Lecture 14-2 Multinomial logit (Maddala Ch 12.2) Research Method Lecture 14-2 Multinomial logit (Maddala Ch 12.2)

Multinomial logit as a random utility model It is useful to understand the multinomial logit model as a random utility model with extreme value distribution. First, we consider a random utility model with two choices. This turns out to be the same as the logit model. This can be naturally extended to more than 2 choices, which becomes the multinomial logit model.

Random utility model with two choices Consider the labor force participation of married women. The woman decides whether to participate or not in the labor force by comparing the utility from participation and non-participation.

where we are using the following vector notation. Consider that the utility from participation, U1, and the utility of non-participation, U2 are given by: where we are using the following vector notation. Transpose to make this a column vector.

It is important to understand what the above vector notation means:

The assumptions We make the following assumptions. The two error terms ε1 and ε2 are independent within a person. These error terms are independent across persons. Both ε1 and ε2 follow type I extreme value distribution, where the density and the cumulative distribution functions are given by

Given these assumptions, the probability of participation is given by:

In a similar way, you can compute the probability of non-participation,

To summarize, These probability is the basis for the estimation. If a person is working, then the likelihood contribution is P1. If the person is not working, the likelihood contribution is P2.

One important thing to note is that, you cannot estimate β1 and β2 separately. You can only estimate the difference (β1- β2). This is fine since our purpose is to investigate the participation probability, and it is determined by the difference. For estimation, we normalize either β1 or β2 to be zero.

So, let us normalize be β2 to be zero: β2 =(β20, β21)=(0,0) So, let us normalize be β2 to be zero: β2 =(β20, β21)=(0,0). In this case, we set the equation 2 to be the base equation. When we set the equation 2 to be the base equation, P1 and P2 can be written as: Notice, these are identical to the logit model. Thus, when there are only two choice, the random utility model is identical to the logit model.

The important things to remember are that You have to normalize the parameters of one equation to be zero. When one equation is normalized in two choice mode, it is identical to the logit model. This model can be extended to more than two choices. Such models are called the multinomial logit model.

Exercise Using Mroz.dta, estimate the following model using STATA mlogit command. This command estimate the multinomial logit model. Set the non-participation outcome as the base outcome.

Answer

Check that mlogit is identical to the logit model

Multinomial logit model (Random utility model with 3 choices) When we extent the model to 3 or more choices, the model is called the multinomial logit model. For simplicity, I only explain the case where we have 3 choices.

As an example, consider the choice of working in either (1) national university, (2) private university and (3) public university. Then, the random utility model is written as: We assume that ε1, ε2 and ε2 are independent and follow type I extreme value distribution.

Then, the probability that a person is working in private university is written as

In a similar way, we can compute the probability of working in national university and public university as follows.

Similarly to the case of two choice cases, we have to normalize the parameters of one equation to be zero. This is because we can estimate only the differences: (β1-β2) and (β2-β3).

So, let us set the parameters for equation 3 to be zero. Then, the probability can be re-written as:

Let’s set equation 3 to be the base equation (set the parameters for the third equation to be zero). This means that we set equation 3 to be the base equation. Then, the probabilities are written as:

If the person is working in private university, the likelihood contribution of that person is P1. If the person is working in national university, the likelihood contribution is P2. If the person is working in public university, the likelihood contribution is P3. This model is called the multinomial logit model.

Exercise Estimate multinomial logit model of the choice among working in Private, National and Public University with experience as the only explanatory variable. Set public university as the base equation. Use univtype.dta

The estimated probability After estimating the multinomial logit model, you can estimate the probability of working in either private, national or public university given the value of explanatory variable as:

The partial effects Partial effect chose “how much the probability of choosing one alternative increase if the explanatory variable increases by one unit”. Let us take the probability of choosing private university as an example. In our example x=(1, x1)T, where x1 = age.

Then, the partial effect is given as: You have to note the use of the vector notation where and . Thus, β11 is the age coefficient for the first equation, and β21 is the age coefficient for the second equation. The partial effect for other alternatives are computed by taking derivatives in a similar way.

The partial effect at average is computed by plugging in the average values of the explanatory variables, and it is computed automatically by STATA. It is extremely important to note that the sign of the partial effect depends not only on the parameters of that equation but also the parameters of other equations. In some case, you may have negative parameter in one equation but the partial effect is positive. Thus you always have to check the partial effect before interpreting the results.

Exercise Estimate multinomial logit model of choice among working in Private, National and Public University with experience as the only explanatory variable. Set public university as the base equation. Use univtype.dta. Then, compute the effect of experience on working in a private university.

Computing the partial effect manually

Computing the partial effect automatically