chks
chks: A Stata Command for Frontier Models
chks is a user-written Stata command designed to estimate Nonlinear Index Models and Zero Stochastic Frontier (ZSF) Models.
Development Status & Disclaimer
Please note: This command was developed in 2018 as part of my doctoral studies. It is provided “as-is” and is no longer under active development. While the code has not been updated since its initial release, it may still be useful for estimating basic models.
Getting Started
Install. You can choose from one of the following two methods to install:
From the Stata command window:
net install chks, from(https://luischanci.github.io/chks/) replaceManual 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\).
Syntaxis.
The general syntaxis is,
chks depvariable xregressors, indx(indexvariables) type() estimation() eoption() maxitera() nonconstant robustwhere,
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(). 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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Model Specification
The general model is like,
\[ log(\eta)_i=x_i\beta'+\epsilon_i \]
where, \(\eta\) is a nonlinear index and \(x_{i}\) is a vector of explanatory variables.
To ilustrate, \(\eta\) in a CES specifiction would be like,
\[ \eta=\left(\sum_{m=1}^M{\delta_mY_m^\rho}\right)^{1/\rho} \]
thus, the model to estimate becomes,
\[ log(Y_1)_i=-(1/\rho)*log\left(1+\sum_r^{M-1}{\delta_r (C_r^{\rho}-1)}\right)+x_i\beta'+\epsilon_i \]
with, \(C_r=(Y_r/Y_1)\), and \(r\neq1\).
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Using the code.
Estimation NLS. Let’s say that \(M=3\) and there are \(k\) regressors,
chks Y1 x1 x2 ... xk, indx(Y2 Y3) t(ces) es(nls)Notes: If
indx()is nonempty, variablesY1,...,Ymshould be in levels.On the other hand, if the residual is such that \(\epsilon_i=v_i-u_i\) with, \(v_i\sim\mathcal{N}(0,\sigma^2_v)\) and \(u_i\sim\mathcal{N}^+(0,\sigma^2_u)\), which is bascally a Nonlinear Stochastic Frontier Model, the command for estimation would be:
chks Y1 x1 x2 ... xk, indx(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, indx(Y2 Y3) t(ces) es(zsf)In this case there are two additional options for the estimation method: Maximum Likelihood Extimation (add
eo(ml)) or Expectation-Maximization Algorithm (addeo(em)).
Final comments. Other specifications are: a Cobb-Douglas index or linear functions. For instance, a version of a linear Zero-Inefficiency Stochastic Frontier model would be:
\[y_i=x_i\beta'+\epsilon_i\]
with \(\epsilon_i=v_i-u_i\), and \(v_i\sim\mathcal{N}(0,\sigma^2_v)\) and \(u_i\sim\mathcal{N}^+(0,\sigma^2_u)\). In this case, the command would be:
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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Examples.
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 -----------------------------------------------------------------------------------
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Example 2: Nonlinear index.
Estimation using NLS
``` 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 term, a similar result could be obtained by 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_prob | .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.
- Github: Repository