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