Does Heckman Model the only method solving MNAR? I am interested in MNAR (missing data not at random).
In some missing data problems, we can use inverse probability of treatment weighting and MLE and multiple imputations. However, it seems those methods can not be proved right when missing data not at random.
The only solution I found is Heckman two-stage model, which also needs a full observable variable that determines the data selection.
Could anyone give me more information or recommend some related paper?
 A: If your data are MNAR, then you will need to make assumptions about the distribution of the missing data that cannot be verified by your observed data. You will have parameters that are not identified by the data. There's no escaping this.
As a nice resource, you might look at Section 3.8 of Stef Van Buuren's book. It provides a nice introduction for methods for dealing with non-ignorable missing data mechanisms.
Roughly, methods can be divided based upon whether they decompose the joint distribution of your data (Y) and missingness indicators (M) as either $f(Y,M)=f(Y)f(M|Y)$, which is the 'selection/Heckman model' approach that I believe you are referencing, or as $f(Y,M)=f(M)f(Y|M)$, which is the 'pattern mixture model approach'. You would then specify the joint distribution by virtue of specifying the constituent marginal and conditional distributions.
The nice thing about pattern mixture models (in my opinion, which is probably biased based upon my training and affiliations) is that you can usually more easily isolate the parameters that are not identified by your data and have a better grasp of the assumptions you are making. Here are some examples of this idea that I have been a part of.
