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 1-1000
Dependent variable: psechoice

Coefficient

Std. Error

z

p-value

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 Log-likelihood -875.8217 Akaike criterion 1757.643

Schwarz criterion 1772.367 Hannan-Quinn 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 p-value = 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 2-year
college attendees are being pushed towards no college and the 4-year college attendees are being pushed toward the 2-year 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 cut-off 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 cut-off 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 4-year 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:k-2]

4 mu1 = $coeff[k-1]

5 mu2 = $coeff[k]

6

6 matrix X = {6.64}

7 scalar Xb = X*b

8 P3a = pdf(N, mu2-Xb)*b

10

її matrix X = 2.635

12 scalar Xb = X*b

13 P3b = pdf(N, mu2-Xb)*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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