help frontierhtail

Title

frontierhtail fits stochastic production frontier models for heavy tail data

Syntax

frontierhtail
depvar [indepvars]
[if] [in] [weight]
[, options]

Model
hetero(varlist) independent
variable to model the variance

constraints(estimation options##constraints():constraints) apply specified linear constraints

exposure(varname:varname_e) include ln(varname_e) in
model with coefficient constrained to 1

offset(varname:varname_o) include varname_o in model with
coefficient constrained to 1

noconstant suppress constant term

nolrtest report the model Wald test

Reporting
level(#) set confidence level; default is level(95)

eform report exponentiated coefficients

SE/Robust
vce(vcetype) vcetype
may be oim, robust, or opg

cluster(varname) adjust
standard errors for intragroup correlation; implies vce(robust)

Max options
maximize_options control
the maximization process; seldom used

fweights and pweights are allowed;
see weight.

by is allowed with frontierhtail; see [D] by for more details on by.

see predict below for features available after estimation.

indepvars and the hetero(varlist) option may contain factor variables; see fvvarlist.

Description

frontierhtail implements stochastic production frontier regression for heavy tail data.
As pointed out by Nguyen (2010), economic and financial data frequently evidence fat tails.
frontierhtail is for use in this case where data evidence heavy tail distribution when
estimating stochastic production frontier. The theory behind the command frontierhtail
is based on the work of Nguyen (2010). frontierhtail estimates a linear model (both
dependent and independent variables must be in logarithmic form) where the disturbance is
supposed to be a mixture of two components: the first, the random shock, is assumed to
follow a normal distribution and the, second, the technical inefficiency, is uniformly distributed.

Options

Model

hetero(varlist) specifies variables to model
heteroscedasticity in the idiosyncratic error. By default frontierhtail fits a
homoscedastic model.

constraints(constraints),
"exposure(varname:varname_e)",
offset(varname_o), and
noconstant;
see estimation options.

nolrtest indicates that the model significance test should be a Wald
test instead of a likelihood-ratio test.

Reporting

level(#); set confidence level; default is level(95).

eform specifies that the coefficient table be
displayed in exponentiated form.

SE/Robust

vce(vcetype); vcetype may be oim, observed information matrix (OIM);
robust, Huber/White/sandwich estimator; or opg, outer product of the gradient
(OPG) vectors. see vce_option for more details.

cluster(varname); adjust standard errors for intragroup correlation; implies vce(robust).

Max options

maximize_options:
difficult,
technique(algorithm_spec),
iterate(#),
[no]log,
trace,
gradient,
showstep,
hessian,
showtolerance,
tolerance(#),
ltolerance(#),
nrtolerance(#),
nonrtolerance;
see [R] maximize.
These options are seldom used.

In addition to these maximization options, you can specify the initial values with the option
init(init_specs). Where init_specs specifies the initial values of the coefficients.
See the examples below. The command frontierhtail automatically seeks the initial values of
the coefficients but you can indicates your own initial values if you desire with the option
init(init_specs).

Options for predict

xb, the default, calculates the linear prediction.

stdp calculates the standard error of the linear prediction.

inef produces estimates of the technical inefficiency via E(u|e)

mode produces estimates of the technical inefficiency via the mode M(u|e)

teff produces estimates of the technical efficiency via E

residuals calculates the residuals.

lnsigma calculates the logarithm of the parameter sigma in v~N(0,s^2).

sigma calculates the value of the parameter sigma in v~N(0,s^2).

lntheta calculates the logarithm of the parameter theta in u~Uniform(0,t).

theta calculates the value of the parameter theta in u~Uniform(0,t).

Saved results

frontierhtail saves the following in e(). Note that these saved results are the same as those
returned by the command [R] maximize since frontierhtail is fitted using [R] ml:

Scalars

e(N) number of observations; always saved

e(k) number of parameters; always saved

e(k_eq) number of equations; usually saved

e(k_eq_model) number of equations to include in a model Wald
test; usually saved

e(k_dv) number of dependent variables; usually saved

e(k_autoCns) number of base, empty, and omitted constraints; saved if command supports constra
> ints

e(df_m) model degrees of freedom; always saved

e(r2_p) pseudo-R-squared; sometimes saved

e(ll) log likelihood; always saved

e(ll_0) log likelihood, constant-only model; saved when
constant-only model is fit

e(N_clust) number of clusters; saved when vce(cluster
clustvar) is specified;
see

e(chi2) chi-squared; usually saved

e(p) significance of model of test; usually saved

e(rank) rank of e(V); always saved

e(rank0) rank of e(V) for constant-only model; saved
when constant-only model is fit

e(ic) number of iterations; usually saved

e(rc) return code; usually saved

e(converged) 1 if converged, 0 otherwise; usually saved

Macros

e(cmd) name of command; always saved

e(cmdline) command as typed; always saved

e(depvar) names of dependent variables; always saved

e(wtype) weight type; saved when weights are specified or
implied

e(wexp) weight expression; saved when weights are specified or
implied

e(title) title in estimation output; usually saved by commands using ml

e(clustvar) name of cluster variable; saved when
vce(cluster clustvar) is specified;
see

e(chi2type) Wald or LR; type of model chi-squared
test; usually saved

e(vce) vcetype specified in vce(); saved when command
allows vce()

e(vcetype) title used to label Std. Err.; sometimes saved

e(opt) type of optimization; always saved

e(which) max or min; whether optimizer is to perform
maximization or minimization; always saved

e(ml_method) type of ml method; always saved by commands
using ml

e(user) name of likelihood-evaluator program; always saved

e(technique) from technique() option; sometimes saved

e(singularHmethod) m-marquardt or hybrid; method used
when Hessian is singular; sometimes saved

e(crittype) optimization criterion; always saved

e(properties) estimator properties; always saved

e(predict) program used to implement predict; usually
saved

Matrices

e(b) coefficient vector; always saved

e(Cns) constraints matrix; sometimes saved

e(ilog) iteration log (up to 20 iterations); usually saved

e(gradient) gradient vector; usually saved

e(V) variance-covariance matrix of the estimators; always
saved

e(V_modelbased) model-based variance; only saved when e(V)
is neither the OIM nor OPG variance

Functions

e(sample) marks estimation sample; always saved

Examples

Before beginning the estimations, we use the set more off instruction to tell
Stata not to pause when displaying the output.

set more off

We first illustrate the use of the command frontierhtail with the Stata
manual dataset frontier1.

use http://www.stata-press.com/data/r11/frontier1, clear

We estimate a Cobb-Douglas production function by regressing log output on
log labor and log capital.

frontierhtail lnoutput lnlabor lncapital

To obtain White-corrected standard errors, we specify the vce(robust) option.

frontierhtail lnoutput lnlabor lncapital, vce(robust)

If we do not want to have a constant and the display of the iterations
log at the beginning of the regression, we type.

frontierhtail lnoutput lnlabor lncapital, nocons nolog

We can specify variables to model heteroscedasticity in the idiosyncratic error. To do this
use the size variable with the hetero(varlist) option.

frontierhtail lnoutput lnlabor lncapital, hetero(size)

If we want to estimate a Cobb-Douglas production function with constant
returns-to-scale, we type.

constraint 1 _b[lnlabor] + _b[lncapital] = 1

frontierhtail lnoutput lnlabor lncapital, constraints(1)

If we want to specify our own initial values instead of using those automatically provided
by the command frontierhtail, we proceed as follows. First, we run an OLS regression of
lnoutput on a constant.

regress lnoutput

Then we put the constant value in the local macro b0.

local b0 = _b[_cons]

Finally, we specify the init(init_specs) option as follows.

frontierhtail lnoutput lnlabor lncapital, init(/xb=`b0')

It is important to note that for the intital values, we give only one value to the
equation /xb.

Let's now illustrate how frontierhtail can be used with predict. First,
we calculate the fitted values of the dependent variable.

frontierhtail lnoutput lnlabor lncapital

predict lnoutputhat, xb

To calculate the standard error of the linear prediction, we type.

predict serlp, stdp

To calculate the technical inefficiency via E(u|e), we type.

predict etechinef, inef

To calculate the technical inefficiency via the mode M(u|e), we type.

predict mtechinef, mode

To calculate the technical efficiency via E, we type.

predict techeff, teff

To calculate the residuals, we type.

predict resids, residuals

You can calculate the other options of the predict command in the
same way as above by specifying: predict new_variable_name, option_name.

Let's now show how to use the command frontierhtail with the Stata
manual dataset greene9.

use http://www.stata-press.com/data/r11/greene9, clear

We estimate a Cobb-Douglas production function by regressing log value added on
log capital and log labor. We specify the option technique(dfp) to obtain convergence.

frontierhtail lnv lnk lnl, technique(dfp)

If we want to test the constant returns-to-scale hypothesis on this model, we type.

test _b[lnk] + _b[lnl] = 1

This result shows that we cannot reject the null hypothesis of constant returns-to-scale
technology in this model.

References

Nguyen, N. B.: 2010, "Estimation of technical efficiency in stochastic frontier analysis"
Dissertation, Graduate College of Bowling Green State University.
Downloadable at: http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1275444079.

Author

Diallo Ibrahima Amadou, zavren@gmail.com

Also see

Online: help for frontier, xtfrontier, regress