ï»¿ WPS6578
Policy Research Working Paper 6578
Growth and Volatility Analysis Using Wavelets
Inga Maslova
Harun Onder
Apurva Sanghi
The World Bank
Poverty Reduction and Economic Management Network
Economic Policy and Debt Department
August 2013
Policy Research Working Paper 6578
Abstract
The magnitude and persistence of growth in gross decomposition techniques, as demonstrated on a small
domestic product are topics of intense scrutiny by sample of countries. In addition to having desirable
economists. Although the existing techniques provide a technical advantages, such as localization in time and
range of tools to study the nature of growth and volatility frequency and the ability to work with non-stationary
time series, these usually come with shortcomings, series, these techniques also make it possible to accurately
including the need to arbitrarily define acceleration decompose the association between growth trajectories
spells, and focus on a particular frequency at a time. of different countries over different time horizons. Such
This paper explores the application of â€œwavelet-basedâ€? â€œco-movementâ€? analysis can provide policy makers with
techniques to study the time-varying nature of growth important insights on regional integration, growth poles,
and volatility. These techniques lend themselves to and how short and long term developments in other
a more robust analysis of short-term and long-term countries affect their domestic economy.
determinants of growth and volatility than the traditional
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Growth and Volatility Analysis Using Wavelets
Inga Maslova, Harun Onder, Apurva Sanghi
JEL Classiï¬?cation: O47, C15, C10.
Keywords: Growth, volatility, growth acceleration, wavelets.
Sector Board: Economic Policy (EPOL).
Inga Maslova is an assistant professor in the American University (maslova@american.edu), Harun
Onder is an economist in the Economic Policy and Debt Department (PRMED) of the World Bank
(honder@worldbank.org), and Apurva Sanghi is a lead economist in the Africa Poverty Reduction and
Economic Management Unit (AFTP2) of the World Bank (asanghi@worldbank.org). The authors would
like to thank Cesar Calderon, Ayhan Kose, Theo Janse van Rensburg, Norman Loayza, Milan Brahmbatt,
Brian Pinto, William Battaile, Stefanie Sieber, Daniel Lederman, Ralph van Doorn, and Amna Raza for
valuable comments and suggestions. This study was made possible by a grant from the Research Support
Budget (RSB) of the World Bank.
1
1 Introduction
An everlasting quest for policy makers is how to promote rapid and sustained growth. In
practice, many economies have grown rapidly for short periods of time. However, sustain-
ing the same performance in a longer time horizon is much less common. The diï¬€erences
between developed and developing economies are particularly striking in this regard. In
many developed countries, a substantial part of the evolution of per capita GDP can be
summarized by a single statistic: the average growth rate over time (Pritchett (2000)).
This is mainly because the growth process is relatively stable in these countries, e.g. the
variation around the long term trend is small. In comparison, growth exhibits signiï¬?cant
volatility and instability in the majority of the developing countries. Frequent breaks in
the long term trend as well as large variations around the trend are common. Therefore,
the average growth rate will only explain a relatively small share of the information in
these cases. We also need to investigate the pattern of ï¬‚uctuations in order to understand
the determinants of growth.
Starting from a similar observation, Hausmann et al. (2005) suggest that identi-
fying the clear shifts in growth (breaks in the trend or volatility around the trend) can
shed light on the relationship between growth and its fundamental determinants. Using
a stylized deï¬?nition of growth acceleration episodes, which is based on the magnitude
and persistence of growth (e.g. an increase in per capita growth of 2 percentage points
or more for 8 consecutive years), they ï¬?nd that the relationship between growth and its
determinants varies on the basis of the time frame of the analysis. For instance, economic
reform for openness, which is measured by a number of factors including structural (e.g.
presence/absence of marketing boards) as well as macroeconomic (e.g. presence/absence
of a large black market premium for foreign currencies) indicators, is found to be a signiï¬?-
cant determinant of growth accelerations that are sustained over the longer term, whereas
externals shocks, deï¬?ned as substantial improvements in the countryâ€™s Terms of Trade,
are found to generate growth accelerations that die out in the short term.
In this paper, we demonstrate a useful methodology to study the time-varying
characteristics of growth in ï¬?ne detail. Using a wavelet-based technique, we decompose the
time series into high frequency (transitory) and low frequency (persistent) components.
This, in turn, enables us to identify the growth acceleration and deceleration phases
2
without using arbitrary restrictions. Therefore, this technique lends itself to a robust
analysis of the short-term and long-term determinants of growth. The same approach
is extended to analyzing the volatility of the GDP series, where the focus is on changes
in the growth rates as well as the levels of GDP. Figure 1 shows a decomposition of
GDP per capita growth series in the United States by using this technique. Changes
in the actual growth series between 1960 and 2010 (the top row) are decomposed into
subcomponents due to variations at 2-4 years frequency (D1), 4-8 years frequency (D2),
8-16 years frequency (D3), and 16-32 years frequency (D4). Finally, the wavelet smooth
(S4) denotes the trend term in the series.
Wavelet-based techniques have certain desirable characteristics that prove to be
useful in growth and volatility analysis. First, wavelet decomposition provides an uncor-
related set of frequency scales, i.e. the sum of components is equal to the original series.
When analyzing the growth ï¬‚uctuations, this ensures that volatility due to diï¬€erent time
scales are fully identiï¬?ed. This is not the case for common ï¬?ltering techniques such as
Hodrick-Prescott, where information â€œleaksâ€? while ï¬?ltering the series consecutively in or-
der to separate the diï¬€erent frequencies. Second, wavelet decomposition is localized both
in time and frequency, and the time domain and frequency domain information of the
original series are preserved (the horizontal and vertical axes in Figure 1). Therefore,
one-oï¬€ events such as crises do not aï¬€ect the decomposition at other points in time. In
contrast, with traditional spectral analysis techniques, such as the Fourier transformation,
the information is spread over the entire period of analysis. Therefore, one-oï¬€ events have
global impacts. Overall, these characteristics suggests that the wavelet techniques can be
employed in several policy related studies including commodity price diagnostics and fea-
sibility studies for economic unions. Table 1 shows a set of potential areas where wavelet
techniques can be employed to enhance the existing analytical approaches.
This paper proceeds as follows. The next section discusses the fundamental char-
acteristics of wavelet-based techniques with a comparison to other frequently used ap-
proaches in a non-technical manner. The third section introduces a basic description of
wavelets for beginners. A more technical desription of wavelet transform with an em-
There is a well established literature that investigates various aspects of volatility and growth rela-
tionship. For the impact of volatility on long term average growth rates, see Burnside and Tabova (2009),
Hnatkovska and Loayza (2004), and Ramey and Ramey (1995); for the impact of openness on business
cycle volatility and synchronization, see Calderon et al. (2007), Kose et al. (2003).
3
USA
âˆ’4 2
Time
âˆ’4 2
D1
Time
âˆ’4 2
D2
Time
âˆ’4 2
D3
Time
âˆ’4 2
D4
Time
âˆ’4 2
S4
1961 1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005 2009
Figure 1: A wavelet-based multiresolution decomposition of income per capita growth
series of USA.
Notes: This multiresolution decomposition is performed using Maximum Overlap Discrete
Wavelet Transformation (MODWT) on ï¬?rst diï¬€erence of the annual series from Penn World
Tables 7.0. It is implemented using the pyramid algorithm shown in Figure 3. The top panel
shows the actual series (growth rate of income per capita). Variations due to 2 â€“ 4 year frequency
oscillations are shown in the second panel (D1 ), others as follows: 4 â€“ 8 year frequencies in the
third panel (D2 ), 8 â€“ 16 year frequencies in the fourth panel (D3 )and 16 â€“ 32 year frequencies
in the ï¬?fth panel (D4 ). The last panel (S4 ) shows the â€œsmoothâ€? component, e.g. all frequencies
lower than 16 years. These components are approximately independent to each other and the
original series can be recovered by aggregating the four sub-components.
4
Table 1: Potential Applications of Wavelets
Applications Countries of Primary
Interest
X Identify acceleration and deceleration phases All countries
Growth Analytics (hills, plateaus, mountains, and plains)
X Identify structural breaks All countries
X Investigate the country resilience by analyzing the All countries
persistence of impacts due to diï¬€erent types of
shocks
Synchronization X Analyze the co-movement of growth between two All countries
Analysis economies
X Investigate the feasibility of economic union Trade/monetary union
formation (monetary union, free trade areas, members or candidates
customs union) by analyzing the cyclical
synchronization among a group of economies
Commodity Price X Analyze the short-term and long-term behavior of Commodity traders,
Diagnostics key commodity prices resource-rich countries
X Investigate the co-movement of commodity prices Commodity traders,
and desired macroeconomic aggregates resource-rich countries
5
phasis on Maximum Overlap Discrete Wavelet Transformation (MODWT) and wavelet
variance analysis is presented in the appendix. The fourth section demonstrates an ap-
plication of the wavelet scalogram using income, consumption, and investment series of a
selected group of countries. The ï¬?fth section investigates the characteristics of variance
and covariance of these series. The sixth section introduces an analysis of co-movement
of growth across the countries. The last section concludes.
2 The Advantages of Using Wavelet Techniques
Economists have long been aware of the time varying characteristics of economic phe-
nomena. Traditionally, these variations have been analyzed by using various spectral
analysis methods, which enable decomposition of the time series into an independent set
of frequency components. However, this is done under relatively strict assumptions in
the spectral analysis. Fourier transformation is used to decompose a series into sinusoidal
components when the series is stationary, and preservation of the information in time is
not required (e.g. Granger (1966) and Nerlove (1964)). In the case of non-stationary data,
however, the original series is ï¬?ltered to be made stationary, which does not preserve all
information from the series. Moreover, a single event in time, or an extension of the series
by including new data points, can change the analysis at all frequencies; hence, the de-
composition is not localized. This could be resolved by using a windowed Fourier analysis.
However, it has weakness similar to the moving window averaging methods. It requires
selection of a window where data are stationary and assumes that volatility range does
not change over time. Wavelet-based techniques provide a robust alternative by allowing
us to perform a volatility analysis in frequency domain with minimal speciï¬?cations of
analysis parameters.
In contrast to classical spectral analysis, wavelet-based techniques provide a decom-
position that is localized both in time and frequency. By combining several combinations
Consumption smoothing over an economic agentâ€™s lifetime is one example. Permanent Income Hy-
pothesis (PIH) suggests that agents consume out of permanent incomes and (dis)save out of transitory
incomes, which implies that the marginal propensity to consume is expected to be greater for the former
than the latter. Corbae et al. (1994) show that the marginal propensity to consume at high frequencies is
lower than at low frequencies. Hence, decomposing the interaction into diï¬€erent time horizon components
provides a better approximation of the true nature of the relationship.
6
of scaled and shifted versions of the mother wavelet (basis function), the wavelet trans-
formation captures the localized information in time domain and presents the associated
frequency information along with it. Therefore, standard time series measures such as
correlation and covariance can be employed to analyze the association of the variables in
the frequency scale of choice.
Another characteristic that makes the wavelet technique appealing in economic
analysis is its ability to work with non-stationary data. In the case of trending data,
detrending techniques like Hodrick-Prescott (HP) and band pass ï¬?lters are used to derive
the variations around the trend. These ï¬?lters require selecting a window width for aver-
aging on which the data are approximately stationary. Therefore, these processes depend
on the assumptions regarding the underlying properties of the data. Unlike the HP ï¬?lter,
wavelet-based ï¬?ltering does not require normality of the errors while extracting periodic
components associated with multiple frequencies. Furthermore, these derived components
are uncorrelated with each other. This enables the original series to be equal to the sum
of the components, which is not the case for HP ï¬?lter. In an attempt to analyze the
medium term business cycles across countries, Comin and Gertler (2006) note that be-
cause the medium and high frequency variations in the data are not independent after
the HP ï¬?ltering, it is not feasible to compare the two components in isolation. Wavelet
ï¬?ltering provides a feasible ï¬?ltering tool in similar conditions.
The class of non-stationarity that can be handled by the wavelet transform is
broader than the existence of a mere unit root process (Ramsey and Lampart (1998).
Time series models typically assume second order stationarity, i.e. the mean and the
covariance of the process do not change over the period of analysis. Therefore, structural
breaks require customized treatment depending on whether the break is considered to
be in the mean or in the variance. Wavelet transform, on the other hand, provides a
straightforward method to test and isolate the breaks. In the case of a sudden change in
variance, the high frequency components in wavelet transform contain the shift and the
low frequency components remain stationary. If the structural break is about the long
term relationship, then all frequency scales in wavelet transform will reï¬‚ect this (Gencay
et al. (2001)). As discussed in Ramsey (1999) the ability of wavelets to represent complex
structures without knowing the underlying functional form of the process is of great value
in economic and ï¬?nancial research.
7
Figure 2: Examples of â€œmother waveletsâ€?: (a) Haar, (b) a wavelet related to the ï¬?rst
derivative of the Gaussian PDF, (c) Daubechies, (d) Morlet (real component)
The next section provides a more formal introduction to the wavelet techniques.
3 An Introduction to the Wavelet Techniques
A wavelet (small wave) is a mathematical function with special characteristics, e.g. inte-
gration to zero and unit energy, that is used to transform a time series into components
corresponding to diï¬€erent frequency ranges. This is done by ï¬?ltering the original series
via a selected algorithm, which uses the scaled and shifted versions (daughter wavelets)
of the basis function (mother wavelet). Figure 2 demonstrates the commonly used basis
functions. The simplest example of a wavelet ï¬?lter is Haar mother wavelet, which is shown
in panel (a) of Figure 2. The mathematical deï¬?nition of this wavelet is the following;
ï£± âˆš
ï£´
ï£´ âˆ’ 1 / 2 âˆ’1 < t â‰¤ 0
ï£² âˆš
Ïˆ (t) = 1/ 2 0