In josephsdavid/tstest: tstest

MA(q) Models

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.

A quote for your thoughts:

"All models are wrong... but some are useful" - George Box

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: $$\psi_0 = 1, \psi_1 = -\theta_1, ... , \psi_q = -\theta_q, \psi_k = 0, k>q$$

Operator Zero-Mean Form

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

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.

Some definitions:

$$E(X_t) = \mu$$

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).

Some examples of MA(1)

Positive theta1

xs  <- gen.arma.wge(n = 200, theta = 0.99)

Lets check this out now

ggacf(xs) +th
parzen.wge(xs)

Lets get some math done too and check out the ACF at lag 1

macf1 <- function(theta) {
    num <- -theta
    denom <- 1 + theta ^ 2
    num/denom
}
macf1(.99)
# [1] -0.4999747

From this we can also see that the maximum possiblevalue of autocorrelation at lag 1 is 0.5.

Negative theta

Now lets try it out with a negative theta:

xs  <- gen.arma.wge(n = 200, theta = -0.99)
ggacf(xs) +th
parzen.wge(xs)
macf1(-.99)
# [1] 0.4999747

Note: the negative autocorrelation of a positive theta, as well as the dip at 0 in the spectral density tell us it is going to oscillate a bit more than the negative theta. Lets check out the true autocorrelations and spectral densities of these models:

plotts.true.wge(theta = 0.99)
plotts.true.wge(theta = -.99)

MA(2)

A MA(2) model behaves pretty similarly, however for our autocorrelations we have: $$\rho_0 = 1$$ $$\rho_1 = \frac{-\theta_1 + \theta_1 \theta_2}{1 + \theta_1^2 + theta_2^2}$$ $$\rho_2 = \frac{-\theta_2}{1 + \theta_1^2 + theta_2^2}$$ $$\rho_k = 0; k>2$$

This weird behavior for rho_k is why we dont really use this on real data.

Invertibility

Consider two MA(1) models, with theta equal to 0.8 and 1.25. Then, let us consider their autocorrelations: for theta = 0.8, $rho_1 = r macf1(0.8)$, and for theta = 1.25, it is r macf1(1.25) (use the macf1 function). These two models can have the same ACF, which is not good. Model multiplicity is undesirable. This happens I believe due to some magic, similar to aliasing. How do we tell them apart?

Criteria for invertibility

A MA model is invertible iff all roots of the model are outside of the unit circle. We can just solve the MA equations as we do with the AR equations, to check for invertibility, we use the characteristic equation trick. We can also use factor.wge

josephsdavid/tstest documentation built on July 3, 2019, 4:48 p.m.