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I'm looking for a proof for the stationarity of an AR(p) process, I know a stationary process $Yt$ must fulfill the following conditions:

$(1)$-$E(Y_t)=m$ for all values of $t$.

$(2)$-$Var(Y_t)=\sigma^2$ for all values of $t$.

$(3)$-$cov(Y_t,Y_s)=\gamma(|t-s|)$ for every $t$ and $s$.

First I'll define an AR(p) process $Y_t$ as follows:

$\phi(B)Y_t=\epsilon_t$ with: $BY_t=Y_{t-1}$, $\epsilon_t$ a white noise and $\phi(B)=1-\alpha_1B-...-\alpha_pB^p$.

The condition that I read about in several posts is:

$(4)$ If the modulus of each root of $\phi$ is strictly greater than 1 then the AR(p) process as defined before is stationary.

So, what I'm looking for is a proof for: $(4)$$<=>$$(1)$&$(2)$&$(3)$

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  • $\begingroup$ Are you sure about strictly greater part? if I recall correctly, as long as the roots are outside the unit disk, the solution is stationary $\endgroup$ Commented Apr 12, 2019 at 17:40

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We first start with AR(1) with $|\phi|<1$. We write $$x_t=\phi^{-1}x_{t-1}+w_{t}=\phi(\phi x_{t-2}+w_{t-1})+w_{t}=\dots=\phi^{k}x_{t-k}+\sum_{j=1}^{k-1}\phi^{j}w_{t-j}$$ We assume $sum_t var(x_t)< \infty$, and we can write $$x_t=\sum \phi^j w_{t-j}$$ now$$E(x_t)=\sum \phi^j E(w_{t-j})=0 $$ autocovariance function is $$cov(x_{t+h},x_t)=E((\sum_{j}\phi^j w_{t+h-j})(\sum_k\phi^k w_{t-k}) )=E((w_{t+h}+\dots+\phi^h w_t+\phi^{h+1}w_{t-1}+\dots))\\(w_t+\phi w_{t-1}+\dots)=\sigma^2_w\sum_j \phi^{h+j}\phi^j=\sigma^2_w\phi^h\sum_j\phi^{2j}=\frac{\sigma_w^2\phi^k}{1-\phi^2} $$

The other way: $$x_t=\phi x_{t-1}+w_t\rightarrow(1-\phi B)x_t=w_t\rightarrow1-\phi B=0\rightarrow x=\frac{1}{\phi}\rightarrow |\frac{1}{\phi}|>1\rightarrow|\phi|<1$$ When $|\phi|>1$, the process is called explosive and one can prove that every explosive process has a stochastically equivalent causal process.

To generalize for AR(p):

$$x_t=\phi_1x_{t-1}+\phi_2x_{t-2}+\dots+\phi_px_{t-p}+w_t$$ We rewrite this as an AR(1) model $y_t=Fy_{t-1}+w_t$

[ \begin{bmatrix} x_{t} \\ x_{t-1} \\ \dots\\ x_{t-h+1} \end{bmatrix}=\begin{bmatrix} \phi_{1} & \phi_{2} & \phi_{3} & \dots & \phi_{p} \\ 1 & 0 & 0 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \end{bmatrix}\begin{bmatrix} x_{t-1} \\ x_{t-2} \\ \dots \\ x_{t-p} \end{bmatrix}+\begin{bmatrix} w_{t} \\ 0 \\ \dots \\ 0 \end{bmatrix} ]

Lets call thus augmented AR(1) notation. We can write the AR model recursively $$y_t=F^ty_0+w_t+Fw_{t-1}+F^2w_{t-2}+\dots+F^{t-1}w_1+F^tw_0$$. Recall from eigenvalue decomposition: $F=Q\Lambda Q^{-1}$ and $F^j=Q\Lambda^jQ^{-1}$ where $\Lambda^j$ is \begin{bmatrix} \lambda_{1}^j & 0 & 0 & \dots & 0 \\ 0 & \lambda_{2}^j & 0 & \dots & 0 \\ \dots \\ 0 & 0 & 0 & \dots & \lambda_p^j \end{bmatrix} and is stable when $|\lambda_i|<1$ for all i.

The actual eigenvalues of F satisfy $$\lambda^p-\phi_1\lambda^{p-1}-\phi_2\lambda^{p-2}-\dots-\phi_{p-1}\lambda-\phi_p=0$$ So the eigenvalues are the reciprocal of the values that solve the characteristic polynomial of the AR(p), so the roots should be outside the unit circle

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  • $\begingroup$ Thanks for answering, I understand what you did for the AR(1)...Although could you please further detail the 'augmented AR(1)' as you called it. First of all shouldn't the last element of $y_t$ be $x_{t−p}$ and why would the eigenvalues of F being lower than 1 in modulus an indication that the process is stationary? Also consider changing in the recursive formula from $y+t$ to $y_t$ for future viewers $\endgroup$
    – YsfEss
    Commented Apr 12, 2019 at 23:30
  • $\begingroup$ yes, you are correct. Typing in Latex syntax is not my forte. Eigenvalues of Fs being less than 1 implies that F is stable which implies that y satisfies causality assumption. Since it satisfies causality, y_t is stationary $\endgroup$ Commented Apr 12, 2019 at 23:42
  • $\begingroup$ I unwrapped all the elements for y to be in terms of $y_0$ and $w_j$s $\endgroup$ Commented Apr 12, 2019 at 23:43
  • $\begingroup$ Could you lead me to good sources where I can read about causality and causality assumption? $\endgroup$
    – YsfEss
    Commented Apr 13, 2019 at 0:00
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    $\begingroup$ Chapter 3of this book covers pretty much everything ARIMA related on undergraduate level $\endgroup$ Commented Apr 15, 2019 at 15:08

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