# Stata/Binomial Outcome Models

## Binomial outcome models

• The logit model can be estimated using logit or glm and the probit model with probit or glm.
```. clear
. set obs 10000
obs was 0, now 10000
. gen u = invnorm(uniform())
. gen x = invnorm(uniform())
. gen ystar = x + u
. gen y = (ystar > 0)
. eststo clear
. eststo : qui : reg ystar x
(est1 stored)
. eststo : qui : glm y x, family(binomial) link(logit)
(est2 stored)
. eststo : qui : logit y x
(est3 stored)
. eststo : qui : glm y x, family(binomial) link(probit)
(est4 stored)
. eststo : qui : probit y x
(est5 stored)
. esttab , se
```

### Scobit

• Scobit (Skewed logistic regression) was developped by Jonathan Nagler 1994 (American Journal of Political Science). The idea is two estimate a skewness parameter of the underlying distribution.
```. global N = 2000
. global alpha = 1
. clear
. set obs \$N
obs was 0, now 2000
. gen u = ln(uniform()^(-1/\$alpha) - 1)
. gen x = uniform()
. global beta = 2
. gen y = (\$beta * x + u > 0)

.
. scobit y x

Fitting logistic model:

Iteration 0:   log likelihood =  -1190.598
Iteration 1:   log likelihood = -1126.9573
Iteration 2:   log likelihood = -1125.9604
Iteration 3:   log likelihood = -1125.9597
Iteration 4:   log likelihood = -1125.9597

Fitting full model:

Iteration 0:   log likelihood = -1125.9597
Iteration 1:   log likelihood = -1125.9459
Iteration 2:   log likelihood = -1125.8543
Iteration 3:   log likelihood = -1125.8241
Iteration 4:   log likelihood = -1125.8008
Iteration 5:   log likelihood = -1125.7376
Iteration 6:   log likelihood =  -1125.731
Iteration 7:   log likelihood =  -1125.724
Iteration 8:   log likelihood = -1125.7185
Iteration 9:   log likelihood = -1125.7158
Iteration 10:  log likelihood = -1125.7144
Iteration 11:  log likelihood =  -1125.714
Iteration 12:  log likelihood = -1125.7139
Iteration 13:  log likelihood = -1125.7139

Skewed logistic regression                      Number of obs     =       2000
Zero outcomes     =        565
Log likelihood = -1125.714                      Nonzero outcomes  =       1435

------------------------------------------------------------------------------
y |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
x |   4.290739   7.563589     0.57   0.571    -10.53362     19.1151
_cons |   1.737008   4.256165     0.41   0.683    -6.604923    10.07894
-------------+----------------------------------------------------------------
/lnalpha |  -1.068748   1.993377    -0.54   0.592    -4.975694    2.838198
-------------+----------------------------------------------------------------
alpha |   .3434382   .6846017                      .0069037    17.08495
------------------------------------------------------------------------------
Likelihood-ratio test of alpha=1:   chi2(1) =     0.49    Prob > chi2 = 0.4832

note: Likelihood-ratio tests are recommended for inference with scobit models.
```