# can two AR(1) processes be cointegrated

Let $$Y_{t}$$, for $$t=1,2, \dots$$ be AR(1) process. $$Y_{t+1} = c_{1} + \phi Y_{t} + \varepsilon_{t}$$

Next, assume that for some $$X_{t}$$ we have $$X_{t} - \beta Y_{t} = u_{t},$$ where $$u_{t}$$ is stationary. Therefore, we can say that $$X_{t}$$ and $$Y_{t}$$ are cointegrated.

Is it possible that $$X_{t}$$ is also AR(1)?

Attempt:

Assume that $$u_{t}$$ is i.i.d. Then, in this simplest case $$X_{t}$$ is $$X_{t} = \beta Y_{t} + u_{t},$$ therefore, it is ARMA(1,1). Though, it is not a proof of the original statement.

• To be co-integrated you have to first be integrated. It is not possible for two AR(1) processes to be cointegrated unless the coefficient is 1 in both cases. This is just the definition of cointegration. Jan 27, 2021 at 15:32
• Dear @ChrisHaug , If I understand correctly, in order to speak about co-integration of the process $Y$ to another process, Y mast be $I(d)$, with $d > 1$. Is this what you mean?
– ABK
Jan 27, 2021 at 15:40
• $d=1$ is fine and is the basic example. Regarding your attempt, the process is not ARMA(1,1) as expressed. For ARMA(1,1) you need lag of $X_t$ and lag of $u_t$ on the right hand side but you have neither of these. You do have $Y_t$, though, and it does not belong in an ARMA(1,1) for $X_t$. @ChrisHaug, why not post a short answer based on the comment? Jan 27, 2021 at 15:43
• Dear @RichardHardy, oh, I see. In order to have ARMA, one would need $X_{t} = \beta Y_{t} + u_{t-1}$, but in the current case we have $X_{t} = \beta Y_{t} + u_{t}$.
– ABK
Jan 27, 2021 at 15:54
• @ABK, you may be right (especially based on the reference), but you have not demonstrated that you are. My comment was about that (before you provided the reference). Jan 27, 2021 at 16:29

Cointegration literally means "to be integrated, together" (see the usual "common trend" interpretation). It cannot logically apply to processes which are not integrated. So if you have $$|\phi| < 1$$ for both AR processes, they cannot be cointegrated by definition.
More precisely, the general definition of cointegration requires $$X_t$$ and $$Y_t$$ to be integrated of order $$d_1$$ (the same order for both), and that there exist a linear combination $$Y_t - \beta X_t$$ which is integrated of order $$d_2$$, with $$d_1 > d_2 \geq 0$$. Explicitly, the order of integration of the linear combination $$d_2$$ must be strictly smaller than the order of integration of the individual processes $$d_1$$, which precludes $$d_1$$ from being zero.