In this chapter, the last tools are developed in order to justify modeling any ARMA(m,l) model with an AR(m) filter: The Yule–Walker equations [and the R functions ar.yw() and ar.mle()], the partial autocorrelation function and plot [and the R function pacf()], and the spectrum for ARMA(m,l) models as well as the relationship between the spectrum and the impulse response function (this will not be particularly useful for modeling, but it is too cool not to mention, and does not take up much space).
The autocovariance function, with the derivation of the impulse response function, can be viewed from a very different perspective. Recall that there is an interest in computing things like E(wrϵt), but now it has been established that ϵt is a linear combination of all white noise components up to, and including, time t. Recall also that E(wjwk) = 0, unless j = k, in which case E(w2j) = σw2.
So and . The only nonzero component of this infinite series is when r = t − j, if that is possible (if r > t, this is not possible). Therefore:
Recall that the autocovariance,