kit)η

mit (8)

ξmit =

Smit (1 − Git exp(ηmit )) +

n

βWmnηnit

Git exp(ηmit )

. (9)

As mentioned earlier, in the MLKT model we approximateuAit and ξit terms by a Taylor series expansion around ηit =

(η1it , η

2it , . . . , η

M−1it )′ = 0 to get a closed form expression for the

log-likelihood function. Kumbhakar and Tsionas (2005) showedthatuA

it ≃ 0 and ξmit ≃

n H

mnit ηn

it , whereHmnit = βWmm−Smit (1−Smit )

ifm = n andHmnit = βWmn+Smit S

nit ifm = n. The distributions for our

model are given by vit ∼ N(0, σ 2v ), uT

it = uTi ∼ N+(0, σ 2

u ), ηit =

ηi ∼ N(0, Ση), and eit ∼ N(0, Σe), where vit , uTi , ηi, and eit are

mutually independent.

3. Simulations

We assume that Kumbhakar’s (1997) exact model is the truemodel and that there are only two inputs. The distributions ofrandom variables are given by vit ∼ N(0, σ 2

v ), uTit = uT

i ∼

N+(0, σ 2u ), ηit = ηi ∼ N(0, σ 2

η ), and eit ∼ N(0, σ 2e ), where

vit , uTi , ηi, and eit are mutually independent. Moreover, uA

it andξmit are generated according to Eqs. (8) and (9). The regressors,Xit = [wityit ]′, are generated by a bivariate VAR model6:Xit = RXi,t−1 + δit , where δit ∼ N(0, σ 2

δ I2) and Xi1 ∼

N(0, σ 2δ (I2 − R2)−1).7 Wemake sure that the regularity conditions

hold: (1) monotonicity with respect to w and y, (2) concavity withrespect to w, (3) homogeneity of the degree 1 in w, (4) well-defined input shares, and (5) uA

≥ 0. The values of yit and wit areshifted by µyi and µwi , respectively. We used µyi ∼ N(µy, σ

2µy

)

and µwi ∼ N(µw, σ 2µw

). At each simulation, we drew a largernumber of firms thanN and discarded those firms that violated the

6 We followed Park et al. (2003, 2007) and Kutlu (2010).7 In contrast to our simulations, Kumbhakar and Wang (2006) considers cross

sectional data.

Page 3: Misspecification in allocative inefficiency: A simulation study

L. Kutlu / Economics Letters 118 (2013) 151–154 153

Table 2Simulation results.

σ 2u = 0.05 (LOW) σ 2

u = 0.2 (HIGH)PARAM FE CSSW MLSE MLSYS MLKT PARAM FE CSSW MLSE MLSYS MLKT

β0 0.4793 0.4466 0.5200 0.5166 0.5167 β0 0.4916 0.4643 0.5214 0.5180 0.5181βw 0.4996 0.4995 0.4997 0.4989 0.4989 βw 0.4996 0.4995 0.4997 0.4988 0.4989βy 0.9003 0.9005 0.9002 0.9052 0.9050 βy 0.9003 0.9005 0.9003 0.9053 0.9051βww −0.1018 −0.1017 −0.1016 −0.0918 −0.0923 βww −0.1018 −0.1017 −0.1017 −0.0918 −0.0923βwy 0.1017 0.1018 0.1016 0.0920 0.0924 βwy 0.1017 0.1018 0.1016 0.0920 0.0924βyy 0.0982 0.0980 0.0984 0.0979 0.0979 βyy 0.0982 0.0980 0.0983 0.0979 0.0979

SIZE FE CSSW MLSE MLSYS MLKT SIZE FE CSSW MLSE MLSYS MLKT

β0 – – 0.2675 0.2720 0.2775 β0 – – 0.2365 0.2405 0.2415βw 0.0530 0.0640 0.0685 0.6615 0.6640 βw 0.0535 0.0640 0.0610 0.6675 0.6655βy 0.0590 0.0730 0.0660 0.1035 0.1035 βy 0.0590 0.0730 0.0650 0.1000 0.0990βww 0.0475 0.0540 0.0575 0.6485 0.6525 βww 0.0475 0.0540 0.0545 0.6495 0.6535βwy 0.0580 0.0590 0.0650 0.6585 0.6645 βwy 0.0580 0.0590 0.0660 0.6560 0.6645βyy 0.0555 0.0635 0.0655 0.0660 0.0675 βyy 0.0555 0.0635 0.0640 0.0665 0.0675

RMSE 0.1253 0.1338 0.1224 0.1476 0.1473 RMSE 0.1253 0.1339 0.1242 0.1487 0.1484MAE 0.0850 0.0908 0.0830 0.1014 0.1013 MAE 0.0850 0.0909 0.0843 0.1023 0.1021RMSEeff 0.0412 0.0822 0.0360 0.0359 0.0478 RMSEeff 0.0380 0.0724 0.0390 0.0389 0.0498MAEeff 0.0329 0.0640 0.0286 0.0285 0.0359 MAEeff 0.0303 0.0553 0.0311 0.0310 0.0378Mean u 0.2313 0.2639 0.1906 0.1916 0.2164 Mean u 0.3976 0.4248 0.3677 0.3688 0.3935

True u 0.2098 True Eff 0.8187 True u 0.3884 True Eff 0.7019

regularity conditions. Then, we randomly picked N firms from theremaining firms. Finally, the sample distributions of µyi , µwi , vit ,and eit were checked by Kolmogorov–Smirnov and Jarque–Beratests. We also separately tested for validity of the mean and thevariance. For uT

i we only used the Kolmogorov–Smirnov test. Onlythose data that passed all these tests for the 5% significance levelwere accepted.

Each simulation was carried out 2000 times. In our basescenario the following parameter valueswere chosen: PV = {β0 =

0.5, βw = 0.5, βy = 0.9, βww = −0.1, βwy = 0.1, βyy =

0.1, µy = 1, σ 2y = 0.2, µw = 0.5, σ 2

w = 0.2, σ 2δ =

0.1, σ 2v = 0.025, σ 2

e = 0.0025, σ 2η = 0.2, σ 2

u = 0.1, R =0.4 −0.05

−0.05 0.4

,N = 100, T = 30}. In addition to the base

scenario, we examined the effects of changes in two parameters,σ 2u , σ 2

η

. In these scenarios we fixed all the parameters in the

base scenario and only changed the relevant parameter value. Foreach parameter we considered a low and a high scenario: σ 2

u =

{0.05, 0.1, 0.2} and σ 2η = {0.1, 0.2, 0.5}.

We present the average of β estimates, the empirical sizes forthe 5% level, and aggregate versions of root mean squared error(RMSE) and mean absolute error (MAE) for the β parameters. TheRMSE and MAE are given by

RMSE =

1simn ∗ parn

i,j

β(i) − βj(i)

β(i)

2

(10)

MAE =1

simn ∗ parn

i,j

β(i) − βj(i)β(i)

, (11)

where simn is the number of simulations, parn is the size of the β

parameter vector, βj is the estimate of the β vector at simulation j,and β(i) and βj(i) are the ith components of β and βj, respectively.Our RMSE and MAE measures exclude the constant term as FE andCSSW do not directly estimate the constant term. The RMSE andMAE for the total efficiency estimates, RMSEeff and MAEeff, arecalculated as follows:

RMSEeff =

1

simn ∗ N ∗ T

i,t,j

TE(i, t) − TE j(i, t)

2(12)

MAEeff =1

simn ∗ N ∗ T

i,t,j

TE(i, t) − TE j(i, t) , (13)

where simn is the number of simulations, TE(i, t) is the trueefficiency at time t for firm i, and TE j(i, t) is the estimate of TE(i, t)at simulation j. We also present the mean of the estimated totalcost of inefficiency (mean of u = uA

+ uT ) and themean of the truetotal cost of inefficiency (mean of u = uA

+ uT ). The simulationresults are given in Tables 1–3. The RMSE and MAE give similarrankings for the models. Hence, we only concentrate on the RMSEwhen we are interpreting the results.

The base scenario parameters were σ 2η = 0.2 and σ 2

u = 0.1.All other parameters were fixed throughout the simulations. Weobserve that all of the estimators give biased estimates. The systemestimators (MLSYS and MLKT) have considerably larger biases.Hence, the approximation in MLKT does not seem to be doing avery good job in fixing the bias. Due to the bias, the empirical sizesfor the constant termarewayoff the 5% level.Moreover, the systemestimators’ empirical sizes are worse than their single equationcounterparts because their confidence intervals are smaller. Interms of RMSE the system estimators perform worse than thesingle equation ML estimator (MLSE). Although MLKT does notperform as well as the other estimators in terms of RMSEeff, it isthe best estimator in terms of estimating themean of the total costof efficiency. Overall FE and MLSE show the best performance.

The variation in the cost of technical inefficiency is determinedby the σ 2

u term. Fixing the other base scenario parameters, weconsider cases of σ 2

u = {0.05, 0.1, 0.2}. An increase in σ 2u is

expected to help identifying the cost of technical inefficiency.Hence, as expected, we observe that for larger σ 2

u values boththe Spearman correlation and the Pearson correlation are larger.8The RMSE values remain pretty much the same in response toan increase in σ 2

u . For the regression based estimators, RMSEeffdecreases. On the other hand, for the ML estimators, the RMSEeffvalues get larger asσ 2

u gets larger. The reason for this is the increasein the variation of the efficiency.

We consider cases of σ 2η = {0.1, 0.2, 0.5}. It turns out that for

the single equation estimators the RMSE is not affected much, butit has a tendency to increase for larger values of σ 2

η . However, thefor system estimators, the RMSE gets much worse as σ 2

η increases.For the ML estimators, an increase in σ 2

η leads to an increase inRMSEeff. This is due to the increased degree ofmisspecification andincrease in the variation of the efficiency. FE and MLSE estimatorsseem to perform better than the remaining estimators.

8 The Spearman and Pearson correlations are available upon request.

Page 4: Misspecification in allocative inefficiency: A simulation study

154 L. Kutlu / Economics Letters 118 (2013) 151–154

Table 3Simulation results.

σ 2η = 0.1 (LOW) σ 2

η = 0.5 (HIGH)

PARAM FE CSSW MLSE MLSYS MLKT PARAM FE CSSW MLSE MLSYS MLKT

β0 0.4806 0.4496 0.5142 0.5124 0.5124 β0 0.4920 0.4636 0.5297 0.5245 0.5247βw 0.5002 0.5001 0.5001 0.4997 0.4997 βw 0.4994 0.4995 0.4994 0.4936 0.4935βy 0.9004 0.9004 0.9002 0.9027 0.9026 βy 0.9001 0.8999 0.9000 0.9106 0.9103βww −0.1009 −0.1007 −0.1008 −0.0958 −0.0961 βww −0.1015 −0.1018 −0.1014 −0.0779 −0.0786βwy 0.1006 0.1006 0.1006 0.0960 0.0962 βwy 0.1023 0.1025 0.1023 0.0820 0.0827βyy 0.0989 0.0989 0.0992 0.0988 0.0988 βyy 0.0981 0.0982 0.0982 0.0973 0.0973

SIZE FE CSSW MLSE MLSYS MLKT SIZE FE CSSW MLSE MLSYS MLKT

β0 – – 0.1495 0.1800 0.1795 β0 – – 0.4070 0.3710 0.3770βw 0.0480 0.0565 0.0570 0.6315 0.6325 βw 0.0530 0.0555 0.0685 0.6500 0.6545βy 0.0505 0.0590 0.0645 0.0840 0.0845 βy 0.0485 0.0580 0.0580 0.1160 0.1210βww 0.0530 0.0645 0.0565 0.6255 0.6275 βww 0.0540 0.0520 0.0605 0.7545 0.7545βwy 0.0415 0.0540 0.0470 0.6080 0.6100 βwy 0.0530 0.0610 0.0625 0.7200 0.7165βyy 0.0475 0.0575 0.0620 0.0595 0.0600 βyy 0.0505 0.0645 0.0625 0.06800 0.0670

RMSE 0.1231 0.1325 0.1202 0.1183 0.1181 RMSE 0.1252 0.1339 0.1235 0.2090 0.2080MAE 0.0830 0.0891 0.0809 0.0810 0.0808 MAE 0.0846 0.0903 0.0836 0.1403 0.1397RMSEeff 0.0405 0.0800 0.0341 0.0339 0.0401 RMSEeff 0.0383 0.0732 0.0438 0.0439 0.0650MAEeff 0.0323 0.0619 0.0271 0.0269 0.0311 MAEeff 0.0305 0.0558 0.0356 0.0354 0.0480Mean u 0.2889 0.3198 0.2553 0.2559 0.2724 Mean u 0.3223 0.3508 0.2847 0.2864 0.3284

True u 0.2693 True Eff 0.7774 True u 0.3137 True Eff 0.7451

For some parameters larger sample size decreases the biasslightly but does not eliminate it.9 For a variety of sample sizesFE performs best in terms of empirical sizes.

4. Conclusion

We implemented an extensive simulation analysis in order toexamine the consequences of ignoring allocative inefficiency forstochastic frontier models. For this purpose we used a variety ofestimators and data generating processes. Our simulation resultsindicate that most of the time the system estimators do notperform as well as the single equation estimators. Althoughsometimes (for some criteria) using a system estimator canimprove the estimates, many times the gain by using a systemestimator is negligible compared to the loss due to using amisspecified model. This is true even when the econometricianuses approximations to the allocative inefficiency related terms.

9 We examined N = {30, 100, 300, 1000} and T = {12, 30, 50, 300}.

References

Brissimis, S.N., Delis, M.D., Tsionas, E.G., 2009. Technical and allocative efficiency inEuropean banking. European Journal of Operational Research 204, 153–163.

Cornwell, C., Schmidt, P., Sickles, R.C., 1990. Production frontiers with time-seriesvariation in efficiency levels. Journal of Econometrics 46, 185–200.

Kumbhakar, S.C., 1997. Modeling allocative inefficiency in a translog cost functionand cost share equations: an exact relationship. Journal of Econometrics 76,351–356.

Kumbhakar, S.C., Tsionas, E.G., 2005.Measuring technical and allocative inefficiencyin the translog cost system: a Bayesian approach. Journal of Econometrics 126,355–384.

Kumbhakar, S.C., Wang, H.J., 2006. Pitfalls in the estimation of a cost function thatignores allocative inefficiency: a Monte Carlo analysis. Journal of Econometrics134, 317–340.

Kutlu, L., 2010. Battese-Coelli estimator with endogenous regressors. EconomicsLetters 109, 79–81.

Park, B.U., Sickles, R.C., Simar, L., 2003. Semiparametric efficient estimation of AR(1)panel data models. Journal of Econometrics 117, 279–309.

Park, B.U., Sickles, R.C., Simar, L., 2007. Semiparametric efficient estimation ofdynamic panel data models. Journal of Econometrics 136, 281–301.