Linear Processes#

Stage 3 · 🔗 Linear & ARMA Processes · Lesson 05 of 18 · intermediate

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Important

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What it is#

A linear process writes a series as a weighted sum of white-noise shocks — a linear filter applied to an underlying noise sequence. It is the general form from which AR, MA and ARMA models all descend.

The white-noise filter#

With white noise \(\{w_t\}\) (mean zero, variance \(\sigma^2\)), a linear process is

\[x_t = \mu + \sum_{j=-\infty}^{\infty} \psi_j\, w_{t-j}, \qquad \sum_{j=-\infty}^{\infty} |\psi_j| < \infty.\]

The absolute summability of the weights \(\psi_j\) guarantees the sum converges and the result is stationary; it is a little stronger than square-summability and is what lets the usual limit theorems apply.

Causal and one-sided#

A linear process is causal when it uses only present and past shocks — the future never appears:

\[x_t = \mu + \sum_{j=0}^{\infty} \psi_j\, w_{t-j}.\]

This one-sided form is the MA(∞) representation. It is exactly the sense in which a causal ARMA model can be “unrolled” into an infinite moving average of past noise.

Why it’s central#

Wold’s decomposition says that every weakly stationary process has a linear-process (MA(∞)) component — so linear processes are not one model among many but the canonical template for stationary series. Causality and invertibility (next lesson) are precisely the conditions under which an ARMA model collapses into, or inverts back out of, this form.

Hint