Trying to evaluate model for pairs trading. Consider classic formula:
$\frac{dP}{P} = adt+b\frac{dQ}{Q}+dX$,
where $P$ and $Q$ are stock prices, and $X$ is a mean reverting process (MRP) and $a$ is close to zero.
Using real world example I would like to evaluate parameters of MRP. Practically we cannot observe MRP, rather we can derive it from $P$ and $Q$. If we go straightforward and calculate $\hat{b}$ as least squares estimator of $\frac{dP}P$ against $\frac{dQ}Q$, then we have residual estimation including MRP., i.e.
$$ \frac{dP/P}{dQ/Q} = \hat{b}, $$
so that our beta $\hat{b}$ captures change in stock prices together with MRP
$$ \frac{dP/P - dX}{dQ/Q} = \hat{b}. $$
This gives skewed estimation of $\hat{b}$.
My question is: how to get estimation of $\hat{b}$ adjusted for MRP?