Let $(x_{t})_{t\in \mathbb Z}$ be a causal AR(p) process with operator $\phi$ such that $\phi(L)=\phi_{0}-\phi_{1}L-...-\phi_{p}L^{p}$ and $(\epsilon_{t})_{t \in \mathbb N_{0}}$ white noise sequence:

By definition, there exists a sequence $(\psi_{t})_{t \in \mathbb N_{0}}$ such that $x_{t}=\psi(L)\epsilon_{t}$ where $\psi(L)=\psi_{0}+\psi_{1}L+\psi_{2}L^{2}+...$ with the Lag operator $L$ and $\sum\limits_{j\in \mathbb N_{0}}\lvert \psi_{j}\rvert<\infty$.

One equivalent property of causality is that all roots of $\phi$ lie outside the unit circle.

Furthermore, introducing the notion of weak stationarity, meaning mean stationarity and covariance stationarity of the time series $x_{t}$. We get that a AR(p) process $x_{t}$ is stationary, if the roots of $\phi$ lie outside of the unit circle.

So by the above thoughts I have espoused, I may assume the following:

causality $\implies$ stationarity with mean $0$.

This leads me to what I think may be a contradiction, because under the assumption that $x_{t}$ is stationary, we have

$\mathbb E[x_{t}]=\frac{\phi_{0}}{1-\phi_{1}-...-\phi_{p}}$

So even if i did assume causality, this would have to mean that

$0=\mathbb E[x_{t}]=\frac{\phi_{0}}{1-\phi_{1}-...-\phi_{p}}$, which is certainly not true, and would only be true if I assume $\phi_{0}=0$.

What am I not understanding?

  • $\begingroup$ "...a stationary AR(p) process $x_{t}$ is stationary, if the roots of $\phi$ lie outside of the unit circle..." is a somewhat sloppy statement. The AR(p) model has a stationary solution if and only if $\phi$ has no roots on the unit circle. A special case is when all roots of $\phi$ lie outside the unit circle, in which case the stationary solution is also causal. When all roots of $\phi$ lie inside the unit circle, the model has stationary non-causal solution. $\endgroup$
    – Michael
    Jun 10 '20 at 22:14
  • $\begingroup$ @Michael OP is talking causality and outside unit circle $\endgroup$
    – Aksakal
    Jun 10 '20 at 22:25

Causality is by definition a special case of stationarity. Stationarity, or causality, does not imply mean 0.

Where you went wrong is you're comparing different AR models---one without intercept and one with. Stationary AR models without intercept have zero mean in general, whether causal or non-causal. Vice versa for those with intercept.

When you defined $x_t$ by $\phi(L) x_t = \epsilon_t$, e.g. $$ (\phi_0 - \phi_1 L)x_t = \epsilon_t, \quad (*) $$ you defined an AR model with no intercept. Such processes necessarily have mean zero. (As you already pointed out, in the stationary case where there is a MA representation, it's a (infinite) sum of mean-zero variables.) $\phi_0$ is customarily taken to be 1 in such formulations.

On the other hand, the expression for the unconditional mean $$ \mathbb E[x_{t}] = \frac{\phi_{0}}{1-\phi_{1}} $$ is for the causal AR model with intercept $$ x_{t+1} = \phi_0 + \phi_1 x_t + \epsilon_t. \quad (**) $$ This is not the same model as $(*)$.

Instead, the lag operator formulation of $(**)$ is $$ (1 - \phi_1 L) x_t = \phi_0 + \epsilon_t. $$ So in the causal case $$ x_t = \frac{\phi_0}{1-\phi_1} + \underbrace{ \psi(L)\epsilon_{t}}_\text{$\sum_{h \geq 0} \psi_h \epsilon_{t-h}$}, \;\; \psi_h = \phi_1^h, $$ which has mean $$ \mathbb E[x_{t}] = \frac{\phi_{0}}{1-\phi_{1}}. $$

  • $\begingroup$ That helps a lot! This leads me to the question, can a process with a 'constant' or 'drift' of the form, e.g. $x_{t}=\phi_{0}+\phi_{1}x_{t-1}+\epsilon_{t}$ still be causal? $\endgroup$
    – MinaThuma
    Jun 10 '20 at 23:52
  • $\begingroup$ @MinaThuma, yes, when $|\phi_1|<1$ $\endgroup$
    – Aksakal
    Jun 11 '20 at 0:08
  • $\begingroup$ @Aksakal But what would my representation of $x_{t}=\sum\limits_{j \in \mathbb N_{0}}\psi_{j}\epsilon_{t-j}$ look like to account for the constant? $\endgroup$
    – MinaThuma
    Jun 11 '20 at 0:15
  • $\begingroup$ @MinaThuma, obviously, not, because mean of the right hand size is clearly zero, while of the left side it's not $\endgroup$
    – Aksakal
    Jun 11 '20 at 0:18
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    $\begingroup$ Michael or Aksakal: Do either of you know of a good reference for this material. By "this material" I mean relations between causaiity-stationarity etc. I have a lot of time series books but I could be missing something good. Thanks. $\endgroup$
    – mlofton
    Jun 11 '20 at 3:26

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