Tommaso Proietti 401two orthogonal stationary processes as follows:
ξt=( 1 +L)rζt+( 1 −L)mκt
φ(L)
, (9.15)whereζtandκtare two mutually and serially independent Gaussian disturbances,
ζt∼NID(0,σ^2 ),κt∼NID(0,λσ^2 ), and:
|φ(L)|^2 =φ(L)φ(L−^1 )=| 1 +L|^2 r+λ| 1 −L|^2 m. (9.16)We assume thatλis known. Equation (9.16) is the spectral factorization of the
lag polynomial on the right-hand side of (9.15); the existence of the polynomial
φ(L)=φ 0 +φ 1 L+···+φq∗Lq
∗
, of degreeq∗=max(m,r), is guaranteed by the
fact that the Fourier transform of the right-hand side is never zero over the entire
frequency range (see Sayed and Kailath, 2001, for details).
According to (9.15), for given values ofλ,mandr, the innovationξtis decom-
posed into two ARMA(2,2) processes, characterized by the same AR polynomial,
but by different MA components. The first component will drive the lowpass com-
ponent ofytand its spectral density is proportional toσ^2 wμ(ω), where wμ(ω)is
the gain of the filter (9.6). Ifr>0 the MA representation is noninvertible at theπ
frequency. Notice that, asmandrincrease, the transition from the pass band to
the stop band is sharper.
Substituting (9.15)–(9.16) into the ARIMA representation, the series can be
decomposed into two orthogonal components:
yt=μt+ψt,φ(L)φ(L)(dμt−β)=( 1 +L)rθ(L)ζt, ζt∼NID(0,σ^2 ) (9.17)φ(L)φ(L)ψt=m−dθ(L)κt, κt∼NID(0,λσ^2 ).The trend or lowpass component has the same order of integration as the series
(regardless ofm), whereas the cycle or highpass component is stationary provided
thatm≥d, which will be assumed throughout.
Given the availability of a doubly infinite sample, the Wiener–Kolmogorov esti-
mators of the components areμ ̃t=wμ(L)ytandψ ̃t=[ 1 −wμ(L)]yt, where the
impulse response function of the optimal filters is given by (9.6). Hence, the signal
extraction filter for the central data points will continue to be represented by (9.6),
regardless of the properties ofyt, but this is the only feature that is invariant to the
nature of the time series and its ARIMA representation. The mean square error of
the smoothed components, as a matter of fact, depends on the ARIMA model for
yt. In finite samples, the estimators and their mean square errors will be provided
by the Kalman filter and smoother associated with the model (9.17), and thus will
depend on the ARIMA model foryt.
Bandpass filters can also be constructed from the principle of decomposing the
lowpass component in (9.17). Let us consider fixed values ofmandrand two cut-off
frequencies,ωc 1 andωc 2 >ωc 1 , with corresponding values of the smoothness