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'''Mixed logit''' is a fully general statistical model for examining [[Discrete Choice|discrete choices]]. The motivation for the mixed logit model arises from the limitations of the standard [[Discrete Choice#F._Logit_with_variables_that_vary_over_alternatives_(also_called_conditional_logit)|logit]] model. The standard logit model has three primary limitations, which mixed logit solves: "It [Mixed Logit] obviates the three limitations of standard logit by allowing for random taste variation, unrestricted substitution patterns, and correlation in unobserved factors over time."<ref name=dca>[http://elsa.berkeley.edu/choice2/ch6.pdf Train, K. (2003) Discrete Choice Methods with Simulation]</ref> Mixed logit can also utilize any distribution for the random coefficients, unlike probit which is limited to the normal 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 an appropriate specification of variables and distribution of coefficients."<ref name=mt-mnl>[[Daniel McFadden|McFadden, D.]] and [[Kenneth E. Train|Train, K.]] (2000). “[http://elsa.berkeley.edu/wp/mcfadden1198/mcfadden1198.pdf Mixed MNL Models for Discrete Response],” Journal of Applied Econometrics, Vol. 15, No. 5, pp. 447-470,</ref> The following discussion draws from [http://elsa.berkeley.edu/choice2/ch6.pdf Ch. 6] of [http://elsa.berkeley.edu/choice2/index.html Discrete Choice Methods with Simulation], by [[Kenneth E. Train|Kenneth Train]] ([[Cambridge University Press]]), to which the reader is referred for more details and citations. See also the article on [[discrete choice]] for information on how the mixed logit relates to discrete choice analysis in general and to other specific types of choice models.

==Random taste variation==

The standard logit model's "taste" cofficients, or <math> \beta </math>'s, are fixed, which means the <math> \beta </math>'s are the same for everyone. Mixed logit has different <math> \beta </math>'s for each person (i.e., each decision maker.)

In the standard logit model, the utility of person n for alternative i is:

:<math> U_{ni} = \beta x_{ni} + \varepsilon_{ni} </math>

with

:<math> \varepsilon_{ni} </math>  ~ iid extreme value

For the mixed logit model, this specification is generalized by allowing <math> \beta_n </math> to be random. The utility of person n for alternative i in the mixed logit model is:

:<math> U_{ni} = \beta_n x_{ni} + \varepsilon_{ni} </math>

with

:<math> \varepsilon_{ni} </math>  ~ iid extreme value

:<math> \quad \beta_n \sim f(\beta_n | \theta) </math>

where ''θ'' are the parameters of the distribution of <math> \beta_n </math>'s over the population, such as the mean and variance of <math> \beta_n </math>.

Conditional on <math>\beta_n</math>, the probability that person n chooses alternative i is the standard logit formula:
:<math> L_{ni} (\beta_{n}) = \frac{e^{\beta_{n}X_{ni}}} {\sum_{j} e^{\beta_{n}X_{nj}}} </math>
However, since <math>\beta_n</math> is random and not known, the (unconditional) choice probability is the integral of this logit formula over the density of <math>\beta_n</math>.

:<math> P_{ni} = \int L_{ni} (\beta) f(\beta | \theta) d\beta </math>

This model is also called the random coefficient logit model since <math> \beta_n</math> 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 <math>f(\beta_n | \theta) </math>. 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.<ref name="rt">David Revelt and [[Kenneth E. Train|Train, K]] (1998). "[http://www.jstor.org/stable/pdfplus/2646846.pdf Mixed Logit with Repeated Choices: Households' Choices of Appliance Efficiency Level]," Review of Economics and Statistics, Vol. 80, No. 4, pp. 647-657</ref><ref name="rec">[[Kenneth E. Train|Train, K]] (1998)."[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.4879 Recreation Demand Models with Taste Variation]," Land Economics, Vol. 74, No. 2, pp. 230-239.</ref> When coefficients cannot logically be unboundedly large or small, then bounded distributions are often used, such as the <math> S_b </math> 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 the probability for one alternative given a percentage change in the ''m''th attribute of another alternative is

:<math> E_{nix_{nj}^m} = -\frac{x_{nj}^m} {P_{ni}} \int \beta^m L_{ni}(\beta) L_{nj}(\beta) f(\beta) d \beta = - x_{nj}^m \int \beta^m L_{nj} (\beta) \frac{L_{ni} (\beta)} {P_{ni}} f(\beta) d \beta </math>

where ''β'' <sup> ''m'' </sup> is the ''m''th element of <math> \beta </math>.<ref name="dca">Train, Kenneth</ref>  It can be seen from this formula that "A ten-percent reduction for one alternative need not imply (as with logit) a ten-percent reduction in each other alternative."<ref name="dca"/> The relative percentages depend on correlation between the likelihood that respondent n will choose alternative i, ''L <sub> ni </sub>'', and the likelihood that respondent n will choose alternative j,  ''L <sub> nj </sub>'', over various draws of ''β''.

==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:

:<math> U_{nit} = \beta_{n} X_{nit} + \varepsilon_{nit} </math>

where the subscript t is the time dimension. We still make the logit assumption which is that <math>\varepsilon</math> is i.i.d extreme value. That means that <math>\varepsilon</math> is independent over time, people, and alternatives. <math>\varepsilon</math> is essentially just white noise. However, correlation over time and over alternatives arises from the common effect of the <math> \beta </math>'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 <math>\bar{\beta}</math> and variance <math> \sigma^2 </math>. Then the [[utility]] equation becomes:

:<math> U_{nit} = (\bar{\beta} + \sigma \eta_{n}) X_{nit} + \varepsilon_{nit} </math>

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

:<math> U_{nit} = \bar{\beta} X_{nit} + (\sigma \eta_{n} X_{nit} + \varepsilon_{nit}) </math>

:<math> U_{nit} = \bar{\beta} X_{nit} + e_{nit} </math>

where the unobserved factors are collected in <math>  e_{nit} = \sigma \eta_{n} X_{nit} + \varepsilon_{nit} </math>. Of the unobserved factors, <math>\varepsilon_{nit}</math> is independent over time, and <math> \sigma \eta_{n} X_{nit} </math> is not independent over time or alternatives.

Then the covariance between alternatives <math> i </math> and <math> j </math> is,

:<math> Cov(e_{nit}, e_{njt}) = \sigma^2 (X_{nit} X_{njt}) </math>

and the covariance between time <math> t </math> and <math> q </math> is

:<math> Cov(e_{nit}, e_{niq}) = \sigma^2 (X_{nit} X_{niq}) </math>

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

Conditional on  <math> \beta_n </math>, 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:

:<math> L_{n} (\beta_{n}) = \prod_{t} \frac{e^{\beta_{n}X_{nit}}} {\sum_{j} e^{\beta_{n}X_{njt}}} </math>

since <math> \varepsilon_{nit} </math> 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 <math> \beta </math>.

:<math> P_{ni} = \int L_{n} (\beta) f(\beta | \theta) d\beta </math>

==Simulation==

Unfortunately there is no closed form for the integral that enters the choice probability, and so the researcher must simulate P<sub>n</sub>. Fortunately for the researcher, simulating  P<sub>n</sub> 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 <math> f(\beta | \theta) </math> and label the draw <math>\beta^r</math>, for <math>r=1</math> representing the first draw.

2. Calculate <math>L_n(\beta^r)</math>. (The conditional probability.)

3. Repeat many times, for <math>r=2,...,R</math>.

4. Average the results

Then the formula for the simulation look like the following,

<math> \tilde{P}_{ni} = \frac {\sum_{r} L_{n}(\beta^r)} {R} </math>

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.

==References==
{{Reflist}}

{{DEFAULTSORT:Mixed Logit}}
[[Category:Statistical models]]