Palgrave Handbook of Econometrics: Applied Econometrics

D.S.G. Pollock 257

The ARMA(2, 2) process has been formed by the additive combination of a
second-order autoregressive AR(2) process and an independent white-noise pro-

cess. The autoregressive polynomial isα(z)= 1 + 2 ρcos(θ)z+ρ^2 z^2 , which has
conjugate complex roots of which the polar forms areρexp{±iθ}. In the example,
the modulus of the roots isρ=0.9 and their argument isθ=π/4 radians.
The spectral density function attains a non-zero minimum atω=π. However,
it is possible to decompose the ARMA(2, 2) process into an ARMA(2, 1) process
and a white-noise component that has the maximum variance compatible with
such a decomposition. This is a so-called canonical decomposition of the ARMA
process. The moving-average polynomial of the resulting ARMA(2, 1) process is
1 +z, which has a zero atω=π. By maximizing the variance of the white-noise
component, an ARMA component is derived that is as smooth and as regular as
possible.
Canonical decompositions are entailed in a method for extracting unobserved
components from data sequences described by autoregressive integrated moving
average (ARIMA) models, which will be discussed in section 6.6.3.
Figure 6.7 also shows a periodogram that has been calculated from a sample of
256 points generated by the ARMA(2, 2)process. Its volatility contrasts markedly
with the smoothness of the spectrum. The periodogram has half as many ordi-
nates as the data sequence and it inherits this volatility directly from the data.
A nonparametric estimate of the spectrum may be obtained by smoothing the
ordinates of the periodogram with an appropriately chosen moving average, or
by subjecting the empirical autocovariances to an equivalent weighting operation
before transforming them to the frequency domain.

6.4.1 The frequency-domain analysis of filtering

It is a straightforward matter to derive the spectrum of a processy(t)formed by
mapping the processx(t)through a linear filter. If:

x(t)=

∫

ω

eiωtdZx(ω), (6.39)

then the filtered process is:

y(t)=

∑

j

ψjx(t−j)

=

∑

j

ψj

{∫

ω

eiω(t−j)dZx(ω)

}

=

∫

ω

eiωt

(∑

j

ψje−iωj

)

dZx(ω).

(6.40)