Understanding ARMA Processes#

Stage 3 · 🔗 Linear & ARMA Processes · Lesson 06 of 18 · intermediate

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Important

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The model#

An ARMA(p, q) process blends two mechanisms: an autoregressive (AR) part, where the value depends on its own p past values, and a moving-average (MA) part, where it depends on the last q white-noise shocks:

\[x_t = \phi_1 x_{t-1} + \dots + \phi_p x_{t-p} + w_t + \theta_1 w_{t-1} + \dots + \theta_q w_{t-q}.\]

Backshift form#

Using the backshift operator \(B\) (with \(B^j x_t = x_{t-j}\)), the model compresses to

\[\phi(B)\, x_t = \theta(B)\, w_t,\]

where \(\phi(B) = 1 - \phi_1 B - \dots - \phi_p B^p\) is the AR polynomial and \(\theta(B) = 1 + \theta_1 B + \dots + \theta_q B^q\) the MA polynomial. This algebra makes the next two properties easy to state.

Causality and invertibility#

Two root conditions govern behaviour. The process is causal — expressible as a one-sided MA(∞) of past shocks — when all roots of \(\phi(z)\) lie outside the unit circle (\(|z| > 1\)). It is invertible — expressible as an AR(∞) in past values — when all roots of \(\theta(z)\) lie outside the unit circle. Causality is what makes an ARMA a proper linear process; invertibility makes its parameters identifiable from the data.

Watch for redundancy#

If the AR and MA polynomials share a common factor, the model is over-parameterised: the factor cancels and a simpler model fits identically. The extreme case \(\phi(B) = \theta(B)\) reduces the whole model to \(x_t = w_t\), plain white noise. Always cancel common roots before trusting a fit; statsmodels exposes the fitted .arroots and .maroots for exactly this check.

Hint