What will be the best measure of the contribution of a variable in multiple linear regression? I was thinking of using the coefficient ratio as a marker of a variable's contribution.
For example:
If the equation is $$Y_{predicted}=a_1X_1+a_2X_2+a_3$$
Then $X_1$'s contribution can be written down as: $$\frac{a_1}{a_1+a_2+a_3}$$ Is there some other method possible to write down the contribution. Since in this case if any coefficient is negative, there is a possibility that a variable's contribution exceeds $100\%$.
Your method doesn't work unless your variables have been normalized to 1, and are positive.
The usual way to do what you describe is to do an "ANOVA", looking at the variance for each term and its contribution to the whole variance. A very easy example (especially with few variables) is the $R^2$, the "fit" of your model.
You can look at the correlation as well (the higher the correlation in absolute value the more explaining power).
Lastly, you can check the principal component analysis, which will help you select the combination of variables that has the most explanatory power.