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I have difficulties fitting a joint model in R. My data consists of two responses X & Y and one predictor variable Z. Now I want to model both X and Y in function of Z (just linear regression: $E(X|Z)=Z\alpha$ and $E(Y|Z)=Z\beta$, both outcomes are normally distributed) but while doing so I also want to estimate the variance covariance matrix since it is the correlation between X and Y that I am interested in.

I already looked into a couple of functions (lm, mcer, lme) but it doesn't seem to do the trick. Is there something that I am overlooking in a certain package or a new suggestions to try?

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By having two linear regression models as defined above, you are implicitly assuming that X and Y are conditionally independent given Z. For starters, you should instead look into multivariate linear regression, which should not be confused with multiple regression.

In the linear regression context (i.e., general linear model), multivariate means multiple outcomes, while multiple means multiple inputs.

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Try to use copula models. Refer this article https://www.researchgate.net/publication/265601371_Joint_analysis_of_mixed_discrete_and_continuous_outcomes_via_copula_models

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