# Unit 16 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.

## 16.1 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, \(rho_1 = -0.4878049\), and for theta = 1.25, it is -0.4878049 (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?

### 16.1.1 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