[U] User's Guide

[ U ] 26.14 Models with time-series data 36526.12 Regression with systems of equationsFor systems of equations with endogenous covariates, use the three-stage least-squares (3SLS)estimator reg3. The reg3 command can produce constrained and unconstrained estimates.When we have correlated errors across equations but no endogenous right-hand-side variables,y 1j = x 1j β + ɛ 1jy 2j = x 2j β + ɛ 2j.y mj = x mj β + ɛ mjwhere ɛ k· and ɛ l· are correlated with correlation ρ kl , a quantity to be estimated from the data. Thisis called Zellner’s seemingly unrelated regressions, and sureg fits such models. When x 1j = x 2j =· · · = x mj , the model is known as multivariate regression, and the corresponding command is mvreg.The equations need not be linear; if they are not linear, use nlsur.26.13 Models with endogenous sample selectionWhat has become known as the Heckman model refers to linear regression in the presence ofsample selection: y j = x j β + ɛ j is not observed unless some event occurs that itself has probabilityp j = F (z j γ + ν j ), where ɛ and ν might be correlated and z j and x j may contain variables incommon.heckman fits such models by maximum likelihood or Heckman’s original two-step procedure.This model has recently been generalized to replace the linear regression equation with anotherprobit equation, and that model is fit by heckprob.Another important case of endogenous sample selection is the treatment-effects model, whichconsiders the effect of an endogenously chosen binary treatment on another endogenous, continuousvariable, conditional on two sets of independent variables. treatreg fits a treatment-effects modelby using either a two-step consistent estimator or full maximum likelihood.26.14 Models with time-series dataARIMA refers to models with autoregressive integrated moving-average processes, and Stata’s arimacommand fits models with ARIMA disturbances via the Kalman filter and maximum likelihood. Thesemodels may be fit with or without confounding covariates.Stata’s prais command performs regression with AR(1) disturbances by using the Prais–Winstenor Cochrane–Orcutt transformation. Both two-step and iterative solutions are available, as well asa version of the Hildreth–Lu search procedure. The Prais–Winsten estimates for the model are animprovement over the Cochrane–Orcutt estimates in that the first observation is preserved in theestimation. This is particularly important with trended data in small samples.prais automatically produces the Durbin–Watson d statistic, which can also be obtained afterregress by using estat dwatson.newey produces linear regression estimates with the Newey–West variance estimates that are robustto heteroskedasticity and autocorrelation of specified order.