chks

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.

(Home)

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/")`
 	
```
- 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\).
Syntaxis.

The general syntaxis is,

chks depvariable xregressors, indx(indexvariables) type() estimation() eoption()

where,

  - `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: CES `type(ces)` and Cobb-Douglas `type(cd)`.

  - `estimation()` is the estimation method: NLS `estimation(nls)`, Stochastic Frontier `estimation(sf)`, or Zero-Stochastic Frontier `estimation(zsf)`.

  - `eoption()` is the estimation option when estimation is ZSF. It could be Maximum Likelihood Estimation `eoption(ml)` or Expectation-Maximization Algorithm `eoption(em)`. EM is the default option.

  - `maxitera()` specifies the maximum number of iterations; the default is 500.

Models.
- The general model is one in which a nonlinear index ( $\eta$ ) is a function of a vector of explanatory variables (x) and a residual term ( $\epsilon$ ):
  
  $log(\eta)_{it}=\mathbf{x}_{it}^{'}\mathbf{\beta}+\epsilon_{it}$
where,
```
 ![formula](https://render.githubusercontent.com/render/math?math=\eta=\left(\sum_{m=1}^M{\delta_mY_m^\rho}\right)^{1/\rho})
```
thus, the equation to estimate is:

Using the code.

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 $\epsilon_{it}=v_{it}-u_{it}$ , with $v_{it}\sim\mathcal{N}(0,\sigma^2_v)$ and $u_{it}\sim\mathcal{N}^+(0,\sigma^2_u)$ , 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 (add eo(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:

$y_{it}=\mathbf{x}_{it}\mathbf{\beta}+\epsilon_{it}$

with $\epsilon_{it}=v_{it}-u_{it}$ ; with $v_{it}\sim\mathcal{N}(0,\sigma^2_v)$ and $u_{it}\sim\mathcal{N}^+(0,\sigma^2_u)$ . 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).

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 + e

Command: chks y x, es(zsf) for EM-algorithm and chks 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
``` ------ </br>

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.

</br>

Website

chks

Author

Luis Chancí

luischanci@santotomas.cl