product of two variables fitting problem

Question

I have two time-varying variables X and Y. The product of which is Z(= X*Y). A curve Z' has been fit on Z. So Z'can be described as a product of some new X' and Y'. Where X' = X + dX and Y' = Y + dY. How do I get these new X' and Y'? It looks like a regression problem to me. But what are the rules I will have to follow to get X' and Y'?

Answer 1

$Z'=X'Y'$

The problem here is that there are (infinitely) many choices of $X'$ and $Y'$ that will give the product you desire.

$Z'=(X+dX)(Y+dY)$

$Z'=XY+YdX+XdY+dXdY$

You could decide that you want the variations $dX$ and $dY$ to be equal.

In that case set $dX=dY=\delta$

Then $Z'=XY+Y\delta+X\delta+\delta^2$

You can solve the resulting quadratic equation to find $\delta$.

If $\bar X= k \bar Y$ then you might prefer to set $dX=k dY$.

Your relationship is then: $Z'=XY+Yk\delta+X\delta+k\delta^2$