AR(2) model is causal

AR(2) model is:

$$X_t=\phi_1X_{t-1}+\phi_2X_{t-2}+W_t$$

Where $W_t\sim N(o,\sigma^2)$

I want to prove AR(2) model is causal. So, I tried as:

$$X_t-\phi_1X_{t-1}-\phi_2X_{t-2}=W_t$$

$$\Rightarrow (1-\phi_1 B-\phi_1 B^2)X_t=W_t$$

where $B$ is back-shift operator, i.e., $B X_t=X_{t-1}$

$$\Rightarrow X_t=(1-\phi_1 B-\phi_1 B^2)^{-1}W_t$$

Then I don't know how can I proceed?

• What is the definition of a causal model? – AdamO Jun 29 '15 at 19:03
• First, there seem two typos in the post, the latter two $\phi_1$s should be $\phi_2$. Second, some conditions about $\phi_1$ and $\phi_2$ need to be imposed to guarantee the process is causal. – Zhanxiong Jun 29 '15 at 19:13
• @AdamO here's a presentation defining – Scott Kaiser Jun 29 '15 at 19:23
• Are you referring to Granger causality here, or something else? – Glen_b -Reinstate Monica Aug 4 '15 at 16:09
• @Glen_b, the term causal has a precise meaning in the context of ARIMA models and it is different from both Granger causality and the casual meaning of causal. The definition can be looked up in a time series textbook. Zhangxiong gives the relevant answer. – Richard Hardy Nov 18 '18 at 20:55

A famous theorem (Theorem 3.1.1., Brockwell, Davis. Time series: theory and application）states that an ARMA($$p$$, $$q$$) process $$\phi(B)X_t = \theta(B) W_t$$ is causal if and only if $$\phi(z) \neq 0$$ for all $$z \in \mathbb{C}$$ such that $$\left|z\right|\leq 1$$.

So in order the AR($$2$$) process to be causal, the coefficients $$\phi_1$$ and $$\phi_2$$ must satisfy $$1 - \phi_1 z - \phi_2 z^2 \neq 0$$ for all $$\left|z\right| \leq 1$$. It is not a causal process for all $$\phi_1, \phi_2$$. For example, $$\phi_1 = 2, \phi_2 = 0$$.

your final equation leads to the MA representation of an AR process.

Pred[X(t)] = cons + a1*W(t-1) + a2*W(t-2) + .... an*W(t-n) reflecting how previous errors "cause" X.

All ARMA models can be presented as pure AR models (weighted average of the past )

or as a pure MA mode ( weighted average of the past errors )

You write: "I want to prove AR(2) model is causal."

Is simply not possible. AR and/or ARMA models are never causal. ARMA models was thinked exactly for describing a process with its own past. These explicitly have merely statistical meaning.

Causality is something the go beyond merely statistical relationship and involve more than one variable. If we are not aware about this, we surely conflate statistical association and causality. At most you can ask about Granger causality (unhappy name) but the univariate nature of ARMA put away this possibility too. For these reasons I disagree with previous answers that give you other informations without warning about causal meaning.

• I agree, and am curious what theorem @Zhanxiong refers to in his answer – Cam.Davidson.Pilon Nov 18 '18 at 16:41
• The term causal has a precise meaning in the context of ARIMA models and it is different from the casual meaning of causal, therefore this answer is beyond the point. Zhangxiong gives the relevant answer. – Richard Hardy Nov 18 '18 at 20:54
• @RichardHardy: I've read enough about the ARIMA models and never i find the property of "causality". However i looking for it and you are right this property exist. However let me say two things: in some references the casuality property conflate with stationarity property; in any case "causality property" is only another misnomer. – markowitz Nov 18 '18 at 22:20
• @markowitz: Regarding the "causality" in "Granger causality", perhaps Clive Granger himself should have the last word (from his Nobel acceptance speech): *At that time, I had little idea that so many people had very fixed ideas about causation, but they did agree that my definition was not “true causation” in their eyes, it was only “Granger causation.” I would ask for a definition of true causation, but no one would reply. However, my definition was pragmatic and any applied researcher with two or more time series could apply it, so I got plenty of citations. Of course, many ridiculous papers – Lionel Barnett Dec 7 '18 at 16:32