282 Investigating Economic Trends and Cycles
6.7.3 The polynomial component
The formula (6.123) tends to conceal the presence of polynomial components
within the sequences that are generated by filtering the nonstationary data. An
alternative procedure, which we have already adopted in detrending the logarith-
mic consumption data of the UK in the example following (6.14), is to extract
a polynomial trend from the nonstationary data before applying a filter to the
residual sequence, which will have the characteristics of a sequence generated by
a stationary process, provided that the polynomial is of a sufficient degree.
Another procedure that can be followed requires the data to be reduced to sta-
tionarity by a process of differencing, before it is filtered. The filtered output can be
reinflated thereafter to obtain estimates of the components of the non-stationary
process. It transpires that, in the context of WK filtering, such a procedure pro-
duces estimates that are identical to those that are delivered by the finite-sample
filter of (6.123).
To demonstrate this result, we shall assume that, withiny=ξ+η, the vectorξ
is generated by a stochastic process withpunit roots driven by a mean-zero white-
noise process. The vectorηis assumed to be from a stationary process. Therefore,
the specifications of (6.125) and (6.127) remain, but we may choose to setE(δ)=0,
if only to confirm that the polynomial component will arise just as surely in the
absence of stochastic drift.
Let the estimates ofξ,η,δ=Q′ξandκ =Q′ηbe denoted byx,h,dandk
respectively. Then the Wiener–Kolmogorov minimum mean square error estimates
of the differenced components are:
E(δ|g)=d=D(δ){D(δ)+D(κ)}−^1 g=(+Q′Q)−^1 Q′y, (6.131)E(κ|g)=k=D(κ){D(δ)+D(κ)}−^1 g=Q′Q(+Q′Q)−^1 Q′y. (6.132)The estimates ofξandηmay be obtained by integrating, or re-inflating, the
components of the differenced data to give
x=S∗d∗+Sd and h=S∗k∗+Sk, (6.133)whereS∗d∗andS∗k∗are vectors of the ordinates of polynomials of degreep. For
this representation, the polynomial parameters, in the form of the starting values
d∗andh∗, are required.
The initial conditions ind∗should be chosen so as to ensure that the estimated
trend is aligned as closely as possible with the data. The criterion is:
Minimize (y−S∗d∗−Sd)′−^1 (y−S∗d∗−Sd) with respect to d∗. (6.134)The solution for the starting values is:
d∗=(S′∗−^1 S∗)−^1 S∗′−^1 (y−Sd). (6.135)The equivalent starting values ofk∗are obtained by minimizing the (generalized)
sum of squares of the fluctuations:
Minimize (S∗k∗+Sk)′−^1 (S∗k∗+Sk) with respect to k∗. (6.136)