User-written Stata command. Nonlinear index and Zero-Inefficiency Stochastic Frontier Model. This code is a beta version and it's been developed for the working paper Chancí, Kumbhakar, and Sandoval, 2019.
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Install. You can choose from one of the following two methods to install:
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From the Stata command window:
`net install chks, from("https://luischanci.github.io/chks/")` -
Manual installation: Download, unzip, and locate all the files into the Stata ado folder (for instance, locate the unzipped ado and other files into
C:\ado\personal\c\).
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Syntaxis.
The general syntaxis is,
chks depvariable xregressors, indx(indexvariables) type() estimation() eoption()where,
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indx()varlist for the index.indx()could be empty, which means that the model is linear rather than a nonlinear index. -
type(). Functional form for the nonlinear index. There are two types: CEStype(ces)and Cobb-Douglastype(cd). -
estimation()is the estimation method: NLSestimation(nls), Stochastic Frontierestimation(sf), or Zero-Stochastic Frontierestimation(zsf). -
eoption()is the estimation option when estimation is ZSF. It could be Maximum Likelihood Estimationeoption(ml)or Expectation-Maximization Algorithmeoption(em). EM is the default option. -
maxitera()specifies the maximum number of iterations; the default is 500.
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Models.
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The general model is one in which a nonlinear index (
) is a function of a vector of explanatory variables (x) and a residual term (
):
where,
thus, the equation to estimate is:
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Estimation NLS. Let's say that
$M=3$ and there are$k$ regressors,chks Y1 x1 x2 ... xk, idx(Y2 Y3) t(ces) es(nls) -
On the other hand, if the residual is such that
, with
and
, similar to a Nonlinear Stochastic Frontier Model, the command for estimation would be:
chks Y1 x1 x2 ... xk, idx(Y2 Y3) t(ces) es(sf) -
Furthermore, if there is a probability that some
$u_i=0$ , known as Zero-Inefficiency Stochastic Frontier model (see Kumbhakar, Parmeter and Tsionas, 2013, "A zero inefficiency stochastic frontier model", in Journal of Econometrics , 172(1), 66-76), the command would be:chks Y1 x1 x2 ...xk, idx(Y2 Y3) t(ces) es(zsf)In this case there are two additional options: Maximum Likelihood Extimation (add
eo(ml)) or Expectation-Maximization Algorithm (addeo(em)). -
Other models or possibilities are simple variations, such as a Cobb-Douglas index, or linear functions. For instance, a version of a linear Zero-Inefficiency Stochastic Frontier model would be:
with
chks y1 x1 x2, es(zsf)
Finally, it is possible to report the robust standard errors (add robust) or omit the constant term (add nocons).
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Example 1: A Linear Z-SF.
Data (simulation). To illustrate the use of the command, I am going to use a simulation proposed by Diego Restrepo (of course, any mistake would be my responsability). The code for a cost function would be,
clear all set obs 1000 gen x = rnormal()/10 gen v = rnormal()/10 gen z = rnormal() gentrun double u, left(0) /* Need module GENTRUN */ replace u = u/10 /* u ~ Truncated-left N(0,0.1)*/ replace u = 0 if _n > _N-_N/2 /* For a p=50%, u=0, no inefficiency*/ gen e = v - u /* For production function (not cost)*/ gen y = 1 + x + eCommand:
chks y x, es(zsf)for EM-algorithm andchks y x, es(zsf) eo(ml)for MLE.Results using MLE:
. chks y x, es(zsf) eo(ml) Zero Stochastic Frontier for linear models (ZSF, linear model) . . . Iteration 12: f(p) = 715.46477 Iteration 13: f(p) = 715.46477 (Zero) Stochastic Frontier, linear model. M.L.E. ----------------------------------------------------------------------------------- y | Coef. Std. Err. z P>|z| [95% Conf. Interval] ------------------+---------------------------------------------------------------- x | 1.020781 .0376483 27.11 0.000 .9469912 1.09457 _cons | .9688054 .0050194 193.01 0.000 .9589675 .9786434 lnsigma_u | -1.677142 .3815109 -4.40 0.000 -2.424889 -.9293944 lnsigma_v | -2.170457 .0289882 -74.87 0.000 -2.227272 -2.113641 logist_probabil~y | 3.404257 1.090584 3.12 0.002 1.266751 5.541763 -----------------------------------------------------------------------------------Results using EM-algorithm:
. chks y x, es(zsf) Zero Stochastic Frontier for linear models (ZSF, linear model) . . . (Zero) Stochastic Frontier, linear model. EM-Algorithm. ----------------------------------------------------------------------------------- y | Coef. Std. Err. z P>|z| [95% Conf. Interval] ------------------+---------------------------------------------------------------- x | 1.021142 .0367627 27.78 0.000 .9490882 1.093195 _cons | .9699379 .0036587 265.10 0.000 .9627669 .9771089 lnsigma_u | -1.775768 .1535916 -11.56 0.000 -2.076802 -1.474734 lnsigma_v | -2.175327 .0227074 -95.80 0.000 -2.219833 -2.130821 logist_probabil~y | 3.07854 .1541839 19.97 0.000 2.776345 3.380734 ----------------------------------------------------------------------------------- -
Example 2: Non-linear index.
use https://luischanci.github.io/chks/chksdata1, clear chks Y1 x1 x2, indx(Y2 Y3) es(nls) t(ces) r Iteration 16: f(p) = 12.058803 ------------------------------------------------------------------------------ Y1 | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- x1 | .0158278 .0006196 25.54 0.000 .0146134 .0170422 x2 | .0401204 .0008423 47.63 0.000 .0384695 .0417712 _cons | 2.894777 .0180462 160.41 0.000 2.859407 2.930146 Y2 | .225702 .012979 17.39 0.000 .2002636 .2511403 Y3 | .5136438 .0136537 37.62 0.000 .4868831 .5404046 rho | 2.301384 .1146792 20.07 0.000 2.076617 2.526151 ------------------------------------------------------------------------------In this particular example, in which there is a nonlinear function without additional especifications in the residual component, similar results can be obtained using NLS in Stata:
g ly1 = ln(Y1) g ry2 = Y2/Y1 g ry3 = Y3/Y1 nl (ly1 = -(1/{rho = 0.5})*ln( 1 + /// {delta2 = 0.3}*(ry2^{rho}-1) + /// {delta3 = 0.3}*(ry3^{rho}-1) ) /// + ({b0}+{b1}*x1+{b2}*x2) ), r Iteration 6: residual SS = 12.0588 Iteration 7: residual SS = 12.0588 Nonlinear regression Number of obs = 1,000 R-squared = 0.9839 Adj R-squared = 0.9838 Root MSE = .1101435 Res. dev. = -1580.083 ------------------------------------------------------------------------------ | Robust ly1 | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- /rho | 2.301383 .1085109 21.21 0.000 2.088447 2.51432 /delta2 | .225702 .0124839 18.08 0.000 .2012041 .2501999 /delta3 | .5136438 .0134125 38.30 0.000 .4873237 .5399639 /b0 | 2.894777 .0180631 160.26 0.000 2.85933 2.930223 /b1 | .0158278 .0006212 25.48 0.000 .0146087 .0170469 /b2 | .0401204 .0008432 47.58 0.000 .0384657 .041775 ------------------------------------------------------------------------------ Parameter b0 taken as constant term in model -
Example 3: Non-linear (Index) Stochastic Frontier.
chks Y1 x1 x2, indx(Y2 Y3) es(sf) t(ces) r Iteration 34: f(p) = 793.27611 ------------------------------------------------------------------------------ Y1 | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- x1 | .0155817 .0006264 24.87 0.000 .014354 .0168095 x2 | .0392952 .0009681 40.59 0.000 .0373978 .0411925 _cons | 2.993987 .0264685 113.11 0.000 2.94211 3.045865 Y2 | .2249359 .0125398 17.94 0.000 .2003583 .2495134 Y3 | .5125146 .0132849 38.58 0.000 .4864766 .5385526 rho | 2.296911 .1106526 20.76 0.000 2.080036 2.513786 lnsigma_u | -2.206232 .1435049 -15.37 0.000 -2.487496 -1.924967 lnsigma_v | -2.435302 .0812527 -29.97 0.000 -2.594555 -2.27605 ------------------------------------------------------------------------------ -
Example 4: Non-linear (Index). Zero Stochastic Frontier.
chks Y1 x1 x2, indx(Y2 Y3) t(ces) es(zsf) eo(em)
```
------------------------------------------------------------------------------
Y1 | Coef. Std. Err. z P>|z| [95% Conf. Interval]
------------------------------------------------------------------------------
x1 | .0155953 .0005609 27.81 0.000 .014496 .0166946
x2 | .0392709 .0007659 51.27 0.000 .0377697 .0407721
_cons | 2.935441 .0167206 175.56 0.000 2.902669 2.968213
Y2 | .2252023 .0124254 18.12 0.000 .2008489 .2495557
Y3 | .5117927 .0132434 38.65 0.000 .4858362 .5377493
rho| 2.293806 .1025533 22.37 0.000 2.092805 2.494807
lnsigma_u | -2.299401 .0789157 -29.14 0.000 -2.454073 -2.144729
lnsigma_v | -2.340403 .0252419 -92.72 0.000 -2.389876 -2.29093
logist_probability | .5607503 .0657478 8.53 0.000 .4318871 .6896136
------------------------------------------------------------------------------
di 1/(1+exp(-.5607503))
0.63662613
```
Final notes:
- It's a draft (2019). Please provide me with any comments you may have on bugs, wording, inconsistencies, etc.
- All files available here are for education and/or research purposes ONLY. The code within this repository comes with no guarantee, the use of this code is your responsibility.