Identification on the theory part
For identification issue. When Y is continuous, Tang Little and Raghunathan (2003) mentions that only a certain type of parametric family can be allowed. We have to argue against this.
We might be able to use Manski's things. But we have to combine the identification in choice-based sampling with missing-data. Show that when P(Y|X) is specified, any model is ok.
Show that when H(Y,X) and f(X) are known, the missing-data mechanism is known. So missing-data becomes choice-based sampling problem. Then, when P(Y|X) is specified, all we have to do is to find an objective function that its unique solution is the true theta. In this second step, we may be able to use some material from Manski's chapter about choice-based sampling.