# What is the distribution of the difference between two AR(1) processes?

I am reading a paper published in a good economics journal. An econometric model is presented in the paper. A part of the described model is not very clear to me. Please let me state a couple of equations and ask my question.

$\upsilon_{s}$ and $\omega_{s}$ are both first-order autoregressive processes and take the form

$\upsilon_{s} = \rho \upsilon_{s-1} + \phi_{s}$, $\mathbb{E}_{s-1}(\phi_{s}) = 0$

and

$\omega_{s} = \rho \omega_{s-1} + \psi_{s}$, $\mathbb{E}_{s-1}(\psi_{s}) = 0$

for $s = t+1, \dotsc, S$.

Then $\xi_{s}$ is defined as

$\xi_{s} = \upsilon_{s} - \omega_{s}$.

It is then "assumed" that $\xi_{s}$ is normally distributed with variance $\sigma_{\xi}^{2}$.

How strong is the assumption that $\xi_{s}$ is normally distributed? Under which conditions can one claim that the difference between two AR(1) processes is normal? I know that the difference between two I(1) processes can be I(0) (cointegration), but I do not know the distribution of the difference between two AR(1) processes. The paper does not provide any argument on the assumption. I am wondering if this is a strong assumption, or if it is somewhat easy to argue/expect that $\xi_{s}$ is normally distributed.

Thanks.

Using your notation; $$\xi_{s}= \upsilon_{s}-\omega_{s} = \rho( \upsilon_{s-1}-\omega_{s-1}) + \phi_{s} - \psi_s$$ $$=\rho \xi_{s-1} + \epsilon_s$$ where $\epsilon_s = \phi_{s} - \psi_s$, so $E[\epsilon_s]=0$ and $$var(\epsilon_s)=var(\phi_s)+var(\psi_s)-2cov(\phi_s,\psi_s)$$
So it does seem that $\xi_s$ is probably a stationary AR(1). There are some caveats though.
1. $\epsilon_s$ may not be normally distributed if $(\phi_s,\psi_s)$ are not multivariate normal ($\xi_s$ would still be an AR(1) though, but with non-normal disturbance terms).
2. $\epsilon_s$ may not be mutually independent white noise if there is lagged cross-correlation between $\phi_s$ and $\psi_s$. This will violate the assumptions of the AR(1) if it is estimated via maximum likelihood.
In sum, you are right to suspect that $\xi_s$ may not be normally distributed. In econometrics we sometimes assume things like multivariate normality without stating it (it is wrong, but it happens). Look to see if the author did state an assumption about independence or multivariate normality for the AR(1) disturbance terms.