The concept of bivariate normal distibutions is very familiar to even beginning statistics students. Scatter plots and Pearson corelation are tools for examing bivariate normal distributions. Less familiar for some students might be using bivariate response variables in multivariate analyses. In the case of bivariate probit analysis we have two binary response variables that vary jointly. We want to esitmate the coefficients needed to account for this joint distribution.
As you would expect the likelihood function for bivariate probit is more complex than when there is only one esponse variable,
Example 1
use http://www.gseis.ucla.edu/courses/data/schvote, clear
probit priv years ptax inc
Probit estimates Number of obs = 80
LR chi2(3) = 1.14
Prob > chi2 = 0.7680
Log likelihood = -29.572798 Pseudo R2 = 0.0189
------------------------------------------------------------------------------
priv | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
years | -.0092045 .023364 -0.39 0.694 -.0549971 .0365882
ptax | -.1427311 .6937362 -0.21 0.837 -1.502429 1.216967
inc | .4313241 .5792655 0.74 0.457 -.7040154 1.566664
_cons | -4.40218 4.938369 -0.89 0.373 -14.08121 5.276846
------------------------------------------------------------------------------
probit vote years ptax inc
Probit estimates Number of obs = 80
LR chi2(3) = 13.62
Prob > chi2 = 0.0035
Log likelihood = -45.576114 Pseudo R2 = 0.1300
------------------------------------------------------------------------------
vote | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
years | -.0080101 .015893 -0.50 0.614 -.0391598 .0231395
ptax | -2.013629 .7192403 -2.80 0.005 -3.423314 -.6039441
inc | 1.582937 .5671639 2.79 0.005 .4713161 2.694558
_cons | -1.353637 4.411823 -0.31 0.759 -10.00065 7.293378
------------------------------------------------------------------------------
biprobit priv vote years ptax inc
Bivariate probit regression Number of obs = 80
Wald chi2(6) = 11.91
Log likelihood = -74.171253 Prob > chi2 = 0.0640
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
priv |
years | -.0146627 .0264275 -0.55 0.579 -.0664596 .0371342
ptax | -.0923143 .6922562 -0.13 0.894 -1.449112 1.264483
inc | .3644544 .5588324 0.65 0.514 -.7308371 1.459746
_cons | -4.040363 4.872994 -0.83 0.407 -13.59126 5.510529
-------------+----------------------------------------------------------------
vote |
years | -.008866 .0159739 -0.56 0.579 -.0401742 .0224422
ptax | -2.054462 .7310168 -2.81 0.005 -3.487229 -.6216959
inc | 1.574388 .5638432 2.79 0.005 .469276 2.679501
_cons | -.9732729 4.487075 -0.22 0.828 -9.767779 7.821233
-------------+----------------------------------------------------------------
/athrho | -.3425239 .2536544 -1.35 0.177 -.8396774 .1546297
-------------+----------------------------------------------------------------
rho | -.3297287 .2260769 -.6856382 .1534089
------------------------------------------------------------------------------
Likelihood ratio test of rho=0: chi2(1) = 1.95532 Prob > chi2 = 0.1620
test years
( 1) [priv]years = 0.0
( 2) [vote]years = 0.0
chi2( 2) = 0.69
Prob > chi2 = 0.7079
test ptax
( 1) [priv]ptax = 0.0
( 2) [vote]ptax = 0.0
chi2( 2) = 8.15
Prob > chi2 = 0.0170
test inc
( 1) [priv]inc = 0.0
( 2) [vote]inc = 0.0
chi2( 2) = 8.86
Prob > chi2 = 0.0119
mfx compute
Marginal effects after biprobit
y = Pr(priv=1,vote=1) (predict)
= .05187385
------------------------------------------------------------------------------
variable | dy/dx Std. Err. z P>|z| [ 95% C.I. ] X
---------+--------------------------------------------------------------------
years | -.0019032 .00259 -0.73 0.463 -.006986 .003179 8.77500
ptax | -.110602 .08091 -1.37 0.172 -.269192 .047988 6.93727
inc | .1141173 .06655 1.71 0.086 -.016321 .244556 9.96772
------------------------------------------------------------------------------
Example 2
use http://www.gseis.ucla.edu/courses/data/ms00, clear
describe
Contains data from http://www.gseis.ucla.edu/courses/data/ms00.dta
obs: 200
vars: 7 8 Feb 2001 11:23
size: 6,400 (99.2% of memory free)
-------------------------------------------------------------------------------
storage display value
variable name type format label variable label
-------------------------------------------------------------------------------
id float %9.0g
female float %9.0g fl
honors float %9.0g enrolled in honors
read float %9.0g reading test
write float %9.0g writing test
nss float %9.0g national science scholar
mma float %9.0g mooberry math award
-------------------------------------------------------------------------------
summarize
Variable | Obs Mean Std. Dev. Min Max
-------------+-----------------------------------------------------
id | 200 100.5 57.87918 1 200
female | 200 .545 .4992205 0 1
honors | 200 .525 .5006277 0 1
read | 200 52.23 10.25294 28 76
write | 200 52.775 9.478586 31 67
nss | 200 .165 .372112 0 1
mma | 200 .115 .3198225 0 1
tab1 female honors
-> tabulation of female
female | Freq. Percent Cum.
------------+-----------------------------------
male | 91 45.50 45.50
female | 109 54.50 100.00
------------+-----------------------------------
Total | 200 100.00
-> tabulation of honors
enrolled in |
honors | Freq. Percent Cum.
------------+-----------------------------------
0 | 95 47.50 47.50
1 | 105 52.50 100.00
------------+-----------------------------------
Total | 200 100.00
tabulate nss mma
national |
science | mooberry math award
scholar | 0 1 | Total
-----------+----------------------+----------
0 | 155 12 | 167
1 | 22 11 | 33
-----------+----------------------+----------
Total | 177 23 | 200
biprobit nss mma read write honors female
Bivariate probit regression Number of obs = 200
Wald chi2(8) = 56.75
Log likelihood = -105.31311 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
nss |
read | .0493064 .0156631 3.15 0.002 .0186073 .0800055
write | .0679724 .0208136 3.27 0.001 .0271786 .1087663
honors | -.5517298 .2878803 -1.92 0.055 -1.115965 .0125051
female | -.9337984 .2767985 -3.37 0.001 -1.476313 -.3912834
_cons | -6.748716 1.144578 -5.90 0.000 -8.992047 -4.505385
-------------+----------------------------------------------------------------
mma |
read | .0525179 .0197319 2.66 0.008 .0138441 .0911917
write | .1091292 .0364161 3.00 0.003 .037755 .1805034
honors | .8246593 .4328874 1.91 0.057 -.0237844 1.673103
female | -.1103348 .31435 -0.35 0.726 -.7264495 .5057799
_cons | -11.20763 2.338403 -4.79 0.000 -15.79081 -6.624443
-------------+----------------------------------------------------------------
/athrho | .3552813 .2288336 1.55 0.121 -.0932244 .8037869
-------------+----------------------------------------------------------------
rho | .3410509 .2022167 -.0929552 .6661485
------------------------------------------------------------------------------
Likelihood ratio test of rho=0: chi2(1) = 2.52696 Prob > chi2 = 0.1119
test honors
( 1) [nss]honors = 0.0
( 2) [mma]honors = 0.0
chi2( 2) = 8.31
Prob > chi2 = 0.0157
test female
( 1) [nss]female = 0.0
( 2) [mma]female = 0.0
chi2( 2) = 11.41
Prob > chi2 = 0.0033
mfx compute
Marginal effects after biprobit
y = Pr(nss=1,mma=1) (predict)
= .00303069
------------------------------------------------------------------------------
variable | dy/dx Std. Err. z P>|z| [ 95% C.I. ] X
---------+--------------------------------------------------------------------
read | .0005465 .00051 1.07 0.286 -.000457 .00155 52.2300
write | .0010095 .00083 1.22 0.224 -.000616 .002635 52.7750
honors*| .0034291 .00419 0.82 0.413 -.004778 .011636 .525000
female*| -.0043393 .00507 -0.86 0.392 -.01427 .005591 .545000
------------------------------------------------------------------------------
(*) dy/dx is for discrete change of dummy variable from 0 to 1
The ancillary parameter rho measures the correlation of the residuals from the two
models.
As it turns out, the two equations were not strongly associated, rho = .34, which
was not significant (chi-square = 2.53, df = 1, p =.11)Seemingly Unrelated Bivariate Probit Example
It is also possible to run biprobit as a seemlying unrelated bivariate probit in which each of the equations has different predictors. The equations are not independent since they are computed on the same set of subjects.
biprobit (nss = female write)(mma = read write)
Seemingly unrelated bivariate probit Number of obs = 200
Wald chi2(4) = 50.04
Log likelihood = -113.97205 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
nss |
female | -1.016887 .2584353 -3.93 0.000 -1.523411 -.5103629
write | .0860565 .0175865 4.89 0.000 .0515876 .1205253
_cons | -5.273082 .9810768 -5.37 0.000 -7.195957 -3.350206
-------------+----------------------------------------------------------------
mma |
read | .057744 .0199113 2.90 0.004 .0187187 .0967694
write | .1078431 .0345801 3.12 0.002 .0400674 .1756189
_cons | -10.86132 2.238518 -4.85 0.000 -15.24873 -6.473901
-------------+----------------------------------------------------------------
/athrho | .2028541 .2053828 0.99 0.323 -.1996887 .6053969
-------------+----------------------------------------------------------------
rho | .2001167 .1971579 -.1970762 .5408787
------------------------------------------------------------------------------
Likelihood ratio test of rho=0: chi2(1) = .986217 Prob > chi2 = 0.3207
test write
( 1) [nss]write = 0.0
( 2) [mma]write = 0.0
chi2( 2) = 32.42
Prob > chi2 = 0.0000
test read
( 1) [mma]read = 0.0
chi2( 1) = 8.41
Prob > chi2 = 0.0037
display "chi-square approximation = " 2.90^2
chi-square approximation = 8.41
Again it turns out that these two equations are not stongly correlated, rho = .2, which
is not statistically significant (chi-squar1 = .99, df = 1, p = .32).Instrumental Variable Example
biprobit (nss = female mma)(mma = female read write), nolog
Seemingly unrelated bivariate probit Number of obs = 200
Wald chi2(5) = 90.74
Log likelihood = -118.5046 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
nss |
female | -.580865 .2214549 -2.62 0.009 -1.014909 -.1468214
mma | 2.115765 .3094269 6.84 0.000 1.509299 2.72223
_cons | -.9611232 .152064 -6.32 0.000 -1.259163 -.6630833
-------------+----------------------------------------------------------------
mma |
female | -.222301 .3007045 -0.74 0.460 -.811671 .367069
read | .0675854 .0196951 3.43 0.001 .0289836 .1061871
write | .1076171 .0333168 3.23 0.001 .0423173 .1729168
_cons | -11.19188 2.087091 -5.36 0.000 -15.2825 -7.101252
-------------+----------------------------------------------------------------
/athrho | -1.301447 .7019475 -1.85 0.064 -2.677239 .0743445
-------------+----------------------------------------------------------------
rho | -.8620953 .1802543 -.9905906 .0742078
------------------------------------------------------------------------------
Likelihood-ratio test of rho=0: chi2(1) = 10.6787 Prob > chi2 = 0.0011
Categorical Data Analysis Course
Phil Ender