Moving Average Model

Study guide: Exam MAS-II

A moving average model of order $q$ , MA( $q$ ), writes a Time Series as a weighted sum of the current and last $q$ white-noise shocks. It is the "MA" in ARIMA, and it models a series where a shock has a finite memory of exactly $q$ periods.

$Y_t = \mu + \varepsilon_t + \theta_1\varepsilon_{t-1} + \cdots + \theta_q\varepsilon_{t-q}$

$Y_t = \mu + \theta(B)\varepsilon_t, \qquad \theta(B) = 1 + \theta_1 B + \cdots + \theta_q B^{q}$

Always stationary — a finite sum of white-noise terms has constant mean and variance whatever the $\theta$ 's. The condition on the $\theta$ 's is invertibility (roots of $\theta(B)$ outside the unit circle), needed for the parameters to be identifiable
MA(1): $\mathrm{Var}(Y_t) = \sigma^2(1+\theta_1^2)$ and $\rho_1 = \dfrac{\theta_1}{1+\theta_1^{2}}$ , with $\rho_k = 0$ for $k \ge 2$ . That bounds $\lvert\rho_1\rvert \le 0.5$ — an MA(1) simply cannot produce a lag-1 correlation of 0.7
Identification: the ACF cuts off after lag $q$ , the PACF tails off — the exact mirror of an AR
Beyond $q$ steps ahead the forecast is just the mean: shocks older than $q$ periods have left the model entirely
An unexpected MA(1) term after Differencing is the fingerprint of over-differencing

MA(1) Autocorrelation and Forecast $Y_t = 50 + \varepsilon_t + 0.4\varepsilon_{t-1}$ with $\sigma^2 = 25$ . Find $\mathrm{Var}(Y_t)$ , $\rho_1$ , and the one- and two-step forecasts given $\hat\varepsilon_t = -3$ .

Answer $\mathrm{Var}(Y_t) = 25(1 + 0.16) = 29 \qquad \rho_1 = \frac{0.4}{1.16} = 0.345$ $\hat Y_{t+1} = 50 + 0.4(-3) = 48.8$ $\hat Y_{t+2} = 50$ Two steps out, both $\varepsilon_{t+2}$ and $\varepsilon_{t+1}$ are unknown and forecast as 0 — the shock at time $t$ has dropped out of the model, so the forecast is the mean.