D.S.G. Pollock 2990.000.050.10–0.05
–0.1
–0.15
1960 1970 1980 1990 2000Figure 6.20 A filtered sequence obtained by applying the bandpass filter of Christiano and
Fitzgerald to the logarithms of UK household expenditure
6.10.2 Flexible trends and structural breaks
Over a period of a century or so, one can expect to see occasional disturbances
that disrupt the steady progress of the economy. To highlight the effects of such
breaks, a firm trend function can be fitted to the data to characterize the progress
of the economy broadly over the entire period. Such a trend will not be deflected
by temporary disruptions, which will be seen in the residual deviations of the data
from the trend.
Alternatively, it may be appropriate to absorb the breaks within the trend func-
tion. In that case, the trend will not be thrown off course for long by a break and,
therefore, it should serve as a benchmark against which to measure cyclical varia-
tions when the economy resumes its normal progress. At best, the residual sequence
will serve to indicate how the economy might have behaved in the absence of the
break.
Numerous devices have been proposed by economists for accommodating struc-
tural breaks, which give rise to segmented trend functions. Mills (2003) has
illustrated the effects of some of them by applying them to a common data
sequence, which is annual UK output from 1855 to 1999. He has also provided
references to an extensive literature in economics concerning structural breaks.
A common theme that unites many of the methods is their use of polynomial
segments to represent the trends within sub-intervals of the data period. There
is a problem of how the transition between two adjacent sub-periods should be
modelled. This issue has been discussed by Granger and Teräsvirta (1993) and by
Teräsvirta (1998). Others have focused on devising tests to determine the points
in time when one statistical regime that describes the data should be replaced by
another. Work in this area has been summarized by Perron (2006).
When a smoothing spline is used to interpolate a continuous segmented polyno-
mial function through the data, the smoothness of the function is maintained by
imposing the condition that, at the points where they join, the adjacent segments
should have equal derivatives, up to some specified order.
The most common smoothing spline is that of Reinsch (1976), which is subject
to the condition that the first and second derivatives of adjacent cubic segments