Sum of autoregressive processes? I am working on a research topic where I need to add together two AR processes and I was wondering if the distribution of these processes is of a recognizable form/structure.  More formally, if $x_t$ is a AR(p) process with characteristic polynomial $\Phi_x(u)$ and $y_t$ is a AR(q) process with characteristic polynomial $\Phi_y(u)$, then what is the structure of $z_t=x_t+y_t$?
 A: This was studied by Granger and Morris (1976) who showed that
AR($p$) + AR($q$) = ARMA($p+q,\max(p,q)$).
A: As far as I can tell, the assertion that the sum of two $AR$ processes is an $ARMA$ process is based on the assumption that it belongs to $ARMA$ family and then the auto-covariance structure is matched.
Given two $AR1$ processes
$$(1 - a_xL)x(t) = e_x(t)$$
$$(1 - a_yL)y(t) = e_y(t)$$
their sum (assuming invertebility) is
$$ z(t) = x(t) + y(t) = \frac{e_x(t)}{1 - a_xL} + \frac{e_y(t)}{1 - a_yL}$$
Multiplying by the $AR1$ polynomials, we get
$$ (1 - a_xL)(1 - a_yL)z(t) = (1 - a_yL)e_x(t) + (1 - a_xL)e_y(t)$$
which is
$$ (1 - (a_x + a_y)L + a_xa_yL^2)z(t) = e_x(t) + e_y(t) - \left(a_y e_x(t-1) + a_x e_y(t-1)\right)$$
So, the left-hand side is an $AR(2)$ process, but I don't see how can we express the right-hand side as an $MA(1)$ process, because if we define
$$ e_z(t) = e_x(t) + e_y(t)$$
then there is no way we can express the $t-1$ term on the right-hand side as a scalar multiple of $e_z(t-1)$, i.e.
$$a_y e_x(t-1) + a_x e_y(t-1) \neq be_z(t-1) = b\left(e_x(t-1) + e_y(t-1)\right)$$
for a scalar value $b$ (unless, of course, $a_x = a_y$).
We can construct an $ARMA(2,1)$ process which will have the same auto-covariance signature as the sum of two $AR(1)$ processes, but it won't be identical to the sum of the two processes.
A: You don't need to be able to multiply the (t-1) component by a scalar to get the MA(1) component and although obtaining the same covariance structure in the sense that γk = 0 ∀ k > 1 is necessary, it is not sufficient.
That is to say on the right hand side of Confounded's answer we have
ξt = ex(t) + ey(t) + ayex(t-1) + axey(t-1)
Although there are exceptions (e.g. ay = -ax, var(ey)=var(ex)), we usually have Var(ξt) != 0, E(ξtξt-1) != 0, E(ξtξt-k) = 0 ∀ k>1, which usually corresponds to an MA structure. I say usually because as Granger and Morris (1976) point out, the maximum value of the first order autocorrelation in an MA(1) is ρ1 = γ1/γ0 = (θ^2)/(1 + θ^2) is 0.5.
So if E[ξtξt-1]/E[ξtξt] > 0.5 it cannot be MA(1). The necessary and sufficient conditions for a certain autocovariance structure to yield an MA(Q) process are given by Wold (1953).
In fact, Granger and Morris (1976) show that the summation of two MA(Q) processes yields MA(Q*) with Q*<=Q consistent with Wold's (1953) conditions. Helmut Lütkepohl (1984) generalise this result to the summation of arbitrarily many MA processes and do so using weaker assumptions.
Edit: Thanks @Sycorax for the tip, I didn't realise LaTeX was supported. I have started doing this but cba to go back and change this answer.
Edit: I am revising and just did a relevant supervision question as well as a relevant past paper question. You can, in fact, find the scalar θ that Confounded alludes to.
We know the γ1 of an MA(1) = (θ^2)/(1 + θ^2), so we just solve
(θ^2)/(1 + θ^2) = E[(ex(t) + ey(t) + ayex(t-1) + axey(t-1))(ex(t-1) + ey(t-1) + ayex(t-2) + axey(t-2))]/Var(ex(t) + ey(t) + ayex(t-1) + axey(t-1))
for θ, although the answer is usually disgusting
