Unit 14 MA(q) Models

14.1 Properties and Characteristics

We use MA models to model stationary data. They are not quite as useful as AR models, but in conjunction with them, we can create ARMA(p,q) models and so forth.

14.1.1 A quote for your thoughts:

“All models are wrong… but some are useful” - George Box

14.2 MA(q)

First of all, we need to define the equation for a Moving Average Model of order q: $$X_t = \mu + a_t - \theta_1 a_{t-1} - ... - \theta_q a_{t-q}$$

Inspecting this equation, we see that a MA(q) model is a finite GLP, and therefore Always stationary. reminder: a GLP is defined as : $$X_t = \mu + \sum_{j = 0}^\infty \psi_j a_{t-j}$$

We can now define a MA(q) model as a GLP in which: $$\psi_0 = 1, \psi_1 = -\theta_1, ... , \psi_q = -\theta_q, \psi_k = 0, k>q$$

14.3 Operator Zero-Mean Form

$$X_t = (1 - \theta_1 B- ... - \theta_q B^q)a_t$$

14.3.1 Characteristic Equation

$$1-\theta_1 z - ... - \theta_q z^q = 0$$

We can solve this just like the AR side, except this is with white noise terms.

14.3.2 Some definitions:

$$E(X_t) = \mu$$

14.3.2.1 For MA(1)

$$\sigma_X^2 = \sigma_a^2(1 + \theta_1^2)$$ $$\rho_0 = 1, \rho_1 = \frac{-\theta_1}{1+\theta_1^2}, \rho_k = 0, k>1$$ For pure MA they do not damp exponentially! (theoretically) This is a defining piece of info.

$$S_X(f) = \frac{\sigma_a^2}{\sigma_X^2} \mid 1 - \theta_1e^{-2 \pi i f} \mid ^2$$

From this we will see that MA spectral densities will have dips rather than peaks (again a function of white noise).