Efficiency Measurement William Greene Stern School of Business New York University

Session 3 The Stochastic Frontier Model

Stochastic Frontier Models  Motivation: Factors not under control of the firm Measurement error Differential rates of adoption of technology  Frontier is randomly placed by the whole collection of stochastic elements which might enter the model outside the control of the firm.  Aigner, Lovell, Schmidt (1977), Meeusen, van den Broeck (1977), Battese, Corra (1977)

The Stochastic Frontier Model u i > 0, but v i may take any value. A symmetric distribution, such as the normal distribution, is usually assumed for v i. Thus, the stochastic frontier is  +  ’x i +v i and, as before, u i represents the inefficiency.

Least Squares Estimation Average inefficiency is embodied in the third moment of the disturbance ε i = v i - u i. So long as E[v i - u i ] is constant, the OLS estimates of the slope parameters of the frontier function are unbiased and consistent. (The constant term estimates α-E[u i ]. The average inefficiency present in the distribution is reflected in the asymmetry of the distribution, which can be estimated using the OLS residuals:

Application to Spanish Dairy Farms InputUnitsMeanStd. Dev. MinimumMaximum MilkMilk production (liters) 131,108 92,539 14,110727,281 Cows# of milking cows Labor# man-equivalent units LandHectares of land devoted to pasture and crops FeedTotal amount of feedstuffs fed to dairy cows (tons) 57,94147,9813, ,732 N = 247 farms, T = 6 years ( )

Example: Dairy Farms

The Normal-Half Normal Model

Normal-Half Normal Variable

Error Decomposition

Standard Form

Battese Coelli Formulation

Estimation: Least Squares/MoM  OLS estimator of β is consistent  E[u i ] = (2/π) 1/2 σ u, so OLS constant estimates α+ (2/π) 1/2 σ u  Second and third moments of OLS residuals estimate

A Possible Problem with the Method of Moments  Estimator of σ u is [m 3 / ] 1/3  Theoretical m 3 is < 0  Sample m 3 may be > 0. If so, no solution for σ u. (Negative to 1/3 power.)

Log Likelihood Function Waldman (1982) result on skewness of OLS residuals: If the OLS residuals are positively skewed, rather than negative, then OLS maximizes the log likelihood, and there is no evidence of inefficiency in the data.

Alternative Models: Half Normal and Exponential

Normal-Exponential Likelihood

Normal-Truncated Normal

Truncated Normal Model: mu=.5

Effect of Differing Truncation Points From Coelli, Frontier4.1 (page 16)

Other Models  Other Parametric Models (we will examine gamma later in the course)  Semiparametric and nonparametric – the recent outer reaches of the theoretical literature  Other variations including heterogeneity in the frontier function and in the distribution of inefficiency

Test for Inefficiency?  Base test on  u = 0 = 0  Standard test procedures Likelihood ratio Wald Lagrange  Nonstandard testing situation: Variance = 0 on the boundary of the parameter space Standard chi squared distribution does not apply.

Estimating u i  No direct estimate of u i  Data permit estimation of y i – β’x i. Can this be used? ε i = y i – β’x i = v i – u i Indirect estimate of u i, using E[u i |v i – u i ] This is E[u i |y i, x i ]  v i – u i is estimable with e i = y i – b’x i.

Fundamental Tool - JLMS We can insert our maximum likelihood estimates of all parameters. Note: This estimates E[u|v i – u i ], not u i.

Other Distributions

Technical Efficiency

Application: Electricity Generation

Estimated Translog Production Frontiers

Inefficiency Estimates

Estimated Inefficiency Distribution

Estimated Efficiency

Confidence Region Horrace, W. and Schmidt, P., Confidence Intervals for Efficiency Estimates, JPA, 1996.

Application (Based on Costs)

Sample Selection Modeling  Switching Models: y*|technology = b t ’x + v –u Firm chooses technology = 0 or 1 based on c’z+e e is correlated with v  Sample Selection Model: Choice of organic or inorganic Adoption of some technological innovation

Early Applications  Heshmati A. (1997), “Estimating Panel Models with Selectivity Bias: An Application to Swedish Agriculture”, International Review of Economics and Business 44(4),  Heshmati, Kumbhakar and Hjalmarsson Estimating Technical Efficiency, Productivity Growth and Selectivity Bias Using Rotating Panel Data: An Application to Swedish Agriculture  Sanzidur Rahman Manchester WP, 2002: Resource use efficiency with self-selectivity: an application of a switching regression framework to stochastic frontier models:

Sample Selection in Stochastic Frontier Estimation Bradford et al. (ReStat, 2000):“... the patients in this sample were not randomly assigned to each treatment group. Statistically, this implies that the data are subject to sample selection bias. Therefore, we utilize a standard Heckman two-stage sample-selection process, creating an inverse Mill’s ratio from a first-stage probit estimator of the likelihood of CABG or PTCA. This correction variable is included in the frontier estimate....” Sipiläinen and Oude Lansink (2005) “Possible selection bias between organic and conventional production can be taken into account [by] applying Heckman’s (1979) two step procedure.”

Two Step Selection Heckman’s method is for linear equations Does not carry over to any nonlinear model The formal estimation procedure based on maximum likelihood estimation – Terza (1998) – general results for exponential models with extensions to other nonlinear models – Greene (2006) – general template for nonlinear models – Greene (2010) – specific result for stochastic frontiers

A Sample Selected SF Model d i = 1[  ′z i + w i > 0], w i ~ N[0,1 2 ] y i =  ′x i +  i,  i ~ N[0,   2 ] (y i,x i ) observed only when d i = 1.  i = v i - u i u i = |  u U i | =  u |U i | where U i ~ N[0,1 2 ] v i =  v V i where V i ~ N[0,1 2 ]. (w i,v i ) ~ N 2 [(0,1), (1,  v,  v 2 )]

Alternative Approach Kumbhakar, Sipilainen, Tsionas (JPA, 2008)

Sample Selected SF Model

Simulated Log Likelihood for a Stochastic Frontier Model The simulation is over the inefficiency term.

2nd Step of the MSL Approach

JLMS Estimator of u i

WHO Efficiency Estimates OECD Everyone Else

WHO Estimates vs. SF Model