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Suppose you have some covariates X1, X2, X3, X4, X5 and you want to create a regression model where there are two response variables : Y1 and Y2.

In the past, I have approached this problem by creating two separate regression models:

Y1 ~ f(X1, X2, X3, X4, X5)

Y2 ~ f(X1, X2, X3, X4, X5)

However, now I am interested in learning about statistical models that can jointly model this data together, for example:

P(Y1, Y2) ~ f(X1, X2, X3, X4, X5)

I think that this approach would allow for modelling potential correlation structures with Y1 and Y2.

My Question: Do such models exist that can jointly model multiple responses - is this a popular topic in statistics? Are there any standard references and software implementations (e.g. R programming language) for these kinds of models? The only thing I could find was the following : https://www.jstatsoft.org/article/view/v084i04.

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Multivariate linear regression can be handled directly by the standard lm() function in R. As Fox and Weisberg say (page 3):

Multivariate linear models are fit in R with the lm() function. The procedure is the essence of simplicity: The left-hand side of the model is a matrix of responses, with each column representing a response variable and each row a case; the right-hand side of the model and all other arguments to lm are precisely the same as for a univariate linear model...

That reference illustrates how to use functions provided by the car package to simplify calculations and tests that take the outcome correlations into account.

Things aren't quite so simple with generalized linear models. The mcglm package that you cite helps with such models.

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