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Econometric ReviewsPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713597248

A Cascade Linear Filter to Reduce Revisions and False Turning Points for RealTime Trend-Cycle EstimationEstela Bee Dagum a; Alessandra Luati a

a Department of Statistics, University of Bologna, Bologna, Italy

Online Publication Date: 01 January 2009

To cite this Article Dagum, Estela Bee and Luati, Alessandra(2009)'A Cascade Linear Filter to Reduce Revisions and False TurningPoints for Real Time Trend-Cycle Estimation',Econometric Reviews,28:1,40 — 59

To link to this Article: DOI: 10.1080/07474930802387837

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Econometric Reviews, 28(1–3):40–59, 2009Copyright © Taylor & Francis Group, LLCISSN: 0747-4938 print/1532-4168 onlineDOI: 10.1080/07474930802387837

A CASCADE LINEAR FILTER TO REDUCE REVISIONS AND FALSETURNING POINTS FOR REAL TIME TREND-CYCLE ESTIMATION

Estela Bee Dagum and Alessandra Luati

Department of Statistics, University of Bologna, Bologna, Italy

� The problem of identifying the direction of the short-term trend (nonstationary mean) ofseasonally adjusted series contaminated by high levels of variability has become of relevantinterest in recent years. In fact, major financial and economic changes of global character haveintroduced a large amount of noise in time series data, particularly, in socioeconomic indicatorsused for real time economic analysis.

The aim of this study is to construct a cascade linear filter via the convolution of severalnoise suppression, trend estimation, and extrapolation linear filters. The cascading approachapproximates the steps followed by the nonlinear Dagum (1996) trend-cycle estimator, a modifiedversion of the 13-term Henderson filter. The former consists of first extending the seasonallyadjusted series with ARIMA extrapolations, and then applying a very strict replacement ofextreme values. The nonlinear Dagum filter has been shown to improve significantly the size ofrevisions and number of false turning points with respect to H13.

We construct a linear approximation of the nonlinear filter because it offers severaladvantages. For one, its application is direct and hence does not require some knowledgeon ARIMA model identification. Furthermore, linear filtering preserves the crucial additiveconstraint by which the trend of an aggregated variable should be equal to the algebraic additionof its component trends, thus avoiding the selection problem of direct versus indirect adjustments.Finally, the properties of a linear filter concerning signal passing and noise suppression canalways be compared to those of other linear filters by means of spectral analysis.

Keywords False turning points; Gain function; Smoothing; Symmetric linear filter; 13-TermHenderson filter.

JEL Classification C14; C22; E32.

1. INTRODUCTION

Major financial and economic changes of global character haveintroduced high levels of variability in time series data, particularly in

Address correspondence to Estela Bee Dagum, Department of Statistics, University of Bologna,via Belle Arti, 41, Bologna 40126, Italy; E-mail: [email protected]

Downloaded By: [University of Bologna] At: 15:34 9 January 2009

A Cascade Linear Filter 41

socioeconomic indicators often used for the analysis of current economicconditions. Traditionally, this type of indicators are seasonally adjustedto determine the direction of the short-term trend (that of the currentyear) for an early detection of a turning point. However, current highlevels of variability have created the need for further smoothing in orderto suppress part of the noise without affecting the underlying trend-cyclecomponent.

In a recent study, Dagum (1996) developed a nonlinear smootherto improve on the classical 13-term Henderson (1916) filter which iswidely applied for short-term trend-cycle estimation. The nonlinear Dagumfilter (NLDF) results from applying the 13-term symmetric Hendersonfilter (H13) to seasonally adjusted series where outliers and extremeobservations have been replaced and which have been extended withextrapolations from an ARIMA model (Box and Jenkins, 1970). The mainpurpose of the ARIMA extrapolations is to reduce the size of the revisionsof the most recent estimates whereas that of extreme values replacementis to reduce the number of unwanted ripples produced by H13. Anunwanted ripple is a 10-month cycle (identified by the presence of highpower at � = 0�10 in the frequency domain) which, due to its periodicity,often leads to the wrong identification of a true turning point. In fact,it falls in the neighborhood between the fundamental seasonal frequencyand its first harmonic. On the other hand, a high frequency cycle isgenerally assumed to be part of the noise pertaining to the frequencyband 0�10 ≤ � < 0�50. The problem of the unwanted ripples is specificof H13 when applied to seasonally adjusted series. It differs from thatof spurious cycles such as those addressed by Cogley and Nason (1995)and Harvey and Jaeger (1993) mainly referring to the Hodrick–Prescott(HP) filter (Hodrick and Prescott, 1997) in the context of long-term trendestimation. In the latter case, the basic assumption is that a time series canbe decomposed into the sum of a long-term trend plus a cycle componentbeing the noise incorporated into the cycle. Applied to monthly data,HP can generate cycles where they are not present, often leaving toomuch noise in the cyclical component. This problem can be overcomeeither by applying HP to a seasonally adjusted series where the noise hasbeen suppressed, as suggested by Gomez (2001) or by assuming smoothstochastic cycles, as proposed by Harvey and Trimbur (2003). In thisarticle we do not deal with long term filters (see, e.g., Proietti, 2005)for detrending seasonally adjusted series but with those that can estimatejointly trend and cycle fluctuations. The main reason for this is that weare concerned with smoothing seasonally adjusted data in the context ofrather short series (less than 15 year long) for which their long-term trendis often difficult to identify and estimate accurately.

Studies by Dagum et al. (1996), Chhab et al. (1999), and Darnè(2002) showed the superior performance of the NLDF respect to both


42 E. B. Dagum and A. Luati

structural and ARIMA standard parametric trend-cycle models applied toseries with different degrees of signal-to-noise ratios. The criteria evaluatedwere those of Dagum (1996): (1) number of unwanted ripples, (2) size ofrevisions, and (3) time delay to detect a turning point. In another study,the good performance of the NLDF is shown relative to nonparametricsmoothers, namely, locally weighted regression (loess), Gaussian kernel,cubic smoothing spline, and supersmoother (Dagum and Luati, 2000).

Given the excellent performance of the NLDF according to the threeabove criteria, the aim of this article is to construct a cascade linear filterthat closely approximates it. The cascading is done via the convolution ofseveral noise suppression, trend estimation, and extrapolation linear filters.

The symmetric filter is the one applied to all central observations,i.e., to a series without the first and last six data points. In this case, ourpurpose is to offer a linear solution to the unwanted ripples problem.To avoid the latter, the NLDF largely suppresses the noise in thefrequency band between the fundamental seasonal and first harmonic.In this regard, we derive a cascade linear filter by double smoothingthe residuals obtained from a sequential application of H13 to the inputdata. The residuals smoothing is done by the convolution of two shortsmoothers, a weighted 5-term and a simple 7-term linear filters. The linearapproximation for the symmetric part of the NLDF is truncated to 13 termswith weights normalized to add to one.

The asymmetric filter is applied to the last six data points, whichare crucial for current analysis. It is obtained by the convolution of thesymmetric filter with linear extrapolation filters for the last six data points.The extrapolations are made linear by fixing the ARIMA model and itsparameters values. The latter are chosen such as to minimize the size ofrevisions and phaseshifts. The model is selected among some parsimoniousprocesses found to fit and extrapolate well a large number of seasonallyadjusted series. Such a model turns out to be the ARIMA(0, 1, 1) with� = 0�4. A simple linear transformation (Dagum and Luati, 2004a) allowsto apply the asymmetric filter to the first six observations.

We will call the new filter a cascade linear filter (CLF) and we willdistinguish between the symmetric (SLF) and the asymmetric linear filter(ALF).

A linear filter offers many advantages over a nonlinear one. Forone, its application is direct and hence, does not require knowledge ofARIMA model identification. Furthermore, linear filtering preserves thecrucial additive constraint by which the trend of an aggregated variableshould be equal to the algebraic addition of its component trends, thusavoiding the selection problem of direct versus indirect adjustments.Finally, the properties of a linear filter concerning signal passing and noisesuppression can always be compared to those of other linear filters bymeans of spectral analysis.



We study the properties of the new CLF relative to H13 shown to be anoptimal trend-cycle predictor among a variety of second and higher orderkernels restricted to be of the same length (see Dagum and Luati, 2002).The theoretical properties are analyzed by means of spectral analysis. Theempirical properties of the new estimator are evaluated on a large sampleof real time series pertaining to various socioeconomic areas and withdifferent degrees of variability. It should be noted that the theoreticalproperties of CLF cannot be compared with those of NLDF since the latteris data dependent.

The outline of the article is as follows. Section 2 summarizes in matrixform the various steps used to obtain the nonlinear estimator developedby Dagum (1996). Section 3 presents the estimator that approximates thesymmetric and asymmetric parts of the nonlinear Dagum filter togetherwith the truncation and normalization to obtain the final 13-term cascadelinear filter. The theoretical properties of CLF are also studied in Section 3,and the results of an empirical analysis are reported. Concluding remarksare given in Section 4.

2. THE NONLINEAR DAGUM FILTER (NLDF)

In time series analysis, it is often assumed that a time series �yt�t=1,��,N ,N < ∞, is decomposed as the sum of a nonstationary mean (signal),g (t), plus an erratic component, ut , that is, yt = g (t) + ut , where g (t) canbe either deterministic or stochastic and ut usually follows a stationarystochastic process with zero mean and constant variance �2

u . A commonassumption is that ut is generated by a white noise process but it can alsobe assumed that it comes from an autoregressive moving average (ARMA)process.

The signal g (t) can be estimated by a function of time g�(t) wherethe smoothing parameter � determines the degree of smoothness of theestimated values. For fixed values of the smoothing parameter, say � = �0,any estimator g�0(t) becomes linear and can be interpreted equivalently as:(a) a nonparametric estimator of g (t) and (b) a smooth estimate of thevalue yt , say yt , resulting from a 2m + 1-term symmetric weighted averageof neighboring observations.

In the context of smoothing monthly time series, it is useful to dividethe frequency domain � = �0 ≤ � ≤ 0�50� in two major intervals: (1) �S =�0 ≤ � ≤ 0�06� associated with cycles of 16 months or longer attributedto the signal (nonstationary mean or trend-cycle) of the series, and(2) the frequency band �S = �0�06 < � ≤ 0�50� corresponding to shortcyclical fluctuations attributed to the noise. In this latter interval, it is ofgreat interest to see how much of the power is not suppressed within aneighborhood of � = 0�10 corresponding to 10-month cycles. These intra-annual short cycles are known as unwanted ripples and can be wrongly



interpreted as true turning points. Keeping in mind that the points atboth ends of a series must be estimated with asymmetric filters, an optimalsmoother should be rather short and have a gain G(�) close to one for0 ≤ � ≤ 0�06 and near to zero for 0�10 ≤ � ≤ 0�50. In other words, thesetwo conditions impede the application of long filters. Furthermore, longfilters require that a large number of ending data points is estimatedwith asymmetric filters that introduce phase shifts. It should be notedthat most seasonal adjustment methods will suppress almost all the poweralready present in the frequency band around the fundamental seasonalfrequency, i.e., 0�06 < � < 0�10.

The NLDF basically consists of: (a) extending with ARIMAextrapolations a seasonally adjusted series modified by extreme values,and (b) applying H13 to the extended series where a stricter secondreplacement of extreme values is made.

The extrapolation is performed with the purpose of reducing the sizeof the revisions of the most recent estimates as new observations are addedto the series. On the other hand, the strict replacement of the extremevalues implies a large noise suppression in the input series and has thepurpose of reducing the number of unwanted ripples created by H13.

To facilitate the identification and fitting of simple ARIMA models,Dagum (1996) recommends at step (a) to modify the input series forthe presence of extreme values using ±2�5� as standard limits, where� is a five-year moving standard deviation of the residuals estimated byX 11ARIMA (Dagum, 1988) or X 12ARIMA (Findley et al., 1998) seasonaladjustment software (currently, the NLDF is applied by means of any ofthese software). The modified series is often modelled by a simple and veryparsimonious ARIMA model such as the (0, 1, 1) which provides a goodfit of the data. Concerning step (b), it is recommended to use very strictsigma limits, such as, ±0�7� and ±1�0�.

The NLDF can be described briefly in matrix notation as follows (seeDagum and Luati, 2000, for a detailed description)

y = H[H + W(IN+12 − H)]EA[H + W0(IN − H)]y,where y is the (N + 12)-dimensional vector of smooth estimates of theN -dimensional input series y; H is the N × N matrix (canonically)associated to the 13-term Henderson filter; IK is the identity matrix ofdimension K ; the superscript E indicates that the matrix is applied to theextended series with ARIMA forecast. W0 is a zero-one diagonal matrix,being the diagonal element wii equal to zero when the correspondingelement yi of the vector y is identified as an extreme value with respectto ±2�5� limits, where � is a 5-term moving standard deviation of theresiduals; A is the (N + 12) × N dimensional matrix canonically associatedto an ARIMA(p, d , q)(P ,D,Q )s process producing one year of monthly



extrapolations; W is a diagonal matrix with non-null element wii such that,wii = 0 if the corresponding value yi falls out of the sigma limits ±1�0�;wii = 1 if the corresponding yi falls within the lower bound limits ±0�7�and wii decreases linearly (angular coefficient equal to −1) from 1 to 0 inthe range from ±0�7� to ±1�0�.

Since the values of the matrices W0 and W corresponding to extremevalues replacement, and matrix A pertaining to ARIMA extrapolations aredata dependent, this filter is nonlinear.

3. THE CASCADE LINEAR FILTER (CLF)

To obtain a linear approximation of NLDF it is necessary to linearize(a) the ARIMA extrapolation process and (b) the replacement of extremevalues. Concerning (a), we make recourse to a simple ARIMA modelwhere the parameter values are fixed. As regards (b) we approach thereplacement of extreme values as a strong noise suppression in the input,sequentially applying a 5-term weighted and a 7-term nonweighted movingaverage.

3.1. The Symmetric Linear Filter (SLF)

The smoothing matrix associated to the symmetric linear filter results

H[H + M7(0�14)(IN − H)][H + M5(0�25)(IN − H)], (1)

where M5(0�25) is the matrix representative of a 5-term moving averagewith weights (0�250, 0�250, 0�000, 0�250, 0�250), and M7(0�14) is the matrixrepresentative of a 7-term filter with all weights equal to 0�143.

We have chosen 5- and 7-term filters following the standard filterslength selected in Census X11 and X11/X12ARIMA software for thereplacement of extreme seasonal-irregular values. In these computerpackages, a first iteration is made by means of a short smoother, a 5-term(weighted) moving average, and a second one by a 7-term (weighted)average. In our case, 5- and 7-term filters are applied to the residuals froma first pass of the H13 filter. These two filters have both the good propertyof suppressing large amounts of power at � = 0�10. Figure 1 shows the gainfunction of the filter convolution [M7(0�14)(IN − H)][H + M5(0�25)(IN − H)].

It is apparent that the filter convolution applied to a series consistingof trend plus irregulars suppresses all the trend power and a great deal ofthe irregular variations. Hence given the input series, the results from theconvolution are the modified irregulars needed to produce a new serieswhich will be extended with ARIMA extrapolations, and then smoothed byH13.



FIGURE 1 Gain function of the symmetric weights of the modified irregulars filter.

Equation (1) produces a symmetric filter of 31 terms with very smallweights at both ends. This long filter is truncated to 13 terms, andnormalized such that its weights add up to unity. Normalization is neededto avoid a biased mean output.

To normalize the filter, the total weight discrepancy (in our case the13 truncated weights add up to 1.065) must be distributed over the 13weights, wj , j = −6, � � � , 6, according to a well-defined pattern. This is a verycritical adjustment for different distributions produce linear filters withvery distinctive properties.

The two most commonly applied normalization procedures are theuniform and the proportional. In a uniform distribution, the total weightdiscrepancy is simply divided by the number of weights and hence,a constant amount is added to each of them. The 13-term truncatedsymmetric filter with uniform distribution (only six plus the central in boldare reported) is

−0�025,−0�007, 0�031, 0�084, 0�139, 0�180, 0�195�

In the proportional distribution, the truncated weights are each onedivided by their sum. In this way, the discrepancy is proportionally assignedto the weights, and the corresponding 13-term truncated symmetric filter is

−0�019,−0�002, 0�034, 0�084, 0�135, 0�174, 0�188�

The gain functions of these two filters, shown in Figure 2, clearly indicatethat none of them satisfies simultaneously the two conditions we are



FIGURE 2 Gain functions of cascade filters with two classical normalization procedures and H13.

looking for, namely, good fit to the data (as H13), and significantreduction in the number of unwanted ripples.

It is known in the literature (see, e.g., Hastie and Tibshirani, 1990)that large values of w0 (which correspond to the mid-data point in a giveninterval) will tend to reduce bias or oversmoothing of the signal, andthus the filter will give a better fit. Furthermore, it is also well-known thatonly filters with negative weights such that

∑mj=−m wj = 1,

∑mj=−m jwj = 0,∑m

j=−m j 2wj = 0 can satisfy the condition of reproducing locally second andthird degree polynomials as H13. Therefore to get a linear filter thatprovides a good trade-off between bias and smoothing, we performeda mixed distribution of the small total weight discrepancy (−0.065) asfollows

−0�103,−0�076,−0�076,−0�341,−0�127, 0�042, 0�364�

The total discrepancy is mostly allocated to w0 (+36%), w3, and w−3 (−34%each). To increase the amount given to the central point, the values ofw3 and w−3 have been reduced for it is important to maintain as much aspossible the area under the positive weights without modifying the negativeones. The presence of the latter is a necessary but not sufficient conditionfor a filter to be unbiased respect to a second degree polynomial trend,which is needed to estimate properly points of maxima and minima. Theweights of the SLF with mixed distribution normalization are given by

−0�027,−0�007, 0�031, 0�067, 0�136, 0�188, 0�224�



The filters resulting from the three normalizations are second-orderkernel estimators for they satisfy the following conditions

m∑j=−m

wj = 1,m∑

j=−m

jwj = 0,m∑

j=−m

j 2wj �= 0�

However, the smallest departure from∑m

j=−m j 2wj = 0 is given by the onewith mixed normalized weights.

In general, second-order kernels have been found to producesmoother estimates relative to those given by higher-order kernels. Thisis often reflected by high values of the mean square errors of theestimates. An empirical analysis over 120 real and simulated time seriescharacterized by different degrees of variability confirmed that among thethree normalizations, the mixed discrepancy distribution gives the bestoverall results. It produced the best fitting (root mean square error closeto that of H13), did not oversmooth (sum of squared third differencesslightly smaller than H13) and showed a large suppression of the numberof unwanted ripples (20% less than H13). On the contrary, the other twonormalizations reduce more the unwanted ripples but at a great cost ofoversmoothing and poor fitting (Dagum and Luati, 2004b).

3.1.1. Theoretical Properties of SLFFrom a theoretical viewpoint, the properties of SLF are studied by

means of classical spectral analysis techniques. Figure 3 exhibits the gainfunctions of the symmetric H13 and SLF estimators.

FIGURE 3 Gain functions of symmetric H13 and SLF.



FIGURE 4 Gain functions of second-order kernels and SLF.

Compared to H13, SLF suppresses more signal only in the frequencyband pertaining to very short cycles, ranging from 15 to 24 monthsperiodicity (0�03 < � ≤ 0�06), whereas it passes without modification cycleof three year and longer periodicity. Furthermore, it reduces by 14%the power of the gain corresponding to the unwanted ripples frequency� = 0�10.

Compared to the performance of other second-order kernel estimatorsrestricted to 13 terms, such as the locally weighted regression smootherof degree one (L1) and the Gaussian kernel (GK), SLF appears to be abetter signal predictor with a smoothing power at � = 0�10, close to GK,as Figure 4 shows.

3.1.2. Empirical Properties of SLFTo perform an empirical evaluation of the fitting and smoothing

performances of SLF relative to H13, we apply both to a largesample of 100 seasonally adjusted real time series pertaining to varioussocioeconomic areas, characterized by different degrees of variability. As ameasure of fitting we use the root mean square error (RMSE) calculated by

RMSE =√√√√ 1

N − 12

N−5∑t=7

(yt − ytyt

)2

,

where yt denotes the original observations and yt the predicted values. Ouraim is to assess the extent to which the fitting properties of the SLF areequivalent to those of H13, known to be a good signal predictor but at the



TABLE 1 Empirical RMSE and number of false turning pointsfor SLF and H13 filters applied to real time series (mean valuesstandardized by those of H13)

Empirical measures of fitting and smoothing SLF H13

RMSE/RMSEH 13 1.05 1ftp/ftpH 13 0.80 1

expense of producing many unwanted ripples. We are also interested inreducing the number of unwanted ripples that may lead to the detectionof false turning points. In this regard, we use the accepted definition ofa turning point for smoothed data (see, among others, Zellner et al.,1991) according to which a turning points occurs at time t if (upturn)yt−k ≤ · · · ≤ yt−1 > yt ≥ yt+1 ≥ · · · ≥ yt+m or (downturn) yt−k ≥ · · · ≥ yt−1 <yt ≤ yt+1 ≤ · · · ≤ yt+m for k = 3 and m = 1. An unwanted ripple ariseswhenever two turning points occur within a 10 month period.

Table 1 shows the mean values of the RMSE calculated over the 100series and standardized respect to H13 to facilitate the comparison. Inthe same way, the number of false turning points produced in the finalestimates from SLF is given relative to that of H13.

The empirical results are consistent with those inferred from thetheoretical analysis. Furthermore, SLF reduces by 20% the number of falseturning points produced by H13.

3.2. The Asymmetric Linear Filter (ALF)

The smoothing matrix associated to the asymmetric linear filter for thelast six data points is obtained in two steps:

(1) a linear extrapolation filter for six data points is applied to theinput series. This filter is represented by a (N + 6) × N matrix A∗

A∗ =[

INO6×N−12 �∗

6×12

]

where ∗6×12 is the submatrix containing the weights for the N − 5,

N − 4, � � � ,N data points. ∗6×12 results from the convolution

H[H + M7(0�14)(IN+12 − H)]EA[H + M5(0�25)(IN − H)] (2)

where [H + M5(0�25)(IN − H)] is the N × N matrix representative of trend-filter plus a first suppression of extreme values, [H + M7(0�14)(IN+12 − H)]E



is the N × N + 12 matrix for the second suppression of the irregularsapplied to the input series plus 12 extrapolated values, generated by

A =[

IN�12×N

]�

This N + 12 × N matrix A is associated to an ARIMA(0, 1, 1) linearextrapolations filter with parameter value � = 0�4.

It is well-known that for all ARIMA models that admit a convergentAR(∞) representation, such that (1 − 1B − 2B2 − � � � )yt = at , the j ’scan be explicitly calculated. For any lead time �, the extrapolated valuesyt(�) may be expressed as a linear function of current and past observationsyt with weights (�)

j that decrease rapidly as they depart from the currentobservations (Box and Jenkins, 1970). That is

yt(�) =∞∑j=1

(�)j yt−j+1,

where

(�)j = j+�−1 +

�−1∑h=1

h(�−h)j

for j = 1, 2, � � � and (1)j = j � Strictly speaking, the yt ’s go back to infinite

past, but since the power series is convergent, their dependence on yt−j

can be ignored after some time has elapsed. For this reason, we willconsider (�)

j = 0 for j > 12. Furthermore, to generate one year of ARIMAextrapolation we fix � = 12.

Parsimonious models found to fit and extrapolate well a large numberof seasonally adjusted series are the ARIMA(0, 1, 1), ARIMA(0, 2, 1),ARIMA(0, 2, 2). We used a fixed range of values for �i = 0�1, 0�3, � � � , 0�9for i = 1, 2. Among the selected models with different parametercombinations, we found that only the linear asymmetric filters of theARIMA(0, 1, 1) models with 0�2 ≤ � ≤ 0�6 gave good results in termsof signal passed and noise suppressed. We selected � = 0�4 as thepreferred one. For an ARIMA(0, 1, 1) model, the coefficients are j =(1 − �)�j−1, and it can be shown that (�)

j = j , ∀� ∈ N . Hence, 12×N isthe matrix whose generic row is (0, � � � , 0, 0�001, 0�002, 0�006, 0�015, 0�038,0�096, 0�24, 0�6).

Since we only need to extrapolate six observations, we truncated the12-term filters, and uniformly normalized the weights to obtain the 6 × 12



matrix ∗6×12 given by

−0�023 −0�004 0�034 0�086 0�143 0�187 0�200 0�180 0�130 0�071 0�021 −0�0240�000 −0�023 −0�004 0�033 0�088 0�148 0�166 0�195 0�160 0�116 0�085 0�0170�000 0�000 −0�023 −0�006 0�035 0�093 0�146 0�180 0�166 0�155 0�168 0�0850�000 0�000 0�000 −0�024 −0�003 0�037 0�090 0�141 0�148 0�182 0�255 0�1730�000 0�000 0�000 0�000 −0�021 −0�006 0�034 0�089 0�114 0�196 0�331 0�2640�000 0�000 0�000 0�000 0�000 −0�032 −0�009 0�039 0�075 0�200 0�386 0�342

Hence, the asymmetric filters for the last six data points thatapproximate the nonlinear Dagum filter results from the convolution of:(1) the asymmetric weights of an ARIMA(0, 1, 1) model with � = 0�40, (2)the weights of M7(0.14) and M5(0.25) filters repeatedly used for noisesuppression, and (3) the weights of the symmetric Henderson 13. In orderto further reduce the noise the symmetric filter is applied to the seriesextrapolated by A∗ that is

y = SA∗y

where S is the N × (N + 6) matrix given by

H[H + M7(0�14)(I − H)][H + M5(0�25)(IN − H)]The convolution SA∗ produces 12-term asymmetric filters for the last sixobservations, that we truncate and uniformly normalize in order to obtainthe following final asymmetric linear filters (ALF) for the last observations

−0�026 −0�007 0�030 0�065 0�132 0�183 0�219 0�183 0�132 0�065 0�030 −0�0060�000 −0�026 −0�007 0�030 0�064 0�131 0�182 0�218 0�183 0�132 0�065 0�0310�000 0�000 −0�025 −0�004 0�034 0�069 0�137 0�187 0�222 0�185 0�131 0�0640�000 0�000 0�000 −0�020 −0�005 0�046 0�083 0�149 0�196 0�226 0�184 0�1300�000 0�000 0�000 0�000 0�001 0�033 0�075 0�108 0�167 0�205 0�229 0�1820�000 0�000 0�000 0�000 0�000 0�045 0�076 0�114 0�134 0�182 0�218 0�230

Hence, these asymmetric filters for the last six data points results fromthe convolution of: (1) the asymmetric weights of an ARIMA(0, 1, 1) modelwith � = 0�4, (2) the weights of M7(0�14) and M5(0�25) filters repeatedly usedfor noise suppression, and (3) weights of the final linear symmetric filterSLF.

3.2.1. Theoretical Properties of ALFThe convergence pattern of the asymmetric filters corresponding to

H13 and ALF are shown in Figures 5 and 6, respectively. It is evident thatthe ALF asymmetric filters are very close to one another, and convergefaster to their symmetric one relative to H13. The distance of eachasymmetric filter with respect to the symmetric one gives an indication of



FIGURE 5 Gain functions convergence pattern of H13 asymmetric weights.

the size of the revisions due to filtering changes, when new observationsare added to the series. Figure 7 shows that the gain function of thelast point ALF filter does not amplify the signal as H13 and suppressessignificantly the power at the frequency � < 0�10.

From the viewpoint of the gain function, the ALF is superior to H13concerning the unwanted ripples problem as well as faster convergence tothe symmetric one which implies smaller revisions.

FIGURE 6 Gain functions convergence pattern of the ALF symmetric weights to thesymmetric SLF.



FIGURE 7 Gain functions of the last point filter of H13 and ALF.

On the other hand, the phaseshift of ALF last point filter is muchgreater (near two months at very low frequencies) relative to H13 asexhibited in Figure 8. For the remaining asymmetric filters, the differencesare much smaller (not shown here for space reasons). Nevertheless, theimpact of the phaseshift in a given input cannot be studied in isolation ofthe corresponding gain function. It is well known that a small phaseshiftassociated with frequency gain amplifications may produce as poor resultsas a much larger phaseshift without frequency gain amplifications.

FIGURE 8 Phaseshifts of the last point filter of H13 and ALF.



TABLE 2 Mean values of revisions of last and previous tothe last point asymmetric filters of ALF and H13 appliedto a large sample of real time series

Mean absolute size of revisions ALF H13

|Ds,l | 0.87 1|Dl ,p | 0.60 1Mean square size of revisionsD2s,l 0.42 1

D2l ,p 0.11 1

3.2.2. Empirical Properties of ALFWe evaluated empirically the performance of last and previous to

the last point asymmetric filters applied to a sample of 55 time seriescharacterized by different degrees of variability. The purpose is to reducethe size of the revision of ALF respect to the asymmetric H13 filter. Hence,we compare the absolute mean revisions between the estimates of thelast and previous to the last observations obtained with the 7- and 8-termasymmetric filters, respectively, with the estimates of the same observationsobtained with the symmetric filters when six more observations are addedto the series. We denote with |Ds,l | the mean absolute revision betweenthe final estimates obtained with the symmetric filter, ysk , and the lastpoint estimate obtained with the asymmetric filter for the last point, ylkcalculated over the whole sample. We denote with |Dl ,p | the mean absoluteerror between the previous ypk and the last point ylk . We also calculatethe corresponding mean square errors. The results are shown in Table 2standardized by H13.

FIGURE 9 AWM: original series and H13 last point filter estimate.



FIGURE 10 AWM: original series and ALF last point filter estimate.

It is evident that the size of revisions for the most recent estimates issmaller for ALF relative to H13. This indicates a faster convergence to thesymmetric filter, which was also shown theoretically by means of spectralanalysis. Similarly, the distance between the last and previous to the lastfilters is smaller for ALF relative to H13.

Although ALF was applied successfully to a large number of economicindicators. We do not show all cases due to space reason but the results areavailable to the reader on request. For illustrative purposes we discuss theresults for the Average Workweek in Manufacturing (AWM) series whichsupports well the observations made regarding the joint effect of phaseshiftand gain function values. This is a monthly seasonally adjusted index seriescharacterized by a medium signal-to-noise ratio. Figures 9 and 10 showthat the ALF has a one month constant phaseshift all along the series,

FIGURE 11 AWM: H13 and ALF last point filter estimates.



whereas H13 only produces one month phaseshift at turning points (see,e.g., February 1991). H13 does not introduce phaseshift in those parts ofthe series of steady increases or decreases (e.g., from May 1982 to January1983). Figure 11 exhibits better both phaseshift patterns as well as the factthat the output from ALF is much smoother relative to H13.

4. CONCLUDING REMARKS

The main purpose of this study was to provide a linear approximationof the nonlinear Dagum filter (Dagum, 1996) that has the good propertyof reducing the size of the revisions and the number of unwanted ripplesrelative to the classical 13-term Henderson trend-cycle filter (Henderson,1916).

The new filter was derived via cascading and called CLF. It wasrepresented by the following matrix S ∗, to be applied to an N -dimensionalvector of seasonally adjusted values y

S∗ =

ALFi6×12

O(6×N−12)

SLF(N−12×N )

O(6×N−12)

ALFf(6×12)

where ALFf denotes the 6 × 12 submatrix whose rows are the asymmetricweights for the last six values, SLF is the N − 12 × N band matrixwhose non null row elements are the symmetric weights for the centralobservations and ALFi is the 6 × 12 matrix whose rows are the asymmetricweights for estimating the first six values. The latter was obtained byapplying to ALFf the t transformation defined in Dagum and Luati(2004a). The matrices O are null and their dimensions are intoparentheses. Hence

y = S∗y�

The new filter approximated the strict noise suppression of the Dagumestimator by means of a 5-term weighted, and simple 7-term movingaverages, applied sequentially. The symmetric convolution generated31 weights, being the first and last nine very close to zero. Hence,the filter was truncated to 13 terms and normalized according to amixed distribution. Applied to real and simulated time series, the linearsymmetric estimator reduced by 20% the number of unwanted ripplesproduced by H13, gave an excellent fit and did not oversmooth the data.



The asymmetric part of the NLDF was obtained from an ARIMA(0, 1, 1)model with fixed values of the parameter �. For 0�2 ≤ � ≤ 0�6 theasymmetric linear filters gave better results than H13 concerning the sizeof the revision of the most recent estimates. In particular, for � = 0�4,we obtained an extrapolation filter that combined with the symmetricone gave a set of asymmetric filters which converged rapidly to thecorresponding central one. Furthermore, these asymmetric filters did notamplify the power spectrum at the frequencies associated with the trend-cycle while suppressed most of the power corresponding to the noise.Applied to a large sample of real time series, the new filter reducedsignificantly the mean absolute and mean square revisions of the last andprevious to the last estimates relative to H13.

ACKNOWLEDGMENTS

We would like to thank two anonymous referees for their constructivecomments on earlier versions of this article. We are also indebted toBill Cleveland, David Findley, Siem Koopman, and Benoit Quenneville,for many stimulating discussions and valuable comments on the topic.Financial support from MIUR-Cofin 2004 is gratefully acknowledged.

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