[U] User's Guide

364 [ U ] 26 Overview of Stata estimation commandsIn small samples, exact estimates have better coverage than the asymptotic estimates, and exactestimates are the only way to obtain estimates, tests, and confidence intervals of covariates thatperfectly predict the observed outcome.26.10 Linear regression with heteroskedastic errorsWe now consider the model y j = x j β + ɛ j , where the variance of ɛ j is nonconstant.First, regress can fit such models if you specify the vce(robust) option. What Stata callsrobust is also known as the White correction for heteroskedasticity.For scientists who have data where the variance of ɛ j is known a priori, vwls is the command.vwls produces estimates for the model given each observation’s variance, which is recorded in avariable in the data.If you wish to model the heteroskedasticity on covariates, use the het() option of the archcommand. Although arch is written primarily to analyze time-series data, it can be used with crosssectionaldata. Before using arch with cross-sectional data, set the data as time series, by typing genfaketime = n and then typing tsset faketime.Finally, qreg performs quantile regression, which in the presence of heteroskedasticity is mostof interest. Median regression (one of qreg’s capabilities) is an estimator of y j = x j β + ɛ j whenɛ j is heteroskedastic. Even more useful, you can fit models of other quantiles and so model theheteroskedasticity. Also see the sqreg and iqreg commands; sqreg estimates multiple quantilessimultaneously. iqreg estimates differences in quantiles.26.11 Stochastic frontier modelsfrontier fits stochastic production or cost frontier models on cross-sectional data. The modelcan be expressed aswhereu iy i = x i β + v i − su i{ 1 for production functionss =−1 for cost functionsis a nonnegative disturbance standing for technical inefficiency in the production function orcost inefficiency in the cost function. Although the idiosyncratic error term v i is assumed to havea normal distribution, the inefficiency term is assumed to be one of the three distributions: halfnormal,exponential, or truncated-normal. Also, when the nonnegative component of the disturbanceis assumed to be either half-normal or exponential, frontier can fit models in which the errorcomponents are heteroskedastic conditional on a set of covariates. When the nonnegative componentof the disturbance is assumed to be from a truncated-normal distribution, frontier can also fit aconditional mean model, where the mean of the truncated-normal distribution is modeled as a linearfunction of a set of covariates.For panel-data stochastic frontier models, see [U] 26.15.1 Linear regression with panel data.