Regression - Merging separate linear regression models

Question

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.

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The Question:

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.

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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.

Answer 1

I recommend reading more about multivariate analysis, specifically multivariate multiple regression. "multivariate" means multiple dependent variables. "multiple regression" means that there are multiple independent variables. When models are fit in this context, you will see that the parameter estimates account for the covariance between the dependent variables. See this for an introduction with R. A super-simple example is below:

mlm1 <- lm(cbind(mpg,  hp) ~ cyl + disp + wt + vs, data = mtcars)
summary(mlm1)
vcov(mlm1)

EDIT: Update with additional explanation

The question assumed "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". This is what leads to multivariate regression. $Y_1$ depends on $X_1$, $Y_2$ depends on $X_2$, and they may depend on each other. That dependency shows up in the form of a correlation between $Y_1$ and $Y_2$ and some interdependency on the $X_1$ and $X_2$ independent variables. Multivariate models can include interactions.

Alternate Answer

It is also possible that what is meant by the question is a causal mediation model where $X_2$ and $Y_1$ are causal mediators. using the original $Y_1, Y_2, X_2, X_1$ notation:

$Y_1 = \beta_{10} + \beta_{11} X_1$

$X_2 = \beta_{20} + \beta_{21} Y_1 + \beta_{22} X_1$

$Y_2 = \beta_{30} + \beta_{31} X_2 + \beta_{32} Y_1 + \beta_{33} X_1$

I recommend fitting this with MCMC using Stan.