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. as the mixing distribution. 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.

Random taste variation

's for each person (i.e., each decision maker.)

is:

with

1. ~ iid extreme value

in the mixed logit model is:

with

1. ~ iid extreme value

where θ.

is the standard logit formula:

.

is a random variable. 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. The most widely used distribution is normal, mainly for its simplicity. For coefficients that take the same sign for all people, such as a price coefficient that is necessarily negative or the coefficient of a desirable attribute, distributions with support on only one side of zero, like the lognormal, are used. or triangular distributions.

Unrestricted substitution patterns

The mixed logit model can represent general substitution pattern because it does not exhibit logit's restrictive independence of irrelevant alternatives given a percentage change in the m (the elasticity) is

is the m...

Correlation in unobserved factors over time

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. That is a very unlikely assumption. To take into account both random taste variation and correlation in unobserved factors over time, the utility for respondent n for alternative i at time t is specified as follows:

's, which enter utility in each time period and each alternative.

To examine the correlation explicitly, assume that the β. Then the utility equation becomes:

and η is a draw from the standard normal density. Rearranging, the equation becomes:

is not independent over time or alternatives.

is,

is

By specifying the X's appropriately, one can obtain any pattern of covariance over time and alternatives.

, the probability of the sequence of choices by a person is simply the product of the logit probability of each individual choice by that person:

.

Simulation

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. There are four basic steps to follow

representing the first draw.

. (The conditional probability.)

.

4. Average the results

Then the formula for the simulation look like the following,

where R is the total number of draws taken from the distribution, and r is one draw.

Once this is done you will have a value for the probability of each alternative i for each respondent n.

See also

• Discrete choice

Further reading

• Ch. 6 of Discrete Choice Methods with Simulation, by Kenneth Train (Cambridge University Press)

References