I'm trying to assess if the following intuition has sense or not. Any help will be appreciated.
Consider two simple linear models obtained in the same experimental framework (for example, measurements involve the same $N$ subjects):
Model 1: $ Y_1 = \alpha + \beta X_1 $
Model 2: $ Y_2 = \alpha + \beta X_2 $
Hypothesize that the effect of $X_1$ in the first model and the effect of $X_2$ in the second model are both strongly significant. Now consider a situation in which, for some logical reason, the experimenter may want to assess if the two effects are associated, hypothesizing that the second one may depend on the first one.
Would it be sensed, given the aim, to merge the two models using the following "system" of linear models?
Model 3A: $ X_2 = \alpha + \beta X_1 + \beta Y_1 + \beta X_1 Y_1 $
Model 3B: $ Y_2 = \alpha + \beta X_1 + \beta Y_1 + \beta X_1 Y_1 $
Model 3C: $ X_2 Y_2 = \alpha + \beta X_1 + \beta Y_1 + \beta X_1 Y_1 $
Alternatively, given the interest in the interactions, will the following model be sufficient?
Model 4: $ X_2 Y_2 = \alpha + \beta X_1 Y_1 $
Of course, multivariate regression is not an option here because of the interaction (at least, that was my thought). Any other suggestions are appreciated (e.g., maybe SEM should be applied instead..?). Thanks in advance.
P.S. Note that $Y_1$, $Y_2$, $X_1$, and $X_2$ are different measures so that, for example, the variables $Y_1$ and $Y_2$ can not be concatenated.
P.S. Also note that the various coefficients ($\alpha$ and $\beta$) are not supposed to be equal across models. I just avoided indexing them to simplify the reading.