What is a moving average (MA) model and how does it differ from AR?

Question

AcadiFi · Accepted Answer

A moving average model of order q, MA(q), expresses the current value as a linear combination of the current and past q white noise error terms. Formally, x_t equals mu plus epsilon_t plus theta_1 times epsilon_{t-1} plus and so on plus theta_q times epsilon_{t-q}.

In contrast to AR models, which embed persistent feedback from past observations, MA models have short memory: shocks persist for exactly q periods and then fully disappear. The ACF of an MA(q) cuts off cleanly after lag q, while the PACF decays slowly — the mirror image of an AR pattern.

Example: quantitative researcher Desmond Pritchard-Hale at Ironvale Fund estimates monthly inflation surprises as an MA(2) process: surprise_t equals 0.1 plus epsilon_t plus 0.4 times epsilon_{t-1} minus 0.2 times epsilon_{t-2}. This captures news spillovers that take a few months to fully propagate but then die out.

```mermaid
flowchart LR
    E0[epsilon t] --> X[x t]
    E1[epsilon t-1] --> X
    E2[epsilon t-2] --> X
```

AR models suit series with enduring feedback (asset prices, GDP). MA models fit series with transient shocks (innovation-driven errors, forecast revisions). In practice, many financial series are best captured by ARMA (next question), combining both.

For the CFA, remember the ACF-PACF diagnostic table: AR has decaying ACF and cutoff PACF; MA has cutoff ACF and decaying PACF; ARMA has decaying in both.

What is a moving average (MA) model and how does it differ from AR?

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