1
$\begingroup$

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?

$\endgroup$
2
  • $\begingroup$ "...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
4
$\begingroup$

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

$\endgroup$
11
  • $\begingroup$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.