Moving Average Process

A moving average process of order $q$ expresses the current value as a linear combination of present and past white noise terms.

1. Definition

$\{X_t\}$ is an MA($q$) process if

$$X_t = Z_t + \theta_1 Z_{t-1} + \theta_2 Z_{t-2} + \cdots + \theta_q Z_{t-q}$$

where $\{Z_t\} \sim \text{WN}(0, \sigma^2)$.

Using the Backshift Operator:

$$X_t = \theta(B) Z_t, \quad \theta(B) = 1 + \theta_1 B + \theta_2 B^2 + \cdots + \theta_q B^q$$

Here $\theta_1, \dots, \theta_q$ are the MA coefficients and $q$ is the order of the model.

2. MA(1) Special Case

$$X_t = Z_t + \theta Z_{t-1}$$

Always stationary (regardless of $\theta$).

ACVF:

$$\gamma_X(h) = \begin{cases} \sigma^2(1 + \theta^2) & h = 0 \\ \sigma^2 \theta & |h| = 1 \\ 0 & |h| > 1 \end{cases}$$

ACF:

$$\rho_X(h) = \begin{cases} 1 & h = 0 \\ \frac{\theta}{1 + \theta^2} & |h| = 1 \\ 0 & |h| > 1 \end{cases}$$

3. General MA(q) ACVF

$$\gamma_X(h) = \begin{cases} \sigma^2 \sum_{j=0}^{q-|h|} \theta_j \theta_{j+|h|} & |h| \leq q \\ 0 & |h| > q \end{cases}$$

where $\theta_0 = 1$.

4. Key Properties

  • Always stationary for any choice of $\theta_1, \dots, \theta_q$
  • Always causal because it depends only on present and past noise terms
  • If $\mathbb{E}[Z_t] = 0$, then $\mathbb{E}[X_t] = 0$
  • Finite memory: ACF cuts off after lag $q$ (i.e., $\rho_X(h) = 0$ for $|h| > q$)
  • This cutoff is the defining diagnostic feature distinguishing MA from AR (which tails off)

5. Invertibility

An MA($q$) process is invertible if all roots of $\theta(z) = 0$ lie outside the unit circle. Invertibility ensures a unique MA representation and allows the process to be written as an AR($\infty$).