Test for comparing corresponding coefficients in multivariate regression

Question

This question is related to this but I am hoping for more tangible techniques than a general discussion.

For simplicity, suppose I am trying to linearly model two correlated responses $Y_1$ and $Y_2$ with numerous predictors, $X_1,\ldots,X_p$. I am interested in analysing and applying hypothesis tests to understand whether each predictor $X_i$ is more influential on $Y_1$ or $Y_2$ (or neither).

What techniques/tests are available to do this? As a side note, if a technique other than standard MVR would be more appropriate, I would be happy to change to that instead.

My language of choice is R but I am happy to manually implement any algebra if someone has a resource detailing the formulation of relevant test statistics.

Answer 1

In my opinion, you are asking too much, with many variables, from simple multiple regression analysis. Expect wrong signs, and other unexplainable results. Supporting comments from a source, to quote:

The problems involved in obtaining meaningful coefficients of regression by the method of least squares with such intercorrelated data are well known [2, 13, 21]

A possible solution, employ a more advanced analysis approach. For example, try Factor Analysis to construct variables as in this work 'Use of factor scores in multiple regression analysis for estimation of body weight by certain body measurements in Romanov Lambs' and also a related work here.

In the context of your problem, follow the work in these two articles carefully.

Also, one can also look into Factor Analysis Regression, which may be appropriate to quote a source:

Factor analysis appears to be a particularly appropriate tool in the field of economics where many “independent” variables have high intercorrelation and where there are errors in all the variables .

Here are pertinent comments relating to coefficients, which differ from more facilely generated Least-Squares Regression parameters:

Stochastic linear equations can be obtained from factor analysis which give better coefficients (better from the standpoint of their economic meaning and their theoretical expectation) then do regression equations obtained through a traditional least squares [9].

Better tools, applied with knowledge and skill, may produce better results.

Answer 2

Regression may not be the way to go for the following reasons (note that the regressors would need to be standardised first to allow comparisons of derived coefficients, and this is not a problem Itself):

Regression coefficients are dependent on the other regressors in the model, unless the two models $Y_1$ and $Y_2$ can be modelled with the same set of regressors (unlikely) then comparing the same regressor coefficient between models is possibly not a good idea.

If a regressor is needed in the model more than once as lagged or polynomial values, then it will not be easy to compare it to a regressor that is just present once.

Correlation between regressors may be present thus the influence of a regressor can be between regressors. Any correlated regressors may effect the independent variable in different ways to any non-correlated regressors. Meaning correlated and non-vorrelated regressors cannot be compared without ignoring this complication.

Will be interesting for answers to this question, regression is fairly easy but for the above reasons it may not be an optimal method.