# Confusion on the connection between causality and stationarity and possible implications

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?

• "...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. Jun 10, 2020 at 22:14
• @Michael OP is talking causality and outside unit circle Jun 10, 2020 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 $$\mathbb E[x_{t}] = \frac{\phi_{0}}{1-\phi_{1}}.$$

• 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? Jun 10, 2020 at 23:52
• @MinaThuma, yes, when $|\phi_1|<1$ Jun 11, 2020 at 0:08
• @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? Jun 11, 2020 at 0:15
• @MinaThuma, obviously, not, because mean of the right hand size is clearly zero, while of the left side it's not Jun 11, 2020 at 0:18
• 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. Jun 11, 2020 at 3:26