268 Investigating Economic Trends and Cycles
applying a filterθ(z)/α(z)to a white-noise sequenceε(t)of independently and
identically distributed random variables.
If the trend and the transitory motions that accompany it are due to the same
motive force, which is the white-noise process, then it is difficult to draw a distinc-
tion between them. However, a distinction can be made by attributing the trend to
the unit roots within∇d(z)=( 1 −z)dand by attributing the transitory motions to
the stable roots of the autoregressive operatorα(z). This is what the decomposition
of Beveridge and Nelson (1981) achieves.
Faced with the insistence that the trend and the fluctuations are due to sepa-
rate sources, an obvious recourse is to attribute separate and independent ARIMA
models to each of them. In that case, the aggregate data are also described by
a univariate ARIMA model. Provided that their models have distinct parameters,
WK filters may be used tentatively to extract the independent components from
the data.
The assumption that the components originate from transformations of white-
noise sequences implies that their spectra extend over the entire frequency range
[0,π]. This means that they are bound to overlap substantially. In practice, the
spectral structures of the components are often confined to frequency bands that
are separated by wide spectral dead spaces. In that case, the separation of one
component from another can be achieved in a more decisive manner than the WK
filters will usually allow.
6.6.1 The Beveridge–Nelson decomposition
The Beveridge–Nelson decomposition relates to an ARIMA model with first-order
integration and with a stochastic drift. This can be represented inz-transform
notation by:
y(z)=
μ(z)
∇(z)+
θ(z)
α(z)∇(z)ε(z). (6.74)If the system has a start-up att =0, thenμ(z), which represents the drift, is
thez-transform of a sequence that is constant over the integers 0, 1,...,tand zero-
valued fort<0. The operator associated withε(z)has the following partial-fraction
decomposition:
θ(z)
α(z)∇(z)
=
ρ(z)
α(z)
+
δ
∇(z). (6.75)
Multiplying both sides by∇(z)= 1 −zand settingz=1 givesδ=θ( 1 )/α( 1 ),
where the numerator and the denominator are just the sums of the polyno-
mial coefficients. Substituting the result into equation (6.74) creates an additive
decomposition of the formy(z)=τ(z)+ζ(z), wherein:
τ(z)=1
∇(z)
{μ(z)+δε(z)}, (6.76)ζ(z)=
ρ(z)
α(z)
ε(z), (6.77)