Variable importance across outcomes I would like to know if a variable X3 contributes more to explain outcome Y1 or outcome Y2. In model formula terms, I want to know whether the model improvement between

*

*Y1 ~ X1 + X2 + X3 vs Y1 ~ X1 + X2  or

*Y2 ~ X1 + X2 + X3 vs Y2 ~ X1 + X2

is greater. Is there any way to compare a variable X3 across outcomes?

Comparing predictive utility of an independent variable across predictive equations is tricky and is a topic of discussion in this article.  The article here suggests one way to disentangle the utility of a predictor on two different models is to use a dominance analysis (for example using this Stata implementation).
Consider for instance a model like this:
mpg ~ trunk + price + foreign
and
length ~ trunk + weight
Where our interest is in comparing 'trunk' across outcomes.
This model could be fit using sureg in Stata or a systemfit model in R.
The general idea is to fit a model where you can get a single fit statistic reflecting the entirety of the model in a single value.  In the below, the fit statistic is a McFadden pseudo-R2 ($1 - \frac{loglikelihood_{full}}{loglikelihood_{null}}$).
The implementation fitted below uses the domme module in Stata as it is easier to fit to models like these.
. sysuse auto
(1978 automobile data)

. domme (mpg = trunk price foreign) (length = trunk weight), reg(sureg (mpg = trunk price foreign) (length = trunk weight)) fitstat(e(), mcf) noconditional nocomplete

Total of 31 models/regressions

Progress in running all regression subsets
0%------50%------100%
....................
General dominance statistics: sureg
Number of obs             =                      74
Overall Fit Statistic     =                  0.1496

            |      Dominance      Standardized      Ranking
            |      Stat.          Domin. Stat.
------------+------------------------------------------------------------------------
mpg         |
 trunk      |         0.0111      0.0744            4 
 price      |         0.0112      0.0746            3 
 foreign    |         0.0034      0.0230            5 
length      |
 trunk      |         0.0238      0.1588            2 
 weight     |         0.1001      0.6692            1 
-------------------------------------------------------------------------------------

The idea here is that it appears that 'trunk' predicts 'length' better than it predicts 'mpg' when considering both in the context of the same model (i.e., fitted to the same loglikelihood) which makes the effects across equations more comparable (see the articles above for additional caveats about multi-equation models).
Also possible do implement a model like this using R's {domir} package but it is not yet optimized to do so easily (i.e., takes a lot more programming know-how).
