Dimitriy V. Masterov responded to Silke Humber's query on collinear
variables in probit and appling the "sigma" restrictions to the
coefficients of indicator variables:
>
> If I understand your question correctly, there are two ways of doing it:
>
> (1) Omit the constant, so that you can include the full set of dummies:
> probit outcome d1 d2 d3, nocons
>
> (2) Use the linear combination command to construct coefficients you want:
> probit outcome d2 d3
> lincom _cons+d2
>
To apply the sigma restrictions subtract one of your indicator
variables from all the others. For example:
. webuse auto
(1978 Automobile Data)
. gen byte wt1 = cond(weight<=3190,1,0)
. gen byte wt2 = 1-wt1
* first the over parameterized model
. probit foreign wt1 wt2 mpg
note: wt2 dropped due to collinearity
Iteration 0: log likelihood = -45.03321
Iteration 1: log likelihood = -30.819636
Iteration 2: log likelihood = -29.941532
Iteration 3: log likelihood = -29.897746
Iteration 4: log likelihood = -29.897518
Probit regression Number of obs = 74
LR chi2(2) = 30.27
Prob > chi2 = 0.0000
Log likelihood = -29.897518 Pseudo R2 = 0.3361
------------------------------------------------------------------------------
foreign | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
wt1 | 2.134792 .5714695 3.74 0.000 1.014732 3.254851
mpg | -.0046676 .0385094 -0.12 0.904 -.0801445 .0708094
_cons | -1.847266 .77984 -2.37 0.018 -3.375725 -.3188077
------------------------------------------------------------------------------
* now use the indicator coding that will apply wt1+wt2 = 0
. gen byte w1 = wt1 - wt2
. probit foreign w1 mpg
Iteration 0: log likelihood = -45.03321
Iteration 1: log likelihood = -30.819636
Iteration 2: log likelihood = -29.941532
Iteration 3: log likelihood = -29.897746
Iteration 4: log likelihood = -29.897518
Probit regression Number of obs = 74
LR chi2(2) = 30.27
Prob > chi2 = 0.0000
Log likelihood = -29.897518 Pseudo R2 = 0.3361
------------------------------------------------------------------------------
foreign | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
w1 | 1.067396 .2857348 3.74 0.000 .507366 1.627426
mpg | -.0046676 .0385094 -0.12 0.904 -.0801445 .0708094
_cons | -.7798703 .8444172 -0.92 0.356 -2.434898 .8751571
------------------------------------------------------------------------------
At this point I argue that wt2 = -1.067396. I can use -mprobit-
to verify it using constraints and preventing collinear variables
from being dropped. First -mprobit- will require initial estimates
if we use the collinear option
. mat b = e(b)
. mat b1 = (0,0,b[1,2],b[1,3])
. constraint 1 wt1 +wt2 = 0
. mprobit foreign wt1 wt2 mpg, constraints(1) collinear from(b1,copy) probitparam
Iteration 0: log likelihood = -48.158141
Iteration 1: log likelihood = -30.337043
Iteration 2: log likelihood = -29.900296
Iteration 3: log likelihood = -29.897519
Iteration 4: log likelihood = -29.897518
Multinomial probit regression Number of obs = 74
Wald chi2(2) = 19.44
Log likelihood = -29.897518 Prob > chi2 = 0.0001
( 1) [Foreign]wt1 + [Foreign]wt2 = 0
------------------------------------------------------------------------------
foreign | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
Foreign |
wt1 | 1.067396 .2857411 3.74 0.000 .5073535 1.627438
wt2 | -1.067396 .2857411 -3.74 0.000 -1.627438 -.5073535
mpg | -.0046676 .0385095 -0.12 0.904 -.0801447 .0708096
_cons | -.7798702 .8444202 -0.92 0.356 -2.434903 .8751631
------------------------------------------------------------------------------
(foreign=Domestic is the base outcome)
The estimated _cons is the overall mean and the wt1 and wt2 estimates
are deviations from that mean.
I used only two categories, but this works for as many as you want.
You can get the last estimate using -lincom-.
-Rich
[email protected]
*
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
