Regression wouldmay not be the way to go for the following reasons:
The (note that the regressors would need to be standardised first to allow comparisons of derived coefficients, and this is not a problem. Regression 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 doesfor the above reasons it may not help much herebe an optimal method.