1147
Estimation of Frequency in SCLM Models
Miguel Artiach and Josu Arteche
COMPSTAT 2006 section: Time Series Analysis
Abstract: Seasonal and Cyclical Long Memory Models are defined by two
parameters: the memory parameter or exponent d and the frequency of the
pole ω. This paper analyzes the applicability of an iterative procedure of
estimation of the frequency ω, more commonly used in the context of
Fixed Frequency Effects Models. A recursive algorithm is developed and
its performance is compared to the more usual method of estimation restricted to the Fourier or harmonic frequencies.
Key words: Persistent cycle, frequency domain, hidden frequency
1. Introduction
The search for cyclical components in time series models is of undoubted interest in many scientific fields, such as in Economics [11] or
Engineering [7, 10]. The cyclical behaviour of the series can be deterministic, with a systematic and strictly periodic evolution that repeats every
period, or stochastic with a cyclical behaviour that is not strictly periodic
but evolves slightly over time. Both of them show a peak or pole in the
spectral density function (pseudo-spectral density function in the nonstationary case) at some frequency ω such that the period of the cycle is 2π/ω.
The deterministic cycles are mainly based on the Cyclical Regression or
Fixed Frequency Effects model defined as
Xt = μ + α·cos(ωt) + β·sin(ωt) + εt
(1)
where α and β are two zero mean uncorrelated random variables with the
same variance and εt is a stationary sequence of random variables inde-
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pendent of α and β. Once the initial conditions have been fixed and for any
α and β, this model exhibits a periodic behaviour that remains constant
over time. In this case, the cyclical component can be more or less clear
depending on the sizes of the variance of α and β relative to that of εt. This
model only shows a cycle at frequency ω but can be easily extended by
simple addition of other cycles.
For the stochastic cycles, it is well known that the autoregressive AR(2)
process
(1 - φ1L - φ2L2) Xt = εt
(2)
for εt ∼ iid(0,σ2) and L the lag-shift operator, displays a quasi-periodic behaviour when the roots of the polynomial (1-φ1y-φ2y2) are complex, which
implies φ2<-(φ1)2/4. However, contrary to the fixed frequency effects
model, the cyclical pattern of this process fades out over time. This difference turns up clearly in the frequency domain. Whereas the fixed frequency effects model possesses a spectral distribution function with a
jump at frequency ω, the autoregressive AR(2) process shows a continuous
spectral density function with a peak at frequency ω=cos-1[- φ1(1-φ2)/4φ2].
Between these two extreme categories there is scope for a class of intermediate models whose periodic behaviour is more persistent than that of
the AR(2) processes but at the same time does not remain constant over
time. Here the cyclical behaviour of the model exhibits a certain degree of
memory or persistence, but this memory does not imply a constant recurrence of the cycle but evolves slightly over time. These are the SCLM
(Seasonal and Cyclical Long Memory) models in the terminology of Arteche & Robinson [3], which are characterized by a spectral or pseudospectral (in the nonstationary case) density function satisfying
f x (λ + ω) ≈ C λ
(3)
−2 d
as λ → 0, around some frequency ω∈(0,π] and for C a finite constant. If
d>0 the spectral density function diverges at ω showing a persistent cyclical behaviour. (If d<0 it would display a zero with an antipersistent behaviour). The main example of parametric SCLM models are the GARMA
(Generalized AutoRegressive Mean Average) models of Gray et al. [8].
The simplest form of the model is the GARMA (0,d,0) defined as
(1 − 2L cos ω + L ) X
2 d
t
= εt
(4)
where εt is iid(0,σ2). It can be generalized to GARMA(p,d,q) by simply allowing εt to be an ARMA(p,q) process. These processes are stationary for
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d<0.5 and mean reverting for d<1. The case d=1 implies a cyclical unit
root. Its spectral or pseudospectral density function is
f x (λ) = [2(cos ω − cos λ )]
−2d
f ε (λ )
(5)
which behaves around ω as in Eq. (3). These models assume a symmetric
behaviour of the spectral density function around ω, but although this happens for ω=0,π, it is not necessarily true at any other frequency 0<ω<π as
noted by Arteche and Robinson [4] and Arteche [2], permitting an asymmetric long memory behaviour with different memory parameters on the
right and left of ω.
The recent interest in these models has focused mainly on the estimation
of the memory parameter d in the case of a known ω. This situation is
common in seasonal series, but with any other type of cyclical behaviour
the frequency ω of the cycle needs to be estimated. This has received recent attention by authors like Yajima [12] or Hidalgo and Soulier [9].
Yajima [12] considers that the best estimator of ω is the argument that
maximizes the periodogram. The periodogram is the sampling correlative
of the density function and can be defined as
I(λ ) = (2πT )
−1
2
T
∑
Xte
iλ t
(6)
t =1
where Xt is the time series and T the sample size. He proves Tαconsistency under gaussianity for α ∈ (0,1).
Hidalgo and Soulier [9] focuses on the maximizer of the periodogram at
Fourier frequencies λj = 2πj/T such that the estimator of ω is
ˆT =
ω
2π
arg max I T λ j
T 1≤ j≤ ñ
( )
(7)
where j = 1, ..., ñ, and ñ = [(T-1)/2]. They prove that this estimator is TvT-1
–consistent for vT satisfying
lim v T−2ν log(T ) = 0 with ν ∈ (0,1/2)
T →∞
However, they restrict the candidate values of ω̂ T to the Fourier frequencies. Considering that the spectral density function of SCLM processes is continuous and displays a pole around ω, the Fourier frequency
that maximizes the periodogram can be relatively far from the real ω, especially if the sample size is small, leading to erroneous conclusions about
the cyclical nature of the data. In this paper we consider an iterative proce-
1150
dure that does not restrict to the Fourier frequencies but allows a wider
range of possibilities for ω̂ T . The procedure is described in Section 2. Section 3 analyzes the finite sample performance of the procedures via Monte
Carlo and finally Section 4 shows an empirical application to the wellknown sunspots data.
2. Iterative Estimation of Frequency
The estimation procedure we propose is based on Ebner et al. [7]. It
starts by calculating the periodogram just for the Fourier frequencies, with
low computational cost, and then widens the range of frequencies in an iterative way. The procedure entails the following steps:
1. Initial estimation: Let ω̂(j0) be the Fourier frequency that maximizes
the periodogram.
2. Recursive Refinement:
ˆ (ji−1)
a) Select the previous and following evaluated frequencies to ω
ˆ (ji−−11) y ω
ˆ (ji+−11) .
(where i = 1, ... is the number of iteration), ω
b) Refine the range of frequencies for which the periodogram is calculated by defining λ r = 2πr T ⋅ 10i where r covers the integers
between (j - 1)×10 and (j + 1)×10. The result is that 20 frequency
ˆ (ji−−11) , ω
ˆ (ji+−11) ] where
values are evaluated in the same interval [ ω
only three had been in the former iteration.
c) Select the new frequency where the periodogram reaches its maximum. Let ω̂ (ji ) be this frequency.
ˆ (ji ) − ω
ˆ (ji −1) > δ , where δ is a predetermined stopping criterion,
d) If ω
ˆ (ji ) − ω
ˆ (ji −1) ≤ δ the algorithm stops and ω̂ (ji ) is
go back to (a). If ω
the best estimation of ω.
Iterative algorithms have been extensively used to estimate the hidden
frequency in Fixed Frequency Effects models. All these strategies are supported by the development of asymptotical properties concerning the
maximum of the periodogram, as can be seen in An et al. [1], Chen [5] or
more recently Davies and Mikosch [6]. These papers coincide in the great
stability of the maximum of the periodogram, which supports our choice as
point estimator of the ω parameter.
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3. Monte Carlo Analysis
In order to assess the applicability of these strategies to SCLM models
we generated a number of series with different sample sizes, memory parameters, and periods of the cycle. For this purpose we used the method
described by Arteche and Robinson [4], for the GARMA (0,d,0) model
based on standard normal innovations.
Overall, two different sample sizes, T=75 and T=150, were employed,
together with four memory parameters d1=0.2, d2=1/3, d3=0.45 (stationary
series) and d4=0.8 (nonstationary and mean-reverting series). Four frequencies were used for each sample size, two extreme frequencies (near 0
and π) and two other frequencies in between. They are displayed in table 1.
Table 1. Frequencies for the Monte Carlo analysis
ω175 = 2π2.5 75 = 0.2094395
ω150
= 2π4.5 150 = 0.1884956
1
ω75
2 = 2π17.5 75 = 1.4660766
ω150
2 = 2π34.5 150 = 1.4451326
ω375 = 2π27.5 75 = 2.3038346
ω150
3 = 2π55.5 150 = 2.3247786
ω75
4 = 2π36.5 75 = 3.0578168
ω150
4 = 2π73.5 150 = 3.0787608
The frequencies were chosen equidistantly between two consecutive
harmonic frequencies with the purpose of assessing the ability of the iterative procedure to estimate the real value of ω. Considering only Fourier
frequencies, an estimation error of at least ±2π0.5/T would be expected.
This provides the benchmark to show the improvement of the final estimate. A bias of the estimator smaller than this value indicates that the iterative estimate overcomes significantly its initial (Fourier) value.
1000 samples were generated in every situation resulting from the combination of the frequencies and memory parameters, summing up to a total
amount of 32000 samples.
In order to evaluate the improvement of the iterative Maximum of the
Periodogram (MP) technique over the Fourier frequency based (FF) estimate we compare the Root Mean Squared Errors (RMSE) calculated as
(8)
ˆ ) = VAR (ω
ˆ ) + BIAS(ω
ˆ)
RMSE (ω
2 1
1
ˆ)=
ˆ)= ω
ˆ − ω , VAR (ω
BIAS(ω
∑ [ωˆ − ωˆ ]
1000
2
i
i =1
1000
ˆ =
1000 and ω
∑ ωˆ
i
i =1
1000 .
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As can be seen in table 2, the MP technique gives a smaller bias in almost every situation. They also show that the technique is more appropriate for non-extreme frequencies and to some extent fails in the estimation
of frequencies near 0 or π where larger biases can occur, mainly with low
d. Another statement that can be made is that the accuracy increases with
the value of d, the memory parameter. The biases of the higher values d3
and d4 are generally smaller than those of d1.
Table 2. Biases
ω1
ω2
ω3
T=75
d1 = 0.2
0.0737369
0.0093099
-0.0055839
d2 = 1/3
-0.0011136
-0.0045737
0.0260057
d3 = 0.45
-0.0070675
0.0005373
0.0096694
d4 = 0.8
-0.0014546
0.0003797
0.0005634
In bold the cases with BIAS < 2π0.5/75 = 0.041887902
ω4
-0.0906098
-0.0077746
0.0014576
0.0018661
ω1
ω2
ω3
T=150
d1 = 0.2
-0.0234916
-0.0210249
0.0053823
d2 = 1/3
-0.0076019
0.0018983
0.0147125
d3 = 0.45
-0.0062359
-0.0000236
0.0041530
d4 = 0.8
-0.0001110
0.0000686
-0.0006492
In bold the cases with BIAS < 2π0.5/150 = 0.02094395
ω4
0.0219145
0.0032140
0.0045838
0.0009000
Table 3. Roots of Mean Square Errors
T=150
T=75
ω1
ω2
MP
0.27555
0.45497
d1 = 0.2
FF
0.27536
0.45660
MP
0.08288
0.19282
d2 = 1/3
FF
0.08588
0.19521
MP
0.05217
0.08795
d3 = 0.45
FF
0.06098
0.09485
MP
0.01264
0.00947
d4 = 0.8
FF
0.04255
0.04188
MP
0.19127
0.33152
d1 = 0.2
FF
0.19174
0.33170
MP
0.05858
0.10968
d2 = 1/3
FF
0.05985
0.11069
MP
0.03401
0.05513
d3 = 0.45
FF
0.03771
0.14883
MP
0.00826
0.00750
d4 = 0.8
FF
0.02185
0.02153
In bold the cases with RMSEMP < RMSEFF.
ω3
0.38843
0.38891
0.16150
0.16458
0.08902
0.09697
0.00963
0.04222
0.28258
0.28475
0.10021
0.10131
0.04845
0.05153
0.00836
0.02169
ω4
0.29521
0.29528
0.06721
0.07022
0.03904
0.04870
0.01364
0.04188
0.10418
0.10480
0.03543
0.03790
0.02550
0.02986
0.00759
0.02128
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Similar conclusions can be drawn from the RMSEs (table 3). As expected the MP technique is more accurate than the FF technique in almost
every situation, and the improvement is more significant as d increases.
For example, for d4=0.8, central ω and T=75 the RMSE of the MP estimates is less than one fourth the RMSE of the FF estimates.
4. Empirical Application. Iterative Estimation of the First
Frequency of the Sunspots Series.
The sunspots series is well known and has been re-analyzed by many
authors. This series comprises monthly measurements of the sunspots from
1749 to 1983, displayed in the figure 1.
250
Sunspots
200
150
100
50
0
0
200
400
600
800
1000 1200 1400 1600 1800 2000 2200 2400 2600 2800
Months
Fig. 1. Sunspots series
According to the FF technique the frequency with the largest periodogram value is ωFF1=0.04678968, which corresponds to a period
τFF1=134.2857, that is 11 years, 2 months, 8 days, 13 hours and 42 minutes.
The exact periodogram value at this frequency is I(ωFF1)=1087290.33.
Taking this frequency as the first step, we apply the iterative procedure
of estimation with a result of ωMP1=0.0472674, or τMP1=132.9285, a period
of 11 years, 27 days, 20 hours and 32 minutes. A reduction of around 1.5
months takes place. The periodogram value is in this case
I(ωMP1)=1253648.68, with an increment of 166358.35 points.
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Address: M. Artiach, Departamento de Economía Aplicada III, Universidad del
País Vasco, Bilbao, Bizkaia, Spain, mmartiach@yahoo.es
J. Arteche, Departamento de Economía Aplicada III, Universidad del País Vasco,
Bilbao, Bizkaia, Spain, josu.arteche@ehu.es