frontier models and efficiency measurement lab session 2: stochastic frontier
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William Greene Stern School of Business New York University. Frontier Models and Efficiency Measurement Lab Session 2: Stochastic Frontier. 0Introduction 1 Efficiency Measurement 2 Frontier Functions 3 Stochastic Frontiers 4 Production and Cost 5 Heterogeneity 6 Model Extensions - PowerPoint PPT PresentationTRANSCRIPT
Frontier Models and Efficiency Measurement
Lab Session 2: Stochastic Frontier
William GreeneStern School of BusinessNew York University
0 Introduction1 Efficiency Measurement2 Frontier Functions3 Stochastic Frontiers4 Production and Cost5 Heterogeneity6 Model Extensions7 Panel Data8 Applications
Application to Spanish Dairy Farms
Input Units Mean Std. Dev.
Minimum
Maximum
Milk Milk production (liters)
131,108 92,539 14,110 727,281
Cows # of milking cows 2.12 11.27 4.5 82.3
Labor
# man-equivalent units
1.67 0.55 1.0 4.0
Land Hectares of land devoted to pasture and crops.
12.99 6.17 2.0 45.1
Feed Total amount of feedstuffs fed to dairy cows (tons)
57,941 47,981 3,924.14 376,732
N = 247 farms, T = 6 years (1993-1998)
Using Farm Means of the Data
OLS vs. Frontier/MLE
JLMS Inefficiency EstimatorFRONTIER ; LHS = the variable ; RHS = ONE, the variables ; EFF = the new variable $
Creates a new variable in the data set.
FRONTIER ; LHS = YIT ; RHS = X ; EFF = U_i $
Use ;Techeff = variable to compute exp(-u).
Confidence Intervals for Technical Inefficiency, u(i)
Prediction Intervals for Technical Efficiency, Exp[-u(i)]
Prediction Intervals for Technical Efficiency, Exp[-u(i)]
Compare SF and DEA
Similar, but differentwith a crucial pattern
The Dreaded Error 315 – Wrong Skewness
Cost Frontier Model
1 2 K
1 2 K
Cost=C(Output, Input Prices)C = C(Q, P , P ,... P )Frontier ModellogC = logC(Q, P , P ,... P ) + v + u
Linear Homogeneity Restriction
1 2 1 2
0 1 1 2 2 M M
1 2 M
M
C(Q, aP , aP ,... aP ) = aC(Q, P , P ,... P )Cobb-Douglas FormlogC = logP logP ... logP logQHomogeneity: ... 1Normalized CD Cost Function with Homogeneity ImposedlogC/P =
M M
0
1 1 M 2 2 M M-1 M-1 M
log(P /P ) log(P /P ) ... (P /P ) + logQ
Translog vs. Cobb Douglas
M 0
1 1 M 2 2 M M-1 M-1 M
2111 1 M 222
Normalized TranslogCost Function with Homogeneity ImposedlogC/P = log(P /P ) log(P /P ) ... (P /P ) + logQ +
log (P /P ) Q
2 21 12 M M-1,M-1 M-1 M2 2
12 1 M 2 M
21QQ 2
1 1 M 2 2 M
log (P /P ) ... log (P /P ) +
log(P /P )log(P /P ) ... (all unique cross products)
log
log(P /P )logQ log(P /P )logQ
Q
M-1 M-1 M ... log(P /P )logQ
Cost Frontier Command
FRONTIER ; COST; LHS = the variable
; RHS = ONE, the variables ; TechEFF = the new variable
$
ε(i) = v(i) + u(i) [u(i) is still positive]
Estimated Cost Frontier: C&G
Cost Frontier Inefficiencies
Normal-Truncated NormalFrontier Command
FRONTIER ; COST; LHS = the variable
; RHS = ONE, the variables; Model = Truncation
; EFF = the new variable $ ε(i) = v(i) +/- u(i)
u(i) = |U(i)|, U(i) ~ N[μ,2] The half normal model has μ = 0.
Observations about Truncation Model
Truncation Model estimation is often unstable Often estimation is not possible When possible, estimates are often wild
Estimates of u(i) are usually only moderately affected
Estimates of u(i) are fairly stable across models (exponential, truncation, etc.)
Truncated Normal Model ; Model = T
Truncated Normal vs. Half Normal
Multiple Output Cost Function
1 2 L 1 2 M 1 2 L 1 2 M
0 1 1 2 2 M M 1 l
1 2 M
C(Q ,Q ,...,Q , aP , aP ,... aP ) = aC(Q ,Q ,...,Q , P , P ,... P )Cobb-Douglas Form
logC = logP logP ... logP logQHomogeneity: ... 1Normalized CD Multiple Output Cost
Ll l
M 0
1 1 M 2 2 M M-1 M-1 M
1 l
Function with HomogeneitylogC/P = log(P /P ) log(P /P ) ... (P /P ) +
logQ
Ll l
Ranking Observations
CREATE ; newname = Rnk ( Variable ) $
Creates the set of ranks. Use in any subsequent analysis.