Ordered Probit
In this example, the probabilities of attending no college, a 2 year college, and a 4 year college after graduation are modeled as a function of a student’s grades. In principle, we would expect that those with higher grades to be more likely to attend a 4 year college and less likely to skip college altogether. In the dataset, grades are measured on a scale of 1 to 13, with 1 being the highest. That means that if higher grades increase the probability of going to a 4 year college, the coefficient on grades will be negative. The probabilities are modeled using the normal distribution in this model where the outcomes represent increasing levels of difficulty.
We can use gretl to estimate the ordered probit model because its probit command actually handles multinomial ordered choices as well as binomial choice. Open the nelssmall. gdt data
1 open "@gretldirdatapoenels_small. gdt"
2 probit psechoice const grades
The results in below are very much like the ones in POE4 and those produced by Bayesian estimation provided by MCMCpack, the generation of which is discussed in section 16.10.3.
Model 2: Ordered Probit, using observations 11000
Dependent variable: psechoice
Coefficient 
Std. Error 
z 
pvalue 

grades 
0.306624 
0.0191735 
15.9921 
0.0000 
cut1 
2.94559 
0.146828 
20.0615 
0.0000 
cut2 
2.08999 
0.135768 
15.3938 
0.0000 
Mean dependent var 2.305000 S. D. dependent var 0.810328 Loglikelihood 875.8217 Akaike criterion 1757.643
Schwarz criterion 1772.367 HannanQuinn 1763.239
Number of cases ‘correctly predicted’ = 587 (58.7 percent)
Likelihood ratio test: x2(1) = 285.672 [0.0000]
Test for normality of residual –
Null hypothesis: error is normally distributed Test statistic: x2(2) = 2.96329 with pvalue = 0.227264
The coefficient on grades is negative and significant at 5%. This means that as the grades variable gets larger (grades get worse), the index is getting smaller and at the margins 2year
college attendees are being pushed towards no college and the 4year college attendees are being pushed toward the 2year option. We know for sure that the probability of being in the lowest category goes up and of being in the highest category goes down. Whatever happens in the middle depends on net effects of people being pushed out below and pulled in from above.
The other two parameters are the cutoff points that determine the boundaries between categories. The parameter ц,1 < ^2.
The algebraic expressions for the marginal effects are:
—ф(^1 — в grades) в
[ф(ц.1 — в grades) — ф(ц.2 — в9rades)}в
ф(ц.2 — в grades) в where ф is the probability density function of a standard normal distribution. The parameters ^1 and ^2 are the thresholds (or cutoff points) and в is the coefficient on grades. So, for example if you want to calculate the marginal effect on the probability of attending a 4year college (y = 3) for a student having grades at the median (6.64) and 5th percentile (2.635) use:
1 probit psechoice grades
2 k = $ncoeff
3 matrix b = $coeff[1:k2]
4 mu1 = $coeff[k1]
5 mu2 = $coeff[k]
6
6 matrix X = {6.64}
7 scalar Xb = X*b
8 P3a = pdf(N, mu2Xb)*b
10
її matrix X = 2.635
12 scalar Xb = X*b
13 P3b = pdf(N, mu2Xb)*b
14
їв printf "nFor the median grade of 6.64, the marginal
16 effect is %.4fn", P3a
17 printf "nFor the 5th percentile grade of 2.635, the
18 marginal effect is %.4fn", P3b
This yields
For the median grade of 6.64, the marginal effect is 0.1221
For the 5th percentile grade of 2.635, the marginal effect is 0.0538
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