Multinomial Logit Regression: Probability of Going to College Given Intelligence, Siblings, Parent’s Education and IQ

Based on lecture by Dr. Gill, CSUF

Binary dependent variable models are useful for determining positive or negative results, but a research might be interested in an array of outcomes.  The Multinomial Logit Model is a model that allows the variable of interest to have multiple alternatives.  The probability of these alternatives can be analyzed with explanatory variables and interpreted much like other models where the variable of interest is not continuous.

Suppose one wanted to study the probability of a person dropping out of high school, graduating high school, or having some sort of college education.  This is different from the question, “what is the probability that a person drops out of high school given x set of variables”.  In this case we are examining relative probabilities which would estimate things like, “what is the probability that someone graduates high school as opposed to dropping out or getting some college as opposed to dropping out given x set of variables”.  In both of these case the probability of something occurring is relative to the base outcome which is dropping out of high school.

Explanation of Multinomial Probit Model

Derivation of Beta’s in the Regression

The derivation of the Betas in the Regression is a little tricky, first the relative probability is a ration of exponential functions and second in order to interpret the Betas marginal effects need to be calculated.  This is of course after setting a base alternative that is used to calculate the relative probabilities.  To simply matters even further we make the assumption that the explanatory variables are the same for all individuals this removing the j from the x.

In order to understand this ratio better and know what the output will be in STATA we need to eliminate the exponent and take the partial derivative with respect to x to get the interpretation of beta, which is the relative probability of alternative 3 relative to alternative 1 as it is explained by variable x.

Empirical Estimations of Relative Odds of Educational Attainment

These results are all in relative log odds, so in order to be able to interpret these coefficients the marginal effects command is use

Concluding Remarks and Interpretation of Multinomial Logit Model and Factors Influencing Educational Attainment

Interpretation of the last one can serve as an example of how to interpret the first and second.  The relative probability of going to college instead of dropping out of school increases by two percent for every point of IQ above the average and by 4.2% and 3.3% for every year of education your father and mother have.  Every sibling reduces your probability of going to college relative to dropping out by 2% and living in an urban environment increase your chances by 7.3 %, but the number of siblings and whether or not you live in an urban setting are statistically insignificant in determining these relative odds.