I am trying to prove that a long-short strategy invested according to the cointegrated relationship from Engle-Granger's. So essentially I'm trying to show that the return $r_{XY}$ of the portfolio (X long Y short) has zero $\beta$ (in other words it is market neutral in CAPM sense).
CAPM says that $r_Y = r_m\beta_Y + \alpha_Y$, $r_X = r_m\beta_X + \alpha_X$, hence $$r_{XY}=r_X - br_Y = \beta_Xr_m + \alpha_X - b\beta_Yr_m - b\alpha_Y = r_m(\beta_X-b\beta_Y) + \alpha_X - b\alpha_Y$$
In order to be market neutral we need that $\beta_X-b\beta_Y=0$ but I'm struggeling a bit to prove this.
Since $\beta$ is defined as $$\beta = \frac{Cov(r_p,r_m)}{Var(r_m)}$$ I firstly need to find $r_X$ and $r_Y$ to find the $\beta$'s.
This is were I get stuck. I have been trying to use that, assuming the stocks continues to be cointegrated, we need that $X_t(1+r_X) = bY_t(1+r_Y) + \mu + \epsilon_t$ $(\star)$. Then solve for $r_X$ or $r_Y$ and plug it into our equations, but it doesn't really work. For $\beta_X-b\beta_Y$ to be $0$ we need that $r_X = br_Y+constant$, but this doesn't work with $(\star)$$$$$Is it possible to do it like this, or is there some other way to prove that it is market neutral in CAPM sense? Or maybe it isn't market neutral according to CAPM?