How to combine regression models?

Say I have three data sets of size $n$ each:

$y_1$ = heights of people from the US only

$y_2$ = heights of men from the whole world

$y_3$ = heights of women from the whole world

And I build a linear model for each with factors $x_i$, $i = 1,..., k$:

$\hat{y}_{j} = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \epsilon_{j}$

with $\epsilon$ having the usual properties for OLS. And I may use a factor $x_i$ in more than one regression.

My question is: How could I combine the regressions such that I can obtain estimates for:

$y_{12}$ = height of men from the US only

$y_{13}$ = height of women from the US only

for which I do not have data

I thought of perhaps some sort of weighting:

$\hat{y}_{12} = w_{1} \hat{y}_{1} + (1 - w_{1}) \hat{y}_{2}$

but then I wouldn't know what to use for $w_1$.

• I don't have anything solid enough to be an answer, but as a comment: the first thing that comes to mind is using a single hierarchical (mixed) regression. But I really can't figure out what would be the random effects, so maybe it wouldn't work. Thought I'd throw the idea out there, though. Jan 14 '16 at 14:25
• Thanks for the suggestion. Yes, it would appear that the for hierarchical model you would need to do it on the $y_{12}$ data, no?
– J4y
Jan 18 '16 at 13:59
• My initial thought was on y with an intercept by sex -- something like height ~ f1 + f2 + f3 + (1 | sex) in the R package lmer Jan 18 '16 at 18:33