{smcl} {* *! version 1.1.2 25nov2011}{...} {cmd:help sfcross}{right:also see: {help sfcross postestimation}} {hline} {title:Title} {p2colset 5 16 23 2}{...} {p2col :{hi:sfcross} {hline 2}}Stochastic frontier models for cross-sectional data{p_end} {p2colreset}{...} {title:Syntax} {p 8 17 2} {cmd:sfcross} {depvar} [{indepvars}] {ifin} {weight} [{cmd:,} {it:options}] {synoptset 33 tabbed}{...} {synopthdr} {synoptline} {syntab :Frontier} {synopt :{opt nocons:tant}}suppress constant term{p_end} {synopt :{cmdab:d:istribution(}{opt e:xponential)}}exponential distribution for the inefficiency term, the default{p_end} {synopt :{cmdab:d:istribution(}{opt h:normal)}}half-normal distribution for the inefficiency term{p_end} {synopt :{cmdab:d:istribution(}{opt t:normal)}}truncated-normal distribution for the inefficiency term{p_end} {synopt :{cmdab:d:istribution(}{opt g:amma)}}gamma distribution for the inefficiency term{p_end} {syntab :Ancillary equations} {synopt :{cmdab:e:mean(}{it:{help varlist:varlist_m}} [{cmd:,} {opt nocons:tant}]{cmd:)}}fit conditional mean model; only with {cmd:d(tnormal)}; use {opt noconstant} to suppress constant term{p_end} {synopt :{cmdab:u:sigma(}{it:{help varlist:varlist_u}} [{cmd:,} {opt nocons:tant}]{cmd:)}}specify explanatory variables for the inefficiency variance function; use {opt noconstant} to suppress constant term{p_end} {synopt :{cmdab:v:sigma(}{it:{help varlist:varlist_v}} [{cmd:,} {opt nocons:tant}]{cmd:)}}specify explanatory variables for the idiosyncratic error variance function; use {opt noconstant} to suppress constant term{p_end} {syntab :{help sfcross##sv_remarks:Starting values}} {synopt:{opt svfront:ier()}}specify a {it:1 X k} vector of initial values for the coefficients of the frontier{p_end} {synopt:{opt sve:mean()}}specify a {it: 1 X k_m} vector of initial values for the coefficients of the conditional mean model; only with {cmd:d(tnormal)}{p_end} {synopt:{opt svu:sigma()}}specify a {it: 1 X k_u} vector of initial values for the coefficients of the inefficiency variance function{p_end} {synopt:{opt svv:sigma()}}specify a {it: 1 X k_v} vector of initial values for the coefficients of the idiosyncratic error variance function{p_end} {synopt :{opt nosearch}}no attempt is made to improve on the initial values{p_end} {synopt :{opt restart}}select the random method to improve initial values{p_end} {synopt :{opt repeat(#)}}# of times the random values are tried; the default is 10{p_end} {synopt :{opt resc:ale}}determine rescaling of initial values{p_end} {syntab :Other options} {synopt :{opt cost}}fit cost frontier model; default is production frontier model{p_end} {synopt :{cmdab:const:raints(}{it:{help estimation options##constraints():constraints}}{cmd:)}}apply specified linear constraints{p_end} {synopt :{cmdab:simtype(}{it:{help sfcross##simtype:simtype}}{cmd:)}}method to produce random draws for simulation; only with {cmd:d(gamma)}{p_end} {synopt :{opt nsim:ulations(#)}}# of draws; only with {cmd:d(gamma)}{p_end} {synopt :{opt base(#)}}prime number used as a base for Halton sequences generation; only with {cmd:d(gamma)} and {cmd:simtype(halton)} or {cmd:simtype(genhalton)}{p_end} {syntab :SE} {synopt :{opth vce(vcetype)}}{it:vcetype} may be {opt oim}, {opt opg}, {opt r:obust}, {opt cl:uster} {it:clustvar}, {opt boot:strap}, or {opt jack:knife}{p_end} {synopt :{opt r:obust}}synonym for {cmd:vce(robust)}{p_end}{synopt :{opt cl:uster(clustvar)}}synonym for {cmd:vce(cluster}{it:clustvar}{cmd:)}{p_end} {syntab :Reporting} {synopt :{opt l:evel(#)}}set confidence level; default is {cmd:level(95)}{p_end} {synopt :{opt nocnsr:eport}}do not display constraints{p_end} {synopt :{opt nowarn:ing}}do not display warning message "convergence not achieved"{p_end} {synopt :{opt postscore}}save observation-by-observation scores in the estimation results list{p_end} {synopt :{opt posthess:ian}}save the Hessian corresponding to the full set of coefficients in the estimation results list{p_end} {synopt :{it:{help sfcross##display_options:display_options}}}control spacing and display of omitted variables and base and empty cells{p_end} {syntab :Maximization} {synopt :{it:{help sfcross##maximize_options:maximize_options}}}control the maximization process; seldom used{p_end} {p2coldent:+ {opt coefl:egend}}display coefficients' legend instead of coefficient table{p_end} {synoptline} {synoptset 20}{...} {marker simtype}{...} {synopthdr :simtype} {synoptline} {synopt :{opt ru:niform}}Uniformly distributed random variates{p_end} {synopt :{opt ha:lton}}Halton sequence with {opt base(#)}{p_end} {synopt :{opt genha:lton}}Generalized Halton sequence with {opt base(#)}{p_end} {synoptline} {p2colreset}{...} {p2colreset}{...} {p 4 6 2} {it:indepvars} may contain factor variables; see {help fvvarlist}.{p_end} {p 4 6 2} {opt bootstrap}, {opt by}, {opt jackknife}, and {opt svy} are allowed; see {help prefix}.{p_end} {p 4 6 2}Weights are not allowed with the {helpb bootstrap} prefix.{p_end} {p 4 6 2} {opt aweight}s, {opt fweight}s, {opt iweight}s, and {opt pweight}s are allowed; see {help weight}.{p_end} {p 4 6 2} {title:Description} {pstd} {opt sfcross} fits stochastic production or cost frontier models; the default is a production frontier model. It provides estimators for the parameters of a linear model with a disturbance that is assumed to be a mixture of two components, which have a strictly nonnegative and symmetric distribution, respectively. {opt sfcross} can fit models in which the nonnegative distribution component (a measurement of inefficiency) is assumed to be from a half-normal, exponential, truncated-normal or gamma distribution. In the latter case, maximization is performed through maximum simulated likelihood. {title:Options} {dlgtab:Frontier} {phang} {opt noconstant}; see {helpb estimation options##noconstant:[R] estimation options}. {phang} {opt distribution(distname)} specifies the distribution for the inefficiency term as half-normal ({opt hnormal}), truncated-normal ({opt tnormal}) or {opt exponential}. The default is {opt exponential}. {dlgtab:Ancillary equations} {phang} {cmd:emean(}{help varlist:varlist_m} [,{opt noconstant}]{cmd:)} may be used only with {cmd:distribution(tnormal)}. With this option, {opt sfcross} specifies the mean of the truncated-normal distribution in terms of a linear function of the covariates defined in {it:varlist_m}. Specifying {opt noconstant} suppresses the constant in this function. {phang} {cmd:usigma(}{help varlist:varlist_u} [,{opt noconstant}]{cmd:)} specifies that the inefficiency component is heteroskedastic, with the variance expressed as a function of the covariates defined in {it:varlist_u}. Specifying {opt noconstant} suppresses the constant in this function. {phang} {cmd:vsigma(}{help varlist:varlist_v} [,{opt noconstant}]{cmd:)} specifies that the idiosyncratic error component is heteroskedastic, with the variance expressed as a function of the covariates defined in {it:varlist_v}. Specifying {opt noconstant} suppresses the constant in this function. {dlgtab:Starting values} {phang} {opt svfrontier()} specifies a 1 x k vector of initial values for the coefficients of the frontier. The vector must have the same length of the parameters vector to be estimated. {phang} {opt svemean()} specifies a 1 x k_m vector of initial values for the coefficients of the conditional mean model. This option cab be specified only with {cmd:distribution(tnormal)}. {phang} {opt svusigma()} specifies a 1 X k_u vector of initial values for the coefficients of the technical inefficiency variance function. {phang} {opt svvsigma()} specifies a 1 X k_v vector of initial values for the coefficients of the technical inefficiency variance function. This option cannot be specified with {cmd:distribution(gamma)}. {phang} {opt nosearch} determines that no attempts are made to improve on the initial values via a search technique. In this case, the initial values become the starting values. {phang} {opt restart} determines that the random method of improving initial values is to be attempted. See also {help mf_moptimize##init_search} {phang} {opt repeat(#)} controls how many times random values are tried if the random method is turned on. The default is 10. {phang} {opt rescale} determines whether rescaling is attempted. Rescaling is a deterministic method. It also usually improves initial values, and usually reduces the number of subsequent iterations required by the optimization technique. {dlgtab:Other options} {phang} {opt cost} specifies that {opt sfcross} fits a cost frontier model. {phang} {opt constraints}({it:{help estimation options##constraints():constraints}}) applies specified linear constraints. {phang} {opt simtype(simtype)} specifies the method to generate random draws with {cmd:distribution(gamma)}. {opt runiform} generates uniformly distributed random variates; {opt halton} and {opt genhalton} create respectively Halton sequences and generalized Halton sequences where the base is expressed by the prime number in {opt base}(#). {opt runiform} is the default. See also {help mf_halton} for more details on Halton sequences generation. {phang} {opt nsimulations(#)} specifies the number of draws for simulation when {cmd:distribution(gamma)} is specified. The default is 250. {phang} {opt base(#)} specifies the number, preferably a prime, used as a base for the generation of Halton sequences and generalized Halton sequences when {cmd:distribution(gamma)} is specified. The default is 7. Note that Halton sequences based on large primes (#>10) can be highly correlated, and their coverage worse than that of pseudorandom uniform sequences. {dlgtab:SE} {phang} {opt vce(vcetype)} specifies the type of standard error reported, which includes types that are derived from asymptotic theory and that use bootstrap or jackknife methods; see {helpb vce_option:[R] {it:vce_option}}. {dlgtab:Reporting} {phang} {opt level(#)}; see {helpb estimation options##level():[R] estimation options}. {phang} {opt nocnsreport}; see {helpb estimation options##nocnsreport:[R] estimation options}. {phang} {opt nowarning} specifies whether the warning message "convergence not achieved" should not be displayed when this stopping rule is invoked. By default the message is displayed. {phang} {opt postscore} saves an observation-by-observation matrix of scores in the estimation results list. Scores are defined as the derivative of the objective function with respect to the {help mf_moptimize##def_parameter:parameters}. This option cannot be used when the size of the scores' matrix is greater than Stata matrix limit; see {helpb limits:[R] limits}. {phang} {opt posthessian} saves the Hessian matrix corresponding to the full set of coefficients in the estimation results list. {marker display_options}{...} {phang} {it:display_options}: {opt noomit:ted}, {opt vsquish}, {opt noempty:cells}, {opt base:levels}, {opt allbase:levels}; see {helpb estimation options##display_options:[R] estimation options}. {marker maximize_options}{...} {dlgtab:Maximization} {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. {pstd} {phang} {opt coeflegend}; see {helpb estimation options##coeflegend:[R] estimation options}. {marker sv_remarks}{...} {title:Remarks} {pstd} {cmd:sv{it:eqname}()} specifies initial values for the coefficients of {it:eqname}. You can specify the initial values in one of three ways: 1) by specifying the name of a vector contained in the initial values (e.g. {cmd:sv{it:frontier}(b0)}, where {cmd:b0} is a conformable vector); 2) by specifying coefficient names with the values in the same order as they appear in the command syntax (e.g. {cmd:sv{it:frontier}(x1=.5 x2=.3 _cons=1)}, if {cmd: sfcross y x1 x2}); 3) or by specifying a list of values (e.g. {cmd:sv{it:frontier}(.5 .3 1)}. {title:Examples} {hline} Setup {phang2}{cmd:. webuse frontier1}{p_end} {pstd}Cobb-Douglas production function with exponential distribution for inefficiency term{p_end} {phang2}{cmd:. sfcross lnoutput lnlabor lncapital}{p_end} {pstd}Cobb-Douglas production function with half-normal distribution for inefficiency term{p_end} {phang2}{cmd:. sfcross lnoutput lnlabor lncapital, d(h)}{p_end} {pstd}Cobb-Douglas production function with quartiles of {cmd:size} as explanatory variable in variance function for idiosyncratic error{p_end} {phang2}{cmd:. xtile qsize = size , nq(4)}{p_end} {phang2}{cmd:. sfcross lnoutput lnlabor lncapital, vsigma(i.qsize)}{p_end} {pstd}Cobb-Douglas production function with gamma distribution for inefficiency term{p_end} {phang2}{cmd:. sfcross lnoutput lncapital lnlabor, d(gamma) rescale simtype(genha) nsim(100)}{p_end} {pstd}Cobb-Douglas production function with {cmd:lnlabor}, {cmd:lncapital} and quartiles of {cmd:size} as explanatory variable of respectively the variance of the inefficiency term, the variance of the idiosyncratic error and the truncated mean{p_end} {phang2}{cmd:. sfcross lnoutput lnlabor lncapital, d(tn) emean(i.qsize) usigma(lnlabor) vsigma( lncapital)}{p_end} {hline} Setup {phang2}{cmd:. webuse frontier2}{p_end} {pstd}Cost frontier model with exponential distribution for inefficiency term with constraints on prices{p_end} {phang2}{cmd:. cons def 1 lnp_l+ lnp_k=1}{p_end} {phang2}{cmd:. sfcross lncost lnout lnp_l lnp_k, cost constr(1)}{p_end} {hline} {title:Saved results} {pstd} {cmd:sfcross} saves the following in {cmd:e()}: {synoptset 20 tabbed}{...} {p2col 5 20 24 2: Scalars}{p_end} {synopt:{cmd:e(N)}}number of observations{p_end} {synopt:{cmd:e(rank)}}rank of {cmd:e(V)}{p_end} {synopt:{cmd:e(k)}}number of parameters{p_end} {synopt:{cmd:e(k_eq)}}number of equations{p_end} {synopt:{cmd:e(k_dv)}}number of dependent variables{p_end} {synopt:{cmd:e(df_m)}}model degrees of freedom{p_end} {synopt:{cmd:e(ll)}}log likelihood{p_end} {synopt:{cmd:e(ll_c)}}log likelihood for H_0: sigma_u=0{p_end} {synopt:{cmd:e(converged)}}{cmd:1} if converged, {cmd:0} otherwise{p_end} {synopt:{cmd:e(ic)}}number of iterations{p_end} {synopt:{cmd:e(iterations)}}number of iterations, including initiali step{p_end} {synopt:{cmd:e(rc)}}return code{p_end} {synopt:{cmd:e(chi2)}}chi-squared{p_end} {synopt:{cmd:e(p)}}significance{p_end} {synopt:{cmd:e(chi2_c)}}LR test statistic{p_end} {synopt:{cmd:e(z)}}test for negative skewness of OLS residuals{p_end} {synopt:{cmd:e(p_z)}}p-value for z{p_end} {synopt:{cmd:e(k_autoCns)}}number of base, empty, and omitted constraints{p_end} {synopt:{cmd:e(sigma_u)}}standard deviation of technical inefficiency{p_end} {synopt:{cmd:e(sigma_v)}}standard deviation of V_i{p_end} {synopt:{cmd:e(g_shape)}}Shape parameter of the Gamma distributed inefficiency{p_end} {synopt:{cmd:e(avg_sigmau)}}average standard deviation of technical inefficiency{p_end} {synopt:{cmd:e(avg_sigmav)}}average standard deviation of V_i{p_end} {synopt:{cmd:e(lambda)}}signal to noise ratio{p_end} {synoptset 20 tabbed}{...} {p2col 5 20 24 2: Macros}{p_end} {synopt:{cmd:e(cmd)}}{cmd:sfcross}{p_end} {synopt:{cmd:e(cmdline)}}command as typed{p_end} {synopt:{cmd:e(title)}}title in estimation output{p_end} {synopt:{cmd:e(function)}}{cmd:production} or {cmd:cost}{p_end} {synopt:{cmd:e(depvar)}}name of dependent variable{p_end} {synopt:{cmd:e(covariates)}}name of independent variables{p_end} {synopt:{cmd:e(crittype)}}optimization criterion{p_end} {synopt:{cmd:e(dist)}}distribution assumption for U_i{p_end} {synopt:{cmd:e(het)}}heteroskedastic components{p_end} {synopt:{cmd:e(Emean)}}{it:varlist} in {cmd:emean()}{p_end} {synopt:{cmd:e(Usigma)}}{it:varlist} in {cmd:usigma()}{p_end} {synopt:{cmd:e(Vsigma)}}{it:varlist} in {cmd:vsigma()}{p_end} {synopt:{cmd:e(simtype)}}method to produce random draws{p_end} {synopt:{cmd:e(base)}}base number to generate Halton sequences{p_end} {synopt:{cmd:e(nsim)}}number of random draws{p_end} {synopt:{cmd:e(predict)}}program used to implement {cmd:predict}{p_end} {synopt:{cmd:e(opt)}}type of optimization{p_end} {synopt:{cmd:e(user)}}name of likelihood-evaluator program{p_end} {synopt:{cmd:e(ml_method)}}type of {cmd:ml} method{p_end} {synopt:{cmd:e(singularHmethod)}}{cmd:m-marquardt} or {cmd:hybrid}; method used when Hessian is singular{p_end} {synopt:{cmd:e(technique)}}maximization technique{p_end} {synopt:{cmd:e(which)}}{cmd:max} or {cmd:min}; whether optimizer is to perform maximization or minimization{p_end}{synopt:{cmd:e(wtype)}}weight type{p_end}{synopt:{cmd:e(wexp)}}weight expression{p_end} {synopt:{cmd:e(chi2type)}}{cmd:Wald}; type of model chi-squared test{p_end} {synopt:{cmd:e(vce)}}{it:vcetype} specified in {cmd:vce()}{p_end} {synopt:{cmd:e(vcetype)}}title used to label Std. Err.{p_end} {synopt:{cmd:e(contraints)}}list of specified constraints{p_end} {synopt:{cmd:e(properties)}}{cmd:b V}{p_end} {synoptset 20 tabbed}{...} {p2col 5 20 24 2: Matrices}{p_end} {synopt:{cmd:e(b)}}coefficient vector{p_end} {synopt:{cmd:e(Cns)}}constraints matrix{p_end} {synopt:{cmd:e(ilog)}}iteration log (up to 20 iterations){p_end} {synopt:{cmd:e(gradient)}}gradient vector{p_end} {synopt:{cmd:e(V)}}variance-covariance matrix of the estimators{p_end} {synopt:{cmd:e(V_modelbased)}}model-based variance{p_end} {synopt:{cmd:e(postscore)}}observation-by-observation scores{p_end} {synopt:{cmd:e(posthessian)}}Hessian corresponding to the full set of coefficients{p_end} {synoptset 20 tabbed}{...} {p2col 5 20 24 2: Functions}{p_end} {synopt:{cmd:e(sample)}}marks estimation sample{p_end} {p2colreset}{...} {title:Authors} {pstd}Federico Belotti{p_end} {pstd}Centre for Economic and International Studies, University of Rome Tor Vergata{p_end} {pstd}Rome, Italy{p_end} {pstd}federico.belotti@uniroma2.it{p_end} {pstd}Silvio Daidone{p_end} {pstd}Centre for Health Economics, University of York{p_end} {pstd}York, UK{p_end} {pstd}silvio.daidone@york.ac.uk{p_end} {pstd}Vincenzo Atella{p_end} {pstd}Centre for Economic and International Studies, University of Rome Tor Vergata{p_end} {pstd}Rome, Italy{p_end} {pstd}atella@uniroma2.it{p_end} {pstd}Giuseppe Ilardi{p_end} {pstd}Economic and Financial Statistics Department, Bank of Italy{p_end} {pstd}Rome, Italy{p_end} {pstd}giuseppe.ilardi@bancaditalia.it{p_end} {title:Also see} {psee} {space 2}Help: {help sfcross_postestimation}, {help sfpanel}, {help sfpanel_postestimation}. {p_end}