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

Schlicht, Ekkehart

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Trend Extraction From Time Series With Structural Breaks

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Ekkehart Schlicht:Trend Extraction From Time Series With StructuralBreaksMunich Discussion Paper No. 2007-17Department of EconomicsUniversity of MunichVolkswirtschaftliche FakultätLudwig-Maximilians-Universität MünchenOnline at http://epub.ub.uni-muenchen.de/1926/Trend Extraction From Time Series WithStructural BreaksEkkehart SchlichtAbstract: Trend extraction from time series is often performed by using the filterproposed by L (), also known as the Hodrick-Prescott filter. A practicalproblem arises, however, when the time series contains structural breaks (such asproduced by German unification for German time series, for instance). This noteproposes 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 IntroductionTrend 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 Germanunification for German time series, for instance). This note proposes a method for copingwith this problem. The Leser FilterThe idea proposed by L () for the case where all data are available is to look for atrend y ∈ RT such that deviationu = x− y ()is “small” and the trend is “smooth.” The size of the deviation is measured by the sum ofsquared residuals u′u, and the smoothness of the trend is measured by the sum of squares ofchanges in the direction of the trend v′v where the trend disturbances v ∈ RT−2 are definedasvt = ((yt − yt−1)− (yt−1 − yt−2)) t = 3, 4, ... , Torv = P y ()withP :=1 −2 1 01 −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 byminimizing the weighted sum of squaresV = u′u+ α · v′v = (x− y)′ (x− y) + α · y′P ′P y () The formailization below follows S ().with respect to y. This gives the first-order condition(IT + α · P ′P ) y = x. ()As (I + αP ′P ) is positive definite, the second order condition is satisfied in any case.Equation () has the unique solutiony = (IT + αP′P )−1 x ()which defines the Leser-Filter. It associates a trend y with the time series x, depending onthe smoothing parameter α.From () and () we obtainV = x′(IT − (IT + α · P ′P )−1)x ()as the value of the criterion function (). Structural BreaksA practical problem arises when the time series contains structural breaks. Equation ()interprets these wrongly as disturbances. The obvious way to generalize the filter in orderto cope with this problem is to look for a smooth trend by introducing dummy variablesto capture these effects, and to select these variables such that the criterion function () isminimized. This can be done as follows.Consider a time series x ∈ RT withm ≤ T − 2 structural breaks. The break points areindicated by a vector m ∈ RT with components mt = 1 for all t ∈ {1, 2, ...T} where abreak point occurs, andmt = 0 otherwise. Denote the break points by t1, t2, .., tm. Definethe (T ×m)-matrix with elements bi,j = 0 for i < tj and bi,j = 1 for i ≥ tj . For T = 5and break points at t1 = 2 and t2 = 4 we would have, for exampleB =0 01 01 01 11 1 .Define further the vector of dummies dor the break points d ∈ Rm, where di gives thedummy for the i-th break point.The adjusted vector of observations isx∗ = x+Bd. ()It is a function of the values assumed for the dummies d. These values can now easily bedetermined by replacing x by x∗ in (), and minimizing this expression. Thus we obtain thepositive definite quadratic formV = (z′ + s′B′)(IT − (IT + α · P ′P )−1)(z +Bd) ()that is to be minimized with respect to s.The first-order conditions for a minimum is∂V∂s= 2B′(IT − (IT + α · P ′P )−1)(z +Bs) = 0. ()and the second-order condition is that B′(IT − (IT + α · P ′P )−1)B is positive definite.As(IT + αP′P )−1 = IT − αP ′P + (αP ′P )2 − (αP ′P )3 + ...we can write (IT − (IT + α · P ′P )−1)= αP ′ (IT−2 + α · PP ′)−1 P.Hence(IT − (IT + α · P ′P )−1)is non-negative definite of rank T − 2 andB′(IT − (IT + α · P ′P )−1)B has full rank and is positive definite. Therefore equa-tion () defines the unique maximizing choice of the dummy terms asd∗ = −(B′(IT − (IT + α · P ′P )−1)B)−1B′(IT − (IT + α · P ′P )−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 5024681012with breakoriginalcorrected(a)10 20 30 40 5024681012with breakoriginalcorrected(b)Figure : (a) The original US unemployment rate - has been augmentedby adding, beginning with period , two percentage points to the original se-ries. The corrected series overcorrects the structural break in this case. (b) Thesmoothed series reproduce this pattern. (Smothing constant is α =, data fromthe Bureau of Labor Statistics.)in ():y∗ = (IT + αP ′P )−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 withperiod , a structural break has been introduced by adding  percentage points to theoriginal time series. The correction obtained by the method sketched above overcorrects thisbreak by subtracting . percentage points. The corrected trend estimation is overcorrectedas 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 muchdifferent, or would look even worse if the adjustment is made such that the adjacent datapoints  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 CommentsThe 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 theunderlying trend. Yet there seems no better way available to do such corrections. If the Lesermethod is interpreted stochastically as in S (), a theoretical justification is thatit can be interpreted as giving Maximum-Likelihood estimates for the dummies that capurestructural breaks.ReferencesL, C. E. V. , “A SimpleMethod of Trend Construction,” Journal of the Royal StatisticalSociety. 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 MethodDerived 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 .

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Trend Extraction From Time Series With Structural Breaks

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