274 Investigating Economic Trends and Cycles
structure of the residual component. More elaborate filters are available that also
take account of a seasonal component.
Consider, therefore, a seasonal ARIMA model of the form:
y(z)=
θ(z)
φ(z)ε(z)=
θ(z)
φS(z)φT(z)ε(z), (6.99)whereφS(z)contains the seasonal autoregressive factors andφT(z)contains the
non-seasonal factors.
The denominator contains both an ordinary differencing operator∇d(z)and a
seasonal differencing operator∇sD(z)=∇D(z)SD(z). The operator∇s(z)= 1 −zs=
( 1 −z)S(z)forms the differences between the data from the same season (or month)
of two successive years. Its factors are the ordinary difference operator and a sea-
sonal summation operatorS(z)= 1 +z+z^2 +···+zs−^1. The factorization of the
seasonal operator implies that the overall degree of differencing within the ARIMA
model isd+D. The factor∇d+D(z)is assigned toφT(z), whereasSD(z)belongs to
φS(z).
On the assumption that the degree of the moving-average polynomialθ(z)is at
least equal to that of the denominator polynomialφ(z), there is a partial-fraction
decomposition of the autocovariance generating function of the model into three
components, which correspond to the trend effect, the seasonal effect and an
irregular influence. Thus:
θ(z−^1 )θ(z)
φS(z−^1 )φT(z−^1 )φT(z)φS(z)=
QT(z)
φT(z−^1 )φT(z)+
QS(z)
φS(z−^1 )φS(z)+R(z). (6.100)Here, the first two components on the right-hand side represent proper rational
fractions, whereas the final component is an ordinary polynomial. If the degree
of the moving-average polynomial is less than that of the denominator polyno-
mial, then the irregular component is missing from the decomposition in the first
instance.
To obtain the spectral density function ofy(t), we setz=e−iω, whereω∈[0,π].
(This function is more properly described as a pseudo-spectrum in view of the
singularities occasioned by the unit roots in the denominators of the first two
components.) The spectral decomposition corresponding to equation (6.100) can
be written as:
f(ω)=f(ω)T+f(ω)S+f(ω)R, (6.101)wheref(ω)=θ(eiω)θ(e−iω)/{φ(eiω)φ(e−iω)}.
LetνT=min{f(ω)T}andνS=min{f(ω)S}. These correspond to the elements
of white noise embedded inf(ω)Tandf(ω)S. The principle of canonical decom-
position is that the white-noise elements should be reassigned to the residual
component. On defining:
γT(z)γT(z−^1 )=QT(z)−νTφT(z)φT(z−^1 ),γS(z)γS(z−^1 )=QS(z)−νSφS(z)φS(z−^1 ),and ρ(z)ρ(z−^1 )=R(z)+νT+νS,(6.102)