Lesson 2.2: Moving Average (MA) Models

Modeling how past forecast errors or 'shocks' influence the present value of a series.

Part 1: The Core Idea - Memory of Past Shocks

A Moving Average (MA) model proposes that the value today is influenced by the random, unpredictable "shocks" ( $\epsilon_t$ ) from previous periods.

The Core Analogy: Ripples in a Pond

A shock, $\epsilon_{t-1}$ , is like a pebble dropped into a pond. An MA(q) model describes the ripples from that pebble, which persist for `q` periods and then disappear. It's a model with a finite memory of shocks.

Part 2: The MA(q) Model Specification

The MA(q) Model

The value of the series $Y_t$ is a linear function of the current shock and $q$ past shocks.

Y_t = c + \epsilon_t + \theta_1 \epsilon_{t-1} + \dots + \theta_q \epsilon_{t-q}

$\theta_i$ are the moving average coefficients.
Finite-order MA models are always stationary.

Part 3: Model Identification

The Signature of an MA(q) Process

The **ACF plot** will **cut off sharply** after lag $q$ .
The **PACF plot** will show a pattern of **gradual decay**.

The ACF plot is the primary tool for identifying the order `q` of an MA model.

What's Next? Combining the Models

We've modeled memory of past values (AR) and memory of past shocks (MA). What if a process has both?

In the next lesson, we will synthesize these two ideas to create the flexible **Autoregressive Moving Average (ARMA) Model**.

Autoregressive (AR) Models

ARMA Models