Multinomial logistic regression where two choices are pooled/censored in the data?

Question

I am looking for a lead on estimating a specific type of multinomial choice model.

Specifically, assume that I see $N$ people and those people have some vector of characteristic $X$. For ten periods, individuals choose between four choices: $\{A,B,C,D\}$ based on their $X$ and their taste shock which we assume is distributed iid Weibull (which makes the model a multinomial logit).

So far this could be modelled as 10 standard multinomial logistic regressions, but here is what I am trying to add to the model. In my data, I do not observe $\{A,B,C,D\}$, rather I only observe $\{A,B,E\}$, where I see $E$ if the agent chooses $C$ or the agent chooses $D$, but I do not know which.

Given I have 10 repeated samples, it seems like it may be possible to use something like an expectation-maximization (EM) framework to estimate the probability each individual chose $C$ or chose $D$ given I observe that they chose $E$ (i.e. I know they chose one of the two) and to estimate the coefficients on $X$ for choice $C$ and choice $D$.

I also have another set of ten 10 linear models for each individual where the loadings on the observables depend on the choice the agent made above: $$Y_{i,j,k} = X_i\beta_{j,k} + \epsilon_{i,j,k}$$ for $j \in \{1,...,10\}$ and $k \in \{A,B,C,D\}$. Assume that the $\epsilon$ are uncorrelated with the errors from the choice equation so that selecion is not an issue. Then I can estimate $\hat{\beta}_{j,k}$ using the EM algorithm for those who chose $E$, where I assume two latent types. This would also let me estimate the probability each individual was one of the two types.

Looping back to the choices, I have a multinomial model, where rather than observing the particular choice, I observe the probabilities the person made each of the FOUR choices. If they choose $A$ or $B$, the probabilities will be $\{1,0,0,0\}$ or $\{0,1,0,0\}$, but if they choose $E$, I would now see the probabilities $\{0,0,.23,.77\}$ for example. This means that from the 10 conditional outcomes I observer I have some information about if the agent chose $C$ or $D$ when I observe a choice of $E$.

Does anyone know if it may be possible to consistently estimate the coefficients associated with choice $C$ and choice $D$ using the data laid out above? Is there a way to do this estimation jointly rather than in two steps as I have described it? Any related references would be very helpful.

Answer 1

Without any information on whether they choose $C$ or $D$ and a multinomial likelihood, the parameters for choosing $C$ or $D$ are non-identifiable. But you can get around this with a Bayesian model and a strong prior on whether $C$ or $D$ is preferred given that $E$ was selected. Alternatively, you may consider an ordered probit model if there is some natural ordering such that $D$ can always be considered "higher" in some sense than $C$.

Your 10 repeated samples and censoring is reminiscent of a multi-state mark-recapture model from ecology. Here you would be estimating the parameters for a transition matrix for each time period (probability a person first picks $A$, then switches to $B$, back to $A$, etc.) In ecology these might be states such as "nesting" and "foraging", with each state having an associated probability of observing an animal if it present during (this is the censoring bit). Check out Bayesian Population Analysis Using WinBUGS for some very good practical examples of how to fit such models. Many of the examples models in that book were recently translated into Stan, if you'd like to go that route.