help frontierhtail-------------------------------------------------------------------------------

Title

frontierhtailfits stochastic production frontier models for heavy tail data

Syntax

frontierhtaildepvar[indepvars] [if] [in] [weight] [,options]

optionsDescription ------------------------------------------------------------------------- Modelhetero(varlist)independent variable to model the varianceconstraints(constraints)apply specified linear constraintsexposure(varname_e)include ln(varname_e) in model with coefficient constrained to 1offset(varname_o)includevarname_oin model with coefficient constrained to 1noconstantsuppress constant termnolrtestreport the model Wald testReporting

level(#)set confidence level; default islevel(95)eformreport exponentiated coefficientsSE/Robust

vce(vcetype)vcetypemay beoim,robust, oropgcluster(varname)adjust standard errors for intragroup correlation; impliesvce(robust)Max options

maximize_optionscontrol the maximization process; seldom used -------------------------------------------------------------------------fweights andpweights are allowed; see weight.byis allowed withfrontierhtail; see[D] byfor more details onby. seepredictbelow for features available after estimation.indepvarsand thehetero(varlist)option may contain factor variables; see fvvarlist.

Description

frontierhtailimplements stochastic production frontier regression for heavy tail data. As pointed out by Nguyen (2010), economic and financial data frequently evidence fat tails.frontierhtailis for use in this case where data evidence heavy tail distribution when estimating stochastic production frontier. The theory behind the commandfrontierhtailis based on the work of Nguyen (2010).frontierhtailestimates 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 defaultfrontierhtailfits a homoscedastic model.

constraints(constraints),exposure(varname_e),offset(varname_o), andnoconstant; see estimation options.

nolrtestindicates that the model significance test should be a Wald test instead of a likelihood-ratio test.+-----------+ ----+ Reporting +--------------------------------------------------------

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

eformspecifies that the coefficient table be displayed in exponentiated form.+-----------+ ----+ SE/Robust +--------------------------------------------------------

vce(vcetype);vcetypemay beoim, observed information matrix (OIM);robust, Huber/White/sandwich estimator; oropg, outer product of the gradient (OPG) vectors. seevce_optionfor more details.

cluster(varname); adjust standard errors for intragroup correlation; impliesvce(robust).+-------------+ ----+ Max options +------------------------------------------------------

maximize_options:difficult,technique(algorithm_spec),iterate(#), []nolog,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). Whereinit_specsspecifies the initial values of the coefficients. See the examples below. The commandfrontierhtailautomatically seeks the initial values of the coefficients but you can indicates your own initial values if you desire with the optioninit(init_specs).

Options forpredict

xb, the default, calculates the linear prediction.

stdpcalculates the standard error of the linear prediction.

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

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

teffproduces estimates of the technical efficiency via E{exp(-u)|e}

residualscalculates the residuals.

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

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

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

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

Saved results

frontierhtailsaves the following ine(). Note that these saved results are the same as those returned by the command[R] maximizesincefrontierhtailis fitted using[R] ml:Scalars

e(N)number of observations; always savede(k)number of parameters; always savede(k_eq)number of equations; usually savede(k_eq_model)number of equations to include in a model Wald test; usually savede(k_dv)number of dependent variables; usually savede(k_autoCns)number of base, empty, and omitted constraints; saved if command supports constra > intse(df_m)model degrees of freedom; always savede(r2_p)pseudo-R-squared; sometimes savede(ll)log likelihood; always savede(ll_0)log likelihood, constant-only model; saved when constant-only model is fite(N_clust)number of clusters; saved whenvce(clusterclustvar)is specified; see[U] 20.20 Obtainingrobust variance estimatese(chi2)chi-squared; usually savede(p)significance of model of test; usually savede(rank)rank ofe(V); always savede(rank0)rank ofe(V)for constant-only model; saved when constant-only model is fite(ic)number of iterations; usually savede(rc)return code; usually savede(converged)1if converged,0otherwise; usually savedMacros

e(cmd)name of command; always savede(cmdline)command as typed; always savede(depvar)names of dependent variables; always savede(wtype)weight type; saved when weights are specified or impliede(wexp)weight expression; saved when weights are specified or impliede(title)title in estimation output; usually saved by commands usingmle(clustvar)name of cluster variable; saved whenvce(clusterclustvar)is specified; see[U] 20.20 Obtainingrobust variance estimatese(chi2type)WaldorLR; type of model chi-squared test; usually savede(vce)vcetypespecified invce(); saved when command allowsvce()e(vcetype)title used to label Std. Err.; sometimes savede(opt)type of optimization; always savede(which)maxormin; whether optimizer is to perform maximization or minimization; always savede(ml_method)type ofmlmethod; always saved by commands usingmle(user)name of likelihood-evaluator program; always savede(technique)fromtechnique()option; sometimes savede(singularHmethod)m-marquardtorhybrid; method used when Hessian is singular; sometimes savede(crittype)optimization criterion; always savede(properties)estimator properties; always savede(predict)program used to implementpredict; usually savedMatrices

e(b)coefficient vector; always savede(Cns)constraints matrix; sometimes savede(ilog)iteration log (up to 20 iterations); usually savede(gradient)gradient vector; usually savede(V)variance-covariance matrix of the estimators; always savede(V_modelbased)model-based variance; only saved whene(V)is neither the OIM nor OPG varianceFunctions

e(sample)marks estimation sample; always saved

ExamplesBefore beginning the estimations, we use the

set more offinstruction to tellStatanot to pause when displaying the output.set more off

We first illustrate the use of the command

frontierhtailwith theStatamanual datasetfrontier1.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

sizevariable with thehetero(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 oflnoutputon 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

frontierhtailcan be used withpredict. 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{exp(-u)|e}, we type.

predict techeff, teff

To calculate the residuals, we type.

predict resids, residuals

You can calculate the other options of the

predictcommand in the same way as above by specifying:predict new_variable_name, option_name.Let's now show how to use the command

frontierhtailwith theStatamanual datasetgreene9.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.

ReferencesNguyen, N. B.: 2010, "Estimation of technical efficiency in stochastic frontier analysis"

Dissertation, Graduate College of Bowling Green StateUniversity. Downloadable at: http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1275444079.

AuthorDiallo Ibrahima Amadou, zavren@gmail.com

Also seeOnline: help for

frontier,xtfrontier,regress