Can I maximize the correlation between a linear combination of variables and some other variable?

Question

For example, If I have 3 sets of data and I want to see what is the best combination between these 3 sets of data and a stocks price, is there a way to optimize a linear combination between the 3 variables to have it maximally correlated with price? I am thinking some sort of cplex optimization but I am not sure theoretically if this works.

Answer 1

Some observations when $n=1$ w.o. intercept to guess what is going on.

Let's fix $X$ our predictor and $Y$ the target.

Than:

$cov(\alpha X,Y)=\alpha cov(X,Y)$

is a linear function of $\alpha$. Not much to be maximized here, just a change of scale changes the covariance in a linear (unbounded) fashion.

Instead:

$corr(\alpha X,Y)=corr(X,Y)$

is scale independent. Again, not much to be maximized here.

This shows that you are not looking at a sensible metric, I think.

LR instead would look at $E[(\alpha X-Y)^2]$ ( one may be more used to the form of this expression as the sum of the squared residuals ), that is quadratic in $\alpha$ and can therefore be minimized.