A FREQUENCY DOMAIN
METHODOLOGY FOR
TIME SERIES MODELING
Hens Steehouwer
OFRC Methodological paper No. 2008-02
May 2009
A FREQUENCY DOMAIN METHODOLOGY FOR
TIME SERIES MODELING1
Hens Steehouwer2,3
Methodological Working Paper No. 2008-02
May 2009
Ortec Finance
Research Center
P.O. Box 4074, 3006 AB Rotterdam
Boompjes 40, The Netherlands, www.ortec-finance.com
ABSTRACT
In this paper we have describe a frequency domain methodology for time series modeling. With this
methodology it is possible to construct time series models that give a better description of the
empirical long term behavior of economic and financial variables, bring together the empirical behavior
of these variables at different horizons and observation frequencies and get insight in and
understanding of the corresponding dynamic behavior. We introduce the most important frequency
domain techniques and concepts, describe and illustrate the methodology and finally give the
motivation for doing so. This methodology can contribute to a higher quality of investment decision
making, implementation and monitoring in general and for Central Banks and Sovereign Wealth
Managers in particular.
1
This paper has been published as Steehouwer, H. (2010), “A Frequency Domain Methodology for Time Series Modeling”, in
Interest Rate Models, Asset Allocation and Quantitative Techniques for Central Banks and Sovereign Wealth Funds, edited by
Berkelaar, A., J. Coche and K. Nyholm, Palgrave Macmillan.
2
The author is Head of the Ortec Finance Research Center and affiliated with the Econometric Institute of the Erasmus
University Rotterdam. Please e-mail comments and questions to hens.steehouwer@ortec-finance.com
3
Copyright © 2011 Ortec Finance bv. All rights reserved. No part of this paper may be reproduced, in any form or by any
means, without permission from the authors. Shorts sections may be quoted without permission provided that full credit is given
to the source. The views expressed are those of the individual author(s) and do not necessarily reflect the views of Ortec
Finance bv.
2
1 Introduction
Determining an optimal Strategic Asset Allocation (SAA) in general, and for Central Banks and
Sovereign Wealth Managers in particular, is essentially a decision making problem under uncertainty.
How good or bad a selected SAA will perform in terms of the objectives and constraints of the
stakeholders will depend on the future evolution of economic and financial variables such as interest
rates, asset returns and inflation rates. The uncertainty about this future evolution of these variables is
traditionally modeled by means of (econometric) time series models. The book by Campbell and
Viceira (2002) is an example of this approach. They estimate Vector AutoRegressive (VAR) models on
historical time series and derive optimal investment portfolios from the statistical behavior of the asset
classes on various horizons as implied by the estimated VAR model.
It is also known that the results from (SAA) models that take the statistical behavior of asset classes
as implied by these time series models as input, can be very sensitive for the exact specifications of
this statistical behavior in terms of for example the expected returns, volatilities, correlations, dynamics
(auto- and cross-correlations) and higher order moments. Section 1.2 of Steehouwer (2005) describes
an example of this sensitivity in the context SAA decision making for a pension fund. Besides the
academic relevance, this observation also has an enormous practical impact since many financial
institutions around the world base their actual (SAA) investment decisions on the outcomes of such
models. Therefore, it is of great importance to continuously put the utmost effort in the development
and testing of better time series models to be used for SAA decision making. This paper intends to
make a contribution to such developments.
If we now turn our attention to the methodological foundations of these time series models, virtually all
model builders will agree that empirical (time series) data of economic and financial variables is (still)
the primary source of information for constructing the models. This can already be seen from the
simple fact that virtually all time series models are being estimated on historical time series data. On
top of that, of course also forward looking information can be incorporated in the models. This is
desirable if some aspects of the behavior observed in the (historical) time series data is considered to
be inappropriate for describing the possible evolution of the economic and financial variables in the
future.
Furthermore, it is known that the empirical behavior of economic and financial variables is typically
different at different horizons (centuries, decades, years, months, etc.) and different observation
frequencies (annual, monthly, weekly, etc.). A first way of seeing this is by thinking about typical and
well known economic phenomena such as long term trends, business cycles, seasonal patterns,
stochastic volatilities, etc. For example, on a 30 year horizon with an annual observation frequency,
long term trends and business cycles are important while on a 1 year horizon with a monthly
observation frequency, seasonal patterns need to be taken into account and on a 1 month horizon
with a daily observation frequency modeling stochastic volatility becomes a key issue. A second way
of seeing the relevance of the horizon and observation frequency is by thinking about the so called
‘term structure of risk and return’ as described by Campbell and Viceira (2002). This simply means
that expected returns, volatilities and correlations of and between asset classes are different at
different horizons. For example, the correlation between equity returns and inflation rates is negative
on short (for example 1 year) horizons while the same correlation is positive on long (for example 25
year) horizons.
If we now combine these observations with the described sensitivity of real world (investment)
decision making, for the statistical behavior of the time series models, we see that a first important
issue in time series modeling for (investment) decision making is how to describe the relevant
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empirical behavior as best as possible for the specific problem at hand. So if, for example, we are
modeling for the purpose of long term SAA decision making, what does empirical data tell us about the
statistical properties of long term trends and business cycles and how can we model this correctly?
A second important issue follows from the fact that the results from time series models are also used
in more steps of an investment process than ‘just to determine the SAA. Once the core SAA has been
set for the long run, other (time series) models may be used for a medium term horizon in order to
further refine the actual asset allocation (also called portfolio construction), for example by including
more specialized asset classes or working with specific views that induce timing decisions. Once an
investment portfolio is implemented, also monitoring and risk management are required, for example
to see if the portfolio continues to satisfy (ex ante) short term risk budgets. Because, as mentioned
before, the empirical behavior of economic and financial variables is different at different horizons and
observation frequencies, it is typically the case that different (parametric and non-parametric) time
series models are used in these various steps of an investment process. That is, the best model is
used for the specific problem at hand. This is fine in itself, provided that the different steps in the
investment process do not need to communicate with each other which obviously they do need to do.
The SAA is input for the tactical decision making and portfolio construction while the actual investment
portfolio is input for the monitoring and risk management process. If in these different steps different
time series models are used, portfolios that where good or even optimal in one step may no longer be
good or optimal in the next step, just by switching from one time series model to another. It is not hard
to imagine the problems that can occur because of such inconsistencies. A second important issue in
time series modeling for (investment) decision making is therefore how to bring together the empirical
behavior of economic and financial variables that is observed at different horizons and different
observation frequencies in one complete and consistent modeling approach.
In response to the two important issues described above, this paper puts forward a specific frequency
domain methodology for time series modeling. We will argue and illustrate that by using this
methodology it is possible to construct time series models that
1. Give a better description of the empirical long term behavior of economic and financial variables
with the obvious relevance for long term SAA decision making.
2. Bring together the empirical behavior of these variables as observed at different horizons and
observation frequencies which is required for constructing a consistent framework to be used in
the different steps of an investment process.
In addition, by using frequency domain techniques, the methodology supports
3. Better insight in and understanding of the dynamic behavior of economic and financial variables at
different horizons and observation frequencies, both in terms of empirical time series data and of
the time series models that are used to describe this behavior.
The methodology combines conventional (time domain) time series modeling techniques with
techniques from the frequency domain. It is fair to say that frequency domain techniques are not used
very often in economics and finance, especially when compared to the extensive use of these
techniques in the natural sciences. We explain this from the non-experimental character of the
economic and finance sciences and the therefore in general limited amount of data that is available for
analysis. We show how the corresponding problems of conventional frequency domain techniques can
be solved by using appropriate special versions of these techniques that work well in the case of
limited data.
The methodology builds on the techniques and results described in Steehouwer (2005) as well as
subsequent research. Its applications are not limited to that of (SAA) investment decisions as
described above but cover in principle all other types of applications of time series models.
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Furthermore, the methodology leaves room for and even intends to stimulate the inclusion and
combining of many different types of time series modeling techniques. By this we mean that the
methodology can accommodate and combine classical time series models, VAR models, London
School of Economics (LSE) methodology, theoretical models, structural time series models, (G)ARCH
models, Copula models, models with seasonal unit roots, historical simulation techniques, etc. in one
consistent framework.
This paper does not intend to give a full in-depth description of every aspect of the methodology.
Instead, the objective of the paper is to give a rather high-level overview of the methodology and
provide the appropriate references for further information. The remainder of the paper has the
following setup. We proceed in Section 2 with an introduction to some basic concepts from the
frequency domain together with what we feel is the fundamental reason why frequency domain
techniques do not have the widespread use within economics and finance they deserve. In Section 3
we continue with a description of the proposed methodology followed by the main points of motivation
for proposing this methodology in Section 4. This motivation consists of a combination of technical as
well as more methodological issues. This paper does not include (yet) one complete example
application of the described methodology. Instead, separate examples are given throughout the text to
illustrate individual concepts. Section 5 closes the paper by summarizing the main conclusions.
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2 Frequency Domain
The methodology for time series analysis and modeling that we propose in this paper is based on
concepts and techniques from the frequency domain, also known as spectral analysis techniques.
Frequency domain techniques are not that well known and are not applied very often in economics
and finance. Furthermore, these techniques require a rather different view on empirical time series
data and stochastic processes. Therefore, this section briefly introduces some key concepts from
frequency domain analysis such as spectral densities, frequency response functions and the leakage
effect. We will argue that the latter is the key reason why frequency domain techniques are not used
more often in economics and finance. These concepts are used in the description of the proposed
methodology in Section 3. Those interested in a historical overview of the development of frequency
domain and spectral analysis techniques are referred to Section 2.2.11 of Steehouwer (2005). Further
details, proofs and references on the concepts discussed here can be found in Chapter 4 of the same
reference. Well known classical works on spectral analysis techniques and time series analysis are
Bloomfield (1976) and Brillinger (1981).
2.1 Frequency domain versus time domain
All frequency domain techniques are built on the foundations of the Fourier transform. With the Fourier
transform any time series {xt, t=0,…,T-1} can be written as a sum of cosine functions
T −1
xt = ∑ R j cos(ω j t + φ j )
(2.1)
j =0
The parameters {Rj, ωj and φj, j = 0,…,T-1} represent the amplitudes, frequencies and phases of the T
cosine functions. The conventional representation {xt, t=0,…,T-1} of the time series is referred to as
the representation in the time domain. The representation {Rj, ωj and φj, j = 0,…,T-1} is referred to as a
representation in the frequency domain. An important property of this frequency domain
representation is
1 T −1 2 T −1 2
Rj
∑ xt = ∑
T t =0
j =0
(2.2)
So, if we assume the time series xt to have an average value of zero, then this relation tells us that the
frequency domain representation decomposes the total variance of the time series into the squared
amplitudes of the set of cosine functions. The higher the Rj for a certain ωj, the more this frequency
contributes to the total variance of the time series.
2.2 Spectral densities
A periodogram plots the variance per frequency from (2.2) as a function of the frequencies and
thereby shows the relative importance of the different frequencies for the total variance of the time
series. If one would calculate the periodogram for different samples from some stochastic time series
process, this would result in different values and shapes of the periodogram. Doing this for a great
number of samples of sufficient length and calculating the average periodogram on all these samples
results in what is called the spectral density (or auto-spectrum) of a (univariate) stochastic process. A
spectral density describes the expected distribution of the variance of the process over periodic
fluctuations with a continuous range of frequencies. The word “spectrum” comes from the analogy of
decomposing white light into colors with different wave lengths. The word “density” comes from the
analogy with a probability density function. A probability density function describes the distribution of a
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probability mass of 1 over some domain while a spectral density describes the distribution of a
variance mass over a range of frequencies. It can be shown that the spectrum and the traditional autocovariances contain the same information about the dynamics of a stochastic process. Neither can
give information that cannot be derived from the other. The only difference is the way of presenting the
information. An auto-spectrum specifies the behavior of a univariate stochastic process. However,
economic and financial variables need to be studied in a multivariate setting. The dynamic relations
between variables are measured by the cross-covariances of a stochastic process. A cross-spectral
density function (cross-spectrum) between two variables can be derived in the same way as the autospectrum for a single variable. The only difference is that the cross-covariances need to be used
instead of the auto-covariances. In the form of the coherence and phase spectra, these cross-spectral
densities ‘dissect’ the conventional correlations at the various frequencies into a phase shift and the
maximum correlation possible after such a phase shift. Note that various auto- and cross-spectra can
also be combined in a straightforward manner into multivariate spectral densities.
2.3 Filters
If a linear filter G(L) is applied on a time series xt we obtain a new time series
b
 b

yt = ∑ g l xt −l =  ∑ g l Ll  xt = G ( L) xt
l =a
 l =a

(2.3)
The Fourier transform of this filter is called the Frequency Response Function (FRF) of the filter
because for each frequency ω it specifies how the amplitude and phase of the frequency domain
representation of the original time series xt are affected by the filter. The effects of the filter can be split
up into two parts. First, the squared Gain gives the multiplier change of the variance of the component
with frequency ω in a time series. The squared Gain is therefore often called the Power Transfer
Function (PTF). Second, the Phase of a linear filter gives the phase shift of the component of
frequency ω in a time series, expressed as a fraction of the period length. Although it is often not
recognized as such, probably the most often applied linear filter consists of calculating the first order
differences of a time series. Its squared Gain (i.e. PTF) and Phase are shown in Figure 2.1.
Figure 2.1: Squared Gain or Power Transfer Function (PTF) and Phase of the first order
differencing operator. The PTF (left panel) shows how the first order differencing operator
suppresses the variance of low frequency fluctuations in a time series while it strongly
enhances the variance of high frequency fluctuations in a time series. The Phase (right panel)
shows that that the first order differencing operator also shifts these fluctuations back in time
by a maximum of 0.25 times the period length of the fluctuations.
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Assume t is measured in years. The PTF on the left shows that the variance at frequencies below
approximately 1/6 cycles per year (i.e. with a period length of more than six years) is being reduced by
the first order differencing filter. This explains why the filter is often used to eliminate trending behavior
(i.e. very long term and low frequency fluctuations) from time series data. First order differencing also
(strongly) emphasizes the high frequency fluctuations. This can be seen from the value of the PTF for
higher frequencies. The variance of the highest frequency fluctuations is multiplied by a factor 4.
Besides changing the variance at the relevant frequencies, which can be directly thought of in terms of
changing the shape of spectral densities, the first order differencing filter also shifts time series in the
time domain which corresponds to phase shifts in the frequency domain. The Phase on the right
shows that the lowest frequencies are shifted backwards in time by approximately a quarter of the
relevant period length while the phase shift for the higher frequencies decreases towards zero for the
highest frequency in a linear fashion.
2.4 Leakage Effect
The previous sections already demonstrate some clear intuitive appealing properties of a frequency
approach for time series analysis and modeling. Spectral densities very efficiently give information
about the dynamic behavior of both univariate and multivariate time series and stochastic time series
processes. Gains and Phases clearly show what linear filters do to spectral densities at different
frequencies. Nevertheless, frequency domain techniques do not have the widespread use within
economics and finance one would expect based on these appealing properties, especially given the
extensive use of these techniques in the natural sciences. We feel that the fundamental reason for this
lies in the fact that in economics and finance the available amount of (historical) time series data is in
general too limited for conventional frequency domain techniques to be applied successfully, which in
turn is caused by the fact that these techniques typically require large amounts of data. If these
conventional techniques are applied anyway on time series of limited sample sizes this can for
example result in disturbed and / or less informative spectral density estimates. Another example is
that filtering time series according to some FRF can give disturbed filtering results. Fortunately, there
exist special (parametric) versions of frequency domain techniques for the estimation of spectral
densities and filtering of time series that are especially adapted to (also) work well on short sample
time series data and can therefore be successfully applied on economic and financial time series data.
These techniques avoid spurious spectral analysis and filtering results in ways that are described in
Sections 3.1 and 3.2. The fundamental disturbing consequences of applying standard frequency
domain techniques on time series of limited size are caused by what is called the leakage effect. This
effect can best be understood by thinking of the Fourier transform of a perfect cosine function of some
frequency. Obviously in the periodogram of this cosine function 100% of the variance should be
located at the specific frequency of the cosine function. However, if one only has a limited sample of
the cosine function available for the Fourier transform, this turns out not to be the case. Instead, a part
of the variance at the specific frequency will have ‘leaked’ away to surrounding frequencies in the
periodogram. As the sample size increases, the disturbing effects of leakage decrease and the
periodogram gets better and better at revealing the true identity of the time series by putting a larger
and larger portion of the variance at the specific frequency of the cosine function. In Section 3.1 we
will explain how in small samples the leakage effect can result in disturbed and less informative
spectral density estimates while in Section 3.2 we will explain how it can cause disturbed filtering
results. In both cases we will also describe the appropriate solutions to these problems.
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3 Methodology
In this section we first describe the proposed frequency domain methodology for time series analysis
and modeling. In Section 4 we will give various motivations for proposing this methodology. We use
this aberrant order of presentation because knowing the methodology makes it easier to fully
understand and appreciate its motivations. This means that in this section we deliberately say little
about why the methodology works in the proposed way and instead limit ourselves to explaining how it
works. The methodology consists of the consecutive steps described in the following sub sections and
builds on the fundamental frequency domain concepts described in Section 2. Various examples are
given throughout the text to illustrate individual concepts. These examples are taken from different
sources and are therefore not necessarily consistent
3.1 Time series Decomposition
After having collected the appropriate time series (and possibly also cross section) data, the first step
of the methodology is to zoom in on the different aspects of the time series behavior by decomposing
the time series. The different components of the time series can then be analyzed and modeled
separately by zooming in on the behavior of the time series in the different frequency regions. As an
example consider Figure 3.1 which shows a decomposition of the long term nominal interest rate in
4
the Netherlands in a trend, low frequency and high frequency component. This decomposition is such
that the three components add up to the original time series. The trend component consists of all
fluctuations in the time series with a period length longer than the sample length (194 years) which is
very natural definition for a trend. The low frequency component consists of all fluctuations with a
period length shorter than the sample length but longer than 15 years. 15 years is a wide upper bound
on business cycle behavior. The third component consists of all fluctuations with a period length
between 15 and 2 years, the shortest period length possible on annual data.
Decomposing a time series in such a way is also called filtering the time series. It is not hard to
imagine that the way this filtering is implemented is of crucial importance for the subsequent analysis
and modeling. Therefore we proceed by giving more information on the available and required filtering
techniques. An overview of filtering techniques and their properties can be found in Chapter 5 of
Steehouwer (2005).
4
End of year values of Netherlands 10-year Government Bond Yield 1814 - 2007. 1918 and 1945 values based on beginning of
next year. Source: GlobalFinancialDatabase (GFD) code IGNLD10D.
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Figure 3.1: Decomposition of a long term interest rate time series for the Netherlands.
Source: GlobalFinancialDatabase (GFD). The time series is decomposed into the three dotted
lines in such a way that adding these component time series results in the original time series.
The component time series distinct themselves by the period length of the fluctuations from
which they are constituted. The first component indicated by ‘Trend’ captures all fluctuation in
the interest rate time series with a period length between infinity (i.e. a constant term) and the
sample length of 193 years. The second component indicated by ‘Low frequencies’ captures all
fluctuations with a period length between 193 and 15 years. The third component indicated by
‘High frequencies’ captures all fluctuations with a period length between 15 and 2 years.
3.1.1 Filter Requirements
What do we require from a filtering technique that is to be applied to decompose time series in a way
as shown in Figure 3.1? An ideal filter should allow for / result in:
1. User defined pass-bands: By this we mean that the user of the filter should be able to freely
specify which period lengths (frequencies) of fluctuations to include and exclude in the filtered time
series and should not be restricted by the properties of the filter.
2. Ideal pass-bands: The filter should exactly implement the required user defined pass-bands. Often
filters do this only in an approximating sense.
3. No phase shifts: The filter should not move time series back or forth in time as this could influence
inferences about the lead / lag relations between variables. This means that the Phase of a filter
must be zero for all frequencies.
4. No loss of data: The available amount of data in economics and finance is in general limited.
Therefore, an ideal filter should not lose observations at the beginning or end of the sample.
3.1.2 Zero Phase Frequency Filter
Many conventional filters fail one or more of the four described requirements. There are two main
reasons for this. A first reason is that many filters were originally defined in the time domain instead of
in the frequency domain in terms of the properties of their PTF and Phase. As an example consider
the properties of the simple first order differencing filter described in Section 2.3 which fails on all of
the four requirements. To a lesser, though still significant, extent this also holds for other well known
filters such as the exponential smoothing and Hodrick Prescott filter. The second reason is the
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leakage effect from Section 2.4 which causes that ideal filtering results cannot be achieved when
filters are applied on time series of a limited sample size, even when the filters have been explicitly
designed to achieve the required ideal frequency domain properties. To see how this works think a of
a conceptually simple direct frequency filter that starts by transforming a time series into the frequency
domain. Next, based on the required PTF of the filter, the weights at certain frequencies are set to
zero while other are preserved (i.e. the ones that lie within the pass-band of the filter). And finally, the
adjusted frequency domain representation of the time series is transferred back into the time domain.
For time series of limited sample sizes this approach does not work well because, as explained in
Section 2.4, the true frequency domain representation of the (limited sample) time series is disturbed
in the first step. If, in the second step, the ideal PTF is applied to this erroneous frequency domain
representation, some (frequency) parts of the time series behavior will be deleted that should have
been preserved and vice versa.
Section 5.4 of Steehouwer (2005) describes a special Zero Phase Frequency Filter that does meet all
four requirements by focusing on a solution for the second problem of filtering time series of finite
sample sizes as caused by the leakage effect. This filtering technique is very much based on the ideas
from Bloomfield (1976) and the filtering approach of Schmidt (1984). For example Hassler et al.
(1992), Baxter and King (1999) and Christiano and Fitzgerald (1999) describe different approaches to
deal with the problems of filtering finite time series in the frequency domain. The filter algorithm comes
down to the iterative estimation of a number of periodic components and multiplying each of these
components by the value of the PTF to obtain the filtered time series. The key trick of this filter is that it
avoids the disturbing leakage effects by skipping the transformation into the frequency domain but
filtering estimated periodic components (sine and cosine functions) directly instead. This is possible
because in case we would have had an infinite sample size, the filtering result of a periodic component
of a certain frequency would be known exactly beforehand. As an example think of the frequency
domain representation of the perfect cosine time series described in Section 2.4. In a sense, by
estimating periodic components, the time series is “extrapolated” from the sample size into infinity and
thereby the disturbing effects of leakage can be avoided.
As shown in Section 6.3.2 of Steehouwer (2005), this Zero Phase Frequency Filter results in very
similar filtering results as the popular Baxter and King and Christiano and Fitzgerald filters. The
additional advantages are that compared to these filters, the filter is more precise by filtering directly in
the frequency domain (requirement 2), it causes no phase shifts (requirement 3) and leads to no loss
of data (requirement 4).
3.1.3 Zero Correlation Property
An important advantage of decomposing time series in the frequency domain, based on ideal filters
and implemented by appropriate filtering techniques, is that all filtered components of non-overlapping
pass-bands of some (set of) time series have zero correlation in the time domain, both in a univariate
and multivariate context. This (theoretical) property holds for “all” filters which adequately implement
an ideal pass-band and is the continuous analogue of the orthogonal property of cosine functions. This
theoretical property can also be checked to apply to practical filter output when the Zero Phase
Frequency Filter is applied on actual time series. Tests described in Steehouwer (2007) show that
although non-zero correlations can actually occur in practical filter output, it can still be concluded that
from a theoretical perspective zero correlations between the component time series can (and must)
safely be assumed for time series modeling purposes. This can easily be understood by thinking about
short samples of several low frequency components. On short samples the correlations between such
components can be very different from zero. However, if the time series behavior of these low
frequency components would be modeled and simulated on sufficiently long horizons, the fundamental
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correlations would still need to be zero. Although this zero correlation property greatly simplifies the
time series modeling process, note that zero correlations do not need to imply that component time
series are also independent. In fact, there exist quite complex forms of dependencies between the
component time series which need to be taken into account in the modeling process. The perhaps
counterintuitive fact is that, despite the zero correlation property, the decomposition approach does
not hinder the analysis and modeling of such complex dependencies but actually facilitates it. We say
more about these complex dependencies in Section 3.3.3.
3.2 Time series Analysis
After having decomposed the time series in order to be able to zoom in on the behavior of the time
series variables in the different frequency regions, the second step in the methodology consists of
actually analyzing this behavior of the time series. The resulting understanding of this time series
behavior then forms the basis for an adequate modeling of this time series behavior. For the trend and
low frequency components, as for example shown in Figure 3.1, traditional time series analysis
techniques (mean values, volatilities, correlations, cross correlations, etc.) typically suffice for
obtaining the appropriate understanding. For higher frequency components, again as for example
shown in Figure 3.1, spectral analysis techniques are very powerful for a further unraveling and
understanding of the time series behavior.
3.2.1 Maximum Entropy Spectral Analysis
There exist two groups of methods for estimating (multivariate) spectral densities as defined in Section
2.2. The first are the traditional non-parametric spectral estimators which estimate the spectral density
of a stochastic process by means of its sample counterpart, the periodogram. The periodogram in its
pure form can be shown to be an inconsistent estimator in the sense that it does not converge to the
true spectrum as the sample size increases. This inconsistency can be repaired by applying so-called
spectral windows which comes down to replacing the periodogram values at all frequencies by a
weighted average of the periodogram values at adjacent frequencies. However, the most important
problem in practice of using the periodogram as an estimator for a spectral density is that because of
the finite sample only a limited number of auto-covariances are fed into the formula for the
periodogram while the theoretical spectrum contains all (infinite) auto-covariances of the process. In
Section 2.4 we saw that the disturbing leakage effect is a direct consequence of the finite sample size.
Although the leakage effect can be “reduced” by applying spectral windows, this will always come at
the expense of a lower resolution of the estimated spectrum. By resolution we mean the extent to
which a spectral estimator is able to differentiate between separate, possibly adjacent, peaks in the
theoretical spectrum. A lower resolution is therefore equivalent to a larger bias in the estimate. It is
unavoidable that the averaging of the periodogram over adjacent frequencies causes adjacent peaks
in the spectrum to be melted together which means a possible loss of valuable information about the
dynamic behavior of the time series process under investigation. Especially in economics and finance,
where most of the times only limited samples of data are available, this low resolution of the
conventional non-parametric estimators is a serious problem. In case of small samples there will be a
lot of leakage and hence a lot of smoothing required which leads to a loss of a lot of potentially
valuable information.
A second group of methods for estimating (multivariate) spectral densities are the less well known
parametric spectral estimators. By estimating the spectrum through a cut off sequence of sample autocovariances, implicitly all higher order auto-covariances are assumed to be zero. This also holds if a
spectral window is applied, although in that case additionally the sample auto-covariances that are
available are modified by the weighting function. In fact, as a consequence of cutting off the auto-
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covariances, the spectrum of an entirely different, to some extent even arbitrary, stochastic process is
estimated. Parametric spectral estimators try to circumvent this problem by first estimating the
parameters of some stochastic process on the available sample. Once such a stochastic process is
“known”, its auto-covariances can be calculated for any order up to infinity. These auto-covariances
can then be used to calculate the periodogram of the process as an estimate of the spectral density
function. In a way, such a model ”extrapolates” the auto-covariances observed within the sample into
auto-covariances for orders “outside” the sample. Note that this parametric approach for estimating
spectral densities has a strong analogy with the Zero Phase Frequency filter described in Section
3.1.2 in terms of avoiding the disturbing leakage effects. When estimating a spectral density, the
sample auto-covariances are extrapolated by means of a parametric model. When filtering a time
series, a periodic component of some frequency present in the time series is extrapolated using a
cosine function.
A special case of such parametric spectral estimators are the so called autoregressive or Maximum
Entropy spectral estimators. These consist of first estimating the parameters of a Vector
AutoRegressive (VAR) model on the (decomposed) time series data and then calculating the
(multivariate) spectral densities from this estimated model. This autoregressive spectral analysis leads
to consistent estimators of spectral densities. Furthermore, a theoretical justification exists for
choosing autoregressive models instead of many other possible models (including the non-parametric
approach) to estimate the spectral densities. This justification is based on the information theoretical
concept of Maximum Entropy. Entropy is a measure of “not knowing” the outcome of a random event.
The higher the value of the entropy, the bigger the uncertainty about the outcome of the event. The
entropy of a (discrete) probability distribution has its maximum value when all outcomes of the random
event have an equal probability. The fundamental idea of the maximum entropy approach as first
proposed by Burg (1967) is to select from all possible spectra that are consistent with the available
information, represented by a (finite) sequence of (sample) auto-covariances, the spectrum which
contains the least additional information as the best spectral estimate. All additional information on top
of the information from the available sample is not supported by the data and should therefore be
minimized. This is consistent with choosing the spectrum with the maximum entropy from all spectra
that are consistent with the observed sample of auto-covariances. The solution to the corresponding
optimization problem shows that the best way to estimate a spectrum in terms of this criterion is by
estimating an autoregressive model and using the spectrum of this model as the spectral estimate.
Furthermore, Shannon and Weaver (1949) show that given a number of auto-covariances, a Normal
distributed process has the maximum entropy. So in total, the Maximum Entropy concept comes down
to estimating a Normal distributed autoregressive model on the available time series data and
calculating the spectral densities of this estimated model. Note that the Maximum Entropy concept
itself says nothing about which order to select or which estimation procedure to use. Section 4.7 of
Steehouwer (2005) describes Monte Carlo experiments which are set up to find the answers to these
two questions.
As an example, Figure 3.2 shows a Maximum Entropy spectral density estimate of the high frequency
component of the long term interest rate time series as shown in Figure 3.1. Here, an AR(6) model is
estimated by means of the Yule-Walker estimation technique on the full 1814-2007 sample. In terms
of the further unraveling and understanding of the time series behavior of this long term interest rate
time series, we can learn from this spectral density that on average its high frequency behavior is
composed of fluctuations with a period length of around 10 years which describe approximately 50%
of the high frequency variance. Furthermore, fluctuations with a period length of around 4.5 years
seem to be important. Both observations are consistent with what is known about the business cycle
behavior of many economic and financial time series. By extending the analysis into a multivariate
13/40
framework, also coherence and phase spectra can be used to further unravel the correlations between
variable into frequencies, lead / lag relations and phase corrected correlations.
Figure 3.2: Maximum Entropy (or Auto Regressive) spectral density estimate of the high
frequency component of the long term interest rate time series for the Netherlands as shown in
Figure 3.1. Around 50% of the variance of these high frequency (business cycle type of)
fluctuation in the interest rate time series is described by fluctuations with a period length of
around 10 (1/0.10) years while also pseudo periodic behavior with a period length of around 4
(1/0.25) to 5 (1/0.20) years can be observed.
3.3 Model Specification and Estimation
After having analyzed the decomposed time series in order to obtain an adequate understanding of
the total time series behavior, the third step in the methodology consists of actually modeling the
component time series in line with this understanding. Because of the zero correlation property
described in Section 3.1.3, the trend, low frequency and high frequency (or any other) components
can in principle be modeled separately. In a multivariate context, each of these sub models will be a
multivariate model. Once the separate trend and frequency models have been constructed, these can
be added back together again to model the total time series behavior in terms of for example
forecasts, confidence intervals or scenarios. This modeling process is depicted in Figure 3.3.
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Figure 3.3: Frequency domain time series modeling approach which start at the top by
decomposing multiple time series just as in the example in Figure 3.1. Next, the corresponding
components (f.e. all high frequency components) from all time series are modeled by means of
a suitable (multivariate) time series model. In the final step, these models are combined again
to obtain a model that adequately describes the behavior of the time series at all frequencies.
Historical time series
Frequency domain filter
Trend
Model
Frequency
Model 1
. ..
Frequency
Model n
Forecasts / Confidence intervals / Samples
3.3.1 Trend Model
On purpose, in Figure 3.3 we distinguished the trend model from the models for the other frequency
component time series. The reason for this is that the trend component of time series will typically
require another modeling approach than the modeling of the other frequency components. For
example, the low and high frequency component time series shown in Figure 3.1 can be modeled well
by means of conventional time series modeling techniques because the time series show considerable
variability. The trend component however, will typically be a straight flat or trending line for which it is
clearly of little use to apply conventional time series modeling techniques. What to do then? One
possibility would be just to extrapolate these straight lines into the future as being our trend model.
Although we have only one observation of the trend value for each time series, we know that it could
very well have had a different value and hence could very well have a very different value in our model
for the future also. We should therefore prefer to work with a stochastic rather than a deterministic
(linear) trend model. One single time series does not offer a lot of information for such a stochastic
trend model. After all, even a very long sample time series, only provides us with one observation of
the long term trend value. Therefore other sources of information are needed here. One could for
example use a Bayesian approach, survey data or theoretical macroeconomic models. Also the
distribution of the sample mean estimators could be used. A more direct data oriented approach is to
switch from time series data to cross section data to obtain information about the ultra long trend
behavior of economic and financial variables. As an example consider the statistics in Table 3.1 and
th
the top panel of Table 3.2. These are based on cross section data consisting of 20 century annual
averages for the five indicated variables for 16 OECD countries (excluding Germany because of the
extreme effects of the world wars). The volatilities and correlations observed here could form the basis
for constructing an appropriate trend model. From these tables we can for example see that with a
standard deviation of 2.0% the long term inflation uncertainty is substantial and that in terms of the
trends, a total return equity index is positively correlated (0.55) to this long term inflation rate. Of
course, the fundamental underlying assumption for such a cross section based approach is that the
countries included in the cross section are sufficiently comparable to form a homogeneous group on
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which to base the long term trend model behavior for each of these countries individually. Finally note
that such an approach is very similar to the ones followed in historical long term cross country growth
studies, but here it is proposed to extend this into a forward looking stochastic modeling framework.
th
Table 3.1: 20 century averages and (geometric) average growth rates of annual
GDP (volume), CPI, (nominal) short and long term interest rates and
(nominal) total equity returns (i.e. price changes plus dividend yields) for 16 OECD countries.
Data from Maddison (2006) and Dimson et al. (2002).
Short
Long
Log TRR
Interest Rate
Interest Rate
Equity Index
3.9%
4.5%
5.2%
11.9%
2.2%
5.7%
5.2%
5.1%
8.1%
Canada
3.8%
3.2%
4.9%
4.9%
9.7%
Denmark
2.8%
4.1%
6.5%
7.1%
9.3%
France
2.4%
7.8%
4.4%
7.0%
12.1%
Ireland
2.0%
4.4%
5.2%
5.4%
9.4%
Italy
2.9%
9.2%
5.0%
6.8%
12.1%
Japan
4.0%
7.8%
5.5%
6.1%
13.0%
Netherlands
3.0%
3.0%
3.7%
4.1%
9.1%
Norway
3.3%
3.9%
5.0%
5.4%
8.0%
South Africa
3.2%
4.8%
5.6%
6.2%
12.2%
Spain
3.0%
6.2%
6.5%
7.5%
10.2%
Sweden
2.7%
3.8%
5.8%
6.1%
12.4%
Switzerland
2.6%
2.5%
3.3%
5.0%
7.3%
United Kingdom
1.9%
4.1%
5.1%
5.4%
10.2%
Unites States
3.3%
3.0%
4.1%
4.7%
10.3%
Avg
2.8%
4.8%
5.0%
5.7%
10.3%
Stdev
0.6%
2.0%
0.9%
1.0%
1.8%
Min
1.9%
2.5%
3.3%
4.1%
7.3%
Max
4.0%
9.2%
6.5%
7.5%
13.0%
Country
Log GDP
Log CPI
Australia
2.3%
Belgium
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3.3.2 Frequency Models
As indicated in Figure 3.3, next to the trend model there exist a number of so called frequency models
in the proposed methodology. These model the low and high frequency behavior of the time series
variables around their underlying long term trends. The number of frequency models is the same as
the number of frequency segments used in the decomposition step described in Section 3.1, excluding
the trend component. Ideally the number of frequency segments and their size could be determined on
a spectral analysis of the data. Due to data limitations this will often be difficult. Furthermore, because
in the decomposition step no information is lost or added, we expect the exact choice of the frequency
intervals to have a limited impact on the final model behavior. Instead, the split between the frequency
models can therefore also be determined based on the sample size and observation frequencies of
the data and the economic phenomena that, based on a thorough empirical analysis, need to be
modeled. As an example of how this can work, Figure 3.4 shows five-annual observations of low
frequency filtered times series of five economic and financial variables (the same as in Table 3.1) for
the Netherlands for the 1870-2006 sample period. The second panel of Table 3.2 shows some
statistics of the same low frequency component time series. These time series data capture the very
long term deviations from the underlying trends and also contain information about changes in
economic regimes. The sample needs to be as long as the data allows because here we are
interested in the very low frequency behavior and we simply need a long sample to be able to observe
this behavior adequately. Five-annual (instead of annual) observations are sufficient here to capture
this low frequency behavior and facilitate the modeling of the corresponding time series behavior. If
this is required for combining the various frequency models, the observation frequency of such a
model can be increased to for example an annual observation frequency by means of for example
simple linear interpolation.
Figure 3.4: Five-annual observations of low frequency filtered times series of
five indicated economic and financial variables for the Netherlands for 1870-2006.
Log National Product
Log Consumer Price I ndex
Log TRR Equity Index
Short Interest rate
Long Interest Rate
1870
1890
1910
1930
1950
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1970
1990
2010
Figure 3.5 shows annual observations of high frequency filtered times series of the same five
economic and financial variables for the Netherlands but now for the 1970-2006 sample period. The
third panel of Table 3.2 shows some statistics of the same high frequency component time series.
These time series data capture the business cycle behavior of the variables around the sum of the
trend and the low frequency components. Although empirically speaking business cycle behavior is
surprisingly stable across samples spanning several centuries, a more recent 1970-2006 sample will
be considered more relevant for the business cycle behavior in the near future while also more recent
data will be of a higher quality. Higher (than five-annual) annual observation frequencies are needed
to adequately capture the higher frequency behavior of the time series.
Figure 3.5: Annual observations of high frequency filtered times series of
five indicated economic and financial variables for the Netherlands for 1970-2006.
Log National Product
Log Consumer Price Index
Short Interest rate
Long Interest Rate
Log TRR Equity Index
1970
1975
1980
1985
1990
1995
2000
2005
If also the monthly behavior of the variables would need to be modeled, a third frequency model could
be included that would run on filtered monthly time series data for an even more recent sample, say
1990:01–2005:12, on a monthly observation frequency. These time series data would capture
seasonal patterns and possibly also stochastic volatility patterns. In principle we can continue in this
way by including additional frequency models for weekly, daily or over ticker data if required.
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Table 3.2: Statistics of trend, low frequency and high frequency components of five economic
and financial variables. Trend data are from Table 3.1. Low frequency data are five-annual
observations of low frequency filtered times series for 1870-2006. High frequency data are
annual observations of high frequency filtered time series for 1970-2006.
Original time series are updates of data for the Netherlands from Steehouwer (2005).
Trend
Avg
Corr and stdev
Log GDP
2.8%
0.6%
Log CPI
4.8%
0.07
2.0%
Short Interest Rate
5.0%
0.08
0.31
0.9%
Long Interest Rate
5.7%
-0.01
0.67
0.70
1.0%
Log TRR Equity Index
10.3%
0.19
0.55
0.27
0.44
Low Frequencies
Avg
1.8%
Corr and stdev
Log GDP
0.0%
12.2%
Log CPI
0.0%
0.23
15.7%
Short Interest Rate
0.0%
0.70
0.50
1.6%
Long Interest Rate
0.0%
0.64
0.31
0.86
1.8%
Log TRR Equity Index
0.0%
-0.18
0.39
-0.25
-0.27
High Frequencies
Avg
33.2%
Corr and stdev
Log GDP
0.0%
1.7%
Log CPI
0.0%
-0.37
1.3%
Short Interest Rate
0.0%
0.39
-0.22
1.8%
Long Interest Rate
0.0%
0.10
0.10
0.74
0.9%
Log TRR Equity Index
0.0%
-0.12
-0.20
-0.37
-0.56
24.0%
For each of the frequency models the most appropriate time series modeling and estimation
techniques can be used to model the corresponding economic phenomena and time series behavior
as good as possible. In principle, these models need not to come from the same class of models. For
example, a structural business cycle model could be used for the high frequency components from
Figure 3.5 while a model with seasonal unit roots could be used for the seasonal components in the
monthly time series and a (G)ARCH model could be used to model the stochastic volatility in the
seasonally corrected part of the monthly time series. Other time series modeling techniques to use in
the different frequency models could be classical time series models, VAR models, models from the
London School of Economics (LSE) methodology, theoretical models, Copula models, historical
simulation techniques, etc.
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3.3.3 State Dependencies
In Section 3.1.3 we already described the usefulness of the zero correlations between the component
time series that are obtained from the filtering process in the sense that it simplifies subsequent time
series modeling. However, zero correlations do not need to imply that component time series are also
independent. An example of a complex dependency between component time series is the so called
‘level effect’ in high frequency interest rate volatility. In the left panel of Figure 3.6 we see again the
long term nominal interest rate in the Netherlands together with the filtered low and high frequency
components. The low frequency component is shown here as the sum of the trend and low frequency
components from Figure 3.1. If we define this sum as the underlying ‘level’ of the interest rate then it is
clear that there exists a positive relation between the volatility of the high frequency component and
the level of the interest rate. A similar effect can be found in short and long term interest rates for other
countries and also in inflation rates.
Figure 3.6: Level effect in the high frequency volatility of interest rates. The left panel shows
the empirical positive relation between the underlying (low frequency) level of the nominal long
term interest rate in the Netherlands from Figure 3.1 and its short term (high frequency)
volatility. The right panel shows three samples from a model that explicitly captures this level
effect by dynamically linking the volatility of the high frequency model to the level of the
simulated underlying low frequency model according to a simple linear relation that has been
estimated between the value of the low frequency component and the volatility of the high
frequency component from the left panel.
The perhaps counter intuitive fact is that, despite the zero correlation property, the decomposition
approach actually facilitates the modeling of such complex dependencies between the component
time series. If we can define and estimate some functional relation between the volatility of the high
frequency component and the level or ‘state’ of the lower frequency models that describe the trend
and low frequency components, then it is easy to implement this relation in for example a simulation
framework of the model. What we need to do then is just start by simulating from the trend and low
frequency models and then to simulate from the high frequency model while constantly updating the
volatility of this high frequency model based on the observed level of the trend and low frequency
simulations. The right hand panel of Figure 3.6 shows three example simulations of the long term
interest rate that include this level effect in the high frequency volatility. The functional relation used
here is taken from Section 13.2.1 of Steehouwer (2005). The approach described here can equally
well be applied to more complex types of state dependencies, as for example state (with time as a
special case) dependent business cycle dynamics and state dependent asset correlations.
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3.3.4 Higher Moments and Non-Normal Distributions
Spectral analysis techniques as described in Section 2 focus on the decomposition of the variance of
time series and stochastic processes over a range of frequencies. Spectral densities are calculated as
the Fourier transform of auto- and cross-covariances which shows that spectral analysis focuses on
the second moments of the behavior of time series and stochastic processes. Although these second
moment are of course very important, they do not cover all relevant information. Furthermore, in
Section 3.2.1 we gave a justification for using Normal distributions in the Maximum Entropy spectral
analysis framework while we know that the behavior of economic and financial variables will often be
far from Normal. When studying the low and high frequency properties of decomposed time series
data, one soon finds many forms of non-Normal distributions with aberrant third and fourth moments.
As an example look at the Skewness and (excess) Kurtosis numbers of the cross section trend data in
Table 3.1. Another example are monthly high frequency components of equity returns in which we
often find empirical distributions that are more peaked with thinner tails than the Normal distribution
(leptokurtic). A first way of modeling these kinds of non-Normal distributions in the proposed modeling
framework is an explicit modeling approach. As an example, note that the modeling of the level effect
in high frequency interest rate volatility along the lines as described in Section 3.3.3 will result in a
skewed overall distribution. A second example is modeling a stochastic volatility in the monthly high
frequency component of equity prices which will result in leptokurtic distributions for the corresponding
frequency model. A second way of modeling non-Normal distributions would of course be to use other
than Normal distributions in the trend and frequency models.
3.3.5 Missing Data Solutions
Ideally the methodology described here would run on an abundance of empirical time series data
consisting of samples of hundreds of years on say a daily observation frequency of all economic and
financial variables. Of course this kind of time series data will not be available in most cases.
Especially the trend and low frequency models require cross section or very long time series which will
not be available for all variables. This poses a problem because every variable needs to be specified
in the trend model and in each of the frequency models to be able to describe the complete behavior
of the variables. One standpoint could be that if long term data is not available for certain variables,
one should not try to model the long term behavior of these variables in the first place (also see
Section 4.3) but this is not always a feasible standpoint. Another solution to this missing data problem
is to describe the behavior of variables for which we have insufficient data in certain frequency ranges
as a function of the variables for which we do have sufficient data. The functional form and parameters
of such relations can be determined in two ways. The first is to perform an empirical analysis on a
shorter sample for which data for all the required variables is available and estimate an appropriate
relation. The second possibility is to base the relations on economic theory.
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As examples of this theoretic approach consider
•
•
•
to describe the low frequency behavior of a real interest rate as the low frequency behavior in a
nominal interest rate minus the low frequency behavior of price inflation (Fisher relation),
to describe the low frequency behavior of an exchange rate as the difference in the low frequency
behavior of the involved price indices (Purchasing Power Parity) or
to describe the low frequency behavior in private equity prices as the low frequency behavior of
public equity prices plus some error term.
3.4 Model Analysis
After in the previous steps having decomposed time series data, analyzed the decomposed time
series and modeled the component time series, the fourth step in the methodology consists of
analyzing the constructed models. Such an analysis is required for example to check whether the
models adequately describe the intended behavior or to use the properties of the models as input for
the framework in which they are applied. An example of the latter would be to calculate the relevant
moments of the implied stochastic behavior of the economic and financial variables as input for a SAA
optimization routine across various horizons. In the described methodology, traditional methods to
perform such a model analysis can still be applied. Depending on the types of time series models that
are used, for example unconditional and conditional distribution characteristics such as means,
standard deviations, correlations and percentiles can still be calculated for the trend and various
frequency models separately. The only extension here is that these characteristics have to be
combined to obtain characteristics of the total model. In most cases this is fairly straightforward
because of the zero correlation property described in Section 3.1.3. For example the total covariance
between two variables is simply the sum of the covariances between those two variables in the trend
and each of the frequency models. In addition to the traditional methods, two additional types of
analysis are available in the described frequency domain methodology which consist of spectral
analysis and variance decomposition techniques.
3.4.1 Spectral Analysis
In Section 3.2.1 we described that spectral analysis techniques are very powerful for unraveling and
understanding the behavior of filtered time series components. However, spectral analysis techniques
can also be fruitfully applied to analyze the behavior of the constructed time series models.
Multivariate spectral densities in terms of the auto-, coherence and phase spectra can equally well be
calculated for each of the frequency models separately as for the total model. The resulting model
spectral densities can for example be compared to the spectral densities of the (decomposed) time
series data. Calculating spectral densities for estimated models can be done in two ways. The first is a
direct calculation of the spectral densities based on the parameters of the model. Section 4.5 of
Steehouwer (2005) for example gives the spectral density formulas for a simple white noise process
and moving average and autoregressive models. A second and more flexible possibility is to first
calculate, possibly numerically, the auto- and cross-covariances of a sufficient high order and then to
apply the Fourier transform to transform these auto- and cross-covariances into the corresponding
spectral densities. Because of the zero correlation property, spectral densities for the total model in
terms of the trend and various frequency models can easily be obtained by first summing the autoand cross-covariances across the sub models before transforming these into the frequency domain.
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3.4.2 Variance Decomposition
Just as a spectral density decomposes the variance of one of the frequency models over the whole
frequency range, a similar decomposition can be made of the variance of the total model into the
variance of the individual trend and frequency models. Again because of the zero correlation property,
constructing such a variance decomposition is fairly straightforward. The variance of the sum of the
5
models is simply the sum of the individual variances . A variance decomposition can be constructed
for both the unconditional distributions and the conditional distributions. The latter gives insight into the
contribution of the different sub models to the total variance at different horizons. As an example
consider Figure 3.7 which shows a conditional variance decomposition for a long term interest rate
and a log equity total rate of return index. The model consists of a stochastic trend model, a low
frequency model, a business cycle model, a seasonal monthly model and a seasonally corrected
th
monthly model. For every 12 month in a 35 year horizon, the two panels show in a cumulative
fashion the proportion of the total conditional variance of that month that can be attributed to the
various sub models. From the right hand panel we can see for example that the business cycle model
(m=2, which is here the same as m=3) describes around 80% of the variance of the log equity total
return index on a 1 year horizon while on a 35 year horizon this holds for the trend and low frequency
model (m=1). That is, high frequency models are typically important on short horizons while low
frequency models are important on long horizons.
th
Figure 3.7: Variance decompositions evaluated at every 12 month in a 35 year horizon. Left
panel for a long term interest rate and right panel for a log equity total rate of return index. The
lines show in a cumulative fashion which portion of the conditional variance at the various
horizons is caused by a trend model and four frequency models. In general such variance
decompositions show for models of economic and financial that low (high) frequency behavior
causes a relatively larger part of the variance at long (short) term horizons.
5
Note that some kind of interpolation method might be needed to align all frequency models on the same (highest) observation
frequency.
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4 Motivation
Why do we propose the specific frequency domain methodology for time series analysis and
modeling? The motivation for this consists of several points, some of which have already been
implicitly given in the description of the methodology in Section 3. In the following sub sections we
make these points of motivation more explicit, extend them with a further argumentation and also put
forward a number of new points of motivation.
4.1 Understanding Data and Model Dynamics
A first reason for proposing the frequency domain methodology for time series modeling is that it
provides very powerful tools for understanding the dynamic behavior in historical time series data as
well as for analyzing the dynamic properties of models that describe time series behavior for the
future. If there is one thing about the behavior of economic and financial variables we know, then it is
that they move up and down and never move in straight and stable paths. Therefore, frequency
domain techniques are the most natural to analyze how exactly they move up and down. What types
of fluctuations dominate the behavior of a variable and what are the correlations and lead / lag
relations with other variables at the various speeds of fluctuations? As a first element, decomposing
time series into different components allows us to zoom in on the behavior in various frequency
regions which provides us with a clearer and more focused insight in the corresponding dynamic
behavior. We can for example focus our attention on the business cycle behavior of economic and
financial variables, something which is very common in business cycle research but much less
common in a time series modeling context. As a second element, estimating and analyzing spectral
densities is a very efficient way of summarizing the dynamic behavior of component time series within
a certain (high) frequency region. Although spectral densities contain exactly the same information as
conventional time domain auto- and cross-correlations, spectral densities represent this information in
a more efficient and intuitive manner. These appealing properties of frequency domain approaches
are certainly not new to economists. It has long been recognized that when studying macroeconomics
one has to make a clear distinction as to which aspect of macroeconomics one is interested in. It
seems likely that we will be dealing with very different ‘forces‘ when we are studying for example the
long term growth of economies, comprising many decades, compared to the intra day trading effects
on a stock exchange. If different forces are at work, also different models or approaches may be
needed to adequately analyze and describe the relevant economic behavior. The first formalization of
this idea dates back to Tinbergen (1946) who proposed a decomposition of time series of economic
variables as
Time Series = Trend + Cycle + Seasonal + Random
(4.1)
The first economic applications of spectral analysis date as far back as Beveridge (1922) who used a
periodogram to analyze the behavior of a wheat price index. However, for a successful application of
frequency domain techniques on economic and financial time series, it is crucial to use appropriate
and special versions of these techniques that can deal with the leakage problem of Section 2.4 as
caused by the in general limited sample sizes available in economics and finance. This is exactly what
the Zero Phase Frequency filter from Section 3.1.2 and the Maximum Entropy spectral analysis from
Section 3.2.1 provide us with.
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4.2 Different Economic and Empirical Phenomena
A second reason for proposing the frequency domain methodology for time series modeling concerns
the fact that at different horizons (centuries, decades, years, months, etc.) and different observation
frequencies (annual, monthly, weekly, etc.) the empirical behavior of economic and financial variables
is typically different and dominated by different well known economic phenomena such as long term
trends, business cycles, seasonal patterns, stochastic volatilities, etc. First of all, decomposing time
series into different components allows us to analyze these economic phenomena separately while
spectral analysis techniques can be used for a further unraveling of the corresponding behavior.
Second, by also following a decomposed modeling approach, as summarized by Figure 3.3, we are
also able to adequately model the different economic phenomena simultaneously by using the most
appropriate time series models for each of them. The potential benefits of this decomposition
approach to time series modeling can perhaps best be illustrated by the following extreme example.
Suppose one wants to model realistic behavior of economic and financial variables up to a horizon of
several decades but on a daily basis. Such a model should at the same time give an adequate
description of the trending, low frequency, business cycle, seasonal and daily behavior, each with their
very specific properties and for each of the variables. Obviously, achieving this within one single
conventional time series modeling approach is very difficult. Estimating for example a conventional
Vector AutoRegressive (VAR) model on a sample of daily observations will probably not produce the
intended result. However, with the proposed decomposition approach to time series modeling this is
exactly what is possible in a theoretically well founded, simple and flexible way. Thereby, the proposed
methodology brings together the empirical behavior of economic and financial variables that is
observed at different horizons and different observation frequencies in one complete and consistent
modeling approach. Furthermore, this way of modeling also stimulates the incorporation of economic
reasoning and intuition into what can become purely statistical time series modeling exercises.
4.3 Appropriate Samples and Observation Frequencies
A third reason for proposing the frequency domain methodology for time series modeling is that it
allows us to use appropriate time series data in terms of samples and observation frequencies to
analyze and model the various economic phenomena mentioned in Section 4.2. Furthermore, the
methodology allows us to optimally combine all available sources of time series information. In some
cases it is evident what type of time series data is required for modeling a certain economic
phenomenon. For example, if we want to model seasonal patterns it is clear that monthly data for a
sufficient number of years is required. However, in some cases what data to use seems less evident.
This especially holds in the case of modeling the very long term (low frequency) behavior of economic
and financial variables. As an example consider Figure 4.1 which shows two samples of the same
long term nominal interest rate in the Netherlands as used in Section 3.1. Suppose we would use the
left hand 38 year long annual 1970-2007 sample as a basis for modeling the long term behavior of this
nominal interest rate, suppose up to an horizon of 30 years into the future. A sample of almost 40
years certainly seems like a long sample at first sight. However, from a frequency domain point of
view, we are modeling the behavior at very low frequencies, which may have period lengths of 40
years or more, based on one single sample of 40 years. In a sense, by doing this, we are estimating a
model for the low frequency behavior based on one single observation of this low frequency behavior.
This is clearly inadequate and can lead to something Reijnders (1990) calls perspective distortion
which means that if one looks at too short a sample, one can be misled about the behavior of
economic and financial variables. Based on the left hand panel of Figure 4.1 we could for example
conclude that the average interest rate is somewhere around 7%, the interest rate has a strong
downward trend (which might lead us to the conclusion that the interest rate is a non-stationary
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process that needs to be modeled in terms of the annual changes of the interest rate instead of the
levels) and always has a large amount of short term volatility. However, if we look at the full 194 year
1814-2007 sample, as shown in the right hand panel of Figure 4.1, entirely different information about
the behavior of the long interest rate is revealed. We see for example that the average interest rate is
actually more around 4 to 5%, does not have a downward trend but during the 1970-2007 period was
returning back from ‘exceptionally’ high post war levels to this more regular level of 4 to 5% (which
shows us that the interest rate is actually a stationary process with a very slow rate of mean reversion
and should be modeled as such in terms of the levels) and short term volatility can also be quite low,
especially at low interest rate levels (hence, the ‘level effect’ discussed in Section 3.3.3).
Figure 4.1: The risk of ‘perspective distortion’ by using short samples illustrated on a long term
interest rate time series for the Netherlands. Left hand panel shows sample 1970-2007, right
hand panel shows full 1814-2007 sample. From the left panel one could be lead to believe in a
high level, downward trending and high (short term) volatility behavior of this interest rate
variable while the right panel shows that the true behavior of the interest rate actually consists
of long term fluctuations around a lower level with short term volatilities that are positively
related to the underlying level of the interest rate. Therefore, by considering only a relatively
small part of the available sample one can be misled about the behavior of time series
variables and hence can be led to construct erroneous models to describe this behavior.
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So in short we can say that ‘short’ samples can give insufficient information for modeling the long term
behavior of economic and financial variables. This is a first example of the observation that
conventional modeling approaches tend to model the long term behavior (which should be based on
long samples) of economic and financial variables based on an ‘extrapolation’ of the short term
behavior of these same variables (by using short samples). One could wonder whether this is
important. Is there really a big difference between the long and short term behavior? The answer to
this question is a strong yes. As an example consider the correlation between equity returns and
inflation rates as shown in Table 3.2. We see that this long term correlation in terms of the trend and
low frequency components, is very positive (0.55 and 0.39 respectively) while this short term
correlation in terms of the high frequency component is actually negative (-0.20). For a discussion of
the literature and these data findings about the inflation hedging capacities of equities at different
horizons, we refer to Section 16.4.3 of Steehouwer (2005). Here we just want to stress the potential
danger and impact of modeling long term behavior based on short term data. Conventional annual
equity returns and annual inflation rates will show the same type of negative short term high frequency
correlation. If we estimate a model on this data there is a risk that the implied long term correlation will
also be negative, instead of being positive as the data tells us. It is not hard to imagine that in terms of
for example SAA decision making for a pension plan with inflation driven liabilities, working with a
negative instead of a positive long term correlation between equity returns and inflation rates will have
an enormous negative impact on the amount of equities in the optimal SAA.
To avoid this ‘perspective distortion’ and have sufficient information for modeling both the long term
(low frequency) and short term (high frequency) behavior of economic and financial time series, we
would ideally have very long sample time series (say covering several centuries) with high observation
frequencies (say daily). Based on this data we would then apply the decomposition, time series
analysis, modeling and model analysis steps of the methodology described in Section 3. Of course
such ideal time series data is not available in most cases. However, by accommodating the use of
samples of different sizes and observation frequencies for the long and short term behavior in terms of
the trend and various frequency models, the methodology does allow an optimal use of all the time
th
series data that is available. In the examples given in Section 3.3.1 and 0, we used 20 century cross
section data for 16 countries for the trend model, five-annual 1870-2006 data for the low frequency
model, annual 1970-2006 data for the high frequency (business cycle) model and monthly 1990:012005:12 data for the monthly frequencies. A consistent use of these different data sources in the
methodology is achieved by means of applying an appropriate decomposition approach. For example,
if we use the annual sample above to describe the business cycle behavior as all fluctuation in the
time series with a period length between 15 and 2 years, we should make sure that these types of
business cycle fluctuations are excluded from the monthly sample above by filtering out all fluctuation
longer than 24 months from the monthly time series. Note that some kind of interpolation method
might be needed to align all frequency models on the same (highest) observation frequency.
An obvious comment on the use of using very long samples of data, in some cases covering several
centuries, might be to ask how relevant and representative data this far back still is for the current and
future behavior of economic and financial variables. There are two answers to this question. The first
is that based on thorough empirical analysis of historical time series data, one might be surprised
about the amount of ‘stability’ there actually is in the behavior of economic and financial variables, not
only across time but also across countries. A nice quote to illustrate this in the context of business
cycle behavior comes from Lucas (1977) who states that “though there is absolutely no theoretical
reason to anticipate it, one is led by the facts to conclude that, with respect to the qualitative behavior
of co-movements among series, business cycles are all alike”. This remarkable stability of the
business cycle mechanism has also been reported more recently by for example Blackburn and Ravn
(1992), Backus and Kehoe (1992) and Englund et al. (1992) and Steehouwer (2005). Nevertheless
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also business cycle behavior has gradually changed over time and therefore we would be inclined to
use a relative recent sample (say 1970-2006) to model the business cycle behavior. The second
answer to this question of the relevance of long term data is to realize that all science starts with the
analysis and understanding of data that give us information about the phenomenon that we are
studying. Therefore, to understand and model the long term behavior of economic and financial
variables, by definition we have to start by studying long term time series data. Of course we can start
deviating from what the data tells us by incorporating theoretic or forward looking information but we
should start from the data at the very least. Jorion and Goetzmann (2000) illustrate this approach by
stating that “Financial archaeology involves digging through reams of financial data in search for
answers”. It is known that different economic regimes and historical circumstances are underlying the
behavior observed in long term historical time series data. One could argue that this is not a problem
of using long term data but that in fact such changes in economic regimes and historical
circumstances are exactly what drive the uncertainty and behavior of financial variables in the long
run. So by modeling directly on long term historical data we are taking a kind of stochastic approach to
regime switching. Just as we view annual business cycle observations for a 1970-2006 sample as
realizations of some underlying business cycle process, we can also view five-annual observations of
low frequency behavior for a 1870-2006 sample as realizations of some underlying long term process.
As far as regime changes affect the short term high frequency behavior of economic and financial
variables, this is exactly what the high frequency models should adequately describe. It is for example
well known that at the business cycle frequencies consumer prices have changed from a coincidental
behavior into a lagging behavior when compared to the GDP. We can model this (lagging) behavior
adequately by using a sufficiently recent sample for the business cycle frequencies. In other cases we
might also want to model the dependency of the high frequency behavior on the low frequency
behavior directly, for example in terms of the ‘level effect’ in the short term volatility of interest rates
described in Section 3.3.3. In this way the described methodology allows for an optimal use of the
available time series data and allows us to use the appropriate samples and observation frequencies
for modeling the various long and short term economic phenomena.
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4.4 Equal Importance of all Frequencies
A fourth reason for proposing the frequency domain methodology for time series modeling is that it
considers the behavior of economic and financial variables in all frequency ranges, and thereby also in
terms of the long and short term behavior, of equal importance. Therefore the methodology does not
put the focus on either the long term low frequency behavior or the short term high frequency
behavior, but allows us to focus on the long and short term behavior at the same time. We can best
explain this point by again using the long term interest rate time series introduced in Section 3.1. In
Section 4.3 we argued that a first reason that conventional modeling approaches tend to model the
long term behavior of economic and financial variables based on an ‘extrapolation’ of the short term
behavior is that too short samples are being used to be able to contain fundamental information about
the long term behavior in the first place. However, even if long samples are used, there is still a
second risk of ‘extrapolating’ the short term behavior into the long term. This is so because in addition
to the sample, also the representation of the time series data used for the modeling plays an important
role. Especially, the effects of the often applied first order differencing operator are important here.
This filter is often applied to model for example period to period changes of variables such as interest
rates or period to period returns of variables such as equity prices. From the PTF of the filter in Figure
2.1 we already saw that in terms of a frequency domain point of view, the first order differencing filter
suppresses the low frequency behavior and amplifies the high frequency behavior in time series. This
effect is clearly visible in the left hand panel of Figure 4.2 which shows the original (year end) levels
together with annual changes of the interest rate. In the annual changes, the information about what
maximum and minimum levels the interest rate has achieved or what type of long term fluctuations it
has experienced is missing while at the same time the short term fluctuations dominate the time series
of annual changes.
Because of the clearly different types of fluctuations that dominate the variance of the level and the
annual changes of the interest rate time series (just by using another representation of the data), it is
not hard to imagine that models estimated on the level will do well at describing the long term behavior
while models estimated on the annual changes will do well at describing the short term behavior. So,
by focusing on long sample time series but still using representations of the time series in terms of
period to period changes or returns, conventional approaches are still focusing on the short term high
frequency behavior and still run the risk of not adequately capturing the true long term low frequency
behavior, simply because the latter type of information has been severely suppressed in the time
series data used for estimating the models. Of course the short term behavior of interest rates is
important in some applications, for example for modeling the returns on fixed income portfolios.
However, in other applications the long term behavior of interest rates is important, for example to
determine how large the uncertainly about the mark to market value of accrued pension liabilities is.
That is, the behavior of economic and financial variables at all frequencies is in principle of equal
importance and we should therefore avoid to be (implicitly) putting more importance on the behavior in
one specific frequency range.
How does the proposed methodology solve this problem? The answer is by using an appropriate
decomposition approach as the one described in Section 3.1. The frequency domain filter proposed
for these purposes, does not (unintended) amplify nor suppresses the behavior in certain frequency
ranges. Instead, it just cuts up the behavior into different frequency regions. In technical terms, the
PTF of the filters described that are used, only have values of 0 or 1 and together cover exactly the
range of all possible frequencies. We can see the benefits of this approach in the right hand panel of
Figure 4.2. This is the same filter output as shown in Figure 3.1 with the only exception that we added
the trend and low frequency components. If we add the resulting low and high frequency components
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we again get the original interest rate time series. This is not possible for the annual changes because
in that case there is no second component time series which contains for example the suppressed
part of the low frequency behavior. Comparing the right and left hand panel it is easy to see that the
low frequency component captures the long term behavior of the original level of the interest rate while
the high frequency component captures the short term behavior in terms of the annual changes of the
interest rate. By constructing separate models for the low and high frequency components we are
capable of modeling both the long term low frequency and short term high frequency behavior of
economic and financial variables adequately at the same time instead of focusing on of the two while
we know that both can be of equal importance.
Figure 4.2: The benefits of the decomposition approach. The left hand panel shows the original
level and the annual changes of the long term nominal interest rate for the Netherlands from
Figure 3.1. The right hand panel shows the sum of the trend and low frequency component
together with the high frequency component from the same Figure 3.1. Comparing the two
panels shows how the low frequency component captures the behavior of the long term level
of the interest rate while the high frequency component captures its short term annual
changes. If one models both the low and high frequency component separately, one can
therefore adequately model both the level and annual changes of the interest rate at the same
time. The left hand panel shows that if one for example tries to model the complete behavior in
terms of the annual changes, this becomes rather difficult as the information about the long
term level of the interest rate has been suppressed and is therefore hardly visible in the annual
changes.
4.4.1 Forecasting consequences
What are the possible consequences of using (conventional) modeling approaches that (implicitly)
give an unequal importance to the different frequency ranges? A first possible consequence concerns
the forecasting performance, and especially the long term forecasting performance. To see this,
assume that indeed there is some valuable information in the long term low frequency behavior of
economic and financial time series and let us compare the forecasts of two modeling approaches. The
first is a conventional modeling approach that ‘ignores’ this low frequency information by either using
relatively short samples and / or using (first order differencing) representations of the data that
suppress this low frequency information. The second approach uses the proposed decomposition
approach that explicitly takes the (unchanged) low frequency information into account. For the first
approach we estimated an autoregressive model on the annual (log) GDP growth rates for the
Netherlands and produced forecasts and confidence intervals for a horizon of 50 years into the future.
The results are shown in the left hand panel of Figure 4.3. The right hand panel shows the same
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results but now for the case in which we first decomposed the (log) GDP time series into low and high
frequency components on which we then estimated separate autoregressive models. The most
striking difference between the two forecasts (the median central solid line), is that for the conventional
approach the forecast soon becomes a rather uninformative flat trending line while for the
decomposition approach it still fluctuates and is therefore informative during the complete 50 year
horizon. Although this is a topic for future research, preliminary results from a formal backtesting
procedure have already indicated that the decomposition approach can indeed lead to smaller
forecast errors when compared to conventional approaches in which no decomposition is applied. This
indicates that there is some valuable information in the low frequency behavior of time series that can
be (better) exploited by the decomposition approach.
Figure 4.3: Out of sample forecasts and confidence intervals of log GDP in the Netherlands.
Left hand panel show results of a conventional modeling approach based on annual growth
rates. Right hand panel shows results of a decomposition approach based on separate low and
high frequency components in the log GDP time series. Fluctuations with long period lengths
are suppressed by the PTF of the first order differencing operator and are therefore hardly
present in the growth rate time series. As a result, the forecasts of the model estimated on
these growth rates as shown in the left panel are very uninformative in terms of the low
frequency behavior of the GDP series. The right panel clearly also shows low frequency
information in the forecasts from the decomposition approach.
4.4.2 Monte Carlo experiment
In addition to the backtesting experiments that point in the direction of a superior forecasting
performance of the decomposition approach, we have performed a formal Monte Carlo experiment to
show that the decomposition approach works better in terms of modeling both the long term low
frequency and short term high frequency behavior well at the same time. We already explained the
intuition why this may be so based on the results shown in Figure 4.2 and also the results of the tests
in Chapter 19 of Steehouwer (2005) already pointed in this direction. Here we describe and give the
results of a Monte Carlo experiment which adds formal evidence to this claim.
We assume the Data Generating Process (DGP) for a variable zt defined by (4.2) where t is measured
in years and of which the dynamic properties are inspired by the empirical behavior of the long term
interest rate time series shown in Figure 3.1. This process consists of two independent components, a
low frequency component xt and a high frequency component yt which are both driven by a specific
autoregressive process.
Table 4.1 show the means and variances of these two components from which we see that the low
frequency component describes 80% of the total variance and the high frequency component 20%.
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zt = xt + yt
(4.2)
with
xt = 1.89 xt −1 − 0.90 xt − 2 + ε t
ε t ~ N (0,0.0027 2 )
yt = 0.65 yt −1 − 0.34 yt −2 − 0.26 yt −3 − 0.20 yt −4 + 0.26 yt −5 − 0.46 yt −6 + γ t
γ t ~ N (0,0.08102 )
Table 4.1: Mean and variance of low, high frequency and total model from (4.2). The total model
has a mean of 0 and a variance of 1 which for 80% comes from the low frequency model and
for 20% from the high frequency model.
xt
yt
zt = xt+yt
∆zt
Mean
0.00
0.00
0.00
0.00
Variance
0.80
0.20
1.00
0.19
Table 4.2: Complex roots of low and high frequency models from (4.2). The low frequency
model (xt) describes pseudo period behavior with a period length of around 50 years. The low
frequency model (yt) describes fluctuations with a period length of around 10, 5 and 2.5 years.
Root 1 xt
Root 1 yt
Root 2 yt
Root 3 yt
Modulus
0.95
0.92
0.87
0.85
Frequency
0.02
0.10
0.20
0.40
Period length
50
10
5
2.5
Table 4.2 shows the modulus, frequency and corresponding period length of the complex roots of the
two autoregressive polynomials. From this we see that the low frequency model describes pseudo
periodic behavior with a period length of around 50 years while the high frequency model is composed
of three types of pseudo periodic behavior with a period length of around respectively 10, 5 and 2.5
years. These types of pseudo period behavior are clearly visible in the (non-normalized) spectral
densities in the top two panels of Figure 4.4 which notably integrate to the total variances of 0.80 and
0.20. The bottom left panel shows the implied total spectral density of the DGP (4.2) in terms of the
level of zt. The bottom right hand panel of Figure 4.4 shows the implied spectral density of the annual
changes of zt, that is of ∆zt. Note how the PTF of the first order differencing operator (∆) shown in
Figure 2.1 has rescaled the spectral density by reducing the importance of the low frequencies and
increased the importance of the high frequencies in the process. The latter two spectral densities are
accompanied by the spectral densities of good approximating autoregressive models. This shows that
the DGP of both zt and ∆zt can be well described by (different) autoregressive models. Figure 4.5
shows an example simulation of 200 years from the xt, yt, zt=xt+yt and ∆zt processes from which we
can clearly see the behavior described by the roots and spectral densities of the models.
Based on the DGP (4.2), the Monte Carlo experiment is now set up as follows. We generate 1000
simulations of sample sizes of 50, 100, 200, 500 and 1000 years. In all cases we simulate a presample of 500 years to guarantee that all simulations adequately represent the unconditional
distributions. For each of the individual simulations we use the Yule-Walker estimation technique to try
to back out the original GDP of zt from the simulation. Tests described in Section 4.7 of Steehouwer
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(2005) show that this estimation technique shows the best performance in terms of estimating spectral
densities, especially in small samples. In large samples many estimation techniques show a similar
performance because of identical asymptotic properties. We compare three approaches:
1. Level approach: Estimate an autoregressive model on the simulated level zt
2. Delta approach: Estimate an autoregressive model on the simulated annual changes ∆zt
3. Decomposition approach: Estimate separate autoregressive models on the underlying simulated
low and high frequency components xt and yt
For each simulation we use each of these approaches to estimate the spectral densities of both zt and
∆zt. The spectral densities can be calculated directly from the parameters of the autoregressive
parameters or by applying the Fourier transform on the auto-covariances of the estimated model up to
a sufficiently high order. If the estimation has been done in terms of zt, the spectral density of ∆zt is
calculated by applying the PTF of the first order differencing filter on the estimated spectral density of
zt. If, the other way around, the estimation has been done in terms of ∆zt, the spectral density of zt is
calculated by applying the PTF of the inverse first order differencing filter on the estimated spectral
density of ∆zt. We then calculate the errors between the estimated spectral densities and the true
spectral densities given in the bottom two panels of Figure 4.4 as
b
S error = ∫ Sˆ (ω ) − S (ω ) dω
a
b
∫ S (ω )dω
(4.3)
a
Here S (ω ) and Sˆ (ω ) are respectively the non-normalized auto-spectrum of the known underlying
DGP and its estimated counterpart. The integration interval [a,b] defines the frequency range over
which the error is calculated. The error (4.3) measures the ‘distance’ between the estimated and the
DGP spectra on the indicated frequency range. The smaller this distance, the better the estimated
process corresponds to the original DGP and hence the better is the performance of the relevant
approach. We calculate these errors for in total six cases that are combinations of the DGP
representation in terms of the level zt or the annual changes ∆zt on the one hand and different
frequency regions on the other hand. These combinations are indicated in Table 4.3. The split
between the low and high frequency ranges is made at a frequency of 0.0667 which is equivalent to
fluctuations with a period length of 15 years.
Table 4.3: Six combinations of DGP representation and frequency ranges for which
error (4.3) is calculated. Frequencies hold in terms of cycles per year.
Frequency range
Level zt
Annual change ∆zt
Total
[0.0000, 0.5000]
[0.0000, 0.5000]
Low
[0.0000, 0.0667]
[0.0000, 0.0667]
High
[0.0667, 0.5000]
[0.0667, 0.5000]
To be sure we get the best out of each approach, for each estimation we calculate the errors for
autoregressive models of all orders between 1 and 25 (this range includes the orders needed to
closely approximate the model spectra as shown in the bottom two panels of Figure 4.4) and select
the order which gives the smallest error in terms of the representation of the DGP used in the different
approaches (levels, annual changes or separate low and high frequency components). Finally, we
calculate and compare the mean errors from the calculated errors for each of the 1000 simulations.
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Figure 4.4: Non-normalized spectral densities of xt (top left panel), yt (top right panel), level of zt
(bottom left panel) and annual changes of zt (bottom right panel) for models from (4.2). The
latter two are accompanied by the spectral densities of close approximating autoregressive
models. The complex roots from Table 4.2 are clearly visible in the spectral densities.
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Figure 4.5: Example simulation of 200 years of xt (top left panel), yt (top right panel), level of zt
(bottom left panel) and annual changes of zt (bottom right panel) for models from (4.2).
The results of the experiment are reported in Table 4.4. A mean error of for example 0.50 for the level
approach (second column in each block) says that the level approach on average results in a wrong
allocation of 50% of the variance over the frequencies in the estimated spectral densities. The mean
errors for the delta and decomposition approaches (third and fourth columns in each block) are
reported as a percentage of the mean error of the level approach. A value of for example -0.25% for
the delta approach says that the delta approach results in a mean error which is 25% lower than the
mean error for the level approach. In case of the example, the mean error of the delta approach is
75%×0.50=0.375.
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Table 4.4: Mean errors (4.3) for each of the six combinations in Table 4.3 based on 1000
simulations. The mean errors of the delta and decomposition approaches (third and fourth
columns in each block) are reported as a percentage of the mean error of the level approach
(second column in each block).
Mean error delta (∆zt) total
Mean error level (zt) total
Sample
Level
Delta
Decomp
Sample
Level
Delta
Decomp
50
0.63
6889%
-3%
50
0.65
-26%
-20%
100
0.49
7894%
-6%
100
0.50
-20%
-24%
200
0.37
9655%
-8%
200
0.36
-12%
-26%
500
0.25
13995%
-13%
500
0.25
-5%
-31%
1000
0.18
15372%
-16%
1000
0.19
0%
-34%
Mean error delta (∆zt ) low
Mean error level (zt) low
Sample
Level
Delta
Decomp
Sample
Level
Delta
Decomp
50
0.51
8415%
-1%
50
0.06
12%
-12%
100
0.40
9742%
-3%
100
0.04
32%
-14%
200
0.30
11992%
-4%
200
0.03
68%
-14%
500
0.20
17901%
-8%
500
0.02
132%
-20%
1000
0.14
20003%
-11%
1000
0.01
184%
-22%
Mean error delta (∆zt) high
Mean error level (zt ) high
Sample
Level
Delta
Decomp
Sample
Level
Delta
Decomp
50
0.11
-13%
-14%
50
0.60
-30%
-21%
100
0.09
-10%
-21%
100
0.46
-25%
-25%
200
0.07
-8%
-25%
200
0.33
-19%
-27%
500
0.05
-9%
-32%
500
0.24
-16%
-32%
1000
0.04
-9%
-34%
1000
0.18
-12%
-35%
From the results in Table 4.4 we observe the following:
1. Larger samples lead to smaller errors in terms of reproducing the original DGP spectra.
2. In terms of the total frequency range (top two panels in Table 4.4), the level approach is best at
reproducing the spectral density of the levels while the delta approach is best at reproducing the
spectral density of the annual changes.
3. The delta approach is better at reproducing the high frequency part of the spectral density of the
levels (bottom left panel in Table 4.4) while the level approach is better at reproducing the low
frequency part of the spectral density of the annual changes (middle right panel in Table 4.4).
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4. In virtually all cases, the decomposition approach is better at reproducing the spectral densities of
the levels and annual changes, both in terms of the total as in terms of the separate low and high
frequency ranges.
Observation 1 is of course no surprise. Observation 2 confirms what we explained about the left hand
panel of Figure 4.2. That is, models estimated on the levels will do well at describing the long term
behavior of the levels while models estimated on the annual changes will do well at describing the
short term behavior of the annual changes because of the clearly different types of fluctuations that
dominate the variance of the processes of the levels and the annual changes. Note that the errors for
the delta approach in terms of the low (and total) frequency ranges of the spectral density of the levels
are so large because of the PTF of the inverse first order differencing filter which approaches infinity at
very low frequencies. Another way of seeing this is that in those cases we are modeling a stationary
process in terms of a non-stationary (integrated) process. Observation 3 is somewhat more of a
surprise but only further emphasizes the same point as observation 2. As we saw in Section 2.3, the
PTF of the first order differencing filter suppresses the low frequency behavior and amplifies the high
frequency behavior. Therefore the delta approach performs even better than the level approach itself
at the high frequencies of the level DGP. The other way around, the delta approach itself performs
even worse than the level approach at the low frequencies of the annual changes DGP. Observation 4
obviously closes this experiment by confirming what we explained about the right hand panel of Figure
4.2. That is, by constructing separate models for the low and high frequency components we are
capable of modeling both the long term low frequency (levels) behavior and the short term high
frequency (annual changes) behavior adequately at the same time instead of doing well in terms of the
low frequency behavior and poorly in terms of the high frequency behavior or the other way around.
Thereby, the Monte Carlo experiment gives formal support for the claim that a decomposition
approach can lead to superior modeling results in terms of describing the behavior of economic and
financial variables in all frequency ranges.
4.5 Complex Dependencies between Frequency Ranges
A fifth and final reason for proposing the frequency domain methodology for time series modeling is
that it facilitates the modeling of complex dependencies between the behavior of economic and
financial variables in different frequency ranges. At first sight, the zero correlation property described
in Section 3.1.3 may seem like a restrictive simplifying feature of the decomposition approach. As
explained in Section 3.3.3, by modeling different frequency ranges ‘separately’ we actually get more
rather than less possibilities for modeling complex behavior by explicitly modeling relations between
the properties of the different frequency models. Examples mentioned in Section 3.3.3 were the ‘level
effect’ in the short term volatility of interest and inflation rates, state dependent business cycle
dynamics and state dependent asset correlations.
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5 Conclusions
In this paper we have described a frequency domain methodology for time series modeling. With this
methodology it is possible to construct time series models that in the first place give a better
description of the empirical long term behavior of economic and financial variables which is very
important for SAA decision making. In the second place, the methodology brings together the
empirical behavior of these variables as observed at different horizons and observation frequencies
which is required for constructing a consistent framework to be used in the different steps of an
investment process. In the third place, the methodology gives insight in and understanding of the
corresponding dynamic behavior, both in terms of empirical time series data and of the time series
models used to describe this behavior. In the various parts of the paper we introduced the most
important frequency domain techniques and concepts, described and illustrated the methodology and
finally gave the motivation for doing so. We hope that based on the contents of this paper, more
people will be inclined to explore the possibilities of using the appropriate frequency domain
techniques for analyzing and modeling time series data and time series processes of economic and
financial variables. We are convinced that this can contribute to a higher quality of investment decision
making, implementation and monitoring in general, and for Central Banks and Sovereign Wealth
Managers in particular.
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