What is a causal process in time series analysis?

Question

Hi can someone please explain to me what Causal process means for time series process? I know the formulas that process has to satisfy to be causal but I want to get a better intuition, thanks!!

Answer 1

The answers to questions you asked can be found in any rigorous time series analysis text book such as Brockwell, P. and Davis, R.'s Time Series: Theory and Methods.

Definition. An ARMA$(p, q)$ process defined by the equations $\phi(B)X_t = \theta(B)Z_t$ is said to be causal (or more specifically to be a causal function of $\{Z_t\}$) if there exists a sequence of constants $\{\psi_j\}$ such that $\sum_{j = 0}^\infty |\psi_j| < \infty$ and $$X_t = \sum_{j = 0}^\infty \psi_j Z_{t - j}, \qquad t = 0, \pm 1, \ldots.$$

A theorem that helps check whether an ARMA process is causal is as follows:

Theorem. Let $\{X_t\}$ be an ARMA$(p, q)$ process for which the polynomials $\phi(\cdot)$ and $\theta(\cdot)$ have no common zeros. Then $\{X_t\}$ is causal if and only if $\phi(z) \neq 0$ for all $z \in \mathbb{C}$ such that $|z| \leq 1$.

For your concrete example, it is easy to observe that $\phi(z) = 1 - 7z - az^2$. If $a = 0$, then $\phi(z) = 0$ has one root $z_0 = \frac{1}{7}$, it hence follows by the above theorem that $\{X_t\}$ is non-causal. If $a \neq 0$, by Vieta's theorem, the (complex) roots $z_1, z_2$ of $\phi(z) = 0$ satisfy: $$z_1z_2 = \frac{1}{a}.$$ Hence $|a| = 1/|z_1z_2|$. By the above theorem, $\{X_t\}$ is causal if and only if $|z_1| > 1$ and $|z_2| > 1$, therefore, $0 < |a| < 1$. Under which case: $$z_1 = \frac{-7 + \sqrt{49 + 4a}}{2a}, z_2 = \frac{-7 - \sqrt{49 + 4a}}{2a}$$ are two distinct reals. I will leave the remaining calculation for you to further narrow down the scope of $a$.

---

For your updated question, if you look at the definition of causal process carefully, you may find that $X_t$ is causal if for any current time $t$, we can always express $X_t$ in terms of an infinite linear combination of past innovations $Z_t, Z_{t - 1}, \ldots$. In this sense it is intuitive to say that $X_t$ is causal since $X_t$ is completely determined by past information (by contrast, a non-causal ARMA process does not admit such a representation). The absolutely summable condition further imposes that $X_t$ is a regular process.