264 Investigating Economic Trends and Cycles
On the right-hand side of this equation is a sum of stationary stochastic processes,
which can be expressed as an ordinary first-order moving-average process. Thus:
ε(t− 1 )+ν(t)−ν(t− 1 )=η(t)+θη(t− 1 ), (6.67)whereη(t)is a white-noise process withV{η(t)}=ση^2. Therefore, the sampled
version of the integrated Wiener process is a doubly-integrated IMA(2, 1) moving-
average model.
The essential task is to find the values of the moving-average parameterθ. Thus
is achieved by reference to equation (6.61), which provides the variances and
covariances of the terms on the left-hand side of (6.67), from which the auto-
covariances of the MA process can be found. It can be shown that the variance and
the autocovariance at lag 1 of this composite process are given by:
γ 0 =
2 σε^2
3=ση^2 ( 1 +θ^2 ) and γ 1 =
σε^2
6=ση^2 θ. (6.68)The equations must be solved forθandση^2. There are two solutions forθ, and we
should take the one which fulfils the condition of invertibility:θ= 2 −
√- (See
Pollock, 1999.)
When white-noise errors of observation are superimposed upon values sampled
from an integrated Wiener process at regular intervals, the resulting sequence can
be described by a doubly-integrated second-order moving-average process in dis-
crete time, which is an IMA(2, 2) process. Such a model provides the basis for
the cubic smoothing spline of Reinsch (1976), which can be used to extract an
estimate of the trajectory of the underlying integrated Wiener process from the
noisy data. The statistical interpretation of the smoothing spline is due to Wahba
(1978).
The smoothing spline interpolates cubic polynomial segments between nodes
that are derived by smoothing a sequence of sampled data points. The segments are
joined in such a way as to ensure that the second derivative of the spline function is
continuous at the nodes. An account of the algorithm of the smoothing spline and
of its derivation from the statistical model has been provided by Pollock (1999). It
is shown that the means by which the nodes are obtained from the data amount
to a so-called discrete-time Wiener–Kolmogorov (WK) filter.
The Wiener–Kolmogorov principle can also be used to derive the so-called
Hodrick–Prescott (HP) filter, which is widely employed in macroeconomic
analysis – see Hodrick and Prescott (1980, 1997). The filter, which is presented
in section 6.6.2, is derived from the assumption that the process that gener-
ates the trend is a doubly-integrated discrete-time white noise. When white-noise
errors are added to the sampled values of the process, the observations are
once more described by an IMA(2, 2) model, and the nodes that are gener-
ated by the WK trend-extraction filter are analogous to those of the smoothing
spline.