Joanna Bruzda – Business cycle synchronization according to wavelets – the case of Poland and the euro zone member countries

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Bank i Kredyt 42 (3), 2011, 5–32

www.bankandcredit.nbp.pl www.bankikredyt.nbp.pl

Business cycle synchronization according

to wavelets – the case of Poland and the euro zone

member countries

Joanna Bruzda*

Submitted: 24 January 2011. Accepted: 11 May 2011.

Abstract

In the paper time-frequency analysis in the form of the maximal overlap discrete wavelet transform (MODWT) and its complex variant – the maximal overlap discrete Hilbert wavelet transform (MODHWT) is applied to study changing patterns of business cycle synchronization between Poland and 8 euro zone member countries (France, Germany, Greece, Ireland, Italy, the Netherlands, Portugal and Spain). We also touch upon the endogeneity hypothesis of the optimum currency area criteria and ask about the recent changes in business cycle variability and their influence on the level of synchronization. Wavelet analysis is a very convenient way of studying business cycles as it possesses good localization properties and is highly efficient in extracting time- -varying frequency content of time series. In the paper we make use of these properties and provide a detailed characterization of the degree of business cycle synchronization among the countries under study as well as of the changing amplitudes of business cycles which are measured here as the appropriate frequency components of industrial production indices. In the examination we apply wavelet analysis of variance, wavelet correlation and cross-correlation examination as well as wavelet coherence and wavelet phase angle analysis in their global and (or) local (short-term) versions. The empirical examination points at an increasing synchronization of the Polish business cycle with the euro zone cycles as well as a fairly stable level of business cycle synchronization among the euro zone countries themselves.

Keywords: business cycle synchronization, euro zone, wavelet analysis, maximal overlap discrete wavelet transform, Hilbert wavelet pairs

JEL: C19, E32, E58, O52

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J. Bruzda

6 1. Introduction

There exists a long-standing interest in economics in the causes and mechanisms of business cycles. One of the well-established facts concerning business cycles is their variability over time. This has been stressed already by W.C. Mitchell in his introduction to Business Annals (1926, p. 37): ‘Recurrence of depression, revival, prosperity and recession, time after time in land after land, may be the chief conclusion drawn from the experience packed into our annals; but a second conclusion is that no two recurrences in all the array seem precisely alike. Business cycles differ in their duration as wholes and in the relative duration of their component phases; they differ in industrial and geographical scope; they differ in intensity; they differ in the features which attain prominence; they differ in the quickness and the uniformity with which they sweep from one country to another.’ The last aspect mentioned by Mitchell relates to business cycle synchronization and the phenomenon of an international business cycle.

In the paper time-frequency analysis in the form of the continuous discrete wavelet transform1 is applied to study changing patterns of business cycle synchronization between Poland and 8 euro zone member countries (France, Germany, Greece, Ireland, Italy, the Netherlands, Portugal and Spain) as well as between the euro zone countries themselves. The study is supplemented with an examination of business cycle variability. The main motivations standing behind the empirical analysis are the following. Firstly, business cycle synchronization is an important factor determining costs of adopting the euro and should be taken into account in the decision- -making process concerning entering the euro zone. Secondly, on the experience of the euro zone member countries the endogeneity hypothesis of the optimum currency area (OCA) criteria can be examined, what might constitute another important decision parameter for the candidate countries. Finally, we also ask about the recent changes in business cycle volatility (the end of the Great Moderation) and their influence on the level of synchronization.

Most of the problems mentioned above are not new and are extensively investigated in the literature. Recent studies on business cycle synchronization with(in) the euro zone point at certain diversification among the candidate countries in the degree of synchronization with the euro zone cycle (see, e.g., Fidrmuc, Korhonen 2006; Darvas, Szapáry 2008; Konopczak 2009), a variety in the timing and speed of convergence to the European cycle (Sawa, Neanidis, Osborn 2010), a relatively high degree of synchronization in the case of Poland (see the recent studies of Skrzypczyński 2008; Adamowicz et al. 2009; Konopczak 2009) and usually provide evidence in favour of the endogeneity hypothesis of the optimum currency area criteria (see, e.g., Gonçalves, Rodrigues, Soares 2009, and references therein) with trade intensity being the most important factor standing behind it (see, e.g., de Haan, Inklaar, Jong-A-Pin 2008).

1_{The continuous discrete wavelet transform (CDWT) goes by different names like the non-decimated DWT (NDWT)}

– see e.g. Nason (2008) – and the maximal overlap DWT (MODWT) – see Percival, Walden (2000). For some other names and their origins see, for example, Mallat (1998), Percival and Walden (2000), who mention also the translation invariant DWT, the stationary DWT and the algorithme à trous (algorithm with holes). The acronym CDWT in this context was suggested by Antoniadis and Gijbels (2002) and seems to be a good intuitive reference for the underlying transformation, however we note also that in the wavelet literature it often stands for the complex discrete wavelet transform. Due to this we will mainly use ‘maximal overlap DWT’ throughout the text, partially also – as Percival and Walden (2000, p. 159), notice – ‘because it leads to an acronym that is easy to say (“mod DWT”) and carries the connotation of a “modification of the DWT”’.

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Business cycle synchronization according to wavelets …

7

There is also a growing interest in applying wavelet methodology to examine business cycles and their synchronization and example empirical analyses include Jagrič, Ovin (2004); Raihan, Wen, Zeng (2005); Crowley, Lee (2005); Crowley, Maraun, Mayes (2006); Gallegati, Gallegati (2007); Yogo (2008); Aguiar-Conraria, Soares (2009). Wavelet analysis is a relatively new mathematical concept with a broad range of applications in statistics, image processing and data compression. But wavelets found also their place in modern time series analysis as they make it possible to analyse processes with changing cyclical patterns, trends, structural breaks and other nonstationary characteristics. The distinguishing feature of this technique among other time-frequency methods is an endogenously varying time window, i.e. the ability to analyze short oscillations with narrow time windows (high time resolution) and longer cycles with wider windows (and high frequency resolution). Due to this wavelet methodology is thought of constituting the next logical step in frequency studies, one that elaborates on local time properties of frequency methods. Although the first wavelet was defined in a paper from 1910 (see Haar 1910), wavelet analysis was actually invented in the eighties in France and the United States. The methodology is known to have significant influence on natural sciences (geophysics, oceanography, medicine, etc.), however, with economics and other social sciences it still remains an almost uncharted area with business cycle studies becoming one of the exceptions.

The present study focuses on applications of two continuous discrete wavelet transformations: the MODWT (maximal overlap discrete wavelet transform) and MODHWT (maximal overlap discrete Hilbert wavelet transform). The characteristic feature of the two transformations is that they are continuous in time and discrete in frequency (scales) in the sense that all time units and only octave frequency bands are considered in the analysis. From the point of view of an economist willing to study business cycles the MODWT and MODHWT may offer the following:

– a model-free (nonparametric) approach to examining frequency characteristics of time series, i.e. short, medium and long (as well as other) run features in the series; In particular, due to their nonparametric nature, wavelets enable to examine nonlinear processes without loss of information;

– good time-frequency resolution, and due to this, efficiency in terms of computations needed to extract the features; This enables precise examination of a time-varying frequency content of time series in an efficient way;

– decomposition of variance and covariance of stationary processes according to octave frequency bands (see Percival 1995; Whitcher 1998); In particular, the wavelet variance gives a simplified alternative to the spectral density function, which uses just one value per octave frequency band; The same is true for the wavelet co- and quadrature spectra, which give piecewise constant approximations to the appropriate Fourier cross-spectra on a scale by scale basis;

– precise timing of shocks causing and influencing business cycles;

– low computational complexity; The conventional DWT can be computed with an algorithm that is faster than the well known fast Fourier transform (FFT) – the Mallat’s pyramid algorithm, while the computational complexity of the MODWT is exactly the same as the FFT (see Percival, Walden 2000, p. 159);

– examination of trended, seasonal and integrated time series without prior transformations; In particular, we do not need to deseasonalize the data, as seasonal components are left automatically in lower decomposition levels, unless one is interested in examining very short cycles less than two

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J. Bruzda

8

years in length; Besides, there is no need of any prior elimination of deterministic and stochastic trends due to the fact that wavelet filtering usually embeds enough differencing operations;

– efficient estimation of short-term lead-lag relations for octave frequency bands;

– global and local (short-term) measures of association for business cycle components like the wavelet correlations and cross-correlations, wavelet coherences and wavelet phase angles.

Of course, wavelets are not a panacea and typically are as good as other methods (or worse) and their most important characteristics in the present context seem to be an overall computational efficiency and informativeness. These features together with a certain fresh look at an old problem, i.e. operating on octave frequency bands, seem to be the main reasons, why they are worth considering in business cycle examination. In the paper we make use of these properties and provide a detailed characterization of changing patterns of business cycle synchronization between Poland and 8 euro zone countries as well as changing amplitudes of business cycles measured as the appropriate frequency components of industrial production indices. In the examination we apply wavelet analysis of variance, wavelet correlation and cross-correlation analysis as well as wavelet coherence and wavelet phase angle examination in their global and (or) local versions.

The rest of the paper is structured as follows. Sections 2 and 3 describe the types of wavelet transformations that are used further in the empirical examination and discuss issues connected with building confidence intervals for different wavelet quantities. Section 4 presents results of our empirical examination, while the final section shortly concludes.

2. Conventional and maximal overlap discrete wavelet transformations

Wavelet analysis consists in decomposing a signal into shifted and scaled versions of a basic function, ψ(x), called the mother wavelet. The mother wavelet integrates to zero and has unit energy. The discrete wavelet transform (DWT) of a real-valued function f (x) is defined as follows:

) (x f = – – – _– – dx x x f Wj,t ( ) j,t( ) (1) J j= ,1 …2, , , 1 2 , ,1 , 0 = J j t … ) ( ,t x j ,

(

x t

)

x j j t j, ( )=2 /2 2 (2) ) (x

{

j,t(x)

}

) ( 2 L ) (x ) (x = f x x dx V_j,_t ( ) _j,_t( ) (3) t j W, 1 2 = j j j j+1=2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 , 1 , 1 , 1 , 1 , , , , x D x D x D x S x W x W x W x V x f J J J t t t J t J t Jt Jt t Jt Jt + + + + = = + + + – _– – – + = ) (x S_j ) (x D_j ) (x S_J ) ( 1 x D , D2(x),…, DJ(x) 2 4, 4 8,…, ₂J ₂J+1 } {g_l 1 ,..., 0 } {h_l _l₌ _L _ _ _ _ 1 ,..., 0 = L l _

Σ

φ φ φ λ λ ψ ψ ψ ψ ψ ψ ψ

∫

ψ ℜ

Σ

_t ψ …

Σ

_t ψ ∞ ∞ –

∫

∞ ∞ – (1) where ) (x f = – – – _– – dx x x f Wj,t ( ) j,t( ) (1) J j= ,1 …2, , , 1 2 , ,1 , 0 = J j t … ) ( ,t x j ,

(

x t

)

x j j t j, ( )=2 /2 2 (2) ) (x

{

j,t(x)

}

) ( 2 L ) (x ) (x = f x x dx Vj,t ( ) j,t( ) (3) t j W, 1 2 = j j j j+1=2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 , 1 , 1 , 1 , 1 , , , , x D x D x D x S x W x W x W x V x f J J J t t t J t J t Jt Jt t Jt Jt + + + + = = + + + – – – – + = ) (x Sj ) (x Dj ) (x SJ ) ( 1 x D , D2(x),…, DJ(x) 2 4, 4 8,…, ₂J ₂J+1 } {g_l 1 ,..., 0 } {h_l _l₌ _L _ _ _ _ 1 ,..., 0 = L l _

Σ

∫

ψ ℜ

Σ

_t ψ …

Σ

_t ψ ∞ ∞ –

∫

∞ ∞ – ) (x f = – – – _– – dx x x f Wj,t ( ) j,t( ) (1) J j= ,1 …2, , , 1 2 , ,1 , 0 = J j t … ) ( ,t x j ,

(

x t

)

x j j t j, ( )=2 /2 2 (2) ) (x

{

j,t(x)

}

) ( 2 L ) (x ) (x = f x x dx Vj,t ( ) j,t( ) (3) t j W, 1 2 = j j j j+1=2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 , 1 , 1 , 1 , 1 , , , , x D x D x D x S x W x W x W x V x f J J J t t t J t J t Jt Jt t Jt Jt + + + + = = + + + – – – – + = ) (x Sj ) (x D_j ) (x SJ ) ( 1 x D , D2(x),…, DJ(x) 2 4, 4 8,…, ₂J ₂J+1 } {gl 1 ,..., 0 } {h_l _l₌ _L _ _ _ _ 1 ,..., 0 = L l _

Σ

∫

ψ ℜ

Σ

_t ψ …

Σ

_t ψ ∞ ∞ –

∫

∞ ∞ –

and the wavelet daughters, ) (x f = – – – _– – dx x x f Wj,t ( ) j,t( ) (1) J j= ,1 …2, , , 1 2 , ,1 , 0 = J j t … ) ( ,t x j ,

(

x t

)

x j j t j, ( )=2 /2 2 (2) ) (x

{

j,t(x)

}

) ( 2 L ) (x ) (x = f x x dx Vj,t ( ) j,t( ) (3) t j W, 1 2 = j j j j+1=2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 , 1 , 1 , 1 , 1 , , , , x D x D x D x S x W x W x W x V x f J J J t t t J t J t Jt Jt t Jt Jt + + + + = = + + + – – – – + = ) (x S_j ) (x Dj ) (x SJ ) ( 1 x D , D2(x),…, DJ(x) 2 4, 4 8,…, ₂J ₂J+1 } {gl 1 ,..., 0 } {h_l _l₌ _L _ _ _ _ 1 ,..., 0 = L l _

Σ

∫

ψ ℜ

Σ

_t ψ …

Σ

_t ψ ∞ ∞ –

∫

∞ ∞ –

, are shifted and scaled ver-sions of the mother wavelet with dyadic shifts t and scales j, i.e.:

) (x f = – – – _– – dx x x f W_j,_t ( ) _j,_t( ) (1) J j= ,1 …2, , , 1 2 , ,1 , 0 = J j t … ) ( ,t x j ,

(

x t

)

x j j t j, ( )=2 /2 2 (2) ) (x

{

j,t(x)

}

) ( 2 L ) (x ) (x = f x x dx Vj,t ( ) j,t( ) (3) t j W, 1 2 = j j j j+1=2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 , 1 , 1 , 1 , 1 , , , , x D x D x D x S x W x W x W x V x f J J J t t t J t J t Jt Jt t Jt Jt + + + + = = + + + – – – – + = ) (x S_j ) (x D_j ) (x SJ ) ( 1 x D , D2(x),…, DJ(x) 2 4, 4 8,…, ₂J ₂J+1 } {g_l 1 ,..., 0 } {h_l _l₌ _L _ _ _ _ 1 ,..., 0 = L l _

Σ

∫

ψ ℜ

Σ

_t ψ …

Σ

_t ψ ∞ ∞ –

∫

∞ ∞ – (2) For certain functions ψ(x) with good localization properties

(

x t

)

x j j t j, ( )=2 /2 2 (2) ) (x

{

j,t(x)

}

) ( 2 L ) (x ) (x = f x x dx Vj,t ( ) j,t( ) (3) t j W, 1 2 = j j j j+1=2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 , 1 , 1 , 1 , 1 , , , , x D x D x D x S x W x W x W x V x f J J J t t t J t J t Jt Jt t Jt Jt + + + + = = + + + – – – – + = ) (x Sj ) (x Dj ) (x SJ ) ( 1 x D , D2(x),…, DJ(x) 2 4, 4 8,…, ₂J ₂J+1 } {gl 1 ,..., 0 } {h_l _l₌ _L _ _ _ _ 1 ,..., 0 = L l _

Σ

∫

ψ ℜ

Σ

_t ψ …

Σ

_t ψ ∞ ∞ –

∫

∞ ∞ – is an orthonormal basis in ) (x f = – – – _– – dx x x f Wj,t ( ) j,t( ) (1) J j= ,1 …2, , , 1 2 , ,1 , 0 = J j t … ) ( ,t x j ,

(

x t

)

x j j t j, ( )=2 /2 2 (2) ) (x

{

j_,t(x)

}

) ( 2 L ) (x ) (x = f x x dx Vj,t ( ) j,t( ) (3) t j W, 1 2 = j j j j+1=2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 , 1 , 1 , 1 , 1 , , , , x D x D x D x S x W x W x W x V x f J J J t t t J t J t Jt Jt t Jt Jt + + + + = = + + + – – – – + = ) (x Sj ) (x Dj ) (x S_J ) ( 1 x D , D2(x),…, DJ(x) 2 4, 4 8,…, ₂J ₂J+1 } {gl 1 ,..., 0 } {hl l= L _ _ _ _ 1 ,..., 0 = L l _

Σ

∫

ψ ℜ

Σ

_t ψ …

Σ

_t ψ ∞ ∞ –

∫

∞ ∞ –

. The function ψ(x) is usually defined via another function (the scaling function or father wavelet), φ(x), that applied to the signal after shifting and scaling analogously to (2) produces another set of coefficients in the form:

) (x f = – – – _– – dx x x f W_j,_t ( ) _j,_t( ) (1) J j= ,1 …2, , , 1 2 , ,1 , 0 = J j t … ) ( ,t x j ,

(

x t

)

x j j t j, ( )=2 /2 2 (2) ) (x

{

j,t(x)

}

) ( 2 L ) (x ) (x = f x x dx Vj,t ( ) j,t( ) (3) t j W, 1 2 = j j j j+1=2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 , 1 , 1 , 1 , 1 , , , , x D x D x D x S x W x W x W x V x f J J J t t t J t J t Jt Jt t Jt Jt + + + + = = + + + – – – – + = ) (x S_j ) (x Dj ) (x S_J ) ( 1 x D , D2(x),…, DJ(x) 2 4, 4 8,…, ₂J ₂J+1 } {gl 1 ,..., 0 } {h_l _l₌ _L _ _ _ _ 1 ,..., 0 = L l _

Σ

∫

ψ ℜ

Σ

_t ψ …

Σ

_t ψ ∞ ∞ –

∫

∞ ∞ – (3)

known as scaling coefficients.

For a given j the wavelet coefficients, W_j,t, are computed as differences of moving averages for the previous scale scaling coefficients and are associated with scale

(

x t

)

x j j t j, ( )=2 /2 2 (2) ) (x

{

j,t(x)

}

) ( 2 L ) (x ) (x = f x x dx V_j,_t ( ) _j,_t( ) (3) t j W, 1 2 = j j j j+1=2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 , 1 , 1 , 1 , 1 , , , , x D x D x D x S x W x W x W x V x f J J J t t t J t J t Jt Jt t Jt Jt + + + + = = + + + – – – – + = ) (x S_j ) (x Dj ) (x SJ ) ( 1 x D , D2(x),…, DJ(x) 2 4, 4 8,…, ₂J ₂J+1 } {gl 1 ,..., 0 } {h_l _l₌ _L _ _ _ _ 1 ,..., 0 = L l _

Σ

∫

ψ ℜ

Σ

_t ψ …

Σ

_t ψ ∞ ∞ –

∫

∞ ∞ –

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Business cycle synchronization according to wavelets …

9

contribute to the decomposition of energy of the signal on the time-frequency plane. On the other hand, the level j scaling coefficients are moving averages of scale

(

x t

)

x j j t j, ( )=2 /2 2 (2) ) (x

{

j,t(x)

}

) ( 2 L ) (x ) (x = f x x dx Vj,t ( ) j,t( ) (3) t j W, 1 2 = j j j j+1=2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 , 1 , 1 , 1 , 1 , , , , x D x D x D x S x W x W x W x V x f J J J t t t J t J t Jt Jt t Jt Jt + + + + = = + + + – – – – + = ) (x Sj ) (x Dj ) (x S_J ) ( 1 x D , D2(x),…, DJ(x) 2 4, 4 8,…, ₂J ₂J+1 } {g_l 1 ,..., 0 } {h_l _l₌ _L _ _ _ _ 1 ,..., 0 = L l _

Σ

∫

ψ ℜ

Σ

_t ψ …

Σ

_t ψ ∞ ∞ –

∫

∞ ∞ –

. The two types of coefficients give the multiresolution decomposition of the original function in the form:

(

x t

)

x j j t j, ( )=2 /2 2 (2) ) (x

{

j,t(x)

}

) ( 2 L ) (x ) (x = f x x dx Vj,t ( ) j,t( ) (3) t j W, 1 2 = j j j j+1=2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 , 1 , 1 , 1 , 1 , , , , x D x D x D x S x W x W x W x V x f J J J t t t J t J t Jt Jt t Jt Jt + + + + = = + + + – – – – + = ) (x Sj ) (x Dj ) (x SJ ) ( 1 x D , D2(x),…, DJ(x) 2 4, 4 8,…, ₂J ₂J+1 } {g_l 1 ,..., 0 } {h_l _l₌ _L _ _ _ _ 1 ,..., 0 = L l _

Σ

∫

ψ ℜ

Σ

_t ψ …

Σ

_t ψ ∞ ∞ –

∫

∞ ∞ – (4)

The functions S_j(x) and D_j(x) are known as approximations (smooths) and details. The highest level approximation S_J(x) represents smooth, low-frequency component of the signal, while the details ) (x f = – – – _– – dx x x f W_j,_t ( ) _j,_t( ) (1) J j= ,1 …2, , , 1 2 , ,1 , 0 = J j t … ) ( ,t x j ,

(

x t

)

x j j t j, ( )=2 /2 2 (2) ) (x

{

j,t(x)

}

) ( 2 L ) (x ) (x = f x x dx Vj,t ( ) j,t( ) (3) t j W, 1 2 = j j j j+1 =2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 , 1 , 1 , 1 , 1 , , , , x D x D x D x S x W x W x W x V x f J J J t t t J t J t Jt Jt t Jt Jt + + + + = = + + + – – – – + = ) (x Sj ) (x D_j ) (x SJ ) ( 1 x D , D2(x),…, DJ(x) 2 4, 4 8,…, ₂J ₂J+1 } {gl 1 ,..., 0 } {h_l _l₌ _L _ _ _ _ 1 ,..., 0 = L l _

Σ

∫

ψ ℜ

Σ

_t ψ …

Σ

_t ψ ∞ ∞ –

∫

∞ ∞ –

are associated with oscillations of length ) (x f = – – – _– – dx x x f Wj,t ( ) j,t( ) (1) J j= ,1 …2, , , 1 2 , ,1 , 0 = J j t … ) ( ,t x j ,

(

x t

)

x j j t j, ( )=2 /2 2 (2) ) (x

{

j,t(x)

}

) ( 2 L ) (x ) (x = f x x dx V_j,_t ( ) _j,_t( ) (3) t j W, 1 2 = j j j j+1=2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 , 1 , 1 , 1 , 1 , , , , x D x D x D x S x W x W x W x V x f J J J t t t J t J t Jt Jt t Jt Jt + + + + = = + + + – – – – + = ) (x S_j ) (x Dj ) (x SJ ) ( 1 x D , D2(x),…, DJ(x) 2 4, 4 8,…, ₂J ₂J+1 } {gl 1 ,..., 0 } {h_l _l₌ _L _ _ _ _ 1 ,..., 0 = L l _

Σ

∫

ψ ℜ

Σ

_t ψ …

Σ

_t ψ ∞ ∞ –

∫

∞ ∞ – . In filtering notation the discrete wavelet transform is defined via quadrature mirror filters: the low-pass (scaling) filter

(

x t

)

x j j t j, ( )=2 /2 2 (2) ) (x

{

j,t(x)

}

) ( 2 L ) (x ) (x = f x x dx Vj,t ( ) j,t( ) (3) t j W_, 1 2 = j j j j+1=2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 , 1 , 1 , 1 , 1 , , , , x D x D x D x S x W x W x W x V x f J J J t t t J t J t Jt Jt t Jt Jt + + + + = = + + + – – – – + = ) (x Sj ) (x Dj ) (x SJ ) ( 1 x D , D2(x),…, DJ(x) 2 4, 4 8,…, ₂J ₂J+1 } {gl 1 ,..., 0 } {h_l _l₌ _L _ _ _ _ 1 ,..., 0 = L l _

Σ

∫

ψ ℜ

Σ

_t ψ …

Σ

_t ψ ∞ ∞ –

∫

∞ ∞ –

and the high-pass (wavelet) filter ) (x f = – – – _– – dx x x f Wj,t ( ) j,t( ) (1) J j= ,1 …2, , , 1 2 , ,1 , 0 = J j t … ) ( ,t x j ,

(

x t

)

x j j t j, ( )=2 /2 2 (2) ) (x

{

j,t(x)

}

) ( 2 L ) (x ) (x = f x x dx Vj,t ( ) j,t( ) (3) t j W, 1 2 = j j j j+1=2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 , 1 , 1 , 1 , 1 , , , , x D x D x D x S x W x W x W x V x f J J J t t t J t J t Jt Jt t Jt Jt + + + + = = + + + – – – – + = ) (x Sj ) (x Dj ) (x SJ ) ( 1 x D , D2(x),…, DJ(x) 2 4, 4 8,…, ₂J ₂J+1 } {gl 1 ,..., 0 } {h_l _l₌ _L _ _ _ _ 1 ,..., 0 = L l _

Σ

∫

ψ ℜ

Σ

_t ψ …

Σ

_t ψ ∞ ∞ –

∫

∞ ∞ –

.2_{The two filters} fulfil the quadrature mirror relationship gl =( 1)l+1hL1 l

2 √ J N 2= x=(x0,x1,…,xN 1) 1 , , , ₄ 2 N … N t j W~_, V~j,t 1 2 , , 0 , 1 0 mod mod ] 1 ) 1 ( 2 [ , , = + = J j L l jl t l N t j h x t W j j … (5) 1 , , 0 , ~ 2 /2W, = =₀1h, x( ) t= N j L l jl t l N t j j _… ₍₆₎ 1 2 , , 0 , 1 0 , [2( 1)1 ]mod , = = = j J L l jl t l N t j g x t V j j … (7) 1 , , 0 , ~ 2 /2V, = =₀1g, x( )mod t= N j L l jl t l N t j j _… ₍₈₎ } {hj, l {gj, l} 1 ) 1 )( 1 2 ( + = L L j j ) ~ Var( ) Var( 2 1 ) ( , , 2 t j t j j j t = W = W (9) = = = = 1 1 2 , ) ( ) Var( 1 2 1 ) Var( j j jt j j t W Y (10) 1 2 = j j 2( j) j 2 –₂j+1 ) ~ , ~ Cov( ) , Cov( 2 1 ) ( jX,t Yj,t j,Xt jY,t j j t = W W = W W (11) = = = = 1 , , 1 ) ( ) , Cov( 1 2 1 ) , j j j Y Cov( jXt jt j t t Y W W X (12) ) ( ) ( ) ( ) ( 2 1 j j j j = (13) ) ~ , ~ Cov( ) , Cov( 21 ) ( j = Wj,Xt WYj,t+ = WjX,t WjY,t+ (14) – – – '

Σ

– – = – – – – – – – – + – – – – – – – – – ∞ ∞ ∞ ∞ λ _λ λ λ λ λ λ λ _σ λ_σ λ j λ γ γ γτ τ τ λ λ λ λ λ σ σ σ – ρ ,

, have unit energy and are even-shift orthogonal; the wavelet filter integrates (sums) to zero, while the scaling filter – to

l L l l h g ₌ + 1 1 ) 1 ( 2 √ J N 2= x=(x0,x1,…,xN 1) 1 , , , ₄ 2 N … N t j W~_, V~_j,_t 1 2 , , 0 , 1 0 mod mod ] 1 ) 1 ( 2 [ , , = + = j J L l jl t l N t j h x t W j j … (5) 1 , , 0 , ~ 2 /2W, = =₀1h, x( ) t= N j L l jl t l N t j j _… ₍₆₎ 1 2 , , 0 , 1 0 , [2( 1)1 ]mod , = = = j J L l jl t l N t j g x t V j j … (7) 1 , , 0 , ~ 2 1 0 , ( )mod , 2 / ₌ ₌ = g x t N V Lj l jl t l N t j j _… ₍₈₎ } {hj_{, l} {gj, l} 1 ) 1 )( 1 2 ( + = L L j j ) ~ Var( ) Var( 21 ) ( , , 2 t j t j j j t = W = W (9) = = = = 1 1 2 , ) ( ) Var( 1 2 1 ) Var( j j jt j j t W Y (10) 1 2 = j j 2( j) j 2 –₂j+1 ) ~ , ~ Cov( ) , Cov( 21 ) ( jX,t Yj,t j,Xt Yj,t j j t = W W = W W (11) = = = = 1 Cov( , , , ) 1 ( ) 1 2 1 ) , j j j Y Cov( jXt jt j t t Y W W X (12) ) ( ) ( ) ( ) ( 2 1 j j j j = (13) ) ~ , ~ Cov( ) , Cov( 2 1 ) ( j = WjX,t WjY,t+ = Wj,Xt WjY,t+ (14) – _{– –} '

Σ

– – = – – – – – – – – + – – – – – – – – – ∞ ∞ ∞ ∞ λ _λ λ λ λ λ λ λ σ σ λ λ j λ γ γ γτ τ τ λ λ λ λ λ σ σ σ – ρ , . When processing discrete signals we consider a data vector of length

l L l l h g ₌ + 1 1 ) 1 ( 2 √ J N 2= x=(x0,x1,…,xN 1) 1 , , , ₄ 2 N … N t j W, ~ ~ t j V, 1 2 , , 0 , 1 0 mod mod ] 1 ) 1 ( 2 [ , , = + = j J L l jl t l N t j h x t W j j … (5) 1 , , 0 , ~ 2 /2 _, ₌ ₀1 _, ₍ ₎ ₌ = h x t N W Lj l jl t l N t j j _… ₍₆₎ 1 2 , , 0 , 1 0 , [2( 1)1 ]mod , = = = j J L l jl t l N t j g x t V j j … (7) 1 , , 0 , ~ 2 /2V, = =₀1g, x( )mod t= N j L l jl t l N t j j _… ₍₈₎ } {hj, l {gj, l} 1 ) 1 )( 1 2 ( + = L L j j ) ~ Var( ) Var( 2 1 ) ( , , 2 t j t j j j t = W = W (9) = = = = 1 1 2 , ) ( ) Var( 1 2 1 ) Var( j j jt j j t W Y (10) 1 2 = j j 2( j) j 2 –₂j+1 ) ~ , ~ Cov( ) , Cov( 21 ) ( jX,t Yj,t j,Xt jY,t j j t = W W = W W (11) = = = = 1 , , 1 ) ( ) , Cov( 1 2 1 ) , j j j Y Cov( jXt jt j t t Y W W X (12) ) ( ) ( ) ( ) ( 2 1 j j j j = (13) ) ~ , ~ Cov( ) , Cov( 21 ) ( j = WjX,t WjY,t+ = Wj,Xt WYj,t+ (14) – – – '

Σ

– – = – – – – – – – – + – – – – – – – – – ∞ ∞ ∞ ∞ λ _λ λ λ λ λ λ λ _σ λ_σ λ j λ γ γ γτ τ τ λ λ λ λ λ σ σ σ – ρ , in the form l L l l h g ₌ + 1 1 ) 1 ( 2 √ J N 2= x=(x0,x1,…,xN 1) 1 , , , ₄ 2 N … N t j W~, V~j,t 1 2 , , 0 , 1 0 mod mod ] 1 ) 1 ( 2 [ , , = + = j J L l jl t l N t j h x t W j j … (5) 1 , , 0 , ~ 2 /2W , = =₀1h, x( ) t = N j L l jl t l N t j j _… ₍₆₎ 1 2 , , 0 , 1 0 , [2( 1)1 ]mod , = = = j J L l jl t l N t j g x t V j j … (7) 1 , , 0 , ~ 2 /2V, = =₀1g , x( )mod t= N j L l jl t l N t j j _… ₍₈₎ } {hj, l {gj, l} 1 ) 1 )( 1 2 ( + = L L j j ) ~ Var( ) Var( 21 ) ( , , 2 t j t j j j t = W = W (9) = = = = 1 1 2 , ) ( ) Var( 1 2 1 ) Var( j j jt j j t W Y (10) 1 2 = j j 2( j) j 2 –₂j+1 ) ~ , ~ Cov( ) , Cov( 21 ) ( jX,t jY,t j,Xt Yj,t j j t = W W = W W (11) = = = = 1 , , 1 ) ( ) , Cov( 1 2 1 ) , j j j Y Cov( jXt jt j t t Y W W X (12) ) ( ) ( ) ( ) ( 2 1 j j j j = (13) ) ~ , ~ Cov( ) , Cov( 21 ) ( j = WjX,t WjY,t+ = Wj,Xt WjY,t+ (14) – – – '

Σ

– – = – – – – – – – – + – – – – – – – – – ∞ ∞ ∞ ∞ λ _λ λ λ λ λ λ λ _σ λ_σ λ j λ γ γ γτ τ τ λ λ λ λ λ σ σ σ – ρ , . Then the highest possible decomposition level is J and the numbers of wavelet and scaling coefficients of the

conventional DWT for each level are

l L l l h g ₌ + 1 1 ) 1 ( 2 √ J N 2= x=(x0,x1,…,xN 1) 1 , , , ₄ 2 N … N t j W~_, V~j,t 1 2 , , 0 , 1 0 mod mod ] 1 ) 1 ( 2 [ , , = + = J j L l jl t l N t j h x t W j j … (5) 1 , , 0 , ~ 2 /2W, = =₀1h, x( ) t= N j L l jl tl N t j j _… ₍₆₎ 1 2 , , 0 , 1 0 , [2( 1)1 ]mod , = = = j J L l jl t l N t j g x t V j j … (7) 1 , , 0 , ~ 2 /2V, = =₀1g , x( )mod t= N j L l jl tl N t j j _… ₍₈₎ } {hj, l {gj, l} 1 ) 1 )( 1 2 ( + = L L j j ) ~ Var( ) Var( 2 1 ) ( , , 2 t j t j j j t = W = W (9) = = = = 1 1 2 ,) ( ) Var( 1 2 1 ) Var( j j jt j j t W Y (10) 1 2 = j j 2( j) j 2 –₂j+1 ) ~ , ~ Cov( ) , Cov( 2 1 ) ( jX,t jY,t j,Xt jY,t j j t = W W = W W (11) = = = = 1 , , 1 ) ( ) , Cov( 1 2 1 ) , j j j Y Cov( jXt jt j t t Y W W X (12) ) ( ) ( ) ( ) ( 2 1 j j j j = (13) ) ~ , ~ Cov( ) , Cov( 21 ) ( j = WjX,t WjY,t+ = Wj,Xt WjY,t+ (14) – – – '

Σ

. On the other hand, the maximal overlap discrete wavelet transform (MODWT) produces the same number of wavelet and scaling coefficients ( l L l l h g ₌ + 1 1 ) 1 ( 2 √ J N 2= x=(x0,x1,…,xN 1) 1 , , , ₄ 2 N … N t j W~_, V~_j,_t 1 2 , , 0 , 1 0 mod mod ] 1 ) 1 ( 2 [ , , = + = j J L l jl t l N t j h x t W j j … (5) 1 , , 0 , ~ 2 /2W, = =₀1h, x( ) t= N j L l jl t l N t j j _… ₍₆₎ 1 2 , , 0 , 1 0 , [2( 1)1 ]mod , = = = j J L l jl t l N t j g x t V j j … (7) 1 , , 0 , ~ 2 1 0 , ( )mod , 2 / ₌ ₌ = g x t N V Lj l jl t l N t j j _… ₍₈₎ } {hj_{, l} {gj, l} 1 ) 1 )( 1 2 ( + = L L j j ) ~ Var( ) Var( 21 ) ( , , 2 t j t j j j t = W = W (9) = = = = 1 1 2 , ) ( ) Var( 1 2 1 ) Var( j j jt j j t W Y (10) 1 2 = j j 2( j) j 2 –₂j+1 ) ~ , ~ Cov( ) , Cov( 21 ) ( jX,t jY,t j,Xt Yj,t j j t = W W = W W (11) = = = = 1 Cov( , , , ) 1 ( ) 1 2 1 ) , j j j Y Cov( jXt jt j t t Y W W X (12) ) ( ) ( ) ( ) ( 2 1 j j j j = (13) ) ~ , ~ Cov( ) , Cov( 2 1 ) ( j = WjX,t WjY,t+ = Wj,Xt WjY,t+ (14) – _{– –} '

Σ

– – = – – – – – – – – + – – – – – – – – – ∞ ∞ ∞ ∞ λ _λ λ λ λ λ λ λ σ σ λ λ j λ γ γ γτ τ τ λ λ λ λ λ σ σ σ – ρ , and l L l l h g ₌ + 1 1 ) 1 ( 2 √ J N 2= x=(x0,x1,…,xN 1) 1 , , , ₄ 2 N … N t j W~_, V~j,t 1 2 , , 0 , 1 0 mod mod ] 1 ) 1 ( 2 [ , , = + = J j L l jl t l N t j h x t W j j … (5) 1 , , 0 , ~ 2 /2W, = =₀1h, x( ) t = N j L l jl t l N t j j _… ₍₆₎ 1 2 , , 0 , 1 0 , [2( 1)1 ]mod , = = = j J L l jl t l N t j g x t V j j … (7) 1 , , 0 , ~ 2 /2V, = =₀1g, x( )mod t= N j L l jl t l N t j j _… ₍₈₎ } {hj, l {gj, l} 1 ) 1 )( 1 2 ( + = L L j j ) ~ Var( ) Var( 2 1 ) ( , , 2 t j t j j j t = W = W (9) = = = = 1 1 2 , ) ( ) Var( 1 2 1 ) Var( j j jt j j t W Y (10) 1 2 = j j 2( j) j 2 –₂j+1 ) ~ , ~ Cov( ) , Cov( 2 1 ) ( jX,t jY,t j,Xt jY,t j j t = W W = W W (11) = = = = 1 , , 1 ) ( ) , Cov( 1 2 1 ) , j j j Y Cov( jXt jt j t t Y W W X (12) ) ( ) ( ) ( ) ( 2 1 j j j j = (13) ) ~ , ~ Cov( ) , Cov( 21 ) ( j = WjX,t WjY,t+ = Wj,Xt WjY,t+ (14) – – – '

Σ

, accordingly) at each decomposition level as it does not use downsampling by 2. The coefficients are appropriately scaled in order to retain variance preservation. By definition they are given as follows: l L l l h g ₌ + 1 1 ) 1 ( 2 √ J N 2= x=(x0,x1,…,xN 1) 1 , , , ₄ 2 N … N t j W~, V~j,t 1 2 , , 0 , 1 0 mod mod ] 1 ) 1 ( 2 [ , , = ₊ = J j L l jl t l N t j h x t W j j … (5) 1 , , 0 , ~ 2 /2 _, ₌ ₀1 _, ₍ ₎ ₌ = h x t N W Lj l jl t l N t j j _… ₍₆₎ 1 2 , , 0 , 1 0 , [2( 1)1 ]mod , = = = j J L l jl t l N t j g x t V j j … (7) 1 , , 0 , ~ 2 1 0 , ( )mod , 2 / ₌ ₌ = g x t N V Lj l jl t l N t j j _… ₍₈₎ } {hj, l {gj, l} 1 ) 1 )( 1 2 ( + = L L j j ) ~ Var( ) Var( 2 1 ) ( , , 2 t j t j j j t = W = W (9) = = = = 1 1 2 , ) ( ) Var( 1 2 1 ) Var( j j jt j j t W Y (10) 1 2 = j j 2( j) j 2 –₂j+1 ) ~ , ~ Cov( ) , Cov( 21 ) ( jX,t jY_,t j_,Xt jY_,t j j t = W W = W W (11) = = = = 1 Cov( , , , ) 1 ( ) 1 2 1 ) , j j j Y Cov( jXt jt j t t Y W W X (12) ) ( ) ( ) ( ) ( 2 1 j j j j = (13) ) ~ , ~ Cov( ) , Cov( 21 ) ( j = WjX,t WjY,t+ = Wj_,Xt WjY_,t+ (14) – – – '

Σ

– – = – – – – – – – – + – – – – – – – – – ∞ ∞ ∞ ∞ λ _λ λ λ λ λ λ λ _σ λ_σ λ j λ γ γ γτ τ τ λ λ λ λ λ σ σ σ – ρ , (5) l L l l h g ₌ + 1 1 ) 1 ( 2 √ J N 2= x=(x0,x1,…,xN 1) 1 , , , ₄ 2 N … N t j W~, V~j,t 1 2 , , 0 , 1 0 mod mod ] 1 ) 1 ( 2 [ , , = ₊ = J j L l jl t l N t j h x t W j j … (5) 1 , , 0 , ~ 2 /2 _, ₌ ₀1 _, ₍ ₎ ₌ = h x t N W Lj l jl t l N t j j _… ₍₆₎ 1 2 , , 0 , 1 0 , [2( 1)1 ]mod , = = = j J L l jl t l N t j g x t V j j … (7) 1 , , 0 , ~ 2 /2V, = =₀1g , x( )mod t= N j L l jl t l N t j j _… ₍₈₎ } {hj, l {gj, l} 1 ) 1 )( 1 2 ( + = L L j j ) ~ Var( ) Var( 2 1 ) ( , , 2 t j t j j j t = W = W (9) = = = = 1 1 2 , ) ( ) Var( 1 2 1 ) Var( j j jt j j t W Y (10) 1 2 = j j 2( j) j 2 –₂j+1 ) ~ , ~ Cov( ) , Cov( 21 ) ( jX,t jY,t j,Xt jY,t j j t = W W = W W (11) = = = = 1 Cov( , , , ) 1 ( ) 1 2 1 ) , j j j Y Cov( jXt jt j t t Y W W X (12) ) ( ) ( ) ( ) ( 2 1 j j j j = (13) ) ~ , ~ Cov( ) , Cov( 21 ) ( j = WjX,t WjY,t+ = Wj_,Xt WjY_,t+ (14) – – – '

Σ

– – = – – – – – – – – + – – – – – – – – – ∞ ∞ ∞ ∞ λ _λ λ λ λ λ λ λ _σ λ_σ λ j λ γ γ γτ τ τ λ λ λ λ λ σ σ σ – ρ , (6) l L l l h g ₌ + 1 1 ) 1 ( 2 √ J N 2= x=(x0,x1,…,xN 1) 1 , , , ₄ 2 N … N t j W~, V~j,t 1 2 , , 0 , 1 0 mod mod ] 1 ) 1 ( 2 [ , , = ₊ = J j L l jl t l N t j h x t W j j … (5) 1 , , 0 , ~ 2 /2 _, ₌ ₀1 _, ₍ ₎ ₌ = h x t N W Lj l jl t l N t j j _… ₍₆₎ 1 2 , , 0 , 1 0 , [2( 1)1 ]mod , = = = j J L l jl t l N t j g x t V j j … (7) 1 , , 0 , ~ 2 /2V, = =₀1g , x( )mod t= N j L l jl t l N t j j _… ₍₈₎ } {hj, l {gj, l} 1 ) 1 )( 1 2 ( + = L L j j ) ~ Var( ) Var( 2 1 ) ( , , 2 t j t j j j t = W = W (9) = = = = 1 1 2 , ) ( ) Var( 1 2 1 ) Var( j j jt j j t W Y (10) 1 2 = j j 2( j) j 2 –₂j+1 ) ~ , ~ Cov( ) , Cov( 21 ) ( jX,t jY,t j,Xt jY,t j j t = W W = W W (11) = = = = 1 Cov( , , , ) 1 ( ) 1 2 1 ) , j j j Y Cov( jXt jt j t t Y W W X (12) ) ( ) ( ) ( ) ( 2 1 j j j j = (13) ) ~ , ~ Cov( ) , Cov( 21 ) ( j = WjX,t WjY,t+ = Wj_,Xt WjY_,t+ (14) – – – '

Σ

– – = – – – – – – – – + – – – – – – – – – ∞ ∞ ∞ ∞ λ _λ λ λ λ λ λ λ _σ λ_σ λ j λ γ γ γτ τ τ λ λ λ λ λ σ σ σ – ρ , (7) l L l l h g ₌ + 1 1 ) 1 ( 2 √ J N 2= x=(x0,x1,…,xN 1) 1 , , , ₄ 2 N … N t j W~, V~j,t 1 2 , , 0 , 1 0 mod mod ] 1 ) 1 ( 2 [ , , = ₊ = J j L l jl t l N t j h x t W j j … (5) 1 , , 0 , ~ 2 /2 _, ₌ ₀1 _, ₍ ₎ ₌ = h x t N W Lj l jl t l N t j j _… ₍₆₎ 1 2 , , 0 , 1 0 , [2( 1) 1 ]mod , = = = j J L l jl t l N t j g x t V j j … (7) 1 , , 0 , ~ 2 /2V, = =₀1g , x( )mod t= N j L l jl t l N t j j _… ₍₈₎ } {hj, l {gj, l} 1 ) 1 )( 1 2 ( + = L L j j ) ~ Var( ) Var( 2 1 ) ( , , 2 t j t j j j t = W = W (9) = = = = 1 1 2 , ) ( ) Var( 1 2 1 ) Var( j j jt j j t W Y (10) 1 2 = j j 2( j) j 2 –₂j+1 ) ~ , ~ Cov( ) , Cov( 21 ) ( jX,t jY,t j,Xt jY,t j j t = W W = W W (11) = = = = 1 Cov( , , , ) 1 ( ) 1 2 1 ) , j j j Y Cov( jXt jt j t t Y W W X (12) ) ( ) ( ) ( ) ( 2 1 j j j j = (13) ) ~ , ~ Cov( ) , Cov( 21 ) ( j = Wj,Xt WjY,t+ = Wj_,Xt WjY_,t+ (14) – – – '

Σ

– – = – – – – – – – – + – – – – – – – – – ∞ ∞ ∞ ∞ λ _λ λ λ λ λ λ λ _σ λ_σ λ j λ γ γ γτ τ τ λ λ λ λ λ σ σ σ – ρ , (8) where l L l l h g ₌ + 1 1 ) 1 ( 2 √ J N 2= x=(x0,x1,…,xN 1) 1 , , , ₄ 2 N … N t j W~_, V~j,t 1 2 , , 0 , 1 0 mod mod ] 1 ) 1 ( 2 [ , , = + = j J L l jl t l N t j h x t W j j … (5) 1 , , 0 , ~ 2 /2W, = =₀1h, x( ) t= N j L l jl tl N t j j _… ₍₆₎ 1 2 , , 0 , 1 0 , [2( 1)1 ]mod , = = = j J L l jl t l N t j g x t V j j … (7) 1 , , 0 , ~ 2 /2V, = =₀1g , x( )mod t= N j L l jl tl N t j j _… ₍₈₎ } {hj, l {gj, l} 1 ) 1 )( 1 2 ( + = L L j j ) ~ Var( ) Var( 2 1 ) ( , , 2 t j t j j j t = W = W (9) = = = = 1 1 2 ,) ( ) Var( 1 2 1 ) Var( j j jt j j t W Y (10) 1 2 = j j 2( j) j 2 –₂j+1 ) ~ , ~ Cov( ) , Cov( 2 1 ) ( jX,t jY,t j,Xt jY,t j j t = W W = W W (11) = = = = 1 , , 1 ) ( ) , Cov( 1 2 1 ) , j j j Y Cov( jXt jt j t t Y W W X (12) ) ( ) ( ) ( ) ( 2 1 j j j j = (13) ) ~ , ~ Cov( ) , Cov( 21 ) ( j = WjX,t WjY,t+ = Wj,Xt WjY,t+ (14) – – – '

Σ

– – = – – – – – – – – + – – – – – – – – – ∞ ∞ ∞ ∞ λ _λ λ λ λ λ λ λ _σ λ_σ λ j λ γ γ γτ τ τ λ λ λ λ λ σ σ σ – ρ , and l L l l h g ₌ + 1 1 ) 1 ( 2 √ J N 2= x=(x0,x1,…,xN 1) 1 , , , ₄ 2 N … N t j W~, V~j,t 1 2 , , 0 , 1 0 mod mod ] 1 ) 1 ( 2 [ , , = + = j J L l jl t l N t j h x t W j j … (5) 1 , , 0 , ~ 2 /2W, = =₀1h, x( ) t= N j L l jl tl N t j j _… ₍₆₎ 1 2 , , 0 , 1 0 , [2( 1)1 ]mod , = = = j J L l jl t l N t j g x t V j j … (7) 1 , , 0 , ~ 2 1 0 , ( )mod , 2 / ₌ ₌ = g x t N V Lj l jl tl N t j j _… ₍₈₎ } {h_j_{, l} {g_j_{, l}} 1 ) 1 )( 1 2 ( + = L L j j ) ~ Var( ) Var( 21 ) ( , , 2 t j t j j j t = W = W (9) = = = = 1 1 2 ,) ( ) Var( 1 2 1 ) Var( j j jt j j t W Y (10) 1 2 = j j 2( j) j 2 –₂j+1 ) ~ , ~ Cov( ) , Cov( 21 ) ( jX,t jY,t j,Xt jY,t j j t = W W = W W (11) = = = = 1 Cov( , , , ) 1 ( ) 1 2 1 ) , j j j Y Cov( jXt jt j t t Y W W X (12) ) ( ) ( ) ( ) ( 2 1 j j j j = (13) ) ~ , ~ Cov( ) , Cov( 2 1 ) ( j = WjX,t WjY,t+ = Wj,Xt WjY,t+ (14) – _{– –} '

Σ

– – = – – – – – – – – + – – – – – – – – – ∞ ∞ ∞ ∞ λ _λ λ λ λ λ λ λ σ σ λ λ j λ γ γ γτ τ τ λ λ λ λ λ σ σ σ – ρ ,

are the j-th level wavelet and scaling filters of length

l L l l h g ₌ + 1 1 ) 1 ( 2 √ J N 2= x=(x0,x1,…,xN 1) 1 , , , ₄ 2 N … N t j W~_, V~j,t 1 2 , , 0 , 1 0 mod mod ] 1 ) 1 ( 2 [ , , = + = j J L l jl t l N t j h x t W j j … (5) 1 , , 0 , ~ 2 /2W, = =₀1h, x( ) t = N j L l jl t l N t j j _… ₍₆₎ 1 2 , , 0 , 1 0 , [2( 1)1 ]mod , = = = j J L l jl t l N t j g x t V j j … (7) 1 , , 0 , ~ 2 /2V, = =₀1g, x( )mod t= N j L l jl t l N t j j _… ₍₈₎ } {hj, l {gj, l} 1 ) 1 )( 1 2 ( + = L L j j ) ~ Var( ) Var( 2 1 ) ( , , 2 t j t j j j t = W = W (9) = = = = 1 1 2 , ) ( ) Var( 1 2 1 ) Var( j j jt j j t W Y (10) 1 2 = j j 2( j) j 2 –₂j+1 ) ~ , ~ Cov( ) , Cov( 2 1 ) ( jX,t jY,t j,Xt jY,t j j t = W W = W W (11) = = = = 1 , , 1 ) ( ) , Cov( 1 2 1 ) , j j j Y Cov( jXt jt j t t Y W W X (12) ) ( ) ( ) ( ) ( 2 1 j j j j = (13) ) ~ , ~ Cov( ) , Cov( 21 ) ( j = WjX,t WjY,t+ = Wj,Xt WjY,t+ (14) – – – '

Σ

(L is the length of the basic, first stage wavelet filter).3

The reconstruction part of wavelet analysis utilizes the inverse wavelet transformation in its conventional or maximal overlap versions, what results in a sequence of details and smooths. Though the details and smooths form an additive decomposition of the signal, the lack of translation invariance of the DWT, on the one hand, and the lack of energy preservation of the MODWT details and smooths, on the other, make them somewhat less attractive in studies concerning business and growth cycle synchronization. They can be helpful, however, in extracting cyclical components of economic time series and, when the MODWT details and smooths are used, in dating business cycle turning points.

2_{Here we concentrate on compactly supported orthonormal wavelets – see Daubechies (1992), Chapters VI–VIII.} 3_{For exact definitions of level j wavelet and scaling filters see Percival, Walden (2000), Chapter 4. For higher}

decomposition levels j the following approximate relationships hold: j _j l j l h 2 ψ φ 2 2 , ≈ –

( )

, gj,l≈2–j2

( )

₂lj .