Trend extraction from time series is often performed by using the filter proposed by Leser (1961), also known as the Hodrick-Prescott filter. A practical problem arises, however, when the time series contains structural breaks (such as produced by German unification for German time series, for instance). This note proposes a method for coping with this problem.Trend extraction; structural break; Hodrick-Prescott filter; Leser filter; spline; time-series; smoothing; interpolation.

Schlicht, Ekkehart

English

Research Papers in Economics

Ekkehart Schlicht:
Trend Extraction From Time Series With Structural
Breaks
Munich Discussion Paper No. 2007-17
Department of Economics
University of Munich
Volkswirtschaftliche Fakultät
Ludwig-Maximilians-Universität München
Online at http://epub.ub.uni-muenchen.de/1926/Trend Extraction From Time Series With
Structural Breaks
Ekkehart Schlicht
Abstract: Trend extraction from time series is often performed by using the ﬁlter
proposed by L (), also known as the Hodrick-Prescott ﬁlter. A practical
problem arises, however, when the time series contains structural breaks (such as
produced by German uniﬁcation for German time series, for instance). This note
proposes a method for coping with this problem.
Keywords: Trend extraction, structural break, Hodrick-Prescott ﬁlter, Leser ﬁlter, spline,
time-series, smoothing, interpolation.
Journal of Economic Literature Classiﬁcation: C, C, C, C
 Introduction
Trend extraction from time series is often performed by using the ﬁlter proposed by L
(), also known as the Hodrick-Prescott ﬁlter, or HP-Filter. A practical problem arises,
however, when the time series contains structural breaks (such as produced by German
uniﬁcation for German time series, for instance). This note proposes a method for coping
with this problem.
 The Leser Filter
The idea proposed by L () for the case where all data are available is to look for a
trend y ∈ RT such that deviation
u = x − y ()
is “small” and the trend is “smooth.” The size of the deviation is measured by the sum of
squared residuals u0u, and the smoothness of the trend is measured by the sum of squares of
changes in the direction of the trend v0v where the trend disturbances v ∈ RT−2 are deﬁned
as
vt = ((yt − yt−1) − (yt−1 − yt−2)) t = 3,4,...,T
or
v = P y ()
with
P :=

  

1 −2 1 0
1 −2 1
. . .
0 1 −2 1

  

of order (T − 2) × T.
The decomposition of the original series x into trend y and and residual u is obtained by
minimizing the weighted sum of squares
V = u
0u + α · v
0v = (x − y)
0 (x − y) + α · y
0P
0P y ()
 The formailization below follows S ().
with respect to y. This gives the ﬁrst-order condition
(IT + α · P
0P)y = x. ()
As (I + αP 0P) is positive deﬁnite, the second order condition is satisﬁed in any case.
Equation () has the unique solution
y = (IT + αP
0P)
−1 x ()
which deﬁnes the Leser-Filter. It associates a trend y with the time series x, depending on
the smoothing parameter α.
From () and () we obtain
V = x
0

IT − (IT + α · P
0P)
−1
x ()
as the value of the criterion function ().
 Structural Breaks
A practical problem arises when the time series contains structural breaks. Equation ()
interprets these wrongly as disturbances. The obvious way to generalize the ﬁlter in order
to cope with this problem is to look for a smooth trend by introducing dummy variables
to capture these eﬀects, and to select these variables such that the criterion function () is
minimized. This can be done as follows.
Consider a time series x ∈ RT with m ≤ T − 2 structural breaks. The break points are
indicated by a vector m ∈ RT with components mt = 1 for all t ∈ {1,2,...T} where a
break point occurs, and mt = 0 otherwise. Denote the break points by t1,t2,..,tm. Deﬁne
the (T × m)-matrix with elements bi,j = 0 for i < tj and bi,j = 1 for i ≥ tj. For T = 5
and break points at t1 = 2 and t2 = 4 we would have, for example
B =

  
 

0 0
1 0
1 0
1 1
1 1

  
 

.
Deﬁne further the vector of dummies dor the break points d ∈ Rm, where di gives the
dummy for the i-th break point.
The adjusted vector of observations is
x
∗ = x + Bd. ()
It is a function of the values assumed for the dummies d. These values can now easily be
determined by replacing x by x∗ in (), and minimizing this expression. Thus we obtain the
positive deﬁnite quadratic form
V = (z
0 + s
0B
0)

IT − (IT + α · P
0P)
−1
(z + Bd) ()
that is to be minimized with respect to s.The ﬁrst-order conditions for a minimum is
∂V
∂s
= 2B
0

IT − (IT + α · P
0P)
−1
(z + Bs) = 0. ()
and the second-order condition is that B0  
IT − (IT + α · P 0P)
−1
B is positive deﬁnite.
As
(IT + αP
0P)
−1 = IT − αP
0P + (αP
0P)
2 − (αP
0P)
3 + ...
we can write

IT − (IT + α · P
0P)
−1
= αP
0 (IT−2 + α · PP
0)
−1 P.
Hence
 
IT − (IT + α · P 0P)
−1
is non-negative deﬁnite of rank T − 2 and
B0  
IT − (IT + α · P 0P)
−1
B has full rank and is positive deﬁnite. Therefore equa-
tion () deﬁnes the unique maximizing choice of the dummy terms as
d
∗ = −

B
0

IT − (IT + α · P
0P)
−1
B
−1
B
0

IT − (IT + α · P
0P)
−1
z. ()
The replenished time series is obtained now by inserting () into ().
x
∗ = z + Bd
∗
and the trend is obtained by using the replenished series x∗ instead of the original series x
10 20 30 40 50
2
4
6
8
10
12
with break
original
corrected
(a)
10 20 30 40 50
2
4
6
8
10
12
with break
original
corrected
(b)
Figure: (a) The original US unemployment rate - has been augmented
by adding, beginning with period , two percentage points to the original se-
ries. The corrected series overcorrects the structural break in this case. (b) The
smoothed series reproduce this pattern. (Smothing constant is α =, data from
the Bureau of Labor Statistics.)
in ():
y
∗ = (IT + αP
0P)
−1 (z + Bs
∗).
This gives the trend the time series, adjusted for structural breaks.
 An Example
As an example, consider the time series of US unemployment (Figure ). Beginning with
period , a structural break has been introduced by adding  percentage points to the
originaltimeseries. Thecorrectionobtainedbythemethodsketchedaboveovercorrectsthis
break by subtracting . percentage points. The corrected trend estimation is overcorrected
as well. As can be seen, the correction produces a smoother trend than the original one,
as is implied by the logic of the method. A manual correction would look not very much
diﬀerent, or would look even worse if the adjustment is made such that the adjacent data
points  and  are made to have identical values. (The correction would have been -.
rather than -. in this case.)
 All computations done with the package by L ().
 Concluding Comments
The example illustrates the functioning, as well as the problematic, of introducing dummies,
as these will not only correct for structural breaks, but will also mask changes in the
underlying trend. Yet there seems no better way available to do such corrections. If the Leser
method is interpreted stochastically as in S (), a theoretical justiﬁcation is that
it can be interpreted as giving Maximum-Likelihood estimates for the dummies that capure
structural breaks.
References
L,C.E.V.,“ASimpleMethodofTrendConstruction,”JournaloftheRoyalStatistical
Society. Series B (Methodological), , –.
L, J. , “HPFilter,” Mathematica package, Wolfram Research, online at http:
//library.wolfram.com/infocenter/MathSource/Mathematics/ProbabilityStatistics/.
S, E. , “A Seasonal Adjustment Principle and a Seasonal Adjustment Method
Derived from This Principle,” Journal of the American Statistical Association, (),
–, online at http://www.semverteilung.vwl.uni-muenchen.de/mitarbeiter/es/paper/
schlicht-seasonal_adjustment.pdf.
— , “Estimating the Smoothing Constant in the So-Called Hodrick-Prescott Filter,”
Journal of the Japan Statistical Society, , –, online at http://www.scipress.org/journals/
jjss/pdf//.pdf.


Trend Extraction From Time Series With Structural Breaks

http://epub.ub.uni-muenchen.de/1926/1/schlicht_structura__breaks_DP17.pdf

Trend Extraction From Time Series With Structural Breaks

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