Mixed logit

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. Mixed logit can choose any distribution formula_1 for the random coefficients, unlike probit which is limited to the normal distribution. It is called "mixed logit" because the choice probability is a mixture of logits, with formula_1 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.
The standard logit model's "taste" coefficients, or formula_3's, are fixed, which means the formula_3's are the same for everyone. Mixed logit has different formula_3's for each person (i.e., each decision maker.)
In the standard logit model, the utility of person formula_6 for alternative formula_7 is:
with
For the mixed logit model, this specification is generalized by allowing formula_10 to be random. The utility of person formula_11 for alternative formula_12 in the mixed logit model is:
with
where "θ" are the parameters of the distribution of formula_10's over the population, such as the mean and variance of formula_10.
Conditional on formula_18, the probability that person formula_6 chooses alternative formula_12 is the standard logit formula:
However, since formula_18 is random and not known, the (unconditional) choice probability is the integral of this logit formula over the density of formula_18.
This model is also called the random coefficient logit model since formula_25 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 can be specified for the distribution of the coefficients in the population, i.e., for formula_26. 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. When coefficients cannot logically be unboundedly large or small, then bounded distributions are often used, such as the formula_27 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 (IIA) property. The percentage change in person formula_6's unconditional probability of choosing alternative formula_12 given a percentage change in the "m"th attribute of alternative formula_30 (the elasticity of formula_31 with respect to formula_32) is
where formula_34 is the "m"th element of formula_3. It can be seen from this formula that a ten-percent reduction for formula_31 need not imply (as with logit) a ten-percent reduction in each other alternative formula_37. The reason is that the relative percentages depend on the correlation between the conditional likelihood that person formula_6 will choose alternative formula_39 and the conditional likelihood that person formula_6 will choose alternative formula_41 over various draws of formula_42.
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:
where the subscript t is the time dimension. We still make the logit assumption which is that formula_44 is i.i.d extreme value. That means that formula_44 is independent over time, people, and alternatives. formula_44 is essentially just white noise. However, correlation over time and over alternatives arises from the common effect of the formula_3'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 formula_48 and variance formula_49. 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 formula_53. Of the unobserved factors, formula_54 is independent over time, and formula_55 is not independent over time or alternatives.
Then the covariance between alternatives formula_56 and formula_57 is,
and the covariance between time formula_59 and formula_60 is
By specifying the X's appropriately, one can obtain any pattern of covariance over time and alternatives.
Conditional on formula_10, 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:
since formula_64 is independent over time. Then the (unconditional) probability of the sequence of choices is simply the integral of this product of logits over the density of formula_3.
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
1. Take a draw from the probability density function that you specified for the 'taste' coefficients. That is, take a draw from formula_67 and label the draw formula_68, for formula_69 representing the first draw.
2. Calculate formula_70. (The conditional probability.)
3. Repeat many times, for formula_71.
4. Average the results
Then the formula for the simulation look like the following,
formula_72
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.