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 “wavelet-based� “co-movement� 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 This paper is a product of the Economic Policy and Debt Department, Poverty Reduction and Economic Management Network. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http:// econ.worldbank.org. The authors may be contacted at honder@worldbank.org. The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent. Produced by the Research Support Team Growth and Volatility Analysis Using Wavelets Inga Maslova, Harun Onder, Apurva Sanghi JEL Classi�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 differences 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 signi�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 fluctuations 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 de�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 �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 signi�- cant determinant of growth accelerations that are sustained over the longer term, whereas externals shocks, de�ned as substantial improvements in the country’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 �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 fluctuations, this ensures that volatility due to different time scales are fully identi�ed. This is not the case for common �ltering techniques such as Hodrick-Prescott, where information “leaks� while �ltering the series consecutively in or- der to separate the different 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-off events such as crises do not affect 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-off 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 −4 2 Time −4 2 D1 Time −4 2 D2 Time −4 2 D3 Time −4 2 D4 Time −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 �rst difference 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 – 4 year frequency oscillations are shown in the second panel (D1 ), others as follows: 4 – 8 year frequencies in the third panel (D2 ), 8 – 16 year frequencies in the fourth panel (D3 )and 16 – 32 year frequencies in the �fth panel (D4 ). The last panel (S4 ) shows the “smooth� 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 different 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 �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 �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 speci�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 agent’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 different 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 �lters are used to derive the variations around the trend. These �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 �lter, wavelet-based �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 �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 �ltering, it is not feasible to compare the two components in isolation. Wavelet �ltering provides a feasible �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 reflect 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 �nancial research. 7 Figure 2: Examples of “mother wavelets�: (a) Haar, (b) a wavelet related to the �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 different frequency ranges. This is done by �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 �lter is Haar mother wavelet, which is shown in panel (a) of Figure 2. The mathematical de�nition of this wavelet is the following;  √   − 1 / 2 −1 < t ≤ 0  √ ψ (t) = 1/ 2 0