Mixed logit is a fully general statistical model for examining discrete choices.
It overcomes three important limitations of the standard logit model by allowing for random taste variation across choosers, unrestricted substitution patterns across choices, and correlation in unobserved factors over time.
for the random coefficients, unlike probit which is limited to the normal distribution.
[2] It has been shown that a mixed logit model can approximate to any degree of accuracy any true random utility model of discrete choice, given appropriate specification of variables and the coefficient distribution.
[3] The standard logit model's "taste" coefficients, or
In the standard logit model, the utility of person
is: with For the mixed logit model, this specification is generalized by allowing
in the mixed logit model is: with where θ are the parameters of the distribution of
is random and not known, the (unconditional) choice probability is the integral of this logit formula over the density of
It allows the slopes of utility (i.e., the marginal utility) to be random, which is an extension of the random effects model where only the intercept was stochastic.
Any probability density function can be specified for the distribution of the coefficients in the population, i.e., for
[4][5] When coefficients cannot logically be unboundedly large or small, then bounded distributions are often used, such as the
The mixed logit model can represent general substitution pattern because it does not exhibit logit's restrictive independence of irrelevant alternatives (IIA) property.
given a percentage change in the mth attribute of alternative
need not imply (as with logit) a ten-percent reduction in each other alternative
[1] The reason is that the relative percentages depend on the correlation between the conditional likelihood that person
Standard logit does not take into account any unobserved factors that persist over time for a given decision maker.
This can be a problem if you are using panel data, which represent repeated choices over time.
By applying a standard logit model to panel data you are making the assumption that the unobserved factors that affect a person's choice are new every time the person makes the choice.
However, correlation over time and over alternatives arises from the common effect of the
's, which enter utility in each time period and each alternative.
To examine the correlation explicitly, assume that the β's are normally distributed with mean
Then the utility equation becomes: and η is a draw from the standard normal density.
Rearranging, the equation becomes: where the unobserved factors are collected in
is By specifying the X's appropriately, one can obtain any pattern of covariance over time and alternatives.
Then the (unconditional) probability of the sequence of choices is simply the integral of this product of logits over the density of
Unfortunately there is no closed form for the integral that enters the choice probability, and so the researcher must simulate Pn.
Fortunately for the researcher, simulating Pn can be very simple.
Take a draw from the probability density function that you specified for the 'taste' coefficients.
Average the results Then the formula for the simulation look like the following,