Suppose I have a basket of 3 securities A, B, and C. I believe that the basket is cointegrated and I want to create a mean-reverting trade. I fit the model: $\log(A)=\beta_b*\log(B)+\beta_c*\log(C)+\alpha$ where A, B, and C are the prices of the securities.
This gives me estimates of $\alpha$, $\beta_b$ and $\beta_c$.
Now suppose that I believe that the spread is out of line. I want to sell \$1 of A and buy \$1 of the B and C basket. How should I allocate that dollar to B and C? Is it simply $\beta_b*\$1$ units of B and $\beta_c*\$1$ units of C or is it more complex?
Related, is it more correct to regress log prices or raw prices when fitting the model?
(I know that this is related to How to build a mean reverting basket? but the answers there weren't very detailed and this is a more specific question).