Wavelets for long waves: was Kondratieff’s intuition right?

The validity of the Kondratieff’s theory of long-waves is still a controversial issue among historians, economic, social and political science scholars. In this work, both original Kondratieff’s data (never processed by wavelet analysis in their whole dataset) jointly with additional among the longest and most representative economic indicators are investigated. By the application of this statistical technique, the main purpose of the paper is to furnish an additional information to the literature. We trace evidence on the presence of coherent periodicities in the case of original Kondratieff time series and for other relevant and up-dated economic indicators as well.

consistency of the K-waves hypothesis by the investigation of additional and representative leading economic indicators.
The paper is organized as follows. The next section presents WA and its relative properties in processing non stationary data when specifically compared with HA. Section 3 introduces the whole datasets object of investigation with dedicated unit root and linearity analyses jointly with related considerations about the selection of the variables. Section 4 exhibits and discusses main findings.
Finally, Conclusions are in Section 5.

Preliminary Considerations and the Application of the Methodology
In the present Section a brief overview of the WA related to HA is proposed. WA is a powerful mathematical tool for signal processing in the time-frequency domain able to overcome main drawbacks of HA (Kaiser, 2011).
HA -also known as Fourier Analysis (FA)-is a filtering approach of decomposition of a series y(t) into a sum of sinusoidal (with different frequency) components (Bloomfield, 2000). To convert the time domain into the frequency domain, the Discrete Fourier Transform (DFT) is adopted: for n = 0,1,2….
( −1) 2 wherein: -Zn = the complex number resulting from the DFT formed by a real (a) and an imaginary part identified by a lower-case i (ib); -T = the last term of the discrete series; -e = Euler's number (also known as Nepier's constant equal to 2.71…); -i = is the conventional √−1 for imaginary part; -2 = is the radians representation of the frequency (fnt).
Generally, to speed iteration in elaboration, the Fast Fourier Transform (FFT) dedicated class of algorithm is employed. Through this procedure, then it is possible to perform a spectral analysis of time series data to detect meaningful cyclical periods by implementing the power spectrum (Warner, 1998). Periodogram Intensities (Ik) can be mathematically derived as: where : -ak and bk are the coefficients of the numbers Zn for k = 1, 2, …K (K the last time period until the Nyquist-Shannon frequency; i.e. the minimum sampling period needed in order to identify a possible periodicity, usually represented as : 0 ≤ f ≤ 0.5 f ).
The sum of Periodogram Intensities represents all the variance of the time-series (Box et al., 2016): Nevertheless, such an approach has two inherent disadvantages: the need of a stationary behavior of the variable, and the absence of a correct positioning of the cyclic components over time. For these reasons, data are usually pre-treated with proper detrending procedures (first differencing for example). However, the elimination of trend can create spurious cycles (Beaudry et al., 2020).
Detrending diminishes the original information content of data resulting as not neutral to final findings (Freeman and Louça, 2001). The same holds true for band-pass filtering applications like for example the Baxter and King (1999) and/or Christiano and Fitzgerald (2003) common time domain filters.
As far as the correct positioning of the periodic component is concerned, it must be pointed out that in FA methods, a single disturbance affects all frequencies along the whole datasets through the sum of sine and cosine functions. For the Heisemberg uncertainty principle: the more certainty is related to the measurement of one dimension (for example frequency), the less certainty can be related to the other dimension (time location in time series analysis).
WA is well suited to approach all above mentioned issues. As a matter of fact, instead and differently from FA, its transform is localized both in time and in its functional components (Rhif et al., 2019). Such a feature allows appropriate investigation of non-stationary signals (Torrence and Compo, 1998;Houtveen and Molenaar, 2001;Aristizabal and Glavinovic, 2003;Cazelles et al., 2008 andSleziak et al., 2015). The approximations generated by the procedure are robust to small variations (Gallegati et al., 2017). WA provides an efficient way to deal with variables lasting for a finite time, or showing markedly different behavior in their time-sequence (Crowley, 2007), hence its adoption can be appropriated to analyze time series containing non stationary power at many different frequencies (Magazzino andMutascu, 2019 andDaubechies, 1990). Due to such a flexibility, many disciplines (among the others astronomy, climatology, engineering, medicine) have extensively applied WA, and without doubt, this approach is useful also for research in economics. Pioneering contributions in the analysis of the frequency nature of economic relationships have been proposed by Ramsey and Lampard (1998a,b).
Without entering into excessive mathematical details retrievable in specific references also for economic and finance applications (see for example Percival and Walden, 2000;Gençay et al., 2001;Ramsey, 2002;Schleicher, 2002 as well as Gallegati and Semmler, 2014), wavelets are small waves (or wave packets) representing the varying duration of the components of a time series (Walker, 2008). This allows to obtain an alternative representation in the timescale domain of an original time domain represented function. Several types of wavelet functions exist with proper characteristics: Daubechies, Haar, Mexican Hat and Morlet among the most widespread. In general, we can identify "father" () and "mother" ( wavelets. The first integrates to 1 and the second to 0: and Substantially, the "father" (or scaling function) represents the low-frequency part of the series in the transform calculation, while the "mother" wavelets stand for the high-frequencies. The zero mean and decaying property of the  represent the typical oscillations on the t-axis of the function behaving like a small wave losing its strength as it moves away from the center (Anguiar-Conraria and Soares, 2011). WA allows to a simultaneous estimation of several cyclical components, and its main characteristic lies in the possibility to separate out a variable into inner constituent (multi-resolution) components (Crowley, 2007). Thus, a multi-resolution decomposition (MRD) also termed as multi-resolution analysis (MRA) of y(t) can be represented as: whereas: -the basis functions j,k (t) and j,k (t) are assumed to be orthogonal and represented as: In sum, -the functions  and  satisfy conditions (4) and (5).
-j = 1, 2,… J indexes the maximum scale sustainable with the data to process (each scale represents a fixed interval of frequencies); -k indexes the translation parameter; -, are the trend smooth coefficients in the wavelet transform capturing the underlying behavior of the data at the coarsest scale; -, are the detail wavelet coefficients representing deviations from the smooth behavior.
Also termed as atoms or scale crystals for each scale (the higher the scale, the lower the frequency and/or inversely higher the period length) approximately they are given by the integrals of the following (Bruce and Gao, 1996): In a simpler manner a MRD of the variable yt is given by: wherein, represents the first term on the right side of equation (6), is the second term and so on. Transforms can be seen both in their continuous version in signal processing (CWT) and in their discrete one (DWT). Acting as a filtering approach to extract cycles at various frequencies from the data, DWT uses a given discrete function passed through the series and "convolved" with the data to yield the coefficients labeled as crystals. Convolution is a mathematical procedure to obtain a modified version of the original functions processing the signal (Crowley and Hallet, 2014). The case of discrete signal is very common in economics given that, generally, datasets consist of values sampled at evenly spaced points in time. In sum, with MRA it is possible to decompose any individual time series into its different timescale components (each corresponding to a specific frequency band) to properly isolate the stochastic periodical component of interest. The subsequent representation of the variability of the process on a scale-by-scale basis is then obtained by the energy decomposition analysis. Considering that orthogonal wavelets are energy (thus variance) preserving, and letting E expressing the total energy of a variable yt for j from 1 to J, the corresponding total energy decomposition can be represented as: with: as the energy power of the scaling coefficients (sum of the jth level approximation signal) and representing the energy power of scalar crystals (sum of all detail level signals from the first to jth).
Considering that energy decomposition allows to detect the most contributing scale component, the frequency (or inversely period) having the leading role in approximating the original signal can be derived (for frequency interpretation of MRD and corresponding scale levels see Table 1). proposed by Walden and Cristan (1998), unlike DWT, MODWT skips the downsampling after the filtering of the data producing a more asymptotically wavelet variance estimator (all else being equal). The basic principle of MODWT consists in passing the wavelet function down the series by data observation, rather than moving the wavelet function down the series to cover a completely new data span (Crowley and Hallet, 2014). Using matrix notation, and let x as a vector of N elements, the MODWT wavelet coefficients resulting from the transform represented by a vector (J+1) N is given by: (15) with ̃ representing the (J+1) N × N matrix defining the MODWT (Crowley, 2007).
In order to perform and implement any analysis with WA, some preliminary choices are essential.
Firstly, the family of wavelets must be defined. Secondly, the boundary conditions at the end of the series must be handled. In relation to the first aspect, several families of wavelet filters with different characteristics of the transfer function for what concerns both the filtering and filter length are available (Haar discrete, symmlets, and so on). On these points, the present choice is to apply the Daubechies least asymmetric (LA) having a length L = 8 (wherein eight are the non zero coefficients). In wavelet notation this filter is commonly expressed as LA (8). The LA (8) is the most widely adopted filter for economic applications both because it can be applied to a wide variety of data types and because of its ability to balance the most common desirable features for wavelets basis functions like smoothness, (a)symmetry and length (Gençay et al., 2001). As far as the boundary conditions are concerned, the choice is for a reflecting method (the original signal is reflected about its end points to obtain a series of length 2 N having the same statistical properties in mean and variance).

Datasets and related considerations
Since we are interested in assessing the plausibility of Kondratieff hypothesis and the potential current existence of long-waves in economics through the analysis of appropriate indicators, we use two different datasets.
The first group encompasses all the original data presented by Kondratieff (1926) as they can be retrieved in Gattei (1981). The full list is: -England-Lead production 1855-1920 in t/1,000 inhabitant (L).
All these datasets have been object of exploration through a FA with spectral analysis for the very first time by Focacci (2017). Metz (2011) analyzed just one Kodratieff data series adopting spectral methods, while outliers identification, modelling and bandwidth of filtering process for successfully detection were the main focus by van Ewijk (1982).
In Table 2 the Shapiro-Wilk and Jarque-Bera tests for normality, and four among the most widespread and well-known unit root tests (Augmented Dickey-Fuller: ADF, Augmented Dickey-Fuller Generalized Least Squares Regression: ADF-GLS,Kwiatkowsky-Phillips-Schmidt-Shin: KPSS and Phillips-Perron: PP) are applied. Additionally, since WA is one of the most powerful tool for investigating complex nonlinear data, linearity properties are explored by the Keenan and the Brock-Dechert-Scheinkman (BDS) tests.  Their statistical properties are summarized in Table 3.
The selection of this second group of data is accounted for different reasons. As far as GDP measure is concerned, it must be pointed out it remains the widespread and leading measure internationally employed when approaching economic themes (despite several critics and adverse theoretical opinions arguing again its real effectiveness in representing people's material wellbeing

WA estimation of long waves in the series
In the present Section all findings are proposed. Considering the presence of unit roots and nonlinearity depicted in Table 2 and 3, the adoption of WA is an attractive method to pursue the aim of the research and, contrary to statistical intuition, to strongly relax length issues (observations for original Kondratieff Series from A to L are shorter than for Series from M to P). WA transform proceeds by processing pieces of fixed length for a given period. The overall average power is not related to the length of the series. The shorter the series, the lesser the pieces having the same characterstics (Rösch and Schmidbauer, 2018).
A further aspect to highlight concerns potential outliers. A formal theoretical evaluation can be conducted both to detect and model them. No specific treatment is here applied for three reasons.
Firstly, original data are not normally distributed (as Shapiro-Wilk and Jarque-Bera tests within Tables 2 and 3  However, considering that the highest scale (lower frequency) can only just be resolved, for practical applications it is usually recommended to diminish of an additional unit the number of crystals to produce (Crowley, 2007). Taking this advice into account, crystal determination is reported for each series (Table 4).

At this point, all Kondratieff series except one (Series I) can be investigated by WA producing at
least the scale crystal d5. The d5 detail component covers an annual frequency resolution range between 32 and 64 years, with an average cycle length around 48 years long. Since no assumption has been made on the underlying nature of the data generating process (signal) and a criterion of the same kind of a locally adaptive bandwidth has been adopted, the crystal component d5 represents a nonparametric estimation of the Kondratieff long waves.
The application of the MODWT allows the investigation of the energy distribution within the signal. Corresponding different frequency bands and the relative importance of all cyclical components can be extracted. In the present work, the long term component (S5) has been removed before the application of the MODWT energy decomposition analysis in order to better assessing the relative importance of the different cyclical components. The five crystals energy-related importance in percentage terms (net of the S5 component) for all Series except Series I are resumed in Table 5.

Source: Personal elaboration on Gattei (1981), Williamson (2020), Thomas and Williamson (2020), Hutchinson and Ploeckl (2020), Officer and Williamson (2020).
Two main findings emerge from results in Table 5. First of all, residual energy at each scale tends to diminish with the corresponding level. Secondly, the long wave d5 component concentrates the most of this energy. Substantially, even if the total energy residual appears modest (the S5 components have the predominance), it is possible to conclude that the long waves component carries the most meaningful contribution in terms of the overall variance also considering classical business cycles shorter term components.

Source: Personal elaboration from diagrams
Overall, the analysis of the whole results does confirm the existence of cycles having an about 40-60 years coherent periodicity Especially, the WA applied to the original Kondratieff's series corroborates the idea of the Russian economist on "his own" data. The same conclusions hold when the GDP-per capita and CPI selected series are processed. .

Conclusions
In this paper, the WA approach is applied to original Kondratieff's datasets. Further representative economic up-dated time series are analyzed. The main aim is to explore the plausibility of long waves having an about 40-60 years period according to the Russian economist original conclusions. Spectral analysis methods are generally adopted to investigate the presence of cycles in time series. In economic terms, WA provides the same information content in studying the dynamics of long-term movements. However, non-stationarity and non-linearity properties of data can be better approached by WA in empirical economic and financial applications. Unlike, spectral methods, wavelets have the ability to detect irregularly spaced cyclical components. A relevant fundamental benefit of WA in contrast with Fourier methods (or also spline regression models) lies in the possibility to approach random occurring shocks distorting a dynamical system where statistical properties change among periods. A further distinguishing point of this approach is in the fact that WA does not require to process time series having to respect length pre-requisites in sequence repetitions. Under this point of view, the WA transform is a suitable methodology to investigate historical data that cannot be up-dated. In such cases HA procedures encounter important and crucial binding constraints. In this sense, WA enhances the possibility for researchers to develop their work overcoming several difficulties faced in HA empirical implementation considering that no specific assumption on the characteristics of underlying data generation process is required.
As far as the findings are concerned, the present attempt confirms the presence of the coherent periodical cycles in economic data as long-waves theory would suggest. Curiously, Kondratieff original series seem to give a helping hand to Kondratieff's theory when a WA approach is applied.
Such results can be considered quite interesting, because they acknowledge the important influence Kondratieff's intuitions have exerted (not only) in economic studies (van Duijn, 2006). This work fills a gap present in current literature on the topic. As a matter of fact, WA has not been previously applied to the Russian economist's dataset from which the whole strand of literature started. A wealth of subsequent and related literature is available, but a very limited focus has been proposed on this specific aspect. Well-defined and Kondratieff-coherent patterns depicting endogenous and periodical fluctuations of macroeconomic variables appear plausible in a biological perspective (also for modeling economic activity) as pointed out by Devezas and Corredine (2001). Confirming and similar interpretations can be found in Dalgaard and Strulik (2015) wherein economic growth is related to physiological factors.