4/9/2014. Outline for Stochastic Frontier Analysis. Stochastic Frontier Production Function. Stochastic Frontier Production Function

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1 Productivity Measurement Mini Course Stochastic Frontier Analysis Prof. Nicole Adler, Hebrew University Outline for Stochastic Frontier Analysis Stochastic Frontier Production Function Production function Error term Example: After school programs in Texas (Charnes et al. 1981) Computational and other issues with SFA 2 Stochastic Frontier Production Function Aigner, Lovell & Schmidt (1977) and Meeusen & van den Broeck (1977) independently proposed: y x u, i,...,n ln 0 i i i i 1 i.i.d normal random variable with mean 0 and constant variance i.i.d. function with non-negative random variables 3 1

2 Stochastic Frontier Production Function 4 SFA vs. COLS Alternative functional forms 2

3 Properties of functional forms Flexibility Linear in parameters Regularity Translog is homogeneous of degree r if: Generalized Leontief function is concave if: Parsimony KISS Cobb-Douglas is simple but assumes: Elasticities are constant Returns to scale are constant Elasticity of substitution is unity If data contains zeros, following not relevant: Cobb-Douglas Translog After choosing model, need to check: residuals Hypothesis testing Measures of goodness-of-fit Composed error term Stochastic frontier approach: Error estimation: y x u, i,...,n ln 0 i i i i 1 3

4 Impact of assumptions on estimators Maximum likelihood estimation to solve: use Taylor approximation: which is an iterative process that starts with a guess and may not converge Charnes et al Illustration contd. Input: > msfa<-sfa(matrix(log(x)),matrix(log(y))) > msfa Output Coefficients: (Intercept) x

5 Noise vs. Inefficiency Input: > msfa$lambda Output: Therefore, 59% of the variation is due to inefficiency and 41% is explained by random variation Charnes et al Illustration with STATA one output, multiple inputs. frontier lnread lnmomedu lnoccup lnvisits lntime lnteachers Stoc. frontier normal/half-normal model Number of obs = 70 Wald chi2(5) = Log likelihood = Prob > chi2 = lnread Coef. Std. Err. z P> z [95% Conf. Interval] lnmomedu lnoccup lnvisits lntime lnteachers _cons /lnsig2v /lnsig2u sigma_v sigma_u sigma lambda Likelihood-ratio test of sigma_u=0: chibar2(01) = 2.48 Prob>=chibar2 = Firm specific efficiency therefore need to know u k what do we know? one equation and two unknowns 5

6 Density Estimating individual inefficiency If k is large (>0), what do we know about u k? If k is small (<0), what do we know about u k? Using Bayes theorem, conditional on k, we estimate u k accordingly: where, based on example of a half-normal distribution: we can then compute the expected value of u, EV(u/ ) Conditional expectations given the many estimation methods proposed to date, the one most frequently applied is: which minimizes the mean square error Expected values of technical efficiency for SFA: for plot: > c81<-sfa(log(x), log(y)) > tesfa<-te.sfa(c81) > hist(tesfa,xlim=c(0.5,1),main="",xlab="efficiency", col="gray",cex.lab=1.25,freq=f,breaks=10) > lines(density(tesfa,from=0,to=1),lwd=2) Efficiency 6

7 Relationship between Output (Reading Ability) and Efficiency >plot(c81$y1,tebc,xlab="reading_ability",ylab="efficiency") > lines(lowess(c81$y1,tebc),lwd=1.5) How do DEA & SFA efficiencies compare? for SFA: > c81<-sfa(log(x), log(y)) > tesfa<-te.sfa(c81) for DEA: > c81<-dea(x,y,orientation="out") > tedea<-1/eff(c81) for diagonal line: > abline(0,1,lty="dotted") for plot line: > lines(lowess(tedea,tesfa),lty="solid",lwd=2) Comparing DEA, SFA & COLS 7

8 Stochastic Frontier Analysis (Stone (2002)) SFA has a theoretically imaginative approach that raises the evergreen question of realism. A concept seeded by Farrell (1957) and developed for the single-output case by Aigner et al. (1977), SFA is concerned that uncontrollable variation in output is confounded with inefficiency by deterministic techniques like DEA. The problem it then faces is how to separate these two contributions to the deviation of each unit from the supposed true efficiency frontier. The delicacy and lack of robustness to assumptions of any method of doing this poses a significant challenge to reality. 22 Problems with SFA (Stone (2002)) Anyone adopting this approach must ignore errors in outputs (which would introduce the identifiability problems that are associated with the inevitable errors in both the dependent variable x and the independent variables y). make an arbitrary choice of the joint distribution of u and v, on which the method relies heavily, when the data is (typically) insufficiently informative. assume a functional form for production function also think about environmentals, (true for all methods). 23 Limitations in Stochastic Frontier Analysis Parametric approach that requires many assumptions: Production function Technical inefficiency function Problems with computational solvability and potential endogeneity bias Statistically complicated to solve, although standardized software has helped to some extent Multi-input and multi-output setting has just begun, with the stochastic distance approach to be discussed next 24 8