BINARY CHOICE MODELS: PROBIT ANALYSIS

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BINARY CHOICE MODELS: PROBIT ANALYSIS

In the case of probit analysis, the sigmoid function is the cumulative standardized normal distribution.

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The maximum likelihood principle is again used to obtain estimates of the parameters.

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. probit GRAD ASVABC SM SF MALE

Iteration 0: log likelihood = -118.67769Iteration 1: log likelihood = -98.195303Iteration 2: log likelihood = -96.666096Iteration 3: log likelihood = -96.624979Iteration 4: log likelihood = -96.624926

Probit estimates Number of obs = 540 LR chi2(4) = 44.11 Prob > chi2 = 0.0000Log likelihood = -96.624926 Pseudo R2 = 0.1858

------------------------------------------------------------------------------ GRAD | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .0648442 .0120378 5.39 0.000 .0412505 .0884379 SM | -.0081163 .0440399 -0.18 0.854 -.094433 .0782004 SF | .0056041 .0359557 0.16 0.876 -.0648677 .0760759 MALE | .0630588 .1988279 0.32 0.751 -.3266368 .4527544 _cons | -1.450787 .5470608 -2.65 0.008 -2.523006 -.3785673------------------------------------------------------------------------------

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Here is the result of the probit regression using the example of graduating from high school.


. probit GRAD ASVABC SM SF MALE

Iteration 0: log likelihood = -118.67769Iteration 1: log likelihood = -98.195303Iteration 2: log likelihood = -96.666096Iteration 3: log likelihood = -96.624979Iteration 4: log likelihood = -96.624926

Probit estimates Number of obs = 540 LR chi2(4) = 44.11 Prob > chi2 = 0.0000Log likelihood = -96.624926 Pseudo R2 = 0.1858

------------------------------------------------------------------------------ GRAD | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .0648442 .0120378 5.39 0.000 .0412505 .0884379 SM | -.0081163 .0440399 -0.18 0.854 -.094433 .0782004 SF | .0056041 .0359557 0.16 0.876 -.0648677 .0760759 MALE | .0630588 .1988279 0.32 0.751 -.3266368 .4527544 _cons | -1.450787 .5470608 -2.65 0.008 -2.523006 -.3785673------------------------------------------------------------------------------

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As with logit analysis, the coefficients have no direct interpretation. However, we can use them to quantify the marginal effects of the explanatory variables on the probability of graduating from high school.

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As with logit analysis, the marginal effect of Xi on p can be written as the product of the marginal effect of Z on p and the marginal effect of Xi on Z.


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The marginal effect of Z on p is given by the standardized normal distribution. The marginal effect of Xi on Z is given by i.


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As with logit analysis, the marginal effects vary with Z. A common procedure is to evaluate them for the value of Z given by the sample means of the explanatory variables.


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As with logit analysis, the marginal effects vary with Z. A common procedure is to evaluate them for the value of Z given by the sample means of the explanatory variables.


. sum GRAD ASVABC SM SF MALE

Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- GRAD | 540 .9425926 .2328351 0 1 ASVABC | 540 51.36271 9.567646 25.45931 66.07963 SM | 540 11.57963 2.816456 0 20 SF | 540 11.83704 3.53715 0 20 MALE | 540 .5 .5004636 0 1

Probit: Marginal Effects

mean b product f(Z) f(Z)b

ASVABC 51.36 0.065 3.328 0.068 0.004

SM 11.58 –0.008 –0.094 0.068 –0.001

SF 11.84 0.006 0.066 0.068 0.000

MALE 0.50 0.063 0.032 0.068 0.004

constant 1.00 –1.451 –1.451

Total 1.881

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In this case Z is equal to 1.881 when the X variables are equal to their sample means.



ASVABC 51.36 0.065 3.328 0.068 0.004

SM 11.58 –0.008 –0.094 0.068 –0.001

SF 11.84 0.006 0.066 0.068 0.000

MALE 0.50 0.063 0.032 0.068 0.004

constant 1.00 –1.451 –1.451

Total 1.881

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We then calculate f(Z).

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The estimated marginal effects are f(Z) multiplied by the respective coefficients. We see that a one-point increase in ASVABC increases the probability of graduating from high school by 0.4 percent.



ASVABC 51.36 0.065 3.328 0.068 0.004

SM 11.58 –0.008 –0.094 0.068 –0.001

SF 11.84 0.006 0.066 0.068 0.000

MALE 0.50 0.063 0.032 0.068 0.004

constant 1.00 –1.451 –1.451

Total 1.881



ASVABC 51.36 0.065 3.328 0.068 0.004

SM 11.58 –0.008 –0.094 0.068 –0.001

SF 11.84 0.006 0.066 0.068 0.000

MALE 0.50 0.063 0.032 0.068 0.004

constant 1.00 –1.451 –1.451

Total 1.881

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Every extra year of schooling of the mother decreases the probability of graduating by 0.1 percent. Father's schooling has no discernible effect. Males have 0.4 percent higher probability than females.


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Logit Probit Linear

f(Z)b f(Z)b b

ASVABC 0.004 0.004 0.007

SM –0.001 –0.001 –0.002

SF 0.000 0.000 0.001

MALE 0.004 0.004 –0.007

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The logit and probit results are displayed for comparison. The coefficients in the regressions are very different because different mathematical functions are being fitted.


Logit Probit Linear

f(Z)b f(Z)b b

ASVABC 0.004 0.004 0.007

SM –0.001 –0.001 –0.002

SF 0.000 0.000 0.001

MALE 0.004 0.004 –0.007

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Nevertheless the estimates of the marginal effects are usually similar.


Logit Probit Linear

f(Z)b f(Z)b b

ASVABC 0.004 0.004 0.007

SM –0.001 –0.001 –0.002

SF 0.000 0.000 0.001

MALE 0.004 0.004 –0.007

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However, if the outcomes in the sample are divided between a large majority and a small minority, they can differ.


Logit Probit Linear

f(Z)b f(Z)b b

ASVABC 0.004 0.004 0.007

SM –0.001 –0.001 –0.002

SF 0.000 0.000 0.001

MALE 0.004 0.004 –0.007

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This is because the observations are then concentrated in a tail of the distribution. Although the logit and probit functions share the same sigmoid outline, their tails are somewhat different.


Logit Probit Linear

f(Z)b f(Z)b b

ASVABC 0.004 0.004 0.007

SM –0.001 –0.001 –0.002

SF 0.000 0.000 0.001

MALE 0.004 0.004 –0.007

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This is the case here, but even so the estimates are identical to three decimal places. According to a leading authority, Amemiya, there are no compelling grounds for preferring logit to probit or vice versa.

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Finally, for comparison, the estimates for the corresponding regression using the linear probability model are displayed.


Logit Probit Linear

f(Z)b f(Z)b b

ASVABC 0.004 0.004 0.007

SM –0.001 –0.001 –0.002

SF 0.000 0.000 0.001

MALE 0.004 0.004 –0.007

Logit Probit Linear

f(Z)b f(Z)b b

ASVABC 0.004 0.004 0.007

SM –0.001 –0.001 –0.002

SF 0.000 0.000 0.001

MALE 0.004 0.004 –0.007

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If the outcomes are evenly divided, the LPM coefficients are usually similar to those for logit and probit. However, when one outcome dominates, as in this case, they are not very good approximations.


Binary Response Models: Interpretation II

• Probit: g(0)=.4• Logit: g(0)=.25• Linear probability model: g(0)=1

– To make the logit and probit slope estimates comparable, we can multiply the probit estimates by .4/.25=1.6.

– The logit slope estimates should be divided by 4 to make them roughly comparable to the LPM (Linear Probability Model) estimates.

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BINARY CHOICE MODELS: PROBIT ANALYSIS