16
Stata Technical Bulletin
STB-33
Common derivations
Formulation of the models starts with the basic two-equation system
У H = Xil∕3 + €ц
Уй = Xi2∕3 + Ci2
€il
€i2
Bivariate Normal
.2
I
.P1
f))
The estimation sample is all of the observations for which all of the variables in the two equations are observed in the
bivariate and seemingly unrelated models. For the nested model, the estimation sample is the sample defined by the containing
equation—the contained equation is assumed to be missing for observations where the dependent variable of the containing
equation is zero.
Throughout the next sections, we will use Φ to denote the standard normal cdf, Φ2 to denote the standard bivariate normal
cdf, φ to denote the standard normal pdf, and φ2 to denote the standard bivariate normal pdf.
The bivariate and seemingly unrelated models summarize the 4 possible outcomes such that for a given observation we
have
Pm = P{yn = l,Vi2 = 1)
Pno = P{yn = l,Vi2 = 0)
Pm = P{yn = 0,yi2 = 1)
Pioo = P{yn = 0,yi2 = 0)
For these two models, we have that
Pm = Φ2(xii‰xi2‰p)
Дю = Φ(xiιA) - ¾(xii∕3ι,xi2⅛,p)
Pi0ι = Φ(xi2⅛) - Φ2(xiι∕31, xi2∕32, p)
Pioo = ¾(xii∕31,xi2⅛,p) - Φ(xiι01) - Φ(xi2⅛)
where the bivariate probit has Xiι = Xi2 for all i.
For the nested model, we have that
Pm = Φ2(xiι∕31,xi2⅛,p)
Pno = ¾(-Xii∕31,xi2‰ -p}
Pioi = Φ(-xi2∕¾)
Pioo = Φ(-xi2⅛)
where equation 1 is nested within equation 2; that is, the outcome for y2 is only available when y1 ≠ 0.
Implementation
In fact, all of these models may be implemented with only one command, but two are provided. The only necessary
command is suprob that takes two equations as arguments, but we provide biprob as a convenience so that you are not required
to set up the appropriate equations and may instead use mvreg-type syntax.
The syntax for suprob and biprob are
suprob eq± eq2 [, robust cluster(eltt`trr-aaaaame') score (scoreι score?) nochi
nested level (#) maximize-options ]
biprob depvarr depvar2 ∖aarlist∖ [, robust cluster Cclutter-aanaame) score (score 1 score?) nochi
level (#) maximize-options ]