How to handle missing data in all explanatory variables in linear regression

Question

It's multivariate linear regression meaning multiple dependent variables (Y). Data A has both X (explanatory variables) and Y, but data B has only Y.

I wonder if there is any way to incorporate data A and B into the regression model.

If I remember correctly, Statistical Analysis with Missing Data by Little et al, suggested some kind of iterative approach like EM algorithm where you repeat following until convergence

1. Estimate missing X based on current regression model
2. Compute regression model with full X estimated in #1

But in this case, I have all columns of X matrix is missing. So my questions:

1. Is the approach above still effective? That is, would including data B into the model help in any way?
2. If so, how missing full columns of X can be estimated?

Update: The reason I want to add data B is because data B is application-specific data while data A is more general data. I hope to have training data that has more "weights" for application-specific input space.

Answer 1

In this case, dataset B only provides information about the unconditional distribution of the responses (Y) and nothing about the relationship between the atributes (X) and the responses (Y). So in this case I rather doubt there will be much to gain from using dataset B to estimate the regression parameters, which encode this relationship, without making some stronger assuption about the data generating process.

One condition where it might be of some use is in the presence of covariate shift, where there has been a change in the distribution of the attributes between the time the data were collected and in operational conditions. In this case if B is representative of operational conditions it may be of some use.