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 $D$.

If it is helpful for the EM algorithm, I also have another set of ten binary choices for each individual and 10 linear models for each individual which do not have the censoring/pooling problem laid out above.

Does anyone know if it may be possible to consistently estimate the coefficients and choice probabilities associated with choice $C$ and choice $D$ using the data laid out above? Even related references would be very helpful.

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.