Suggestion: Using your regression model, predict the outcome for each pair of independent variables saving the results. Using the resulting dataset, correlate the two independent variables with the predicted outcome variable. This has been used in the literature to address your question before.

Good luck,

J. McLean

Some people use lasso regression for this purpose. I have attached a screenshot that should help you should you decide to do this.yourself. The second one is a paper that should show you how well lasso works on a common non panel.sitistion

Best wishes David Booth

Harsh -

You say you "...want to rank two of my independent variables on the basis of their degree of influence over the dependent variable." That may not be a viable question to ask. The "independent" variables actually routinely influence one another, most notably through collinearity. Further, the best predicted-y "formula" depends on which set of variables function together, most nearly optimally. Also, it may not be very useful to say that variable A is more important than variable B, just perhaps because there is less omitted variable bias if B is missing than if A is missing, for example. Perhaps A and B together are nearly collinear with C, D, and E, but it is slightly better to use C, D, and E and drop both A and B.

So, I'm saying the question may not be very meaningful.

Why do you want to know? Is it a matter of having resources for collecting one or the other, but not both? If there are other variables also being collected, you have to see how well various possible collections of variables 'work together.' If affordability is the issue, you might want to look at all feasible options.  Sometimes one predictor variable alone is very good.  A "graphical residual analysis" will show you how well your model performs for your sample, including the modeling of heteroscedasticity.   You can compare various model options on the same scatterplot for the same sample.  However, to see if a model holds up rather than just for that sample, consider a "cross-validation." 

Best wishes - Jim

James E. McLean Sir, can you provide me with a reference paper where this has been done?

How's comparing variance explained (R^2) per predictor?