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 is i.i.d extreme value. That means that is independent over time, people, and alternatives. is essentially just white noise. 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 and variance . 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 . Of the unobserved factors, is independent over time, and is not independent over time or alternatives.
Then the covariance between alternatives and is,
and the covariance between time and is
By specifying the X's appropriately, one can obtain any pattern of covariance over time and alternatives.
Conditional on, 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 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 .
Read more about this topic: Mixed Logit
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