{smcl} {cmd:help frontierhtail} {hline} {title:Title} {pstd} {hi:frontierhtail} fits stochastic production frontier models for heavy tail data {title:Syntax} {phang2} {cmd:frontierhtail} {depvar} [{indepvars}] {ifin} {weight} [{cmd:,} {it:options}] {synoptset 26 tabbed}{...} {synopthdr} {synoptline} {syntab:Model} {synopt: {opth het:ero(varlist)}}independent variable to model the variance {p_end} {synopt: {opth c:onstraints(estimation options##constraints():constraints)}}apply specified linear constraints{p_end} {synopt :{opth e:xposure(varname:varname_e)}}include ln({it:varname_e}) in model with coefficient constrained to 1{p_end} {synopt :{opth off:set(varname:varname_o)}}include {it:varname_o} in model with coefficient constrained to 1{p_end} {synopt :{opt noc:onstant}}suppress constant term{p_end} {synopt: {cmd:nolrtest}}report the model Wald test{p_end} {syntab:Reporting} {synopt :{opt l:evel(#)}}set confidence level; default is {cmd:level(95)}{p_end} {synopt :{opt ef:orm}}report exponentiated coefficients {p_end} {syntab:SE/Robust} {synopt :{opth vce(vcetype)}}{it:vcetype} may be {opt oim}, {opt r:obust}, or {opt opg}{p_end} {synopt :{opth cl:uster(varname)}}adjust standard errors for intragroup correlation; implies {cmd:vce(robust)}{p_end} {syntab:Max options} {synopt :{it:{help maximize:maximize_options}}}control the maximization process; seldom used{p_end} {synoptline} {phang} {opt fweight}s and {opt pweight}s are allowed; see {help weight}. {p_end} {p 4 6 2} {cmd:by} is allowed with {hi:frontierhtail}; see {manhelp by D} for more details on {cmd:by}.{p_end} {p 4 6 2} see {hi:predict} below for features available after estimation. {p_end} {p 4 6 2} {it:indepvars} and the {opth hetero(varlist)} option may contain factor variables; see {help fvvarlist}. {p_end} {title:Description} {pstd} {cmd:frontierhtail} implements stochastic production frontier regression for heavy tail data. As pointed out by Nguyen (2010), economic and financial data frequently evidence fat tails. {cmd:frontierhtail} is for use in this case where data evidence heavy tail distribution when estimating stochastic production frontier. The theory behind the command {cmd:frontierhtail} is based on the work of Nguyen (2010). {cmd: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. {title:Options} {dlgtab:Model} {phang} {opth hetero(varlist)} specifies variables to model heteroscedasticity in the idiosyncratic error. By default {cmd:frontierhtail} fits a homoscedastic model. {phang} {opt constraints(constraints)}, {opth "exposure(varname:varname_e)"}, {opt offset(varname_o)}, and {opt noconstant}; see {help estimation options}. {phang} {cmd:nolrtest} indicates that the model significance test should be a Wald test instead of a likelihood-ratio test. {dlgtab:Reporting} {phang} {opt level(#)}; set confidence level; default is {cmd:level(95)}. {phang} {opt eform} specifies that the coefficient table be displayed in exponentiated form. {dlgtab:SE/Robust} {phang} {opt vce(vcetype)}; {it:vcetype} may be {opt oim}, observed information matrix (OIM); {opt r:obust}, Huber/White/sandwich estimator; or {opt opg}, outer product of the gradient (OPG) vectors. see {it:{help vce_option}} for more details. {phang} {opth cluster(varname)}; adjust standard errors for intragroup correlation; implies {cmd:vce(robust)}. {dlgtab:Max options} {phang} {it:maximize_options}: {opt dif:ficult}, {opt tech:nique(algorithm_spec)}, {opt iter:ate(#)}, [{cmdab:no:}]{opt lo:g}, {opt tr:ace}, {opt grad:ient}, {opt showstep}, {opt hess:ian}, {opt showtol:erance}, {opt tol:erance(#)}, {opt ltol:erance(#)}, {opt nrtol:erance(#)}, {opt nonrtol:erance}; see {manhelp maximize R}. These options are seldom used. {phang2} In addition to these maximization options, you can specify the initial values with the option {opt init(init_specs)}. Where {it:init_specs} specifies the initial values of the coefficients. See the examples below. The command {cmd:frontierhtail} automatically seeks the initial values of the coefficients but you can indicates your own initial values if you desire with the option {opt init(init_specs)}. {title:Options for {help predict}} {p 4 8 2}{cmd:xb}, the default, calculates the linear prediction. {p_end} {p 4 8 2}{cmd:stdp} calculates the standard error of the linear prediction. {p_end} {p 4 8 2}{cmd:inef} produces estimates of the technical inefficiency via E(u|e) {p_end} {p 4 8 2}{cmd:mode} produces estimates of the technical inefficiency via the mode M(u|e) {p_end} {p 4 8 2}{cmd:teff} produces estimates of the technical efficiency via E{exp(-u)|e} {p_end} {p 4 8 2}{cmdab:res:iduals} calculates the residuals. {p_end} {p 4 8 2}{cmdab:lns:igma} calculates the logarithm of the parameter sigma in v~N(0,s^2). {p_end} {p 4 8 2}{cmdab:sig:ma} calculates the value of the parameter sigma in v~N(0,s^2). {p_end} {p 4 8 2}{cmdab:lnt:heta} calculates the logarithm of the parameter theta in u~Uniform(0,t). {p_end} {p 4 8 2}{cmdab:the:ta} calculates the value of the parameter theta in u~Uniform(0,t). {p_end} {title:Saved results} {pstd} {cmd:frontierhtail} saves the following in {cmd:e()}. Note that these saved results are the same as those returned by the command {manhelp maximize R} since {cmd:frontierhtail} is fitted using {manhelp ml R}: {synoptset 20 tabbed}{...} {p2col 5 20 24 2: Scalars}{p_end} {synopt:{cmd:e(N)}}number of observations; always saved{p_end} {synopt:{cmd:e(k)}}number of parameters; always saved{p_end} {synopt:{cmd:e(k_eq)}}number of equations; usually saved{p_end} {synopt:{cmd:e(k_eq_model)}}number of equations to include in a model Wald test; usually saved{p_end} {synopt:{cmd:e(k_dv)}}number of dependent variables; usually saved{p_end} {synopt:{cmd:e(k_autoCns)}}number of base, empty, and omitted constraints; saved if command supports constra > ints{p_end} {synopt:{cmd:e(df_m)}}model degrees of freedom; always saved{p_end} {synopt:{cmd:e(r2_p)}}pseudo-R-squared; sometimes saved{p_end} {synopt:{cmd:e(ll)}}log likelihood; always saved{p_end} {synopt:{cmd:e(ll_0)}}log likelihood, constant-only model; saved when constant-only model is fit{p_end} {synopt:{cmd:e(N_clust)}}number of clusters; saved when {cmd:vce(cluster} {it:clustvar}{cmd:)} is specified; see {findalias frrobust}{p_end} {synopt:{cmd:e(chi2)}}chi-squared; usually saved{p_end} {synopt:{cmd:e(p)}}significance of model of test; usually saved{p_end} {synopt:{cmd:e(rank)}}rank of {cmd:e(V)}; always saved{p_end} {synopt:{cmd:e(rank0)}}rank of {cmd:e(V)} for constant-only model; saved when constant-only model is fit{p_end} {synopt:{cmd:e(ic)}}number of iterations; usually saved{p_end} {synopt:{cmd:e(rc)}}return code; usually saved{p_end} {synopt:{cmd:e(converged)}}{cmd:1} if converged, {cmd:0} otherwise; usually saved{p_end} {synoptset 20 tabbed}{...} {p2col 5 20 24 2: Macros}{p_end} {synopt:{cmd:e(cmd)}}name of command; always saved{p_end} {synopt:{cmd:e(cmdline)}}command as typed; always saved{p_end} {synopt:{cmd:e(depvar)}}names of dependent variables; always saved{p_end} {synopt:{cmd:e(wtype)}}weight type; saved when weights are specified or implied{p_end} {synopt:{cmd:e(wexp)}}weight expression; saved when weights are specified or implied{p_end} {synopt:{cmd:e(title)}}title in estimation output; usually saved by commands using {cmd:ml}{p_end} {synopt:{cmd:e(clustvar)}}name of cluster variable; saved when {cmd:vce(cluster} {it:clustvar}{cmd:)} is specified; see {findalias frrobust}{p_end} {synopt:{cmd:e(chi2type)}}{cmd:Wald} or {cmd:LR}; type of model chi-squared test; usually saved{p_end} {synopt:{cmd:e(vce)}}{it:vcetype} specified in {cmd:vce()}; saved when command allows {cmd:vce()}{p_end} {synopt:{cmd:e(vcetype)}}title used to label Std. Err.; sometimes saved{p_end} {synopt:{cmd:e(opt)}}type of optimization; always saved{p_end} {synopt:{cmd:e(which)}}{cmd:max} or {cmd:min}; whether optimizer is to perform maximization or minimization; always saved{p_end} {synopt:{cmd:e(ml_method)}}type of {cmd:ml} method; always saved by commands using {cmd:ml}{p_end} {synopt:{cmd:e(user)}}name of likelihood-evaluator program; always saved{p_end} {synopt:{cmd:e(technique)}}from {cmd:technique()} option; sometimes saved{p_end} {synopt:{cmd:e(singularHmethod)}}{cmd:m-marquardt} or {cmd:hybrid}; method used when Hessian is singular; sometimes saved{p_end} {synopt:{cmd:e(crittype)}}optimization criterion; always saved{p_end} {synopt:{cmd:e(properties)}}estimator properties; always saved{p_end} {synopt:{cmd:e(predict)}}program used to implement {cmd:predict}; usually saved{p_end} {synoptset 20 tabbed}{...} {p2col 5 20 24 2: Matrices}{p_end} {synopt:{cmd:e(b)}}coefficient vector; always saved{p_end} {synopt:{cmd:e(Cns)}}constraints matrix; sometimes saved{p_end} {synopt:{cmd:e(ilog)}}iteration log (up to 20 iterations); usually saved{p_end} {synopt:{cmd:e(gradient)}}gradient vector; usually saved{p_end} {synopt:{cmd:e(V)}}variance-covariance matrix of the estimators; always saved{p_end} {synopt:{cmd:e(V_modelbased)}}model-based variance; only saved when {cmd:e(V)} is neither the OIM nor OPG variance{p_end} {synoptset 20 tabbed}{...} {p2col 5 20 24 2: Functions}{p_end} {synopt:{cmd:e(sample)}}marks estimation sample; always saved{p_end} {title:Examples} {p 4 8 2} Before beginning the estimations, we use the {hi:set more off} instruction to tell {hi:Stata} not to pause when displaying the output. {p_end} {p 4 8 2}{stata "set more off"}{p_end} {p 4 8 2} We first illustrate the use of the command {hi:frontierhtail} with the {hi:Stata} manual dataset {hi: frontier1}. {p_end} {p 4 8 2}{stata "use http://www.stata-press.com/data/r11/frontier1, clear"}{p_end} {p 4 8 2} We estimate a Cobb-Douglas production function by regressing log output on log labor and log capital. {p_end} {p 4 8 2}{stata "frontierhtail lnoutput lnlabor lncapital"}{p_end} {p 4 8 2} To obtain White-corrected standard errors, we specify the {opt vce(robust)} option. {p_end} {p 4 8 2}{stata "frontierhtail lnoutput lnlabor lncapital, vce(robust)"}{p_end} {p 4 8 2} If we do not want to have a constant and the display of the iterations log at the beginning of the regression, we type. {p_end} {p 4 8 2}{stata "frontierhtail lnoutput lnlabor lncapital, nocons nolog"}{p_end} {p 4 8 2} We can specify variables to model heteroscedasticity in the idiosyncratic error. To do this use the {cmd:size} variable with the {opth hetero(varlist)} option. {p_end} {p 4 8 2}{stata "frontierhtail lnoutput lnlabor lncapital, hetero(size)"}{p_end} {p 4 8 2} If we want to estimate a Cobb-Douglas production function with constant returns-to-scale, we type. {p_end} {p 4 8 2}{stata "constraint 1 _b[lnlabor] + _b[lncapital] = 1"}{p_end} {p 4 8 2}{stata "frontierhtail lnoutput lnlabor lncapital, constraints(1)"}{p_end} {p 4 8 2} If we want to specify our own initial values instead of using those automatically provided by the command {hi:frontierhtail}, we proceed as follows. First, we run an OLS regression of {hi: lnoutput} on a constant. {p_end} {p 4 8 2}{stata "regress lnoutput"}{p_end} {p 4 8 2} Then we put the constant value in the local macro {hi: b0}. {p_end} {p 4 8 2}{stata "local b0 = _b[_cons]"}{p_end} {p 4 8 2} Finally, we specify the {opt init(init_specs)} option as follows. {p_end} {p 4 8 2}{stata " frontierhtail lnoutput lnlabor lncapital, init(/xb=`b0')"}{p_end} {p 4 8 2} It is important to note that for the intital values, we give only one value to the equation {hi:/xb}. {p_end} {p 4 8 2} Let's now illustrate how {hi:frontierhtail} can be used with {hi:predict}. First, we calculate the fitted values of the dependent variable. {p_end} {p 4 8 2}{stata "frontierhtail lnoutput lnlabor lncapital"}{p_end} {p 4 8 2}{stata "predict lnoutputhat, xb"}{p_end} {p 4 8 2} To calculate the standard error of the linear prediction, we type. {p_end} {p 4 8 2}{stata "predict serlp, stdp"}{p_end} {p 4 8 2} To calculate the technical inefficiency via E(u|e), we type. {p_end} {p 4 8 2}{stata "predict etechinef, inef"}{p_end} {p 4 8 2} To calculate the technical inefficiency via the mode M(u|e), we type. {p_end} {p 4 8 2}{stata "predict mtechinef, mode"}{p_end} {p 4 8 2} To calculate the technical efficiency via E{exp(-u)|e}, we type. {p_end} {p 4 8 2}{stata "predict techeff, teff"}{p_end} {p 4 8 2} To calculate the residuals, we type. {p_end} {p 4 8 2}{stata "predict resids, residuals"}{p_end} {p 4 8 2} You can calculate the other options of the {hi:predict} command in the same way as above by specifying: {hi:predict new_variable_name, option_name}. {p_end} {p 4 8 2} Let's now show how to use the command {hi:frontierhtail} with the {hi:Stata} manual dataset {hi:greene9}. {p_end} {p 4 8 2}{stata "use http://www.stata-press.com/data/r11/greene9, clear"}{p_end} {p 4 8 2} We estimate a Cobb-Douglas production function by regressing log value added on log capital and log labor. We specify the option {opt technique(dfp)} to obtain convergence. {p_end} {p 4 8 2}{stata "frontierhtail lnv lnk lnl, technique(dfp) "}{p_end} {p 4 8 2} If we want to test the constant returns-to-scale hypothesis on this model, we type. {p_end} {p 4 8 2}{stata "test _b[lnk] + _b[lnl] = 1"}{p_end} {p 4 8 2} This result shows that we cannot reject the null hypothesis of constant returns-to-scale technology in this model. {p_end} {title:References} {pstd} Nguyen, N. B.: 2010, "Estimation of technical efficiency in stochastic frontier analysis" {it:Dissertation, Graduate College of Bowling Green State University}. Downloadable at: {browse "http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1275444079"}. {p_end} {title:Author} {p 4}Diallo Ibrahima Amadou, {browse "mailto:zavren@gmail.com":zavren@gmail.com} {p_end} {title:Also see} {psee} Online: help for {bf:{help frontier}}, {bf:{help xtfrontier}}, {bf:{help regress}} {p_end}