STOCHASTIC FRONTIER ANALYSIS

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MOTIVATION
• Usual textbook presentations treat producers as successful optimizers. They maximize production, minimize cost, and maximize profits. • Conventional econometric techniques build on this paradigm to estimate production/cost/profit function parameters using regression techniques where deviations of observed choices from optimal ones are modeled as statistical noise. • However though every producer may attempt to optimize, not all of them may succeed in their efforts. For example, given the same inputs, and the same technology, some will produce more output than others, i.e., some producers will be more efficient than others. • Econometric estimation techniques should allow for the fact that deviations of observed choices from optimal ones are due to two factors: failure to optimize i.e., inefficiency due to random shocks • Stochastic Frontier Analysis or SFA is one such technique to model producer behavior.

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USEFULNESS OF SFA

•

SFA produces efficiency estimates or efficiency scores of individual producers. Thus one can identify those who need intervention and corrective measures. Since efficiency scores vary across producers, they can be related to producer characteristics like size, ownership, location, etc. Thus one can identify source of inefficiency. SFA provides a powerful tool for examining effects of intervention. For example, has efficiency of the banks changed after deregulation? Has this change varied across ownership groups?

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STRUCTURE OF THIS PRESENTATION

• Part 1: Theory: Illustrate the basics of SFA mainly with analysis of cost efficiency. Concept of efficiency Estimation Identification of sources of inefficiency

• Part 2: Empirics: How to use FRONTIER program to estimate different types of efficiency models An application of SFA to Indian Banking (if time permits) • References 1. Kumbhakar, S.C. and Lovell, C.A.K (2000), Stochastic Frontier Analysis, Cambridge University Press, U.K. 2. Coelli, T.J.; Rao, D.S. Prasada, and Battese, G.E. (1998), An Introduction to Efficiency and Productivity Analysis, Kluwer Academic Publishers, Boston/Dordrecht/London.

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TECHNICAL EFFICIENCY
• Production Function: YM = ƒ(x; β) shows the maximum output YM producible from a given vector of inputs (x). Here β are the production function parameters. ƒ(L,K; β) = 2L0.5K0.5 ƒ(9,16; β) = 2.90.5.160.5 = 24 • Actual output, Y could be less than maximum output. In fact, any output equal to or less than YM can be produced. Y ≤ ƒ(x; β) = YM • Figure 1

• TE = Y/YM

0 ≤ TE ≤ 1

• Y = YM. TE = ƒ(x; β) . TE • Characterization: Y = ƒ(x; β)exp(-u) u≥0 (1)

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STOCHASTIC FRONTIER
• In (1) the frontier is deterministic. All deviations from maximum output are ascribed to inefficiency. • However sometimes maximum output itself might be lower (higher) due to exogenous shocks. The production frontier itself may be shifting. • Figure 2 • Y = ƒ(x; β).exp(v).exp(-u) ƒ(x; β) exp(v) exp(-u) ƒ(x; β).exp(v) v ≤ 0 and u ≥ 0 deterministic kernel effect on output of exogenous shocks inefficiency stochastic frontier

• TE

= = =

Y/ ƒ(x; β).exp(v) ƒ(x; β).exp(v).exp(-u)/ ƒ(x; β).exp(v) exp(-u)

• Y = ƒ(x; β).exp(v – u)

(v – u)

composite error term

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COST EFFICIENCY OR ECONOMIC EFFICIENCY

• Ability to produce observed output at minimum cost, given input prices. • A producer may be technically efficient, but yet cost inefficient because he fails to choose correct input combination. Allocative inefficiency • Figure 3 • Of course, a producer may be both technically inefficient as well as allocatively inefficient. Cost (or economic) inefficiency = technical inefficiency + allocative inefficiency • Figure 4 • Theoretically this notion is well defined but empirically it is involving to segregate these two sources of inefficiency. • Similarly, we can define profit efficiency, but its decomposition into technical, and allocative inefficiency is even more challenging. Will, therefore, concentrate on cost-efficiency in the remainder of the lecture.

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COST EFFICIENCY • Ei = c(yi, wi; β) exp {vi + ui} (1)

– c(yi, wi; β) is the deterministic kernel – vi random noise, takes positive and negative values – ui captures inefficiency, takes only positive values – Note the positive sign before ui • Under this formulation cost efficiency can be calculated as c(yi, wi; β) exp {vi} = exp {−ui} CEi = Ei (2) • 0 ≤ CEi ≤ 1

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Estimation • First rewrite as ln Ei = ln c(yi, wi; β) + ui + vi • Estimating equation (3) requires: – specification of a functional form for the deterministic kernel c(yi, wi; β), – an assumption about the distribution of the random variable vi, and – an assumption about the distribution of the random variable ui. • Given a particular specification for the random variables ui and vi, the Maximum Likelihood (ML) technique is used to estimate the unknown parameters. (3)

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SPECIFICATION • Deterministic Kernel – Cobb-Douglas (in log form) – Translog (a flexible functional form) • Random Variables vi and ui
2 – vi ∼ iidN (0, σv ) 2 – ui ∼ iidN +(0, σu) – vi and ui are distributed independently of each other, and of the regressors.

• Given these assumptions, the log-likelihood function for the sample of size I 1 iλ 2 ln L = K − I ln σ + ln Φ( ) − 2 i. i σ 2σ i (4) 2 2 where i = ui + vi, σ 2 = (σu + σv ), λ = σu σv and Φ(.) is the standard normal cumulative distribution function. We substitute ln Ei − ln c(yi, wi; β) in place of i in the likelihood function.
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DERIVATION OF LIKELIHOOD FUNCTION • The density function of vi is vi2 1 f (vi) = √ exp(− 2 ) 2σv 2πσv • The density function of ui is vi2 2 f (ui) = √ exp(− 2 ) 2σv 2πσv • Given the independence assumption, the joint density function of ui and vi is the product of their individual density function, and so, 2 u2 vi2 i f (ui, vi) = exp(− 2 − 2 ) 2πσuσv 2σu 2σv • Since i = vi + ui, the joint density function for ui and i is: u2 ( i − u i )2 2 i f (ui, i) = exp(− 2 − ) 2 2πσuσv 2σu 2σv
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• The marginal density function of i is then obtained by integrating ui out of f (ui, i) which yields f ( i) =
∞ 0 f (ui , i )dui

− iλ −2 2 i [1 − Φ( )] exp( 2 ) = √ σ 2σ 2πσ • The likelihood function of the sample is then, by independence, the product of the density functions of the individual observations. L(sample) = i=I i=1

f ( i)

• And then taking log of the likelihood function yields the log-likelihood equation. • Battese and Corra (1977) re-parameterization –γ= –λ=
2 σu 2 2 σu +σv σu σv →

→ bounded between 0 and 1 any non-negative value
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• If we use the γ parameterization, the loglikelihood function is given by: 1 ln L = K−I ln σ+ ln[1−Φ(zi)]− 2 i 2σ where zi = γ σ 1−γ i i

2 i.

(5)

• The log likelihood function allows for testing the appropriateness of the SFA
2 2 – γ → 0 when either σu → 0 or σv → ∞ → OLS cost function (i.e., average response function) with no inefficiency. – The test should be done using the Onesided Generalized Likelihood-Ratio Test to ensure correct size. 2 2 – γ → 1 when either σu → ∞ or σv → 0 → deterministic frontier with no noise.

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Calculating Producer Specific Efficiency ˆ • ˆi = ln Ei − ln c(yi, wi; β) is a composite ˆ estimate of ui + vi ˆ • But it contains information about ui. If ˆi is high then chances are ui is high since exˆ pectation of vi is zero. • The conditional distribution of ui given i could be exploited to get estimates of producer specific inefficiency. This was first demonstrate by Jondrow, Lovell, Materov, and Schmidt (1982) and since then this decomposition is known as the JLMS technique. Either the mean or the mode of this conditional distribution can be used. • i φ( σλ ) iλ E(ui | i) = σ∗[ + ( )] (6) σ 1 − Φ( −σiλ )

2 2 2 where σ∗ = σuσv /σ 2 and we have used the parameterization λ.
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• Given a point estimate of E(ui | i) Producer specific efficiency can be calculated as: • ˆ CEi = exp {−E(uˆ | i)}. i or using the point estimator • ˆ CEi = E(exp {−ui} | i) 1 − Φ(σ∗ − µ∗i/σ∗) = [ ] 1 − Φ(−µ∗i/σ∗) 1 2 · exp {−µ∗i + σ∗ } 2
2 where µ∗i = iσu/σ 2.

(7)

• The two formulas won’t give same results as exponential in a non-linear function.

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Alternative Distributional Assumptions
2 • Till now we have assumed that ui ∼ iidN +(0, σu)

• A more general formulation is that of Trun2 cated Normal Distribution ui ∼ iidN +(µ, σu) • Alternative functional forms like exponential and gamma distribution could also be applied. • Does the distributional assumption matter? Yes it does matter in the calculation of the efficiency numbers. • However, the ranking of the prodcuers are much less sensitive to distributional assumptions. • In Panel Data Models, one can estimate efficiency WITHOUT making ANY assumption about the distribution of ui.

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Analyzing Efficiency Behaviour • Two questions: – What is the behavior of efficiencies over time? Are they increasing, decreasing or constant? – What explains the variations in inefficiencies among producers and across time? • Time behavior – Following Kumbhakar (1990), Battese and Coelli (1992) proposed a simple model that can be used to estimate the time behavior of inefficiencies. uit = {exp[−η(t − T )]}ui, (8)

where the ui ∼ N +(µ, σ 2), and η is a parameter to be estimated. – inefficiencies in periods prior to T depend on the parameter η. – uiT = ui reference or benchmark
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– inefficiency prior to period T is the product of the terminal year’s inefficiency and exp {η(T − t)}. If η is positive, inefficiencies fall over time If η is negative, inefficiencies increase over time. – Efficiency behavior is monotonic – Ordering of firms in terms of inefficiencies time-invariant – Good for understanding aggregative behavior – Battese and Coelli “Model 1” in FRONTIER program • Interesting Hypothses to Test – Test η = 0 → Implies Time invariant efficiency – Test η = µ = 0 → Time invariant efficiency with Half-Normal Distribution

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Explaining efficiency • Certain factors influence the environment in which production takes place – degree of competitiveness – input and output quality – network characteristics – ownership form – changes in regulation, – management characteristics • Two ways to handle them – Include them as variables in the production process as control variables. Using this interpretation, these variables influence the structure of the technology by which conventional inputs are converted into outputs, but not efficiency.

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E = c(w, y; γ) exp {v + u}

(9)

The parameter γ now includes cost parameters as well as environmental parameters. – Associate variation in estimated efficiency with variation in the exogenous variables. • Early papers adopted two stage approach – Stage 1: In the first stage a stochastic frontier equation was estimated (excluding the exogenous variables), typically by MLE under the usual distributional and independence assumptions, and the regression residuals were decomposed using the JLMS technique. – Stage 2: The estimated inefficiencies were regressed on exogenous variables to explain/locate the source of inefficiency.

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• Econometrically inconsistent because the iid assumption necessary to use the JLMS technique is contracticted in the second stage. • Kumbhakar, Ghosh, and McGuckin (1991), and Reifcheneider and Setvenson (1991) approach. All the parameters of the stochastic frontier function as well as those of the inefficiency function was estimated together in a single MLE procedure. • The model ln Eit = lnC(wit, yit; β) + vit + uit (10) uit = δ zit + it (11)

uit captures the effect of economic inefficiency, which has a systematic component δ zit associated with the exogenous variables and a random component it.

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• The non-negativity requirement that uit = (δ zit − it) ≥ 0 is modeled as it ∼ N (0, σ 2) with the distribution of it being bounded below by the variable truncation point −δ zit. • This is implemented as ”Model 2” in FRONTIER program. • Interesting hypotheses to Test – Test: γ = δ0 = δ1 = · · · = δm = 0 → Implies no ineffciency – Test: δ1 = · · · = δm = 0 → Implies the Truncated Normal Distribution

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FRONTIER 4.1 -- A PROGRAM FOR ESTIMATING STOCHASTIC FRONTIER PRODUCTION AND COST FUNCTION

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THE FRONTIER PROGRAM
• This was developed by Tim Coelli at the Centre for Efficiency and Productivity Analysis (CEPA). CEPA was initially house at the University of New England, but has recently moved at the University of Queensland. • Website: http://www.uq.edu.au/economics/cepa/software.htm • Read the paragraph on FRONTIER 4.1. Click the link given immediately below • And download the file “front41-xp.zip” • User’s Guide: Coelli, T.J. (1996), “A Guide to FRONTIER Version 4.1: A Computer Program for Stochastic Frontier Production and Cost Function Estimation”, CEPA Working Paper 96/7, Department of Econometrics, University of New England, Armidale NSW Australia. • We are going to use the examples given in this guide to understand how to use the FRONTIER program.

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REQUIREMENTS OF THE PROGRAM

• The FRONTIER program requires you to prepare TWO files for estimating stochastic frontier production and cost functions.

• The Data File

which should preferably have the a .dta

extension like bank.dta

• The Instruction File extension bank.ins

which should preferably have a .ins

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The Data File
• Data should be listed in an text file. Format of this file is very important. • The data must be listed by observation.

• There MUST be 3+k[+p] columns in the following order:

1. Firm number (an integer in the range 1 to N) 2. Period number (an integer in the range 1 to T) 3. Yit 4. to 3 + k regressors 3+k+1 to 3+k+p optional regressors z1it … zpit p number of x1it … dependent variable xkit k number of

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• The program assumes a linear functional form. Thus for estimating a Cobb-Douglas production function, all the input and output quantities should be logged. • The observations can be listed in any order but the columns must be in the stated order. • There must be at least one observation on each of the N firms and there must be at least one observation in time period 1 and in time period T. • If you are using a single cross-section of data, then column 2 (the time period column) should contain the value “1” throughout. N

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EXECUTING THE PROGRAM
• Type “front41” to begin execution. The program will then ask if the instructions will come from a terminal (interactive mode) or from a file (batch mode). • If the interactive (terminal) option is selected, questions will be asked in the same order as they appear in the instruction file.

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THE INSTRUCTION FILE
• We now discuss the structure of the instruction file. • As Coelli write in his Guide, “The best way to describe how to use the program is to provide some examples.” So we discuss the examples given in the users Guide. The data, program, output used in this presentation are all taken directly from the Guide. • The estimation of five SF models are discussed. 1. A Cobb-Douglas production frontier using cross-sectional data and a assuming a half-normal distribution. 2. A Translog production frontier using cross-sectional data and assuming a truncated normal distribution. 3. A Cobb-Douglas cost frontier using cross-sectional data and assuming a half-normal distribution. 4. The Battese and Coelli (1992) specification (Model 1). 5. The Battese and Coelli (1995) specification (Model 2). • For simplicity there are only two production inputs in all cases. In the cross-sectional examples there are 60 firms, while in the panel data examples, there are 15 firms and 4 time periods.

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EXAMPLE 1
• (4.1) A Cobb-Douglas production frontier using cross-sectional data and assuming a half-normal distribution. ln(Qi) = β0 + β1ln(Ki) + β2ln(Li) + (Vi - Ui), • where Qi, Ki and Li are output, capital and labour, respectively, and Vi and Ui are assumed normal and half-normal distributed, respectively.

Table 1a - Listing of Data File EG1.DAT __________________________________________________ 1. 1. 12.778 9.416 35.134 2. 1. 24.285 4.643 77.297 3. 1. 20.855 5.095 89.799 . . . 58. 1. 21.358 9.329 87.124 59. 1. 27.124 7.834 60.340 60. 1. 14.105 5.621 44.218

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__________________________________________________ Table 1b - Listing of Shazam Instruction File EG1.SHA __________________________________________________ read(eg1.dat) n t y x1 x2 genr ly=log(y) genr lx1=log(x1) genr lx2=log(x2) file 33 eg1.dta write(33) n t ly lx1 lx2 stop __________________________________________________

Table 1c - Listing of Data File EG1.DTA _________________________________________________ 1.000000 1.000000 2.547725 2.242410 3.559169 2.000000 1.000000 3.189859 1.535361 4.347655 3.000000 1.000000 3.037594 1.628260 4.497574 . . . 58.00000 1.000000 3.061426 2.233128 4.467332 59.00000 1.000000 3.300419 2.058473 4.099995 60.00000 1.000000 2.646529 1.726510 3.789132 ________________________________________________________

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Table 1d - Listing of Instruction File EG1.INS ________________________________________________________________ 1 1=ERROR COMPONENTS MODEL, 2=TE EFFECTS MODEL eg1.dta DATA FILE NAME eg1.out OUTPUT FILE NAME 1 1=PRODUCTION FUNCTION, 2=COST FUNCTION y LOGGED DEPENDENT VARIABLE (Y/N) 60 NUMBER OF CROSS-SECTIONS 1 NUMBER OF TIME PERIODS 60 NUMBER OF OBSERVATIONS IN TOTAL 2 NUMBER OF REGRESSOR VARIABLES (Xs) n MU (Y/N) [OR DELTA0 (Y/N) IF USING TE EFFECTS MODEL] n ETA (Y/N) [OR NUMBER OF TE EFFECTS REGRESSORS (Zs)] n STARTING VALUES (Y/N) IF YES THEN BETA0 BETA1 TO BETAK SIGMA SQUARED GAMMA MU [OR DELTA0 ETA DELTA1 TO DELTAK] NOTE: IF YOU ARE SUPPLYING STARTING VALUES AND YOU HAVE RESTRICTED MU [OR DELTA0] TO BE ZERO THEN YOU SHOULD NOT SUPPLY A STARTING VALUE FOR THIS PARAMETER. _______________________________________________________________

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Table 1e - Listing of Output File EG1.OUT ________________________________________________________ Output from the program FRONTIER (Version 4.1) instruction file = eg1.ins data file = eg1.dta

Error Components Frontier (see B&C 1992) The model is a production function The dependent variable is logged (… initial estimation outputs based on OLS) the final mle estimates are : coefficient standard-error t-ratio

beta 0 0.56161963E+00 beta 1 0.28110205E+00 beta 2 0.53647981E+00 sigma-squared 0.21700046E+00 gamma

0.20261668E+00 0.27718331E+01 0.47643365E-01 0.59001301E+01 0.45251553E-01 0.11855501E+02 0.63909106E-01 0.33954545E+01

0.79720730E+00 0.13642399E+00 0.58436004E+01

mu is restricted to be zero eta is restricted to be zero log likelihood function = -0.17027229E+02 LR test of the one-sided error = 0.28392402E+01 with number of restrictions = 1 [note that this statistic has a mixed chi-square distribution]

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(… more output ) technical efficiency estimates : firm 1 2 3 . . . 58 59 60 eff.-est. 0.65068880E+00 0.82889151E+00 0.72642592E+00

0.66471456E+00 0.85670448E+00 0.70842786E+00

mean efficiency = 0.74056772E+00 _____________________________________________________

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EXAMPLE 2
• (4.2) A Translog production frontier using cross-sectional data and assuming a truncated normal distribution. ln(Qi) = β0 + β1ln(Ki) + β2ln(Li) + β3ln(Ki)2 + β4ln(Li)2 + β5ln(Ki)ln(Li) + (Vi - Ui),

• where Qi, Ki, Li and Vi are as defined earlier, and Ui has truncated normal distribution.

Table 2a - Listing of Data File EG2.DAT _________________________________________________ 1. 1. 12.778 9.416 35.134 2. 1. 24.285 4.643 77.297 3. 1. 20.855 5.095 89.799 . . . 58. 1. 21.358 9.329 87.124 59. 1. 27.124 7.834 60.340 60. 1. 14.105 5.621 44.218 ________________________________________________

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Table 2b - Listing of Shazam Instruction File EG2.SHA __________________________________________________ read(eg2.dat) n t y x1 x2 genr ly=log(y) genr lx1=log(x1) genr lx2=log(x2) genr lx1s=log(x1)*log(x1) genr lx2s=log(x2)*log(x2) genr lx12=log(x1)*log(x2) file 33 eg2.dta write(33) n t ly lx1 lx2 lx1s lx2s lx12 stop __________________________________________________

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Table 2d - Listing of Instruction File EG2.INS ________________________________________________________
1 eg2.dta eg2.out 1 y 60 1 60 5 y n n 1=ERROR COMPONENTS MODEL, 2=TE EFFECTS MODEL DATA FILE NAME OUTPUT FILE NAME 1=PRODUCTION FUNCTION, 2=COST FUNCTION LOGGED DEPENDENT VARIABLE (Y/N) NUMBER OF CROSS-SECTIONS NUMBER OF TIME PERIODS NUMBER OF OBSERVATIONS IN TOTAL NUMBER OF REGRESSOR VARIABLES (Xs) MU (Y/N) [OR DELTA0 (Y/N) IF USING TE EFFECTS MODEL] ETA (Y/N) [OR NUMBER OF TE EFFECTS REGRESSORS (Zs)] STARTING VALUES (Y/N) IF YES THEN BETA0 BETA1 TO BETAK SIGMA SQUARED GAMMA MU [OR DELTA0 ETA DELTA1 TO DELTAK]

NOTE: IF YOU ARE SUPPLYING STARTING VALUES AND YOU HAVE RESTRICTED MU [OR DELTA0] TO BE ZERO THEN YOU SHOULD NOT SUPPLY A STARTING VALUE FOR THIS PARAMETER. _____________________________________________________________

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EXAMPLE 3
• (4.3) A Cobb-Douglas cost frontier using cross-sectional data and assuming a half-normal distribution. ln(Ci/Wi) = β0 + β1ln(Qi) + β2ln(Ri/Wi) + (Vi + Ui)

• where Ci, Qi, Ri and Wi are cost, output, capital price and labour price, respectively, and Vi and Ui are assumed normal and halfnormal distributed, respectively.

Table 3a - Listing of Data File EG3.DAT __________________________________________________ 1. 1. 783.469 35.893 11.925 28.591 2. 1. 439.742 24.322 12.857 23.098 3. 1. 445.813 34.838 14.368 16.564 . . . 58. 1. 216.558 26.888 7.853 10.882 59. 1. 408.234 20.848 9.411 23.281 60. 1. 1114.369 32.514 14.919 29.672 _________________________________________________

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Table 3b - Listing of Shazam Instruction File EG3.SHA _____________________________________________________ read(eg3.dat) n t c q r w genr lcw=log(c/w) genr lq=log(q) genr lrw=log(r/w) file 33 eg3.dta write(33) n t lcw lq lrw stop ________________________________________________________

Table 3c - Listing of Data File EG3.DTA _______________________________________________________ 1.000000 1.000000 3.310640 3.580542 -0.8744549 2.000000 1.000000 2.946442 3.191381 -0.5858576 3.000000 1.000000 3.292668 3.550709 -0.1422282 . . . 58.00000 1.000000 2.990748 3.291680 -0.3262144 59.00000 1.000000 2.864203 3.037258 -0.9057584 60.00000 1.000000 3.625840 3.481671 -0.6875683 _______________________________________________________

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Table 3d - Listing of Instruction File EG3.INS _______________________________________________________________ 1 1=ERROR COMPONENTS MODEL, 2=TE EFFECTS MODEL eg3.dta DATA FILE NAME eg3.out OUTPUT FILE NAME 2 1=PRODUCTION FUNCTION, 2=COST FUNCTION y LOGGED DEPENDENT VARIABLE (Y/N) 60 NUMBER OF CROSS-SECTIONS 1 NUMBER OF TIME PERIODS 60 NUMBER OF OBSERVATIONS IN TOTAL 2 NUMBER OF REGRESSOR VARIABLES (Xs) n MU (Y/N) [OR DELTA0 (Y/N) IF USING TE EFFECTS MODEL] n ETA (Y/N) [OR NUMBER OF TE EFFECTS REGRESSORS (Zs)] n STARTING VALUES (Y/N) IF YES THEN BETA0 BETA1 TO BETAK SIGMA SQUARED GAMMA MU [OR DELTA0 ETA DELTA1 TO DELTAK] NOTE: IF YOU ARE SUPPLYING STARTING VALUES AND YOU HAVE RESTRICTED MU [OR DELTA0] TO BE ZERO THEN YOU SHOULD NOT SUPPLY A STARTING VALUE FOR THIS PARAMETER. ___________________________________________________________

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EXAMPLE 4
• (4.4) The Battese and Coelli (1992) specification (Model 1). ln(Qi) = β0 + β1ln(Ki) + β2ln(Li) + (Vi - Ui) • Data on 15 firms observed over 4 time periods.

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Table 4a - Listing of Data File EG4.DAT ________________________________________________________ _____________ 1. 1. 15.131 9.416 35.134 2. 1. 26.309 4.643 77.297 3. 1. 6.886 5.095 89.799 4. 1. 11.168 4.935 35.698 5. 1. 16.605 8.717 27.878 6. 1. 10.897 1.066 92.174 7. 1. 8.239 0.258 97.907 8. 1. 19.203 6.334 82.084 9. 1. 16.032 2.350 38.876 10. 1. 12.434 1.076 81.761 11. 1. 2.676 3.432 9.476 12. 1. 29.232 4.033 55.096 13. 1. 16.580 7.975 73.130 14. 1. 12.903 7.604 24.350 15. 1. 10.618 0.344 65.380 data for the 2nd and 3rd firm 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 4. 11.583 4. 31.612 4. 12.088 4. 13.736 4. 19.274 4. 15.471 4. 23.190 4. 30.192 4. 23.627 4. 14.128 4. 11.433 4. 4.074 4. 23.314 4. 22.737 4. 22.639 4.551 36.704 7.223 89.312 9.561 29.055 4.871 50.018 9.312 40.996 2.895 63.051 8.085 60.992 8.656 94.159 3.427 39.312 1.918 78.628 6.177 64.377 7.188 1.073 9.329 87.124 7.834 60.340 5.621 44.218

_____________________________________________________________________

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Table 4b - Listing of Shazam Instruction File EG4.SHA _______________________________________________ read(eg4.dat) n t y x1 x2 genr ly=log(y) genr lx1=log(x1) genr lx2=log(x2) file 33 eg4.dta write(33) n t ly lx1 lx2 stop __________________________________________________

Table 4 c - Listing of Data File EG4.DTA __________________________________________________ 1.000000 1.000000 2.716746 2.242410 3.559169 2.000000 1.000000 3.269911 1.535361 4.347655 3.000000 1.000000 1.929490 1.628260 4.497574 . . . 13.00000 4.000000 3.149054 2.233128 4.467332 14.00000 4.000000 3.123994 2.058473 4.099995 15.00000 4.000000 3.119674 1.726510 3.789132 ____________________________________________________

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Table 4d - Listing of Instruction File EG4.INS ________________________________________________________________ _____ 1 1=ERROR COMPONENTS MODEL, 2=TE EFFECTS MODEL eg4.dta DATA FILE NAME eg4.out OUTPUT FILE NAME 1 1=PRODUCTION FUNCTION, 2=COST FUNCTION y LOGGED DEPENDENT VARIABLE (Y/N) 15 NUMBER OF CROSS-SECTIONS 4 NUMBER OF TIME PERIODS 60 NUMBER OF OBSERVATIONS IN TOTAL 2 NUMBER OF REGRESSOR VARIABLES (Xs) y MU (Y/N) [OR DELTA0 (Y/N) IF USING TE EFFECTS MODEL] y ETA (Y/N) [OR NUMBER OF TE EFFECTS REGRESSORS (Zs)] n STARTING VALUES (Y/N) IF YES THEN BETA0 BETA1 TO BETAK SIGMA SQUARED GAMMA MU [OR DELTA0 ETA DELTA1 TO DELTAK] NOTE: IF YOU ARE SUPPLYING STARTING VALUES AND YOU HAVE RESTRICTED MU [OR DELTA0] TO BE ZERO THEN YOU SHOULD NOT SUPPLY A STARTING VALUE FOR THIS PARAMETER. _______________________________________________________________

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EXAMPLE 5
• (4.5) The Battese and Coelli (1995) specification (Model 2). ln(Qi) = β0 + β1ln(Ki) + β2ln(Li) + (Vi - Ui)

Table 5a - Listing of Data File EG5.DAT ______________________________________________________ 1. 1. 15.131 9.416 35.134 1.000 2. 1. 26.309 4.643 77.297 1.000 3. 1. 6.886 5.095 89.799 1.000 . . . 13. 4. 23.314 9.329 87.124 4.000 14. 4. 22.737 7.834 60.340 4.000 15. 4. 22.639 5.621 44.218 4.000 ________________________________________________________

Table 5b - Listing of Shazam Instruction File EG5.SHA ________________________________________________________ read(eg5.dat) n t y x1 x2 z1 genr ly=log(y) genr lx1=log(x1) genr lx2=log(x2) file 33 eg5.dta write(33) n t ly lx1 lx2 z1 stop ________________________________________________________

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Table 5c - Listing of Data File EG5.DTA _____________________________________________________________________ 1.000000 1.000000 2.716746 2.242410 3.559169 1.000000 2.000000 1.000000 3.269911 1.535361 4.347655 1.000000 3.000000 1.000000 1.929490 1.628260 4.497574 1.000000 . . . 13.00000 4.000000 3.149054 2.233128 4.467332 4.000000 14.00000 4.000000 3.123994 2.058473 4.099995 4.000000 15.00000 4.000000 3.119674 1.726510 3.789132 4.000000 _____________________________________________________________________

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Table 5d - Listing of Instruction File EG5.INS ________________________________________________________________ 2 1=ERROR COMPONENTS MODEL, 2=TE EFFECTS MODEL eg5.dta DATA FILE NAME eg5.out OUTPUT FILE NAME 1 1=PRODUCTION FUNCTION, 2=COST FUNCTION y LOGGED DEPENDENT VARIABLE (Y/N) 15 NUMBER OF CROSS-SECTIONS 4 NUMBER OF TIME PERIODS 60 NUMBER OF OBSERVATIONS IN TOTAL 2 NUMBER OF REGRESSOR VARIABLES (Xs) y MU (Y/N) [OR DELTA0 (Y/N) IF USING TE EFFECTS MODEL] 1 ETA (Y/N) [OR NUMBER OF TE EFFECTS REGRESSORS (Zs)] n STARTING VALUES (Y/N) IF YES THEN BETA0 BETA1 TO BETAK SIGMA SQUARED GAMMA MU [OR DELTA0 ETA DELTA1 TO DELTAK] NOTE: IF YOU ARE SUPPLYING STARTING VALUES AND YOU HAVE RESTRICTED MU [OR DELTA0] TO BE ZERO THEN YOU SHOULD NOT SUPPLY A STARTING VALUE FOR THIS PARAMETER. ________________________________________________________________

25

APPENDIX - PROGRAMMER'S GUIDE A.1 The FRONT41.000 File The start-up file FRONT41.000 is listed in Table A1. Ten values may be altered in FRONT41.000. A brief description of each value is provided below. Table A1 - The start-up file FRONT41.000
____________________________________________________________ KEY VALUES USED IN FRONTIER PROGRAM (VERSION 4.1) NUMBER: DESCRIPTION: 5 IPRINT - PRINT INFO EVERY “N” ITERATIONS, 0=DO NOT PRINT 1 INDIC - USED IN UNIDIMENSIONAL SEARCH PROCEDURE - SEE BELOW 0.00001 TOL - CONVERGENCE TOLERANCE (PROPORTIONAL) 0.001 TOL2 - TOLERANCE USED IN UNI-DIMENSIONAL SEARCH PROCEDURE 1.0D+16 BIGNUM - USED TO SET BOUNDS ON DEN & DIST 0.00001 STEP1 - SIZE OF 1ST STEP IN SEARCH PROCEDURE 1 IGRID2 - 1=DOUBLE ACCURACY GRID SEARCH, 0=SINGLE 0.1 GRIDNO - STEPS TAKEN IN SINGLE ACCURACY GRID SEARCH ON GAMMA 100 MAXIT - MAXIMUM NUMBER OF ITERATIONS PERMITTED 1 ITE - 1=PRINT ALL TE ESTIMATES, 0=PRINT ONLY MEAN TE

26

Deregulation, Ownership, and Efficiency Change in Indian Banking: An Application of Stochastic Frontier Analysis

Subal C. Kumbhakar Department of Economics State University of New York Binghamton, NY 13902, USA E-mail: kkar@binghamton.edu and Subrata Sarkar} Indira Gandhi Institute of Development Research Gen. Vaidya Marg, Goregaon (East) Mumbai 400 065, INDIA E-mail: ssarkar@igidr.ac.in

Table 3: Estimated Parameters of the Translog Cost Function and the Simple Time Varying Inefficiency Function Based on Battese and Coelli Model 1
Period: 19862000 All Banks Coeff. T-stat Sigmasquared Gamma Mu Eta 0.024 8.711 Period: 19861992 All Banks Coeff. T-stat 0.021 8.574 Period: 19932000 All Banks Coeff. T-stat 0.022 5.817

0.767 29.750 0.273 0.016 9.598 3.208

0.829 35.034 0.265 0.019 8.534 1.378

0.845 40.456 0.273 0.016 8.915 1.475

Log-L

765.600

396.420

462.75

LR Test for One571.22 sided Error d.o.f 3

292.22

349.54

3

3

y1 = deposits, y2 = advances, y3 = investments, y4 = branches, t = time, wl = relative wage. All variables, except time, are logged as per translog cost function. The second order terms are obvious: y12 = y1*y2; y33 = y3*y3; etc. .

28

Table 4: Mean Efficiency of Public and Private Banks Based on Battese and Coelli Model 1 Mean Efficiency Year All Banks Public Banks 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 0.689 0.693 0.698 0.702 0.706 0.710 0.715 0.719 0.723 0.726 0.730 0.735 0.739 0.743 0.747 0.652 0.656 0.661 0.666 0.670 0.675 0.679 0.683 0.688 0.692 0.696 0.700 0.705 0.709 0.713 Private Banks 0.745 0.746 0.747 0.751 0.754 0.758 0.761 0.765 0.768 0.772 0.775 0.782 0.790 0.793 0.796

Sample: Period 1986-2000, All banks.

29

Table 7: Estimated Parameters of the Inefficiency Function Based on Battese and Coelli Model 2
Model A Model B Model C Model D Intercept Pvt t t*Pvt Dereg Dereg*Pvt Dereg*t Dereg*t*P vt 0.108 -0.223 0.143 -0.155 -0.006 -0.005 0.054 -0.219 0.320 -0.151 -0.060 -0.253 -0.321 0.051 0.023

0.103 -0.035

Bold numbers denote significant coefficients at the 5% level of significance Pvt (private) and Dereg (deregulation), are dummy variables. Dereg=1 if year > 1992. The variable t stands for time, with t=1 for the year 1986. The interaction between the dummy variables time and private is captured by t*Pvt., and Dereg*t*Pvt is a three-way interaction.

30

Figure 2: Average Efficiency of Banks by Ownership Group
1.020 1.000 0.980 0.960 0.940 eff 0.920 0.900 0.880 0.860 0.840 0.820 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Year

vvv

Public Bank

Privete Bank

Average bank performance

31

Deterministic frontier
Y YM

YA

XA

X

Stochastic Frontier
Y

f(x) exp(vi)
YA

vi < 0

XA

X

X2

(X1A,X2A) Actual expenditure

Minimum expenditure Q = 100

Slope = w1/w2

X1

X2

E XA

C A

Z W Q1 = 100 Q2 = 75

B

D

F

X1

Total inefficiency

: BF

Technical inefficiency : BD Allocative inefficiency : DF