I have a mathematical question, if someone could help :) :

If a have a equation with predifined defined weights as :

$Y= a_0 +a_1X_1 + a_2X_2 + a_3\mathrm{sqrt}(X_3) + a_4\log(X_4)$

and I want to estimate the contribution of each independent variable$ X_1, X_2,$, etc.. on the dependent variable $Y$.

What measures can I use for having these contributions (and what are the formulas to estimate them)?

E.g: I would like to be able to say $X_1$ has a contribution of n% on $Y$

  • $\begingroup$ If the variables are normalized and independent, you can consider the coefficient as its contribution to the Y in terms of the volatility. $\endgroup$ – hbadger19042 Jun 17 '20 at 9:57

One good approach is to find the linear regressions $$Y = a_0$$ $$Y = b_0 + b_1 X_1$$ each with its adjusted $R^2$. Then the difference between those $R^2$ values is the amount additionally explained by $X_1$.

Similarly the differences in adjusted $R^2$ between the regressions with successively more variables

$$Y = c_0 + c_1 X_1+c_2X_2$$

$$Y = d_0 + d_1 X_1+d_2X_2+d_3 \sqrt(X_3)$$

$$Y = e_0 + e_1 X_1+e_2X_2+e_3 \sqrt(X_3)+e_4 \ln(X_4)$$ represent the additional amounts explained by those variables.

Then the amounts explained by each variable will add up to the total $R^2$ explained by the big regression.

For meaningful results, it’s important that the variables avoid multicollinearity. For optimal results, the variables should be arranged in from most important to least important, so that the additional amount explained declines with each additional variable.


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