Recap:
Pros
- Unobserved preference variation
- unrestricted substitution patterns
- correlations in unobserved factors over time
Cons
- Mixed logit choice probabilities are not closed form
- Estimation requires numerical simulation
How:
- not use a set of fixed coeficients for the entire population
- assumes the distribution of coefficients throughout the population
The distribution of coefficients overcome three limitations:
Mixed Logit choice probabilities:
Therefore, the mixed logit choice probabilities is a weighted average of logit choice probabilities
- evaluated at different values of β
- weighted by the density of β
可以看出，标准的logit模型是mixed logit的一种特殊形式
Random coeefficients
Error components
在这种条件下，不同方案的残差项存在相关关系：
substitution pattern：残差项不同的相关关系可以整合出不同的替代模式，如nest内部均为1，nest间均为0，则可以整合为nested logit
mixed logit elasticities do nothave closed-form expressions
Panel data
mixed logit model allows for unobserved preference variation through random coefficients, which yields correlations in utility over time.
Some “dynamics” can be represented in a mixed logit model using panel data
- past and future exogenous variables can be included
- Lagged dependent variables can be included
Empirical consideration
cause the choice probabilities do not have a closed-form expression, we cannot estimate the model using MLE as log-likelihood function can’t be calculated
approximate the probabilities through numerical simulation, calculate the simulated log-likelihood function, and estimate using maximum simulated likelihood