If two series $\{x_y\}$ and $\{y_t\}$ are not stationary but their a linear combination of them, say $u_t = \beta x_t - y_t$, is a stationary process, then we say $\{x_t\}$ and $\{y_t\}$ are cointegrated.
I have two questions.
- If I want to check the cointegration of two series that do not have the same unit, e.g. sales data and exchange rates, do I have to normalize them, for example? I think yes.
- How do I determine which variable should be multiplied by $\beta$ in the model? From what I observed, if I swap the order of the variables I get different results.
For a quantity $X$, let $[X]$ denote its units. If $\beta$ is in front of $x_t$, then $[\beta] = \frac{[y_t]}{[x_t]}$, in which case the residual series has units $[y_t]$. If $\beta$ is instead in front of $y$, $[\beta] = \frac{[x_t]}{[y_t]}$ and so the residuals have unit $[x_t]$.
In either case, normalisation and the choice of which variable is regressed on which (assuming they are centred) does not matter, as your conclusion of whether they are co-integrated will be the same. That is, the residuals of the two regressions are either both stationary or both non-stationary.