A Complete Introduction To Time Series Analysis (with R):: Estimation of ARMA(p,q) Coefficients (Part II)

The ARMA(p,q) model implies that X_{t} can be expressed in the form above.

In the last article, we learned about two algorithms to estimate the AR(p) process coefficients: the Yale-Walker equations method, and Burg’s algorithm. In this article, we will now see a very simple way to determine the MA(q) process coefficients, and a first approach to estimate the ARMA(p,q), jointly. Let’s see how this works:

Estimation of MA(q) (Innovations)

As you may guess by the title, the way to estimate the MA(q) coefficients is… the Innovations Algorithm we saw before. Recall that the MA(q) process can be written as

The idea is as follows:

• Recall that during the iterations of the Innovations algorithm, we obtain the Theta matrix

which in turn provides the coefficients used in the recursive Innovations formula:

. It turns out that these are actually also consistent estimators of the MA(q) process coefficients!

• We can then apply the Innovations algorithm by substituting the sample ACVF instead of the actual ACVF, that is