### Example 22.7 Stochastic Frontier Models

This example illustrates the estimation of stochastic frontier production and cost models.

First, a production function model is estimated. The data for this example were collected by Christensen Associates; they represent a sample of 125 observations on inputs and output for 10 airlines between 1970 and 1984. The explanatory variables (inputs) are fuel (LF), materials (LM), equipment (LE), labor (LL), and property (LP), and (LQ) is an index that represents passengers, charter, mail, and freight transported.

The following statements create the dataset:

title1 'Stochastic Frontier Production Model';
data airlines;
input TS FIRM NI LQ LF LM LE LL LP;
datalines;
1 1 15 -0.0484 0.2473 0.2335 0.2294 0.2246 0.2124
1 1 15 -0.0133 0.2603 0.2492 0.241 0.2216 0.1069
2 1 15 0.088 0.2666 0.3273 0.3365 0.2039 0.0865
3 1 15 0.1619 0.3019 0.4573 0.3532 0.2346 0.0242

... more lines ...



The following statements estimate a stochastic frontier exponential production model that uses Christensen Associates data:

/*-- Stochastic Frontier Production Model --*/
proc qlim data=airlines;
model LQ=LF LM LE LL LP;
endogenous LQ ~ frontier (type=exponential production);
run;


Output 22.7.1 shows the results from this production model.

Output 22.7.1: Stochastic Frontier Production Model

 Stochastic Frontier Production Model

The QLIM Procedure

Model Fit Summary
Number of Endogenous Variables 1
Endogenous Variable LQ
Number of Observations 125
Log Likelihood 83.27815
Number of Iterations 19
Optimization Method Quasi-Newton
AIC -150.55630
Schwarz Criterion -127.92979
Sigma 0.12445
Lambda 0.55766

Parameter Estimates
Parameter DF Estimate Standard Error t Value Approx
Pr > |t|
Intercept 1 -0.085048 0.024528 -3.47 0.0005
LF 1 -0.115802 0.124178 -0.93 0.3511
LM 1 0.756253 0.078755 9.60 <.0001
LE 1 0.424916 0.081893 5.19 <.0001
LL 1 -0.136421 0.089702 -1.52 0.1283
LP 1 0.098967 0.042776 2.31 0.0207
_Sigma_v 1 0.108688 0.010063 10.80 <.0001
_Sigma_u 1 0.060611 0.017603 3.44 0.0006

Similarly, the stochastic frontier production function can be estimated with (type=half) or (type=truncated) options that represent half-normal and truncated normal production models.

In the next step, stochastic frontier cost function is estimated. The data for the cost model are provided by Christensen and Greene (1976). The data describe costs and production inputs of 145 U.S. electricity producers in 1955. The model being estimated follows the nonhomogenous version of the Cobb-Douglas cost function:

All dollar values are normalized by fuel price. The quadratic log of the output is added to capture nonlinearities due to scale effects in cost functions. New variables, log_C_PF, log_PK_PF, log_PL_PF, log_y, and log_y_sq, are created to reflect transformations. The following statements create the data set and transformed variables:

data electricity;
input Firm Year Cost Output LPrice LShare KPrice KShare FPrice FShare;
datalines;
1  1955  .0820  2.0  2.090  .3164  183.000  .4521  17.9000  .2315
2  1955  .6610  3.0  2.050  .2073  174.000  .6676  35.1000  .1251
3  1955  .9900  4.0  2.050  .2349  171.000  .5799  35.1000  .1852
4  1955  .3150  4.0  1.830  .1152  166.000  .7857  32.2000  .0990
5  1955  .1970  5.0  2.120  .2300  233.000  .3841  28.6000  .3859

... more lines ...


/* Data transformations */
data electricity;
set electricity;
label Firm="firm index"
Year="1955 for all observations"
Cost="Total cost"
Output="Total output"
LPrice="Wage rate"
LShare="Cost share for labor"
KPrice="Capital price index"
KShare="Cost share for capital"
FPrice="Fuel price"
FShare"Cost share for fuel";
log_C_PF=log(Cost/FPrice);
log_PK_PF=log(KPrice/FPrice);
log_PL_PF=log(LPrice/FPrice);
log_y=log(Output);
log_y_sq=log_y**2/2;
run;


The following statements estimate a stochastic frontier exponential cost model that uses Christensen and Greene (1976) data:

/*-- Stochastic Frontier Cost Model --*/
proc qlim data=electricity;
model log_C_PF = log_PK_PF log_PL_PF log_y log_y_sq;
endogenous log_C_PF ~ frontier (type=exponential cost);
run;


Output 22.7.2 shows the results.

Output 22.7.2: Exponential Distribution

 Stochastic Frontier Production Model

The QLIM Procedure

Model Fit Summary
Number of Endogenous Variables 1
Endogenous Variable log_C_PF
Number of Observations 159
Log Likelihood -23.30430
Number of Iterations 21
Optimization Method Quasi-Newton
AIC 60.60860
Schwarz Criterion 82.09093
Sigma 0.30750
Lambda 1.71345

Parameter Estimates
Parameter DF Estimate Standard Error t Value Approx
Pr > |t|
Intercept 1 -4.983211 0.543328 -9.17 <.0001
log_PK_PF 1 0.090242 0.109202 0.83 0.4086
log_PL_PF 1 0.504299 0.118263 4.26 <.0001
log_y 1 0.427182 0.066680 6.41 <.0001
log_y_sq 1 0.066120 0.010079 6.56 <.0001
_Sigma_v 1 0.154998 0.020271 7.65 <.0001
_Sigma_u 1 0.265581 0.033614 7.90 <.0001

Similarly, the stochastic frontier cost model can be estimated with (type=half) or (type=truncated) options that represent half-normal and truncated normal errors.

The following statements illustrate the half-normal option:

/*-- Stochastic Frontier Cost Model --*/
proc qlim data=electricity;
model log_C_PF = log_PK_PF log_PL_PF log_y log_y_sq;
endogenous log_C_PF ~ frontier (type=half cost);
run;


Output 22.7.3 shows the result.

Output 22.7.3: Half-Normal Distribution

 Stochastic Frontier Production Model

The QLIM Procedure

Model Fit Summary
Number of Endogenous Variables 1
Endogenous Variable log_C_PF
Number of Observations 159
Log Likelihood -34.95304
Number of Iterations 22
Optimization Method Quasi-Newton
AIC 83.90607
Schwarz Criterion 105.38840
Sigma 0.42761
Lambda 1.80031

Parameter Estimates
Parameter DF Estimate Standard Error t Value Approx
Pr > |t|
Intercept 1 -4.434634 0.690197 -6.43 <.0001
log_PK_PF 1 0.069624 0.136250 0.51 0.6093
log_PL_PF 1 0.474578 0.146812 3.23 0.0012
log_y 1 0.256874 0.080777 3.18 0.0015
log_y_sq 1 0.088051 0.011817 7.45 <.0001
_Sigma_v 1 0.207637 0.039222 5.29 <.0001
_Sigma_u 1 0.373810 0.073605 5.08 <.0001

The following statements illustrate the truncated normal option:

/*-- Stochastic Frontier Cost Model --*/
proc qlim data=electricity;
model log_C_PF = log_PK_PF log_PL_PF log_y log_y_sq;
endogenous log_C_PF ~ frontier (type=truncated cost);
run;


Output 22.7.4 shows the results.

Output 22.7.4: Truncated Normal Distribution

 Stochastic Frontier Production Model

The QLIM Procedure

Model Fit Summary
Number of Endogenous Variables 1
Endogenous Variable log_C_PF
Number of Observations 159
Log Likelihood -60.32110
Number of Iterations 4
Optimization Method Quasi-Newton
AIC 136.64220
Schwarz Criterion 161.19343
Sigma 0.37350
Lambda 0.70753

Parameter Estimates
Parameter DF Estimate Standard Error t Value Approx
Pr > |t|
Intercept 1 -3.770440 0.839388 -4.49 <.0001
log_PK_PF 1 -0.045852 0.176682 -0.26 0.7952
log_PL_PF 1 0.602961 0.191454 3.15 0.0016
log_y 1 0.094966 0.071124 1.34 0.1818
log_y_sq 1 0.113010 0.012225 9.24 <.0001
_Sigma_v 1 0.304905 0.047868 6.37 <.0001
_Sigma_u 1 0.215728 0.068725 3.14 0.0017
_Mu 1 0.477097 0.116295 4.10 <.0001

If no (Production) or (Cost) option is specified, the stochastic frontier production model is estimated by default.