Title
frontierhtail fits stochastic production frontier models for heavy tail data
Syntax
frontierhtail depvar [indepvars] [if] [in] [weight] [, options]
options Description ------------------------------------------------------------------------- Model hetero(varlist) independent variable to model the variance constraints(constraints) apply specified linear constraints exposure(varname_e) include ln(varname_e) in model with coefficient constrained to 1 offset(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_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{exp(-u)|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 [U] 20.20 Obtaining robust variance estimates 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 [U] 20.20 Obtaining robust variance estimates 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{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 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