A Cascade Linear Filter to Reduce Revisions and False Turning Points for Real Time Trend-Cycle Estimation by
Estela Bee Dagum and Alessandra Luati Department of Statistics, University of Bologna, Italy via Belle Arti, 41, 40126 Bologna
[email protected]
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[email protected]
Abstract The problem of identifying the direction of the short-term trend (non stationary mean) of seasonally adjusted series contaminated by high levels of variability has become of relevant interest in recent years. In fact, major …nancial and economic changes of global character have introduced a large amount of noise in time series data, particularly, in socioeconomic indicators used for real time economic analysis. The traditional approach has been that of taking di¤erences (usually of …rst order) of seasonally adjusted series but due to the presence of high volatility, this approach is no longer su¢ cient to provide an indication of the short-term trend, and, in particular of the upcoming of a turning point. The aim of this study is to construct a cascade linear …lter via the convolution of several noise suppression, trend estimation and extrapolation linear …lters. The cascading approach approximates the steps followed by the non linear Dagum (1996) trend-cycle estimator, a modi…ed version of the 13-term Henderson …lter. The former consists of …rst extending the seasonally adjusted series with ARIM A extrapolations, and then applying a very strict replacement of extreme values. The nonlinear Dagum …lter has been shown to improve signi…cantly the size of revisions and number of false turning points with respect to H13. We construct a linear approximation of the nonlinear …lter because it o¤ers several advantages. For one, its application is direct and hence, does not require some knowledge on ARIM A model identi…cation. Furthermore, linear …ltering preserves the crucial additive constraint by which the trend of an aggregated variable should be equal to the algebraic addition of its component trends, thus avoiding the selection problem of direct versus indirect adjustments. Finally, the properties of a linear …lter concerning signal passing and noise suppression can always be compared to those of other linear …lters by means of spectral analysis. Keywords: Symmetric linear …lter, smoothing, false turning points, gain function, 13-term Henderson …lter.
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1
Introduction
Major …nancial and economic changes of global character have introduced high levels of variability in time series data, particularly, in socioeconomic indicators often used for the analysis of current economic conditions. Traditionally, these type of indicators are seasonally adjusted to determine the direction of the short-term trend (that of the current year) for an early detection of a turning point. However, current high levels of variability have created the need for further smoothing in order to suppress part of the noise without a¤ecting the underlying trend-cycle component. In a recent study, Dagum (1996) developed a non linear smoother to improve on the classical 13-term Henderson …lter (1916) which is widely applied for short-term trendcycle estimation. The non linear Dagum …lter (NLDF) results from applying the 13-term symmetric Henderson …lter (H13) to seasonally adjusted series where outliers and extreme observations have been replaced and which have been extended with extrapolations from an ARIM A model (Box and Jenkins, 1970). The main purpose of the ARIM A extrapolations is to reduce the size of the revisions of the most recent estimates whereas that of extreme values replacement is to reduce the number of unwanted ripples produced by H13. An unwanted ripple is a 10-month cycle (identi…ed by the presence of high power at ! = 0:10 in the frequency domain) which, due to its periodicity, often leads to the wrong identi…cation of a true turning point. In fact, it falls in the neighborhood between the fundamental seasonal frequency and its …rst harmonic. On the other hand, a high frequency cycle is generally assumed to be part of the noise pertaining to the frequency band 0:10
! < 0:50. The problem of the unwanted ripples is speci…c of H13 when
applied to seasonally adjusted series. It di¤ers from that of spurious cycles such as those addressed by Cogley and Nason (1995) and Harvey and Jaeger (1993) mainly referring to the Hodrick-Prescott (HP) …lter (1997) in the context of long-term trend estimation. In the latter case, the basic assumption is that a time series can be decomposed into the sum of a long-term trend plus a cycle component being the noise incorporated into the cycle. Applied to monthly data, HP can generate cycles where they are not present, often leaving too much noise in the cyclical component. This problem can be overcome either by applying HP to a seasonally adjusted series where the noise has been suppressed, as suggested by Gomez (2001) or by assuming smooth stochastic cycles, as proposed by Harvey and Trimbur (2003). In this paper we do not deal with long term …lters (see, e.g. Proietti, 2005) for detrending seasonally adjusted series but with those that can estimate jointly trend and cycle ‡uctuations. The main reason for this is that we are concerned with smoothing seasonally adjusted data in the context of rather short series (less than 15 year long) for which their long term trend is often di¢ cult to identify and estimate 3
accurately. Studies by Dagum, Chhab and Morry (1996), Chhab, Morry and Dagum (1999) and Darnè (2002) showed the superior performance of the NLDF respect to both structural and ARIM A standard parametric trend-cycle models applied to series with di¤erent degrees of signal-to-noise ratios. The criteria evaluated were those of Dagum (1996): (1) number of unwanted ripples, (2) size of revisions and (3) time delay to detect a turning point. In another study, the good performance of the NLDF is shown relative to non parametric smoothers, 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 three above criteria, the aim of this paper is to construct a cascade linear …lter that closely approximates it. The cascading is done via the convolution of several noise suppression, trend estimation and extrapolation linear …lters. The symmetric …lter is the one applied to all central observations i.e., to a series without the …rst and last six data points. In this case, our purpose is to o¤er a linear solution to the unwanted ripples problem. To avoid the latter, the NLDF largely suppresses the noise in the frequency band between the fundamental seasonal and …rst harmonic. On this regard, we derive a cascade linear …lter by double smoothing the residuals obtained from a sequential application of H13 to the input data. The residuals smoothing is done by the convolution of two short smoothers, a weighted 5-term and a simple 7-term linear …lters. The linear approximation for the symmetric part of the NLDF is truncated to 13 terms with weights normalized to add to one. The asymmetric …lter is applied to the last six data points, which are crucial for current analysis. It is obtained by the convolution of the symmetric …lter with linear extrapolation …lters for the last six data points. The extrapolations are made linear by …xing the ARIM A model and its parameters values. The latter are chosen such as to minimize the size of revisions and phaseshifts. The model is selected among some parsimonious processes found to …t and extrapolate well a large number of seasonally adjusted series. Such a model turns out to be the ARIM A(0; 1; 1) with
= 0:4. A simple
linear transformation (Dagum and Luati, 2004a) allows to apply the asymmetric …lter to the …rst six observations. We will call the new …lter cascade linear …lter (CLF) and we will distinguish between the symmetric (SLF) and the asymmetric linear …lter (ALF). A linear …lter o¤ers many advantages over a non linear one. For one, its application is direct and hence, does not required knowledge of ARIM A model identi…cation. Furthermore, linear …ltering preserves the crucial additive constraint by which the trend of
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an aggregated variable should be equal to the algebraic addition of its component trends, thus avoiding the selection problem of direct versus indirect adjustments. Finally, the properties of a linear …lter concerning signal passing and noise suppression can always be compared to those of other linear …lters by means of spectral analysis. We study the properties of the new CLF relative to H13 shown to be an optimal trendcycle predictor among a variety of second and higher order kernels restricted to be of the same length (see Dagum and Luati, 2002). The theoretical properties are analyzed by means of spectral analysis. The empirical properties of the new estimator are evaluated on a large sample of real time series pertaining to various socio-economic areas and with di¤erent degrees of variability. It should be noted that the theoretical properties of CLF cannot be compared with those of NLDF since the latter is data dependent. The outline of the paper is as follows. Section 2 summarizes in matrix form the various steps used to obtain the non linear estimator developed by Dagum (1996). Section 3 presents the estimator that approximates the symmetric and asymmetric parts of the non linear Dagum …lter together with the truncation and normalization to obtain the …nal 13term cascade linear …lter. The theoretical properties of CLF are also studied in Section 3 and the results of an empirical analysis are reported. Concluding remarks are given in Section 4. 2
The Non Linear Dagum Filter (NLDF)
In time series analysis, it is often assumed that a time series fyt gt=1;:::;N , N < 1, 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) can be either deterministic or stochastic and ut usually follows a stationary stochastic process with zero mean and constant variance
2 u.
A common assumption is that ut is generated by a white noise process but it can also be assumed that it comes from an autoregressive moving average (ARM A) process. The signal g (t) can be estimated by a function of time gb (t) where the smoothing
parameter
determines the degree of smoothness of the estimated values. For …xed
values of the smoothing parameter, say
=
0,
any estimator gb 0 (t) becomes linear and
can be interpreted equivalently as: (a) a nonparametric estimator of g (t) and (b) a smooth estimate of the value yt , say ybt , resulting from a 2m + 1-term symmetric weighted average of neighboring observations.
In the context of smoothing monthly time series, it is useful to divide the frequency
domain
= f0
!
0:50g in two major intervals: (1)
S
= f0
!
0:06g associated
with cycles of 16 months or longer attributed to the signal (non stationary mean or trendcycle) of the series, and (2) the frequency band 5
S
= f0:06 < !
0:50g corresponding
to short cyclical ‡uctuations attributed to the noise. In this latter interval, it is of great interest to see how much of the power is not suppressed within a neighborhood 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 at both ends of a series must be estimated with asymmetric …lters, an optimal smoother should be rather short and have a gain G(!) close to one for 0 and near to zero for 0:10
!
!
0:06
0:50. In other words, these two conditions impede
the application of long …lters. Furthermore, long …lters require that a large number of ending data points is estimated with asymmetric …lters, that introduce phase shifts. It should be noted that most seasonal adjustment methods will suppress almost all the power already present in the frequency band around the fundamental seasonal frequency, i.e. 0:06 < ! < 0:10. The NLDF basically consists of: (a) extending with ARIMA extrapolations a seasonally adjusted series modi…ed by extreme values, and (b) applying H13 to the extended series where a stricter second replacement of extreme values is made. The extrapolation is performed with the purpose of reducing the size of the revisions of the most recent estimates as new observations are added to the series. On the other hand, the strict replacement of the extreme values implies a large noise suppression in the input series and has the purpose of reducing the number of unwanted ripples created by H13. To facilitate the identi…cation and …tting of simple ARIM A models, Dagum (1996) recommends at step (a), to modify the input series for the presence of extreme values using
2:5 as standard limits, where
is a …ve-year moving standard deviation of the
residuals estimated by X11ARIM A (Dagum, 1988) or X12ARIM A (Findley et al., 1998) seasonal adjustment software (currently, the NLDF is applied by means of any of these software). The modi…ed series is often modelled by a simple and very parsimonious ARIM A model such as the (0; 1; 1) which provides a good …t of the data. Concerning step (b), it is recommended to use very strict sigma limits, such as,
0:7 and
1:0 .
The NLDF can be described brie‡y in matrix notation as follows (see Dagum and Luati 2000, for a detailed description) b = H [H + W (IN +12 H)]E A [H + W0 (IN H)] y y
b is the (N + 12)-dimensional vector of smooth estimates of the N -dimensional where y input series y; H is the N
N matrix (canonically) associated to the 13-term Henderson
…lter; IK is the identity matrix of dimension K; the superscript E indicates that the
matrix is applied to the extended series with ARIM A forecast. W0 is a zero-one diagonal
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matrix, being the diagonal element wii equal to zero when the corresponding element yi of the vector y is identi…ed as an extreme value with respect to
2:5 limits, where
a 5-term moving standard deviation of the residuals; A is the (N + 12)
is
N dimensional
matrix canonically associated to an ARIM A(p; d; q)(P; D; Q)s process producing one-year of monthly extrapolations; W is a diagonal matrix with nonnull element wii such that, wii = 0 if the corresponding value yi falls out of the sigma limits the corresponding yi falls within the lower bound limits (angular coe¢ cient equal to
1:0 ; wii = 1 if
0:7 and wii decreases linearly
1) from 1 to 0 in the range from
0:7 to
1:0 .
Since the values of the matrices W0 and W corresponding to extreme values replacement, and matrix A pertaining to ARIM A extrapolations are data dependent, this …lter is non linear. 3
The Cascade Linear Filter (CLF)
To obtain a linear approximation of NLDF it is necessary to linearize (a) the ARIM A extrapolation process and (b) the replacement of extreme values. Concerning (a) we make recourse to a simple ARIM A model where the parameter values are …xed. As regards (b) we approach the replacement of extreme values as a strong noise suppression in the input, sequentially applying a 5-term weighted and a 7-term non weighted moving average. 3.1
The Symmetric Linear Filter (SLF)
The smoothing matrix associated to the symmetric linear …lter 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 average with weights (0:250; 0:250; 0:000; 0:250; 0:250), and M7(0:14) is the matrix representative of a 7-term …lter with all weights equal to 0:143. We have chosen 5- and 7-term …lters following the standard …lters length selected in Census X11 and X11/X12ARIMA software for the replacement of extreme seasonalirregular values. In these computer packages, a …rst 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 …lters are applied to the residuals from a …rst pass of the H13 …lter. These two …lters have both the good property of suppressing large amounts of power at ! = 0:10. Figure 1 shows the gain function of the …lter convolution M7(0:14) (IN H) H + M5(0:25) (IN H) :
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Figure 1. Gain function of the symmetric weights of the modi…ed irregulars …lter. It is apparent that the …lter convolution applied to a series consisting of trend plus irregulars suppresses all the trend power and a great deal of the irregular variations. Hence given the input series, the results from the convolution are the modi…ed irregulars needed to produce a new series which will be extended with ARIMA extrapolations, and then smoothed by H13. Eq. (1) produces a symmetric …lter of 31 terms with very small weights at both ends. This long …lter is truncated to 13 terms, and normalized such that its weights add up to unity. Normalization is needed to avoid a biased mean output. To normalize the …lter, the total weight discrepancy (in our case the 13 truncated weights add up to 1.065) must be distributed over the 13 weights, wj , j =
6; : : : ; 6,
according to a well de…ned pattern. This is a very critical adjustment for di¤erent distributions produce linear …lters with very distinctive properties. The two most commonly applied normalization procedures are the uniform and the proportional. In a uniform distribution, the total weight discrepancy is simply divided by the number of weights and hence, a constant amount is added to each of them. The 13-term truncated symmetric …lter with uniform distribution (only six plus the central in bold are 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 one divided by their sum. In this way, the discrepancy is proportionally assigned to the weights, and the corresponding 13-term truncated symmetric …lter is -0.019, -0.002, 0.034, 0.084, 0.135, 0.174, 0.188.
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The gain functions of these two …lters, shown in Figure 2, clearly indicate that none of them satis…es simultaneously the two conditions we are looking for, namely, good …t to the data (as H13), and signi…cant reduction in the number of unwanted ripples.
Figure 2. Gain functions of cascade …lters with two classical normalization procedures and H13.
It is known in the literature (see e.g. Hastie and Tibshirani, 1990) that large values of w0 (which corresponds to the mid data point in a given interval) will tend to reduce bias or oversmoothing of the signal, and thus the …lter will give a better …t. Furthermore, it is also well known that only …lters with negative weights such that m m m P P P wj = 1; jwj = 0; j 2 wj = 0 can satisfy the condition of reproducing j= m
j= m
j= m
locally second and third degree polynomials as H13. Therefore to get a linear …lter that provides a good trade-o¤ between bias and smoothing, we performed a mixed distribution of the small total weight discrepancy (-0.065) as follows -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 of w3 and w
3
have been
reduced for it is important to maintain as much as possible the area under the positive weights without modifying the negative ones. The presence of the latter is a necessary but not su¢ cient condition for a …lter to be unbiased respect to a second degree polynomial trend, which is needed to estimate properly points of maxima and minima. The weights of the symmetric linear …lter with mixed distribution normalization (SLF) are given by -0.027, -0.007, 0.031, 0.067, 0.136, 0.188, 0.224. 9
The …lters resulting from the three normalizations are second-order kernel estimators for they satisfy the following conditions m P
j= m
wj = 1;
m P
jwj = 0;
j= m
However, the smallest departure from
m P
j= m
m P
j 2 wj 6= 0.
j 2 wj = 0 is given by the one with mixed
j= m
normalized weights.
In general, second-order kernels have been found to produce smoother estimates relative to those given by higher-order kernels. This is often re‡ected by high values of the mean square errors of the estimates. An empirical analysis over 120 real and simulated time series characterized by di¤erent degrees of variability con…rmed that among the three normalizations, the mixed discrepancy distribution gives the best overall results. It produced the best …tting (root mean square error close to that of H13), did not oversmooth (sum of squared third di¤erences slightly smaller than H13) and showed a large suppression of the number of unwanted ripples (20% less than H13). On the contrary, the other two normalizations reduce more the unwanted ripples but at a great cost of oversmoothing and poor …tting (Dagum and Luati 2004b). 3.1.1
Theoretical Properties of SLF From a theoretical viewpoint, the properties
of SLF are studied by means of classical spectral analysis techniques.Figure 3 exhibits the gain functions of the symmetric H13 and SLF estimators.
Figure 3. Gain functions of symmetric H13 and SLF. Compared to H13, SLF suppresses more signal only in the frequency band pertaining to very short cycles, ranging from 15 to 24 months periodicity (0:03 < ! 10
0:06), whereas
it passes without modi…cation cycle of 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 estimators restricted to 13 terms, such as the locally weighted regression smoother of degree one (L1) and the Gaussian kernel (GK), SLF appears to be a better signal predictor with a smoothing power at ! = 0:10, close to GK, as Figure 4 shows.
Figure 4. Gain functions of second-order kernels and SLF. 3.1.2
Empirical Properties of SLF
To perform an empirical evaluation of the
…tting and smoothing performances of SLF relative to H13, we apply both to a large sample of 100 seasonally adjusted real time series pertaining to various socio-economic areas, characterized by di¤erent degrees of variability. As a measure of …tting we use the root mean square error (RM SE) calculated by v u N u 1 X5 RM SE = t N 12 t=7
ybt
2
yt yt
where yt denotes the original observations and ybt the predicted values. Our aim is to assess the extent to which the …tting properties of the SLF are equivalent to those of H13, known to be a good signal predictor but at the expense of producing many unwanted ripples. We are also interested in reducing the number of unwanted ripples that may lead to the detection of false turning points. On this regards, we use the accepted de…nition of a turning point for smoothed data (see, among others, Zellner et al., 1991) according to which a turning points occurs at time t if (downturn) yt 11
k
yt
1
> yt
yt+1
yt+m or (upturn) yt
yt
k
1
< yt
yt+1
yt+m for k = 3 and m = 1.
An unwanted ripple arises whenever two turning points occur within a 10 month period. Table 1 shows the mean values of the RM SE calculated over the 100 series and standardized respect to H13 to facilitate the comparison. In the same way, the number of false turning points produced in the …nal estimates from SLF is given relative to that of H13. Empirical measures of …tting and smoothing
SLF
H13
RM SE=RM SEH13
1.05
1
f tp=f tpH13
0.80
1
Table 1. Empirical RM SE and number of false turning points for SLF and H13 …lters applied to real time series (mean values standardized by those of H13). The empirical results are consistent with those inferred from the theoretical analysis. Furthermore, SLF reduces by 20% the number of false turning points produced by H13. 3.2
The Asymmetric Linear Filter (ALF)
The smoothing matrix associated to the asymmetric linear …lter for the last six data points is obtained in two steps: (1) a linear extrapolation …lter for six data points is applied to the input series. This …lter is represented by a (N + 6)
where points.
N matrix A 2
6 A =6 4
6 12 6 12
3
IN
O6
N 12
6 12
7 7 5
is the submatrix containing the weights for the N
5; N
4; :::; N data
A H + M5(0:25) (IN H)
(2)
results from the convolution H H + M7(0:14) (IN +12 H)
where H + M5(0:25) (IN H) is the N
E
N matrix representative of trend-…lter plus a …rst
suppression of extreme values, H + M7(0:14) (IN +12 H)
E
is the N
N + 12 matrix for
the second suppression of the irregulars applied to the input series plus 12 extrapolated values, generated by
2
6 A=6 4
IN 12 N
12
3
7 7. 5
This N + 12
N matrix A is associated to an ARIM A(0; 1; 1) linear extrapolations …lter
with parameter value
= 0:4.
It is well known that for all ARIM A models that admit a convergent AR(1) representation, such that 1
1B
2B
2
::: yt = at , the
j ’s
can be explicitly calculated.
For any lead time , the extrapolated values yt ( ) may be expressed as a linear function of current and past observations yt with weights
( ) j
that decrease rapidly as they depart
from the current observations (Box and Jenkins, 1976). That is yt ( ) =
1 X
( ) j yt j+1
j=1
where ( ) j
=
j+
1
+
X1
( h j
h)
h=1
for j = 1; 2; ::: and
(1) j
=
j:
Strictly speaking, the yt ’s go back to in…nite past, but
since the power series is convergent, their dependence on yt time has elapsed. For this reason, we will consider
( ) j
j
can be ignored after some
= 0 for j > 12. Furthermore, to
generate one year of ARIM A extrapolation we …x r = 12. Parsimonious models found to …t and extrapolate well a large number of seasonally adjusted series are the ARIM A (0; 1; 1), ARIM A (0; 2; 1), ARIM A (0; 2; 2). We used a …xed range of values for
i
= 0:1; 0:3; : : : ; 0:9 for i = 1; 2. Among the selected models with
di¤erent parameter combinations, we found that only the linear asymmetric …lters of the ARIM A (0; 1; 1) models with 0:2
0:6 gave good results in terms of signal passed
and noise suppressed. We selected
= 0:4 as the preferred one. For an ARIM A (0; 1; 1)
model, the coe¢ cients are Hence,
12 N
j
= (1
)
j 1
, and it can be shown that
( ) j
=
j,
8 2 N.
is the 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 the 12-term …lters and uniformly normalized the weights to obtain the 6
12 matrix
6 12
given by
(2) The symmetric …lter is applied to the series extrapolated by A that is b = SA y y 13
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 …lters for the last six observations, that we truncate and uniformly normalize in order to obtain the following …nal asymmetric linear …lters (ALF) for the last observations
. Hence, the asymmetric …lters for the last six data points results from the convolution of: (1) the asymmetric weights of an ARIM A(0; 1; 1) model with
= 0:4, (2) the weights of
M7(0:14) and M5(0:25) …lters repeatedly used for noise suppression, and (3) weights of the …nal linear symmetric …lter SLF. 3.2.1
Theoretical Properties of ALF The convergence pattern of the asymmetric
…lters corresponding to H13 and ALF are shown in Figures 5 and 6, respectively. It is evident that the ALF asymmetric …lters are very close one another, and converge faster to their symmetric one relative to H13. The distance of each asymmetric …lter with respect to the symmetric one gives an indication of the size of the revisions due to …ltering changes, when new observations are added to the series. Figure 7 shows that the gain function of the last point ALF …lter does not amplify the signal as H13 and suppresses signi…cantly the power at the frequency ! < 0:10: From the viewpoint of the gain function, the ALF is superior to H13 concerning the unwanted ripples problem as well as faster convergence to the symmetric one which implies smaller revisions.
14
Figure 5. Gain functions convergence pattern of H13 symmetric weights to the symmetric.
Figure 6. Gain functions convergence pattern of the ALF symmetric weights to the symmetric SLF.
15
Figure 7. Gain functions of the last point …lter of H13 and ALF.
On the other hand, the phaseshift of ALF last point …lter is much greater (near two months at very low frequencies) relative to H13 as exhibited in Figure 8. For the remaining asymmetric …lters, the di¤erences are much smaller (not shown here for space reasons). Nevertheless, the impact of the phaseshift in a given input cannot be studied in isolation of the corresponding gain function. It is well known that a small phaseshift associated with frequency gain ampli…cations may produce as poor results as a much larger phaseshift without frequency gain ampli…cations.
Figure 8. Phaseshifts of the last point …lter of H13 and ALF.
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3.2.2
Empirical Properties of ALF
We evaluated empirically the performance of
last and previous to the last point asymmetric …lters applied to a sample of 55 time series characterized by di¤erent degrees of variability. The purpose is to reduce the size of the revision of ALF respect to the asymmetric H13 …lter. Hence, we compare the absolute mean revisions between the estimates of the last and previous to the last observations obtained with the 7- and 8-term asymmetric …lters, respectively, with the estimates of the same observations obtained with the symmetric …lters when six more observations are added to the series. We denote with jDs;l j the mean absolute revision between the …nal estimates obtained with the symmetric …lter, ybks , and the last point estimate obtained
with the asymmetric …lter for the last point, ybkl calculated over the whole sample. We
denote with jDs;p j the mean absolute error between the previous ybkp and the last point ybkl .We also calculate the corresponding mean square errors. The results are shown in Table 2 standardized by H13.
Absolute size of revisions
ALF
H13
jDs;l j
0.87
1
2 Ds;l
0.42
1
Mean squared size of revisions
1
jDs;p j
0.60
1
2 Ds;p
0.11
1
Table 2. Mean values of revisions of last and previous to the last point asymmetric …lters of ALF and H13 applied to a large sample of real time series.
It is evident that the size of revisions for the most recent estimates is smaller for ALF relative to H13. This indicates a faster convergence to the symmetric …lter, which was also shown theoretically by means of spectral analysis. Similarly, the distance between the last and previous to the last …lters is smaller for ALF relative to H13. Although ALF was applied successfully to a large number of economic indicators. We do not show all cases due to space reason but the results are available to the reader on request. For illustrative purposes we discuss the results for the Average Workweek in Manufacturing (AWM) series which supports well the observations made regarding the joint e¤ect of phaseshift and gain function values. This is a monthly seasonally adjusted index series characterized by a medium signal-to-noise ratio. Figures 9 and 10 show that the ALF has a one month constant phaseshift all along the series, whereas H13 only 17
produces one month phaseshift at turning points (see e.g. February 1991). H13 does not introduce phaseshift in those part of the series of steady increases or decreases (e.g. from May 1982 to January 1983). Figure 11 exhibits better both phaseshift patterns as well as the fact that the output from ALF is much smoother relative to H13.
Figure 9. AWM: original series and H13 last point …lter estimate.
Figure 10. AWM: original series and ALF last point …lter estimate.
Figure 11. AWM: H13 and ALF last point …lter estimates.
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4
Concluding Remarks
The main purpose of this study was to provide a linear approximation of the non linear Dagum …lter (1996) that has the good property of reducing the size of the revisions and the number of unwanted ripples relative to the classical 13-term Henderson trend-cycle …lter (1916). The new …lter was derived via cascading and called CLF. It was represented by the following matrix S , to be applied to an N -dimensional vector of seasonally adjusted values y
2
6 6 6 6 6 6 6 6 6 S =6 6 6 6 6 6 6 6 6 4
where ALFf denotes the 6
ALFi
O
6 12
(6 N 12)
SLF (N 12 N )
O
ALFf
(6 N 12)
(6 12)
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
12 submatrix whose rows are the asymmetric weights for the
last six values, SLF is the N
12
N band matrix whose non null row elements are the
symmetric weights for the central observations and ALFi is the 6
12 matrix whose rows
are the asymmetric weights for estimating the …rst six values. The latter was obtained by applying to ALFf the t transformation de…ned in Dagum and Luati (2004a). The matrices O are null and their dimensions are into parentheses. Hence b = S y. y
The new …lter approximated the strict noise suppression of the Dagum estimator by means of a 5-term weighted, and simple 7-term moving averages, applied sequentially. The symmetric convolution generated 31 weights, being the …rst and last nine very close to zero. Hence, the …lter was truncated to 13 terms and normalized according to a mixed distribution. Applied to real and simulated time series, the linear symmetric estimator reduced by 20% the number of unwanted ripples produced by H13, gave an excellent …t and did not oversmooth the data. The asymmetric part of the NLDF was obtained from an ARIM A(0; 1; 1) model with …xed values of the parameter . For 0:2
0:6 the asymmetric linear …lters gave
better results than H13 concerning the size of the revision of the most recent estimates. In
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particular, for
= 0:4, we obtained an extrapolation …lter that combined with the sym-
metric one gave a set of asymmetric …lters which converged rapidly to the corresponding central one. Furthermore, these asymmetric …lters did not amplify 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 …lter reduced signi…cantly the mean absolute and mean square revisions of the last and previous to the last estimates relative to H13.
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