Business cycle synchronization according

Ba n k i K r e d yt (3 ), 1 , –3 www.bankikredyt.nbp.pl www.bankandcredit.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 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 * Nicolaus Copernicus University, Department of Logistics; e-mail: bruzdaj@umk.pl J B ruzda 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 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., Gonỗalves, 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 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”’ Business cycle synchronization according to wavelets … 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 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 J B ruzda 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 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 and 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 presents results of our empirical examination, while the final section shortly concludes ) Conventional and maximalf (xoverlap discrete wavelet transformations f (x) ∞ f (x ) f (x) (1) Wj, t = ∫ –a signal f ( x)ψ jinto ,t ( x) dx Wavelet analysis consists in decomposing shiftedf and ∞ versions of a basic (x) scaled f (x) ∞ Wj, t∞ = ∫ – f ( x) ψ j ,t ( x) dx ∞ ∞to zero and has unit function, f (x)the mother wavelet The mother waveletWintegrates Wj, t = ∫ – ψ(x) f ( x), ψcalled j(1) , t = ∫ – ∞ f ( x) ψ j ,t ( x) dx j ,t ( x) dx j = 1, 2,…∞ , J , ∞ f ( x) ψ ( x) dx ∞ W = ∫ j , t energy The discrete wavelet transform (DWT) of as follows: ) ∞ (1) ,j ,t Wf j(x f ( xa)real-valued ψ j ,t ( x) dx function j = 1– ∞,f (x) 2,…is , Jdefined ,t =∫ – f (x) (1) ∞ W f ( x ) ( x ) dx = ψ – J j ∫ –∞ j, t j ,t , j , , … , = J j = 1, 2,…, J , – ,2 t = 0, 1, … , , 2J – j – ∞ j = 1,t2=,… ∞ 0, ,1J, … (1) (1) W f ( x ) ( x ) dx = ψ , j = , , … , J ∫ – j f ( x) ψ (1) J j , t j , t W = ( x ) dx ∫ –∞ – 1j = 1j,,t 2,…, J, j, t , t = 0, 1, …, JJ––jj – 0, 1, … t = ψj, t ( x) ,– ∞ , –1 t = 0,ψ1, … J –j j, t ( x) , J –j – , , , t = … , j , , … , = J , , – ( x ) ψ x ( ) ψ , t = , , … , j = , , … , J where and the wavelet daughters, j, t , are shifted and scaled verj, t (2) ψ j, t ( x) = – j / ψ – j x – t ψj, t ( x) , 2–j x – t ψ j, t ( x)– j=/ 22 – j / ψ sions of the mother wavelet with dyadic shifts t and scales j, i.e.: – J j – J j , –j/2 j – ( x) , x ( ) ψ –j – – j , t , , , t = … ψ ψ(2) ψ j, t ( xt )==02, 1, …ψ, 22 xj, t–1t j, t ( x) = – j / 2ψ – j x – t ψ (x) x)(x = )2 ψ x – t ψ j, t (ψ – j / j (2) , ψ – x – t ψ (2) (2) x jj,–,tt (t(xx)) = ψ (x) ψ (x)ψj, t ( x) , ψ j, t ( x) = – j / ψ – jψ ψj, t ( x) ψ (x{) ψ ( x)} j, t –j {{ψψj, t ((xx)(2) } {ψj, t (ψxj),}t (x) = – j / 2ψψ(x2)– j x – t ψ2j(x (2) ()x) = – j / ψ 2properties x–t ψ For certain functions ψ(x) with good j, t )} is an orthonormal basis L , (t ℜlocalization ) L ( ℜ) 2 {ψj, tψ(x) ( x)} is usuallyψψ L 2(ℜ ) scaling function or father in L (ℜψ The defined (the ) (x ) function j, t )( x) via another function (x L (ℜ )(x) ψ (x ) ψ wavelet), φ(x), that applied to the signal after shifting and scaling analogously to (2) produces 2 ψ ψ (x) {ψj, t ( x)} (x ) L (ℜ) L ( ) ψ ( x ) ℜ j , t ψ (x ) φ (x ) φ another set of coefficients in the form: (x) φ(x) φ(x) L2 ( ) ψ (x) 2(x) ℜ ψ ∞ ∞ φ(x) L (ℜ ) (3) V Vj, t∞= ∫ – f ( x)φj ,t ( x) dx ∞ j, t = ∫ – ∞ f ( x)φj ,t ( x) dx (3) ∞ Vj,(3) Vj, t =ψ∫ –(x)f ( x)φj ,t ( xφ)(x dx) t = ∫ – ∞∞ f ( x)φj ,t ( x) dx φ (x ) Vj, t = ∫ – f ( x)φj ,t ( x) dx ψ (x) known as ∞scaling coefficients ∞ W j, t∞ W ∞ j , t φ (x ) Vj, t = ∫ – fcoefficients, ( x)φj ,t ( x)φ W , are computed as differences of moving(3) averages for W W Fjor (3) φ Vdx = f ( x ) ( x ) dx (x ) j , t ∞ , t a given j the wavelet j,t ∫ j, t j ,t –∞ W j, t j –1 –1 ∞ j λ = the previous scale scaling coefficients and are associated with scale , while their squares λj = ∞ j φj ,t ( x) dx λ j = jj––(3) λ j = 2Vjj,–t1 = ∫ – ∞ f ( x)W j, t (3) Vj,j,tt = ∫ – f ( x)φj ,t ( x) dx W λ = ∞ j j j = λ + j – λ j +1 = λj = j λ j + = jj λ j +1 =W2j,jt λ j j,=t j –1 W λ j +1 = f ( x))+= ΣW VJ , t φJψ x) + ψ Σ WJ… ΣW(Jx– j –1 , t (x ) + , t ψ+J, tΣ(W VJ , t φ( xJ, )t +(x… ) ++ΣΣW Σ t J , tψ J –1 , t J – 1, t ( xt) + 1, t t1, t ψ ψ ψJ1,,)tt (=(xx)Σ f ( x )λ=j =Σ2VJ , t φJ, t (λxj)+1+=Σ2Wj J, t ψJ, t ( x ) +fΣ( xW)J = W =VJ ,Σ –j1–,j1t t J – 1, t tt φJ, t (x ) + Σ WJ, t ψ J, t ( x ) +t ΣWJ –1, t 1,ft ( x t t V t W t t t ψJ f ( x) = Σ ) + t WJ, t ψJ, t ( x ) + Σ λjj+t1==22 J , t φJ, t (x = SJ Σ (t x) + DJ ( x) + DtJ – J(–x1), t + J t ( ) ( ) ( { ) } { { } } ( ) ( ( ) ) (( ( )) ) Vj, t = ∫ – f ( x)φj ,t ( x) dx –∞ (3) W j, t φ(x(3) ) Vj, t = ∫ – f ( x)φj ,t ( x) dx W j, t W j, t φ(x) ∞ λ j = j –∞1 Vj, t = ∫ – f j(–1x)φj ,t ( x) dx λ j… =∞2 W j, t Businessj –1cycle synchronization according to wavelets ∞ λ = λ j +1 = j (3) Vj, t = ∫ – f ( x)φj ,jt ( x) dx W j –1 ∞ j j, t λ λj = j +1 = λ j +1 = j f ( x ) =j –1Σ VJ , t φJ, t (x) + Σ WJ, t ψJ, t ( x ) + ΣWJ W j, t j λ j = 2f ( x )t = Σ V φ (x)t + Σ W ψ ( x )t + Σ decomposition of energy of the signal on the time-frequency λ j +to , t Jthe =the contribute … +t Σ ψ f ( x ) = Σ VJ , t φJ, t (x) + Σ WJ, t ψJ, t ( x ) + ΣWJ –1, t J –1,t ( x) +plane W ( xD) J=t ( x)J,+t DJ,Jt – ( x) + =JOn S J1,t(,ψ xt )1,t+other j –1 t t t t j λ j = 2j scaling coefficients are moving averages of scale λ = The two= Stypes hand, thef (level of ψ1 1,t ( x) = x ) = Σ VJ , t φJ, t (x) + Σ WJ, t ψJ, t=(Sx ) (+x)Σ+WJD–1, t( xψ)J +–1,tD( x) +( x… J ( x ) + DJ ( x ) + DJ – ( Σt +WS1D,tj +(x ) ++ function ( x) t t J J of the original J –1 j ) in the form: coefficients give tthe multiresolution decomposition λ j +1 = j = S ( x) + D ( x) + D ( x) + + D ( x) f ( xS)j (x = )Σ VJ , t φJ, t (x ) + Σ WJ, t ψJ, t ( x ) + ΣWJ J J –1 t t t SJ j (x) D j (x) VJ , t φJ, t (x) + Σ WJ, t ψJ, t ( x ) + ΣWJ –1, t ψJ –1,t ( x) + … + Σ W1,t ψ1,t ( x=) =SJ ( x) + DJ ( x) + DJ – ( x) + S j (xf) ( x ) = Σ t t t tD j (x) D j (x) (4) S (x SJj ) = S J ( x) + DJ ( x) + DJ – ( x) + + D1 ( x) D j (x) S J (x) S J (x) S j (x) D1j ((x x)), D2 ( x) ,…, DJ (x) The Sfunctions Sj (x) and Dj (x) are known as approximations (smooths) D and details The highest J (x) ( x) , D2 ( x)_,…, DJ (x) _ _ J +1 , ,…, D ( x ) D ( x ) D (x ) level approximation SJ (x) 1represents smooth, low-frequency component of4 the signal, the J 2S 4, J 2while D j (x) ) 8,…, J (x_ _ J _ J +1 details D1 ( x) , D2 ( x) ,…, D_J (x) are associated with oscillations of length 4, 8,…, 2 l +1 _ 4, _ 8,…, J _ J +1 g l {=g(l }–l 1= 0), , LhL 1– 1– l In filtering the discrete wavelet transform is defined via quadrature the D1 ({xg) ,l }lD (mirror x)_,…, Dfilters: S (x_notation ) _ 4, g l = ( –1) l +1 hL– 1– l J (x) = 02, , L J _ J +1 1h)Ll–J+141–hl L8,…, g l =g(l–1=)2(l +–1(scaling) { } –1) l +1 hL– 1–filter _ The two filters g l = ((wavelet) – 1– l filter { g l }l = , , L _ and the high-pass h low-pass l √ l l = , ,gL 1= ( –1) l +1 h _ 4, J _ LJ–+11– l l {h4l }_l =8,…, √2 _ g = ( –1) l +1 hL– 1– l , have unit2 energy { g } _ fulfil the quadrature mirror relationship 0and , , L 1are even-shift l l l = , , L , ,…, D ( x ) D ( x ) D (x ) J l +1 {hl }l = 0J, , √2 √2 _ll++11 √ N = L) l1+1 h – g = ( ) hL– 1– l – g = ( – gglll integrates ==(( –11)) hhLLL(sums) – – √ 2_ l When processing orthogonal; the wavelet filter zero, while the scaling filter {g l }–l =to N––111–=–lll J √to , , L J {hJ } _ J _ + J J l l = , , L1 l +1 N =N = 2signals , x )' Then the discrete we ( –h1L8,…, )–consider h 2–√l 22a2data vector of length N = in the form x = ( x0 , x1N,√… g l =2(g–l14,)=l +4 =2 JN –1 1– l L– 1√ J, x √2 ' ) x = ( x0 ,Nx1=,… { } – _ N h highest decomposition level is J and the 2numbers l = , , L 1coefficients of the (xx10,… , x' )' x = (xx0=,possible N)' , Nlscaling J N –1 ) N_–1 ,…, x N –1and x = ( x0 ,ofx1wavelet 1N {, √ gx1l,2},xl… , … ,x JJJ = , , L , x N –1 )' x02, x1 ,…discrete = (=overlap √ N N N = N = , , … , conventional DWT for each level are On the other hand, the maximal N = ' ( , , , ) x x x … x = N N N , N ,, … ,,1… , 0N N N –1 ~ ~ 4 , , …,1 xN= ( x0 ,coefficients wavelet Ntransform of wavelet W j, t and V j, t Nscaling ,x11 ,…, x N –1 )' _ ~ the ~ ' same 2number hNJ} = J (MODWT) ={2 2, 4, … , x ,… … =(((xxxproduces xxx== 000,, xx111,,… ,1 W,j,,, txxxNNNV–––111jN),))'t'2 , N , … ~ ~~ ~ l l = , , L ~ ~ W V (W j, t and at each decomposition level as it does not use downsampling by The Vj, tj, t j, t ,xaccordingly) L j – 1~N , N~4 , … , , xNN ,–,1NNN)' ,, … … x = ( x0 ,=x(1x,… , ,x1x,N Whj,j,tl xV j,j t – – order L~ … ,1 –1 )'NN ~W j, t V j, t , t = 0, … , J – W j, t = Σ l =By 222 , scaled 444 , …,,1 j – 1retain variance preservation J0– j definition coefficients are appropriately in to t l ] mod are N + [ ( ) W V W j, t J=– jΣJl–= 0j j, th j, l xj, [t j (t +1) –1– l ] mod N , t = 0, … , –~1 ~(5) 1they Lj –1 Lj –1 (5) L – W , as NΣ , Nt ,=~~ W hN N xh,[ 2Nj, jl,4(1tx,+[1… t = ~mod ~0t ,=…0,,2… , 2– – 1(5) J –j V L –j,1t j, t = jΣ –1– lN 2) –j 1(t–,+1 ]~ l1])mod l =follows: 02 , l =j4,0l,2… given (5) , t = ~ , WL j,–tΣ WLj, t– 1= Σ l =j0 h j, l x[ j (t +j1/)2–1–~l ] modW ,j … h, x – 11) –1–tl ]=mod W V N j, t t=j = W V j j , t j , t –l j= x j, l [ j (t + , – Jh W V L ~ j,j,tt j,j,tt W = 0N, … , N – j j / Σ (5)N – – = , W h x j , t j , l ( t l ) mod , , t = … j ~ Σ l=0 –~ – ~ j l , j , t L (6) ~ – W t , … N = h x , = , – – – L t l N [ ( + ) ] mod = ~ l L j j / Σ j j/2 j j ,t – j, l ( t – l ) mod N – l = W V W (6) V = …–, N Lj = l = Lhj j–, 1l x[ j (t +1) –1– l ] mod N , t = W 2W j ,t W= jj,Σ xjh, t x(t – lN) mod , Nt ,= L0LLtj,j––… ,,N 1 (6)j / ~ –11 j, th ,tt = Σ j / 2j,tt~= Σ l = l =j,0l ( tj,–ll ) mod (6)N , t =J –0j j 1h 2mod NW h22JJJj,–l––jjjx–––(t1–11l 0, … , N – (5)(t – l ) mod (5) L j (5) 1, W =Σ Wj,j,tt == x 2–(5) 0,,,… … 1= ,,, jL,ttjtt–= W ) mod N Σ 0Σ W h j ,t = Σ l = h j, l x l = 0,,,2 j,jj,,lll xx[[22j jj((tLt++jj1/–1)12)–––11–~ –ll]]mod = … l===000 h Σ N – l V g x , t (7) = 0, …, j , t J j = – – t l N [ ( + ) ] mod (6) Σ l W , … = h x jN t ,, …t, = j1 ,0 , ,l N[ 2–j (1 Lj –1 Lj –1– l ]Lmod Σ – t N ) + l = – J j – – j , t j , l ( t l ) – mod – V = g x , t L J j – ~ = – L j =W 0– l ] mod =–1 j j (5)N (7) (7) j ,= t ,J0– ,Σ j, ,l –21 j/2 tj2… =J1 [ j (t – )(5) ,= W h jl, =lj 0x[h2 jj(, tl+1x)[–2,1j–(lt]+tmod +1l1 l =,0 0j2,–… t Σ j,x tl == g x , … , – j, t g Σ –t –10 l N ) ] mod V j ,t V = jΣ … , , 2 W = h x – N = Σ L j – – j l ,t = , Σ – Lj j––11 ~ –1– lN J j j j, l ( t – l ) mod N , t = ] mod N LL ~~ l = l =j0, l [ (t +[12) –1(t–+l1])mod j 1h , L –t1=J –0lLj=j,j,t0… V g,j,2l xl =[02 j–(t , t–= (6) 2– jjj///222W W =Σ x((tt–L–ljl))–Vmod , Σtttl ====0 000g,,,… … ,[N Nj (2–t––+j11/)112–1V~– l ] mod Ljj,t–1N= j… ,l x (6) (6) = h x , , j ,t j= Σ Σ N(6) 2N ~ j , t j , l N , N(7) mod + mod W = h x , , j , t j , l N j / g x , = 0,… l = Σ – (–t – l ) mod N 1) –t1–=l ](8) ~ – Lj ll==00V V , , jN ,… j ,t j,=lj ,t = ( t –Σ l ) mod Ngxj, l x j Σ ,tt =–01 l = 0, j ,l L … , –j / 2L j –1 L j L~ tN+1,) –1–tl ]= N ((6) mod lg = 0– Σ – l )[mod j , t j , l ( t j / ~j / 22~j / W j = j – W = h x , t = , … , N l = (6) h x , t = , … , N Σ – (8) g x , t … N , , –N= Σ – j , t j , l ( t l ) N mod (8) 2V j, t V x , t … N , , = jΣ – j ,t g j , l ( t l ) mod V = g x , t = Σ – – l = – ,t = j , l ( t l ) mod N = ~ Lj t~ Σ l = L j 1j, l [ j (t +1) –1– l ] mod N l = l =j0,l (lt =– 0l ) mod N Lj j–––111 LL – g1(7) (7) j / V,L,jj, t–1tt===Σ00,l,=… g,22jJ,JJl{–––xhjjj(–t–––1l1}) mod …Σ, N 2, j}/ t2j ,V = ~ j (7) j0 , t ,= j ,l x ( t – l ) mod N , t = , … N j / V = g x … { g j V = g x , l = Σ j j t j l , , (7) Vjj,t,t =Σ , j, l1 j, l (8) mod {j(2((tgtt+++111j),))–l––J1V }11––––ljjll,]]]tmod = –NNNj l,= tg=j ,l0x, (… Σlll===000, g{thj,j,llj,,l0x}[[[,222t… L j –1 L j –1 t – l ) mod N , t = , …, N –~1 –1 ,–mod 21J Σ {h j, l{}h {j,Vlg}j ,jt, {l =}gVΣ (7) (7) j / V = Σ L j –1 g x = ,02, … g lj,=l0x[g2 jj(, tl +1x)[–21j–(lt]+mod jj, ,lt}= Σ – l ] mod N= t= 1) –1N l=0 – l ) mod N , j , t j , l ( t l = – ~ Lj j––11 {h j, l } { g j, l } j {h j, l } { g j, l } LL jjj///222 ~ ~ jx j 1g { + – – = L L ( )( ) – (8) V = , t … N , , – } h { g } (8) = VVj,j,tt ==Σ – 1N), + 1t = ,…, N j ΣlLl==0j0 =gg(j2jj,l,,ll x–x(((1tjtt–,)(––llll)))Lmod mod (8)(8) j j / ~ L j –1 L j –2 12 Σ modNNj,,l t = , …, N – –j, t 1=) + (2j =2j –j(/122)(V~–L L j =L –1 2j,1t–)(=1L V)Σ+ t =,0N,… – 1, N (8) (8) {h j, l }j { g j, l } g1 x g x(j,t Nt– l ), modt N=l =,00 ,… j l = Σ lj =,l0 ( t –jl,)l mod – – = ( )( ) +1 ~ L L L j j = (2 – 1)(L – 1) + – 1)(L – 1) + ~ σ ( λ ) = 1j Var( W ) = Var(W {{{hhhj,jj,,lll}}} {{{~ggg j,jj,,lll}}}σ ( λ ) L= j =1 (2Var( (9) ) Var( W = W ) j j, t t j j, t ) h{ gj, l }Var( ~ Wj-th) level t jwavelet and scaling j, t j, t of { g j),W where filters length l } )are λ2L~j j = (2 –11)(L – 1) + j, l } (9) 2(9) Var( W Wthe λj σ t2 ( λσjt )({=λhj j), l =}{and t = Var( ~ j, t =j,Var( j, t ) j, t λ j λ j of the basic, Var((9) )= Wj, t ) = Var(Wj, t ) σW t (j,λ )~= Var( = W Var( Wj, tW σ=t2 (31λ j )Var( (L is the length +σ11 –11 –11 =(((22 Ljj =first L–wavelet 2jjj–stage 1)()( )(LL 1))) ++ 12 ( filter) LL t )j j – – (9) = λ ) Var( ) = ) λ j t j, t j, t j1 L––11)()L+–11)j + ∞ ~ ∞ L = (L2j –=1()( j ∞ ∞jutilizes λ j the2 λinverse Var(W ) wavelet ) = Wj, tVar( Wσ The jreconstruction part∞ of∞ wavelet analysis its σ t2 (1λ j Var( ∞ 1∞ )in= Σ ( λj ) j, t ) = j, t (10) Var( Wj, t ) = Σ σ Var( ( λ j )Yt ) = transformation Var(Yt ) = Σ Σ λ 1λ j ) (10)2(10) Var( = ΣVar(Var( σ ~ Var( Yt ) =Yt ) Σ Wj, t )W =j,Σ σΣ(versions, j1 2∞ j =1 smooths ~~ λ=j )(11 λj ∞ Though 2= λj in ∞ t2) j =1 j == Var( ∞ j =1 2 conventional or maximal overlap what results a sequence of details and (9) λ Var( ) ( ) W W ) ~ σ (9) ~ λ Var( Var( W W σσtjtt=)(1(=W ∞ 2 λ ∞ (9) λVar( Var( Var( =W Wj,jj,,ttt))==Var( W)j,jj,,t1=tt)) 1Y jjj=))1)= (9) Var( Wj, t )Var( (10) Wj, t ) = Σ σ ( λ Var( Wσj, t2)(=λVar( )2 Σ λ Var( σY2t ()λ=j(10) σ t2 ( λj =σj1)tλ=(jj=λ1 j ) j=Var( W , t =2 t λλλj, jtjj) decomposition Σ Σ j, t Y2t )(9) = of Var( W )=Σ 22Var( j, tthe j) j =1 ∞ j j =1∞ Σ the details and smooths an jadditive the signal, lack of invariance j –1 – translation λ j λ form λ j j j = j = j λ j = , σ ( λ j ) j =1 λ j σ ( λYjt)) = Σ Var( Wj, t ) = Σ σ ( λ j j ==1 , Var( j – j – 12, ( ) σ = λ λ , ( λthe σ on = 2j DWT, λ j the j of one hand, and the of the MODWT detailsj –1and22smooths, λ j) ∞ 1of energy preservation ∞ j =1 ∞∞ ∞∞ 11 lack λ j = , σ j(=1λ j )j j –2122 ( ∞1 ∞ Var( YY )∞= ∞ Var( 2σ jt)–λ 1= (10) Var( Y ) = W ) = ) σ λ (10) , ) = Var( W ( ) ( ) σ = λ λ t j , j Σ Σ (10) Var( ) = Var( W ) = ( ) , σ ( ) t j t j , σ λ = λ (10) λ j Σ Σ j + j Var( Y Var( W ) = ( ) j σ on the Var( other, somewhat attractive business and jgrowth cycle Y )make = t Σthem W )t = j,2t2less (–λλ ) λ j(10) σj =Σ j , tin studies Σ=111 j concerning ΣVar( j – j +1 j +1 t j j2Σ =11 λ jjj== λ2jjj j j j =1 λ2j j =1 λj j, t Σ λ = –1 , σ ( λtime j =1 j =1j =1 j j – +–1 j) synchronization They can be helpful, however, in extracting cyclical components j j ofj +economic –2 j +1 j j + j – j 1 – ~ ~Y 2 used, 2 λ =details j –1 ~ X ~Y σand λsmooths series and, are business turning X points Y )))=Y2 – Cov( λλjjjjY)== 222jj––1~1,,, Xσσ W j,Xt , WinjY, tdating ) = Cov( W j,Xtj ,)W ) Cov( ~t2~((X((Yλλλ 2the 1=when σ (,MODWT 2λ1jj –1=,Cov( λσj )XY,(W +W j jjjj)~ = j, tcycle t (λ j,(11) t , W j, t ) = Cov( W j, t , W j, t ) – j(11) ( λλj j) = ) λj Cov(W j,Xt W W j, t W ,W ,W ) j, t j, tj, ,t W t ) = Cov( j, tj, t ) =j, Cov( ~ t ( λ j t) = λ j ) =~ Cov( λj λj W j,Xt , W jY, t ) = Cov(W X Y X Y ~ tX( λ ~ X ~Y j Y j + j Cov( W W ) Cov( W W ( ) = , = , ) ( ) = Cov( W , W ) = Cov( W , W ) j +1 t j1 j, t j, t t λj j++11 j, t j, t 2j, λ j, t j, t t λj j, t 2 Here we 222jj –––222jsupported – 22j +j1– on compactly j concentrate orthonormal wavelets Daubechies (1992), Chapters VI–VIII 2– λsee λj j Cov(W j,Xt , W jY, t ) = Cov( t ( λj ) = 3 For exact definitions of level j wavelet and scaling filters see Percival, Walden (2000), Chapter For higher λ j ~WXXX X ,~–~ ~W 11 l ∞ l ∞ 111Y Cov( ~~ ~XX relationships ~XXX)YX= 1~YYY Y∞ 1hold: – j 2∞ j Y2Y Y ∞ ∞j the X YX approximate ∞) = ∞1 X(11) (12) Y γ , Y Cov( ) = ( ) Y λ ≈ ≈ , decomposition levels following h g 2 1 ψ φ Cov( W Cov( W ( ) = , , ) ( ) = Cov( W , W ) = Cov( W , W ) X Y λ λ (11) Σ Cov( W W ) Cov( W W ( ) = , = , ) tWj,jt,ttj,, ,tW W j, lj,j,ttj,, ,tW j, l j λ,λW t= j, t ) Cov( ,)) j , tj,))t j Cov(W Cov( =λCov( Cov( W Wj,jj(12) jγ , )tW j,ttttX((W j,jj,,tttjΣ jjjj),)tY= Cov( ,= W ( = λ t , Yt ) = ∞ 1j, t , W j, t ) = Σ γ ( λ γ Cov(Cov( X t , YtXt )tλ,=jYtt) =Σ Cov( W ) ( = Σ j, t (12) t 2jX ,tCov( jj, )t W λ 2 Σ j t j t , , Σ λ λ Σ j t j t , , j j = j = λ j λ 22 λjjj j =1 X Y ∞ 2Xj =,1Yλ)j∞= j =12λj j =j λ jj ∞ 11 ∞ X j =1 t t j, t ) Σ Cov(W j, t j,=W Y Cov( X (12) Cov(X tCov( , Yt ) =X t , Y Cov( W j,Cov( WγjY,(t λ) j=) Σ γ2(1jλ=1j )∞λ j1 t , W j,W t ) j= Σ t)= , t ,Σ Σ X Y j =1 λ j j =1 λ j j =1 X , Yj =)1 = Cov( t t Σ λ Cov(W j, t , W j, j =1 j γ∞ ( λ j ) ∞ ∞ ∞ Y∞1 ∞∞ X YX1 ∞ ( ) ( λ ,j )Y1 ) ∞= Cov( =Cov( ( λ, Yjγ)X ρ1 )∞=Σ Cov( Cov( ( λXjXX),,W Cov(γX = ( λΣj )γW t t Σ Cov( j, t1)γCov( (12) (13) Σ XXW Cov( W WYYY ))== γγ((λλ ))(12) (12) γ λ j t t)= (12) ,,YjY, t ),)W ==(j,1λt ),j W ψ (x)∞ ∞ ( ) ( ) 10 J B ruzda Among the most popular real wavelet filters are the compactly supported orthonormal Daubechies filters: the extremal phase (dL) and the least asymmetric (laL) filters (see Daubechies 1992) The two wavelet families are characterized by the smallest filter length L for a given number of vanishing moments (embedded difference operators) Besides, the extremal phase scaling filters have the fastest build-up of the energy sequence, while the least asymmetric filters are approximately linear phase Figure gives examples of mother and father wavelets corresponding to two of the Daubechies filters Squared gain functions for filters based on these wavelets are given in Figure They illustrate the fact that by increasing the support length better approximations to ideal bandpass filters are obtained The maximal overlap discrete wavelet transform (MODWT) helps to remove certain deficiencies of the conventional discrete wavelet transformation and in our application can be thought of as an improvement over the DWT Among the distinguishing features of the conventional DWT and the MODWT are the following: • The MODWT can handle any sample size, while the Jth order partial DWT – only multiplies of J • The MODWT is translation invariant, what means that circularly shifting the time series is equivalent to analyzing its circularly shifted wavelet and scaling coefficients or details and g l = ( –1) l +1 hL– 1– l smooths •  As is true for the conventional DWT, the MODWT enables variance and covariance √2 decomposition But the MODWT provides better estimators of the wavelet variance and N = 2J covariance (see Whitcher 1998) • The MODWT approximations are 'associated with zero phase filters, what makes x = (and x0 , xdetails , …, x N –1 ) it possible directly to align their features with those from the original series N , N , …,1 • A n additive decomposition 2of the time series in terms of its details and approximations is ~ ~ valid for both the DWT anW the MODWT Contrary to the DWT, however, the MODWT details j, t V j, t and approximations not form an energy (and covariance) decomposition L –1 –j (5) as well as –stationary W j, t = Σ l =j0 h j, l decorrelates x[ j (t +1) –1– l ] mod Na ,broad • The conventional DWT approximately t = ,range … , Jof nonstationary processes (see Whitcher 1998; Percival, Walden 2000) L j –1 j/2 ~ (6) W = h t = 0, … ,covariance, N–1 Further in the text we concentratej ,ton Σ wavelet coherence and j, analysis l x( t – l ) mod Nof, variance, l=0 phase angle, which are the most important wavelet methods in business cycle studies To this L j –1 J– j – (7) V = g x , t … , , = j Σ j t j l , , – – aim we work with the maximal overlapl =transformation N offers better estimators of our wavelet [ (t +1) l ]as modit quantities A more detailed description of Lthe –1 discrete wavelet transform can be found in many ~ (8) j / V j, t = Σ l =j0 g j ,l x(t – l ) mod N , t = 0,…, N – mathematical and engineering books (see, e.g., Daubechies 1992; Mallat 1998; Białasiewicz 2004; Nason 2008), while interesting introductions {h j, l } { g j, l } to wavelet analysis from an economic point of view offer Genỗay, Selỗuk, Whitcher (2002), Ramsey (2002), Schleicher (2002) and Crowley (2007) = (2 j – 1)(L – 1) + 1wavelet variance is defined as: L j time-dependent For a stochastic process Yt the ~ (9) Var( Wj, t ) = Var(Wj, t ) (9) σ t2 ( λ j ) = 2λ j ∞ we arrive at Assuming that (9) does not depend on1 time, ∞ a variance decomposition according (10) Var(Yt ) = Σ Var( Wj, t ) = Σ σ ( λ j ) to scales λj in the form: j =1 λ j j =1 4 The λ j = j –1 , σ ( λ j ) assumption is fulfilled also for nonstationary processes provided that they are integrated of order d and the width of the wavelet filter, L, is sufficient to eliminate nonstationarity In the case of the Daubechies wavelet filters ~ j +1 j –to2 have the condition is L ≥ 2d Further, in order E{Wj,t} = we assume that L > 2d t ( λj ) = ~ ~ Cov(W j,Xt , W jY, t ) = Cov(W j,Xt , W jY, t ) λj (7)(8) V j ,j2,t t = Σ 00,,… mod NN,, tt= V lj,l=t=00=gΣjj,,lll =x0[[22jg((ttj++,1l1))–x––11(––tl–l]]lmod …,,2Nl =–0 11 N {h = L j ) mod J– j j, l } { g j, l } – (7) V g x , t … = , , = j Σ ~{h } {Lgj jL–,1tjj,–l1} l = j, l [ (t +{1h) –1j,–l l}] mod { gNj, l }– j/2 ~ (6) Wj,j,ljt},t =j,={l Σ , t = , , N gΣj, l l}= 0g hj ,lj, xl (xt –(tl–) mod j… – (8) , , 22 j /{2hV , t … N l ) mod N = N L j = (2 – 1)(L – 1) + l=0 L j –1 j/2 ~ (8) , N –1 VLj, t –=1Σ , 1)(tL=–01,)… L j L=–j(122j – 1)( ) +l1= g j ,l xL(t j– l=) mod +1 (2 Nj –to J– j j Business cycle synchronization according wavelets … 11 + – – = ( )( ) L – } { g } (7) V{hj ,tLj, = g x , t … , , 1 = ~ j l j Σ l = 0j, l j, l [ (t +1) –1– l ] mod N (9) Var( Wj, t ) = Var(Wj, t ) σ t2 ( λ~j ) = {h j, l1} { g j, l } ~ 2 λ1j Var( W(9)) = Var(W Var( ) Var( W = W ) (9) σ – λ ( ) = ) j ( λ j )L= σ ~ t j , t j , t ~ j / j j, t j, t – 1=)(Lj –2g1Var( – l ) mod (8) , N2–λ1j 2L j σ=Vt2((j2, λt = , t = t0,… λ)jj,+ (9) l x( tW l=0 j )Σ j, t ) =NVar(Wj, t ) j ∞ 2Lλj j = (2 – 1)(L – 1) + 1 ∞ (10) Var( Var( Wj, t ) = Σ∞σ ( λ j ) ∞ ~ ∞Yt ) =2 1Σ 1 ∞ W1 ) = Var(W {σh2j,(lλ} ){=g j, l1} Var( (10) (10) λj(9)Var( j =1 σ ( )Σ Yσt ) (=λ jj =)Σ Var( W ) = ) (10) λ Var( Yt )1= ∞ Σ Var( W ) = t j j , t j , t j , t j Σ ∞ , j t λ ~j =1 λj(10) j 22 j =1 Var(Yjt ) 2=λ σ WVar( j =1 Var( λ–1j ) 2W (9) Wj,σtj )=1=(jVar( j, t ) = Σ Σ t ( λ j ) =j j, t ) , λ ( λj ) σ = λ λ L j = (2 – 1)(L j–=11) +j j =j1 j ∞ 12 j∞–1 , 1σ ( λ ) to scale λ j ), informs about variation λ2j = j –1 , σ ((10) The wavelet varianceVar( at level Ytλ)j =j=corresponding j – Σ Var(j Wj,∞t ) = Σ σ ( λj +j 1) ∞ j 1 ~ , ( ) σ = λ λ λinj Ythe j = interval j =1 Var( 12 – of oscillations of length approximately Similarly, (10) covariance ( λ j )the wavelet Var(j =Var( W σ t2 ( jλ j ) = Var( t )W j, t ) = j, t )j Wj, tj)+1= Σ σ (9) Σ jλ +1j j λ – 2 j = j = j and wavelet correlation are introduced For the stochastic processes Xt and Yt the time-varying –2 ~ ~ λ j 2=j –2 2j –j1+,1 σ ( λ j ) (λ ) = Cov(W j,Xt , W jY, t ) = Cov(W j,Xt , W jY, t ) wavelet covariance is defined as follows: ~ ~Y ∞λ j 11= j –1 , σ 2X( λ j∞) Y t ( j ) =2 λ~1j X Cov( ~ Y W j,Xt , W jY, t ) = Cov(W j,Xt , W (10) Var(Yt ) (=λ )Σ Var( WW σ) t=(λCov( λjj ) W j, t , W j, t ) j, t j), t = Σ Cov( W = , ) j t j , t , t j j + λ j ~ X2 j~ Y j 1=1 2λλj j X Y j =1 – 2 ( ) = Cov( W , W ) = Cov( W , W ) λ j, t j, t j, t t j j, t (11) 22λjj – j +1 j –1 ∞ ~ ~ Y1 ∞ λ j = , 1σ ( λ j ) X Cov(W j, t , W jY, t ) = Cov( Cov(W X t j,,XtY,t W ) =j, t ) Σ∞ Cov(W j,Xt , W jY, t ) = Σ∞γ ( λ j ) t ( λj ) = As in the case of the variance decomposition (10), if the wavelet covariances not depend on 1 ~ ~ λj ( λ ) 1= ∞ 1Cov(WCov( X γ(λ ) XjY, ttY,)Y=t )Cov( =2∞ j =1Σ Wλjj,Xt , Cov( W jY, t )W j,Xt , W jY, t ) =j =Σ j t )= j j, tX, W j + j γ Cov( X , Y Cov( W , W ) = ( ) λ time, they produce decomposition of tthet covariance Y2t according to scales λj: j =1 (12) 21 λj between j, t Xtj,and t ∞ Σ ∞ Σ j =1 jλ j –2 X Y λ j (12) Cov(X t , Yt ) = Σ j =1 Cov( W j, t , W j, t ) = Σ γj =(1λ j ) j =1 λ j j =1 ~ ~ X∞ γ ∞ W1 j,Xt , W jY, t ) =X Cov( ) =, Y ) =Cov( , WγjY(, t()λ )j ) t ( λ jX ( λjY,jt W )) ==j, tΣ (12) Cov( Cov(W j∞, t ,ρW λ j λ (12) t 2t λj Σ j()λ ) ∞ ( λ γ) (σ j =1X j =11 X ,j W Y2 ) =j ( λ j ) =σ γ(λt ,λjYj t)) = Σ 1ρ Cov( (12) γ Cov( W ( ) λ Σ j t j t , , j ρ ( λ j ) = γ( λ ) (13) σ1 ( λ j ) σ2 ( λ jj =)1 j =1 λ j j σ1 ( λ j ) σ2 ( λ j ) ρ ( λ j )correlation (13) = The (time invariant) wavelet for scale λj is defined via: (λ j ) σ1 (1λ j )∞σ21coefficient ∞1 ~ Y X ~Y j) (12) , t + τ ) Cov( , Y ) =γ( λΣ Cov(W j,Xt ,γW λ j ) W j,Xt , W jY, t + τ ) = Cov(W(13) = Σ γ (Cov( , t j)) = , t , W j~ j~ τ (jλ ρ (λj X ) =t t λ X Y X W , W ) = Cov( W , W jY, t + τ ) λ j ) =j2=1 j ~ XCov( σ1 ( λρj ()λσ1j =21)(=λλjj ) γX( λ j ) Y γτ ( , , + j t j t j, t(13) τ ~Y λj , W (14) (13) W , W ) = Cov( W ) γτ ( λ j ) = j Cov( j, t~ j, t + τ σ1 X( λ jj, t) σY (j,λt +jτ) ~ λ X Y j (14) Cov( W j, t , W j, t + τ ) = Cov(W j, t , W j, t + τ ) γτ ( λ j ) = 2γλ(j λ ) j ~ Y dependence between strength ρ (13) The above quantity measures the ofX linear two X and Y direction ~ (14) W γτ( (λλj )j )== σ1 ( λ Cov( j, )t , W j, t + τ ) = Cov(W j, t , W j, t + τ ) ~ ) ( σ λ j j ~ 2λ jlevel X the Ylag τ wavelet Xcross-covariance Y processes for a given decomposition (scale) Finally, and (14) Cov(W j, t , W j, t + τ ) = Cov(W j, t , W j, t + τ ) γτ ( λ j ) = 2λ j cross-correlation are given by: ~ ~ (14) Cov(W j,Xt , W jY, t + τ ) = Cov(W j,Xt , W jY, t + τ ) (14) 2λ j γτ ( λ j ) γτ ( λ j ) γτ ( λ j ) (15) ρ τ (λ j ) = (15) ρ = ( ) (15) ρ λ ( ) = j λ τ ( ) ( ) λ σ σ λ j τ γτ ( λ j ) j j (15) ( ) ( ) λ σ σ λ ( ) ( ) λ σ σ λ j j γ ( ) j j (15) λ ρ τ j γτ ( λ j ) (15) τ (λ j ) = σρ1τ((λλjj))σ=2 ( λ j ) (15) ρ N –1 (~λ j ) –1 σ22 ( λ ) σ (1λ )N –1 ~ σ1 ( λ j ) σσˆ2 (2ρλ( λj )( ) =) =1τ 1( λ jγ)Nτ =W (16) ~j,1t(2λ j) = j Σ W (15) 2τ jλ j Σ ˆ (16) σ ~ ~ j j, t (16) σˆ of ( λ the Wvariance An unbiased and efficient wavelet as: ~1t =(LλΣ j ) =N , tλ j ) N ist =defined N –1 ~ estimator j j–)1 σ2 j( L j –1 jσ j (16) N – σˆ ( λ j ) = ~2 Σ W j2,t1 N –1 ~ N = t L – j j ~ σ Nˆ j (tλ= Lj )j –1= ~ Σ W (16) σˆ (1λ j )N=–~ W j2,(16) ~ j, t 1~ Σ t ~ Nj t = L j –W ~ 2 N W t = L j –1 jσ ,ˆt ( λ ) = j j , t (16) W Wj, t j ~ Σ j, t ~ (16) ~ Wj, t ~ Nj t = L j –1 j j Wj, t L j = (2 –j W 1)( L – 1)L+ 1= (2 – 1)( L – 1) + L~j = (2 – j1, t)( L – 1)j + j – – + L ( )( L ) = W j ~ ~ j L j =wavelet (2 j – 1)(coefficients, L – 1λ) + 1j, t ~~ where Wj,t are MODWT of the wavelet fil1)j +=1 Nis–the L j +length NjL j==N(2– Lλ– jj1)( +L –N j ~ λ – L +1 N = Nboundary –~1) j+ wavelet coefficients (i.e coefficients ter for scale λλjj and Nj = N – L~j + is theLj jnumber = (2 j –j 1of )( L λj Nj = N – L j + λj N (1j –=αN)%– L j + (1 at – α)%end unaffected by circularity or reflection α)% ~ of the sample) (1 –the (1 – α)% λj N = N σ– 2L(jλ +) 1is computed as follows: A n approximate (1 – α)% confidence j α)% (1 j– for σ (2λinterval j) σ (λ j ) σ (λ j ) 2 0.5 σ (λ j ) (1 – α)%σ ( λ j ) 0.5 ˆ ( 0) f ˆ W, j 0.5 f (20) ς α 5~ (17) (17) fˆW, j (0) ˆ σˆ 2σ(22λ( λ0).5j )± ς α Wfˆ,Wjσ,ˆj ((0λ)j ) ± (17) 2 Nj ~ j ˆ ( ) f ˆ (17) (17) λ σ σˆ ( λ j ) ± ς α2 ( ) ± ς ( ) f α W j , ~± ς j Nj~ W, j (17) σˆ ( λ j )N α (17) ~ σˆ (2 λ j ) ς±N ας 0~ j j α 2 N ς ˆ α j N ( ) f j ςα ς2 α W, j ςα (17) σ2ˆ ( λ j )ς±α ς α ~ 2 N (1 –j α /2) (1 –ς αα /2) (1 – α /2) (1 –2 α /2) (1 – α /2) (1 – α /2) fˆW, j (0) ˆ ˆ f (0) f (0) γτ ( λ j ) = σ ( λ0.j5) (1 – α)% σ (σσλ jj ()(λλjj)) j fˆW, j (0) ˆ ( 0) σ ( λ ) 0.5 f 00 55 W j , α (1 – )% j ˆ λ ˆ σ ( ) ± ς α ˆ ~ f W, j (0) j (17)ffˆˆ ˆ((00)) 0.25 0.5 σˆ ( λ j ) ± ς α σ ( λ j )fˆ (0) ~ ˆ Nj (17) N ~ ˆσˆˆ 2222(((ˆλλλ2j2j))) ±±±0.ς5ςςαα ffˆWWWW,~,,,~jfjjjfW((W00,, j))j ((0σ0ˆ)) ( λ(17) (17) j ) ± ςα (17) σ ( λ j ) j σˆ ( λ j ) ± ς α W,~j σσ (17) (17) σσˆλ(0j(()λλ±jj))ς0±2α2±.5 ςςαα N~ ~~ (17) Nςj α f ˆ (17) σ ( ςα α W j , ~j Nj 22 N N 12 f~ˆW, jj (0J) B22 ruzda (17) σˆ ( 2λ j )0.±5 ςNαj j ςα Njjj N ˆ(0)( λ j ) ±2ςςαςςα2αα N (17) ς α fˆ σ ~ ς2 α22 ςςjα2αN W, j j (17) σˆ ( λ j ) ± ς α ς~α (1 – α /2) (1 – α /2) N2jς α (1 – α /2) –––(1 αα the standard and fˆW, j (0) is an estimator of (1 ˆ (0) ς α2 is the (1 – α /2) quantile of(1 (1 /2) α –/2) αα/2) fwhere (1 –/2) /2) normal distribution (1 – /2) α W, j ˆ (1 – α /2) fW,atj (frequency 0) the spectral density λj squared wavelet coefficients fˆW, jof (0)scale fffˆWˆˆ, jfˆ(ˆ((00 ))) (0) (1 – α /2) N –1 ~ X ~ Y N –1 f ( ) ˆ W , j ~ofX wavelet ~Y ˆ ( λ j ) = via W, jand E stimates covariances wavelet correlations are computed theWfollowing W , j ˆ f ( ) (18) W , j N –1 γ ~ j, t W j, t f ( ) γˆ ( λ j ) = (1 W W – /2) α W , j ~ X ~ Y Nj t =Σ ~ Σ j, t j, t W, ˆj N – (18) L j –1 ––11 N ˆ γ W W ( ) = f ( ) λ N ~ ~ N formulas:j t = L j –1 ˆ ~ Σ j, t j, t j (18) Y 111 1N –– N~N~ ––X 1X YY ~ ~~ (18) γ( λ j ) = ~W, j Σ W j,XtNW ~ ~ (18) ~ N N ˆ X Y ~ ~ γ W W ( ) = ˆ j t , t L = – –ˆ1(λ Y (18) j (18) j γ W (18) (18) ~Σj,jj,,XtW ~j,XXjt,jjY,,tW fW, j~(0) ˆ (~λ Σ–1W W W = )~~ γ ˆjˆXjj ()))(λλ~= ttW t Y (18) ~1== t~t=~=Σ Σ (18) γ~( λ j ) j, t tWj,j,tt =jjjY),N (–λ λ1γγ L γˆ (Nλjj )t ==L j ~–1 ΣγˆNW N Σ jj –W λ ( ) γ L 11L j,–t1 W j, t j,j~ tW t~jYjj N N ~ ρ j t L = – ˆ N = ( ) = t λ ~ X j t L = – jj –1 j j j Nj t(19) ~ (λ ) ρˆ ( λ j) = ~ N =L γ ( λ(18) W W j j) j~) = j ~t = L j –Σ σ~1 ( λ j ) σ ~ (1λ )N – ~ Xγˆ (~λYγ j ~~ ρˆ ( λ j) = ~ (19) ( λ jN) j t = L j –1 j, t j, t γ σγˆ1((λλ j))=σ (18) j λ ( ) ~ ~ ~ W W λ (( λγγ~ γ jj () (19) ~ ρˆ (Σ j λ ) =j, t j, t ~ ~(ρˆλ( λ) ) = σ1 ( λ j ) σ2 ( λ j ) γ λλjj)) ~ ( j) (19) γ λ ( ) ρ (19) γ Nj t = L j j–1 σ~1 ( λ j ) σ ˆ ρρ~)ˆ ((j λλρρˆjˆjj))((λ= (19)(19) j ~~ ~~ ==( λ~~ ) σ (19) (19) ρˆ ( λ j) = ~ ( λ jγ (19)(1 – α)% ~ ~ (19) ˆ~( λjj) = ~λ~λ =λjj)σ)σ λλjjj ())(λσ (1 – α)% 11 ( 22)(( λ2j2jj)()))(λλ(1 σ σ ) σ σ ~ ~ λ ) σ σ 1( 2j )( j j λ λ ( ) ( ) σ σ j ρ λ λ ( ) ( σ σ (19) – )% α ˆ ~ ) ( λ j) = ~ j ~ j j j j (1γ– (αλ)% γ( λ j) λ j –) σα2)% ( λj ) σ1 ((1 (19) γ ( λ j ) ρˆ ( λ j) = ~ (1 – )% α ~ (1 – )% α (1 – )% α (1 – )% α λ λ ( ) ( ) σ σ γ ( ) (1 – )% α A n approximate confidence interval fore λ j is obtained as previously with γ (j λ j )2 (1j – α)% (1 – α)% fˆW, j (0) γγ ( ) fˆW, j (0) being an estimate of the spectral density function for the product of scale λj wavelet λ ( ) λ jj() ˆ γ ( γ ) λ γ ( ) γ ( ) λ λj (1 – α)% ˆ f W, j (0) j ) λjj γ ( λ j f ( ) coefficients at frequency confidence intervals for the wavelet correlation the γ0 ( λ To ) construct cj c j W, j fˆ (0)j fffˆWˆˆ , j ˆ(ˆ((00 ))) (0) g (a j , b j , c j ) = = ρˆ( λ j ) f (see Whitcher 1998): ρˆ( λWjbe c f ( ) ˆ W , j (20) gfollowing (a j , b j , cγj )(function =λ j ) =must ), ˆj considered j f W, j W(W0,, )jj a j bj (2 g (a j , b j , c j ) = = ρˆ( λ j ) a j bgj (a , b , cf W), =j (0)c j =Wρ,ˆj( λ ) (20) ˆ c j j j j a b c j f W, j (0) c j j c j j cc ρ (20) ρˆˆˆ=(= (20) a j ,(20) bj, c j g(((aaaggjjjj,(,,c(ababbjjj=,,,, bcccbρˆjjj(,),))λc=c= ((λλ g (a j , b j , caj )j b=gjg (20) =j j)j))γ==τc(jλ j )j=j= =ρ (20) λρρjˆjjˆ)()))(λλ jj)) (20) (20) a j , b j , cj j jj , cjj ) = ρ ˆ (20) g a b ( , = ( λ a b (15) ρ = ( ) j j j j ρ λ ˆ a b (20) jj b jj ba , b , c g (a j , b j , c j ) = j τ j j = ( λ j )a a b a c j j j a b jb j j( j j ( ) ) λ σ σ λ j j a – N 1 j j j2 j a j bj (20) g (a j , b j , cajN)j–,=1b j , c j = ρˆ( λ j ) ~ ~ a jN –=1 σˆ X2 ( λ j ) = ~ Σ (W j,Xt ) aaajj,,, abbbjj,,,, bcccjj, c N –1 a j = σ ˆ X2 ( λ j ) = ~ Σ (Waj,Xtjj),b2jb j , c j ~ X Nj t = L j –1 ~ ab~ˆjjj2, bcjjj, cjj ) Na–1jj ,defined = L j –1 Nj t values where the mean as:~ Σ W j2,t a j = σˆ X ( λ j ) = ~ Σ (W j, t(16) σ jX( λ2 jj ) = a j = σˆ X2 a( λj ,j )b=j , c~j are (W = L j –1 t N Σ j,Nt ) –1 – N j = L j –1N –1 a j, bj, c j ~ N2j 1t1 t = L –1 N –1 NN~ –~ N –1 ~ Y ~–11j,XXXt )~)~222XX 22 11Σ N –1 a j = σˆ X2 (Nλjj ) =aaj ~ N –1((W ˆˆˆΣX2X22N(((–W σσ λ1λλ22jjj,X)())~t λ)== = = W ˆ ~ jj = ~ σ b = ( ) = (W j, t ) ~ λ Σ Y a = ( W ) σ – ~ , j t X 2 ˆ N ~ j Y j a = ) = ( W ) σ ~ X at =jˆL=jX–1(σˆλXX(jW ( λjj)N )~= t~ j = ˆ Y ( λ j ) = ~ Σ (W j, t ) aNj –1= σˆ X ( λNj )–a1N LLj ––11(Σ Σ1jj,,1(ttW)σˆj,j2,tt() ) = (W~ Y ) Nj t =Σ b j = σ W = tt=~ ~jjj N X j)= ~jσ j, t N =Σ L jjt–t=1=b Lj –1 N LLjj––= ~ 1X =j jW j, t t =2Σ N λ ~ t = L –1 12 j N ~ j Y j j , t j Y Σ t = L – – L Nj j a j = σˆ Xj ( λbj j)j = σ~ (W~j, t ) Σ (WNj,jNt –)1 j ˆ ( λΣ) = Nj t = Lj –1 11 NYj t = Lj j –12 N ~jY 11 NNN–––111 N~–~–1Y – t L = N –1 ~ X ~ Y j N 1Y ~2 j – – ˆ + L = )( L ) σ N –1 b = ( ) = ( W ) λ – ~ N – N j ˆ ~ j Y j j , t Σ Y ~ σ b = ( ) = ( W ) ˆ 2j2 ) YY 22 c = ( ) = W j, t W j, t γ λ ~ X ~Y λ ˆ σ b ( ( W ) = = ~ ~ j Y j , t – λ Σ N 1 ~ j XY j ~ ˆ Y b=– jjj = (Σ W ) 2j,j,tt)) ˆλ j ()(λ jj( ttW Y,,(W bσ )))~ =YY2–1((σσ ~ X ~ Y Nj t =Σ cj = ~== Σ ~LΣ jLˆj= Σ )~λ= = [[aaγˆj j,XY ,bb(j j,λ,cjc)j j]=] 2~ Σ W1j, t WbNjj–, t1= ~σˆ YY 1( λ2 j )NbN 1j = L j –1 λˆY(YjjW ~bt =jXσ j,jtj N tt=~ N Σ jj N ~ Y~ j,= t )γ ˆ jj –––111(W c ( ) = W W λ L = N ~ Σ N ~ , , j XY j j t j t N t L = Y j – t L = – [ a , b , c ] = t L Nλj j tW a j,,~t b) j , cΣj ]W Nj t–=jLjLj –t1=+Ljj1–1 b j = σˆ Y j(j λ jcj)j j==jγ = L –1 = λ )([W ~ˆ (Σ Nj t = L j –1 j, tN – jj, t Nj = N [a j , b j , c j ] NjXYt = Lj j–1 N ~ X ~ Y NNj–––1j1 ~ ~ ~~––11 t = L j –1 j N NN ˆ ––1 c j = γ XY~ (–1λ j ) =c ~= ˆΣN(W X1X ~ YY NNj j SSabc ((00)~),,–1 ~ ~ – j, t W j, t abc , , j j ~ – ~ ~ N 1W X Y ~ ~ ) = W γ λ ~ ~ X Y ˆ X Y Σ X Y c = ( ) = W W γ λ ~ , , j XY j j t j t , 1== ~Σ W Nj S abc, j (0) cN –=1 Nγˆj S(abc ~ jj,,XW ~ ttWj, t (jj 0(1 ((ˆXY =t jW W ˆ1)% ~ jj ()(λ ttW λλXYW c)c,tγ=ˆ–L=XY ))~ j= =jα–γγ = λ j,)cj N ~ XY ΣΣ jN j, t [a j , b j , c j ] j,λ Wj,j,tjjY,,tW tt=~ LLj ––1W j ~XY Nj –1 S abc, j (0) , X ~ Y c j [= ~jj,jjt N j )] = N aN~jjγjjˆ, XY bt =jΣ N jjt–t=11=LLj j–,–1t1 j, t t ==Σ L N ˆ –1j L, j c c = ( ) = W W γ λ j SSabc ( ( ) ) j j Σ N ~ , , j XY j j t j t abc, ,j j j t = L j –1 S abc, j ( ) Nj t = L j –1S abc, j ( ) σ ( λ j ) ~ –1 ~ [ a , b , c ] –1 sample variance given by Nj S abc, j (0) , where S abc, j ( ) is T~~hen the vector has large j j j ~ ~ ~ ~ ~ ~ XX 22 XX YY YY 22 , N S ( ) [([(W Wj,j,t t)) ,, ((W Wj,j,t t~)) ,X, W Wj,j,t W 0.5 ~ X~ Y2 ~jY 2abc~, j X ~ Y tW ~j,j,t t]] [(~W ˆ 0) δ technique5 the large sample variance ~ j–Y,1tfor the × spectral ~ ~ ~ ~ [(W j, t matrix ) 2, (W ) , W(j,Xt)jW , t )j, t, ](W j, t ) , W j, t W j, t ].f By (the W j , S abc, j ( ) , N S [(W j,Xt ) 2, (W jY, t ) 2, W j,Xt W jY, t ~~ j abc, j ˆ (17) λ σ ( ) ± ς S ( ) α ˆ ˆ ρ ρ ~ j ( ( ) ) R R ( ( ) ) / / N N abc , j λ λ of ~ j j is abc abc, ,j j 00 j j, where: ~ Nj ρˆ ( λj ) Rabc, j (0) / Nj ~ ~ ~ ~ ~ ρˆ ( λj ) R , j (0) / N j S abc [(W j,Xt ) 2, (W jY, t ) 2, W j,Xt Wρˆ jY(, tλ]j ) Rabc, j (0) / Nj ~ ~ Y 2~2 X ~ Y 22 abc, j ( ) TTα X ς[( 22 22 W ) ( W ) W W ] , , (21) (21) j , t j , t j , t j , t RRabc ( ( ) ) = = g g ( ( ), ), ( ( ), ), ( ( ) ) S S ( ( 0 ) ) g g ( ( ), ), ( ( ), ), ( ( ) ) σ σ σ σ σ σ σ σ ∇∇ XX λλj j YY λλj j γγXYXY λλj j ∇∇ XX λλj j YY λTλj j γγXYXY λλj j abc, ,j j 00 abc , ,j j ~ ~, j Y(0)22= ∇ ~ σ ~abc ( λ j ), σTY2 ( λSj ), γXY ( λ j ) S2 abcρˆ, j (),0))σ 2∇(gλRσ),X2 γ( λ(0j )), σ~Y)2 ( λ j ),(21) γ XY ( λ j ) (21) ) Y, (W Rabc, j (0) = [( g j,Xσt )X22R,(abc λ(W λ jg,X), γjXY,XY ∇W j Y j, tσ tW t ] ( λj ) j ), abc,~ j (0) ∇g σ X ( λ(j λ j abc, j XY ( /λN j j Rabc (0(21) ) = ∇g σ X ( λ j ), σ Y , j 00 5 ˆ ρ ( λj ) Rabc, j (0) / Nj ( ( 0 ) ) RˆRˆabc ~ (1 –ˆ α /2) abc, ,j j T 0.5 ρˆ ( λˆ j ) (22) ρˆρˆ((λλj j))±±ςςαα Rabc0.,5j (0) / R Nabc ~~ j , j (0) Rabc, j (0T) = ∇(22) g σ X2 ( λ j ), σ Y2 2( λ j ), γXYRˆ2( λ j ) ((22) ( ) R 22 0) S abc, ˆ ρ 2 ( ) ± N N , abc j ς λ α , abc j ~ and ∇g is theρˆ (gradient of g Then, large sample confidence interval for the wavelet j an approximate jj ˆ ( ), ( ), ( ) ( ) ( ), ( ), R ( ) = g S g σ σ σ σ λ λ λ λ λ γ γ ∇ ∇ (22) ) ± ς fabc λj ρˆ ( λXj ) ± jς α Y ~j , j (0) X j T Y j XY j abc, j XY ( λ α ~ W, jN j 0.5 correlation is: Rabc, j N (0)j = ∇00.g5.5 σ X2 ( λ j ), σ Y2 ( λ j ), γXY ( λ j ) S abc, j (0) ∇g σ X2 ( λj ), σRˆY2 ( λ j(),0)γ XY ( λ j ) (21)N j abc, j 0.5 ρˆ ( λ j ) ± ς α RˆN – ~ (X0)~ 0Y.5 11 ~ (18) , j j, t W j, t abcW ˆ γ ( 0j 5) 0=.5 ~ λ (23) (23) tanh tanh––11[[ρˆρˆ((λλj j))]]±±ςςαα Nj Σ ˆ –ρ(1ˆ0[(ρ)ˆλ(jλ) ±)]ς±Nας t = L –~1 ˆˆ (22) –1 22 N R (23) – – N 3 , abc j α j j j ρˆ ( λ j )] ± ς α [ tanh (23) (22) tanh–ρˆ1 (ρλˆ (j )λj j ±j )ς ±α2 ς α ~ NˆNj j– 2 N Nˆj j – γ~( λ j ) jj – ˆ ˆ – – 0.5 NNj O LLj j =NN 22approach = ~that the ( λ j)fact ther takes advantage of ρˆthe decorrelates [ρˆ ( λ j )(19) ] ± ςprocesses tanh j = α ~ DWT approximately ˆ – 3j σ–11( λ j )0σ Nˆj = N j – Lj ( λ j ) ˆ N j j – Lj ρ ˆ ˆ N z-transformation intanh ordertanh to produce bounded by ±1 Under the Gaussian and uses the N Fisher ± ςα – Lj j = N ( λ j )intervals –1 j = N LLj j ˆ ρˆ ( λ j )] ± ς α (23) [ tanh – N j assumption this leads to the L following interval for scale coefficients: (1 – 2α)% j j –3 Nˆ j confidence ˆ = Nλj 2wavelet – L N L j j j j)( ((LLj j ==((LL––2L2)( 11––22––j j)))) (=λ–N ) –j j ( Lj = (L –N 2ˆγ j 2) )– Lj j)(1 ˆ works 5 The delta technique here Lj Taylor series approximations N L = N as ( Lj = (Lto– 2)(1 – – j ) ) central limit theorem which uses – j a– generalized j j ( Lj = (L – N2N–)( ) ) –ττ1 ––11– nonlinear functions11of estimators ~~ ~~YY L fˆW, j (0) N – τ – ~~ for ττ1==00,,11,,… …,,N –j W Wj,Xj,Xt t W Wj,j,t t++ττ jfor Σ Σ ~~ ~ ~Nj j ––1 ~1 L N– j NNj j –– ττ t t==LLj j––11 W j,Xt W jY, t + τ for (τ L=j 0=, 1(,L…–, 2N)(j 1––12 ) ) Σ ~ N – τ –1 ~ ~ X N j~–Y τ t = L j – c j – j Σ ~ – 2)(1for – τ )==)0ρ,ˆ1(,λ… , Nj – (24) –L Nj – τ t = L b(L (j aj), t)j=,W (24) γˆγˆττ((λλj j))== jj,,t +cτj ) = j) N –(20) τ –1 ––21(gW 11 ( LNNj ––1=1N~~(~L–XX–τ2~~)(YtY1=Σ ~ ~ L ~ X (24) ~Y a… γˆτ ( λjj ) =for ΣΣ WWj WW ~~ for τ1τ==––11,,–N–2–21,,… ~j b,,j––(~(NN ––11)) ~ ( λj, tj )W = j, t + τ for τN γˆ W NNj j –– ττ t t==LLj –j –11––ττ j,j,t t j,j,t t++ττ– W j,Xt WNjY, tj–+jττ–1 for τ = –1, – 2, …, –N~(j N – 1t)=Σ Σ ~ τ – L j – 1τ j (24) N γˆ ( λ ) = NN – –τ –1τ – – (( )) ( ( ) ] [ )) ( )( ( ( [ (( ) ] ( ) () ) ( ( ) ) ρˆ ( λj ) Rj, tabc, j (0) j/, t Nj j, t j, t ρˆ ( λj ) Rabc, j (0) / Nj ~ Rabc, j (0) / Nj T T 2 2 2 ~ (21) ( ), ( ) ( ) ( ), ( ), ) j ), γ2XY Rabc, j (0) = ∇g σRXabc(, λj (j0),) σ S g σ σ λ λ λ λ γ γ (21) ∇ (λj ) = Y∇ S abcX,Tj (0j) ∇gY σ Xj ( λj ), σjY, j((λ0j)),/2abc γNXY XYσ , j λj ) ρˆ )g( λ=σjj )X g( λXY abc 2( Yλ(j λ j ), R 2σ ( λ ), 2γ j ), γ ( ), ( ( ) ( ) ( ), ( R ( S g σ σ σ λ λ λ λ ) λ ∇ ∇ ( ), ( ), R ( ) = g σ σ λ λ abc , j X j Y j XY j abc j X j Y j XY , ∇ T X j Y j γjXY ( λ Rabc, j (0) = ∇g σ X2 ( λ j ), σ Y2 ( λ j ), γ S abc0.,5j (0) ∇g σ X2 ( λj ), σ Y2 ( λ j ), γ XY ( λ j )abc, j (21) 5XY ( λ j ) T ) synchronization Rˆ abc, j (0cycle Business ( λ ), σ ( λto),wavelets RRˆ abc,(j ()0)= ∇g σ02.5according λj ),0.5σ Y2 ( λ j ),(22) γ ( λ ) … S abc, j (0) ∇gˆ σ X2 (13 γ XY ( (22) ρˆ ( λ jˆ) ± ς α 0.5 ρˆ~( λ j ) ± ς α abc, ~j 0Rˆ abc, j (0) X j Y j XY j ( ) R , abc j Rabc, j (20) N j ρˆ ( λ j )2 ± ς αN j ~ ˆ ( λj ) ± ς α ρ ~ 0.5 (22)2 ρˆ ( λ j ) ± ς α N j Rˆ Nj ~ Nj abc, j (0) 0.5 ρˆ ( λ j ) ± ς0.5α ~ 0.5 N 0.5 – 1 j (23) (23) tanh–1 ρˆ ( λ j ) ±tanh ς α 0.5 [ρˆ ( λ–1j )] ± ς α2 Nˆ – –1 ρ ˆ [ ] λ tanh ± ( ) ˆ ςα tanh j (23) Nj – 3ρˆ ( λ j ) ± ςj α ˆ (23) 0.5 tanh–1 [ρˆ ( λ j )] ± ς α Nˆ j – Nj – Nˆ j – – j ˆ tanh ρˆ ( λ j ) ± ς α – Nˆj = N j – Lj Nj = ˆN Lj j Nˆ j – Nˆ = N j – Lj – Lj Nj = of N 2scale where Nˆj = N j – Lj is the Lnumber λj DWT coefficients unaffected jby the boundary, j Lj here as a measure j which is treated scale-dependent sample size, and Lj is the number of – Lj Lj ofNˆthe j = N – j2 Lj boundary DWT coefficients ( Lj –=j (L – 2)(1 – ) ) ( Lj = (L –– j2)(1 – ) ) L ( Lj = (L – 2)(1 – – j ) ) An (unbiased is defined as: Lj = (Lj –cross-covariance 2)(1 – – j ) ) Lj = (L – 2estimator )(1 – ) ) of the( wavelet N – τ –1 ~ ~ X ~Y N – τ –1 N – τ –1 – jW for ~τ = 0, 1, …, Nj – Σ ~ 1 ( Lj~N=X (–L~τ–Y 2t =)( ~ ~ 1– 1–W N2–j, τt – 1) )j, t + τ N – τ –1 L τ – N , , , for … = W W j W j,Xt W jY, t + 1 ~~ X ~ Y ~ ~ ~ X ~ΣY Σ ~ j j,jt j, t + τ τ – N , , , for … = τ – W W N , , , for … = τ N – – τ N – – W W = t L = t L Σ ~ Σ j ~ j j, t j,j t + j j (24) [a j ,[baj j,, cbjj], c j N ] j – τ t = L j –j γˆτ ( λ j ) = τ N j –N –τ1 t~= L j – 1~ Nj,–tτ –1 j, t + τ ~ X Y (24) ~ ~ ~ N – (24) N W W X τ Y= –1, – 2, …,γ ~X ~ Y Σ j, t j, tΣ for 1), N1 – –1 ~ X ~ Y γˆ ( λ ) = N –11 ˆ +τ W for τˆ–τ=( 0λN,jj1),–=… ~N j, t W j, t + τ ~ (24) –Wτ t1= L j – 1~ ~γ – τ N –1 ( λ–1j ) = τ 1j –1τ~ ˆ ( ) = W j, t + γ j λ Σ W j, t(24) ~ ~ W ~~ j – – – – Σ τ j ~ X Y ~ ττ ~=t =XL–j1–,1~ Y2–, …, ( Nj 1) j, t τ Nfor Nj N S abc (0,)j,(~0) , j ,Sj abc –1,j ––Σ for~j, τt + = Nj, t + τ1) for τ = –1, – 2, …, – (NN 2, … ,j, t (W t = L j – 1– τ NΣj –Wτj, t tW W j –– τ = L jj,–t1+–ττ ) j Nj – τ t = L j – 1– τ = L j – 1– τ N –1 otherwise γˆτ ( λNj )j =– τ totherwise ~ ~ S abc,Sj abc ( ), j ( ) ~ 0 W j,Xt W jY,otherwise Σ ~ for τ = –1, – 2, …, – ( Nj – 1) t+τ γˆτ ( λotherwise j) N – – τ t= Lj –1 τ j ~ X ~2 X ~2 Y ~2 Y ~2 X ˆ~ρˆYXτ (~ Yj ) = (λj ) γˆτ(25) )j, t ]theσˆ 1wavelet [(W j[( ) ,j, t()Wγ )( λ,j,estimate ) j,, tγ Wτ j(, tλW ]j of ,ˆj, (tW ˆ ( λcross-correlation ( ) ) ˆ σ λ and the corresponding via: , tW tW ) ρ ( ) = j j τ j ˆ τ j (25) σˆ ( λ ) σˆ ( λ ) γˆτ ( λ j ) otherwise ρˆτ ( j ) =ρ τ ( j ) = ˆ (25) j j (25) σ (~jλ) j )~σ0ˆ ρ(ˆλτ (j 0) j ) = λ jR) σ ρˆ ( λρjˆ)( λjσ)ˆ 1R(abc (ˆ02,)(j/1λ(N 0)j {/ hNl j } {g l } σˆ ( λ j ) σˆ γ(ˆλ (j )λ ) , j abc {hl0 } {g l0 } 0 ρˆτ ( j 0)T=2 T τ j 0 (25) 0 l +1 {hl0 } {g l{0 }hl } {g l 2} (25) 2= 02; (h)(l λ) )S= hλl +j∇ ;Yj2),( λγ g l ),=γ )),σˆj h(2∇ σˆabc λj abc lσ L –1 – l(21) 2)ng ),gΣ (l λ} (j0Σ 0l ()g (σλXj ),( λσnj Y2),≠(σ0 ( λ(XY–j 1)()λ0 j )h(21) Rabc,Rj abc (0), j=(0∇) g= ∇ σ gX (σλXj ),( λΣ σl j Y2),h({ σ λh=Yj2l),(0}λγ;j{XY λ γ 11(;,S XY j j X j XY l l Σl hl = ; Σl (hl ) = 1; Σl hl0 hl0+2n = 0 0 l +1 0 l +1 00 ≠ – ≠ – = ; ( h ) = ; h h = ; n ; g = ( ) h ; hΣ ( h ) = ; h h = ; n ; g = ( ) h Σ Σ Σl hl0 = 0Σ Σ – – –1 – l l l l0+ n0 2n l 0 l l h 0l += ) =DWT-based 1l ; Σ hl0 hl0l+L2estimators n ≠L 10of;l the g l0 =wavelet ( –1) l +1 hcrossg l (}hl of l Σ l {h0 ˆ (0)l ({0h)the n = 0; L –1 – l ll ;} {Σ Decimationl byl Rˆ2abcaffects lag resolution ,R j abc, j l gl } l} { l l 1 ρˆ ( j)( j) ~ ~ {hl } Genỗay, {g (22) } (22) l +1 0 be used in practice 12 cross-correlations -covariances they should not N ≠ 0; l g l0 = Selỗuk, ~ and ~h Nj ~ h h = 0; (see n ( –1) hL –1 – l {hl1 } {gand 1l = ; Σ (hl ) = 1; Σ Σ l{}hl } {g l }j + l l n 1 h j, l ={hh j,}l +{gli h}j, l l l ~by applying ~0 ~similar Whitcher 2002, p 252) Confidence for these quantities are obtained l intervals l h j, l = h j, l + i h j1,(26) ~ ~ ~ ~ 1~ l ~ 1~ ~ +0.5i g ~01.5 h h i h = + g = g = h h i h + j , l j , l j , l ~ ~ ~ arguments to presented about constructing (26) confidence intervals 11 j, l j, l above j, l j, l More j, l ~ (26) ~ + ig ~ 1in the 1j, l j,=l1h{information h } { g } –1 –1 h i h + g = g l l j, j, l j, l(23)(23) j, l ~tanh ( λ0[ρjˆ)(]λ±j ~ ς)]α1± and ς α L covariance ~l + of waveletg~tanh analysis can be found in Percival (1995),j, l Whitcher (1998), i [g~ρˆ1~jvariance (26) ˆ –Nˆ–13~– 30 j, l = g j,g ,l ~ j, l = g j, l + i g 2j~ , l N2~ ~1 ~1 ~ j j ~ ~ L –1 (27a) W = h X = W + i W g = g + i g – Σ j , t l , j t j j , t j , t j , l j , l j , l = h h i h + Percival, Walden (2000), Genỗay, Selỗuk,j, l Whitcher (2002) In the next section we~ turn to~ a complex ~ ~ j, l j, l L –1 W j, t = Σ hl, j X t – j = W j0,t + i W j1,(26) ~ ~ 1~l = L –1 ~ t ~called ~~l, jj X ~ (27a) jj = overlap ==NˆΣ h W + i W variantWofN MODWT the maximal discrete Hilbert wavelet transform (MODHWT) ~ ~ ~ – – L jˆ , t the t j , t j , t l = – Ll,j j X t – j = W (27a) NW =2N–=2LΣ +gi jW ~ ~ j j h j, t g j, l + i g~j,0l –j1, t , l~= L l =j j, t ~ ~ hl, j=XV~t –0 janalysis l = to perform j, t the j, t which makes it possible and phase angle (27b) (27a) V j, t =WΣ g~=l, jΣ Xwavelet +=iW V j1j,,tt + iofWcoherence L –1 – ~ l = t j L –1 j, t ~ X = V~ + i V~1 –1 L L L l = V = g ~ ~ j j ~ ~ ~ ~ ~ j, t j, t Σ l, j t – j j, t (27b) ~1 = Σ hl, j X t – j = W j0,t + i W j1, t V j, t = Σ g~~l, j X t L– –j 1=~V j0,t + i V j1,t~ LW (27a) l=0 – 1j, t (27b) + i V ~ ~ ~ – j =V , t j , t t j l = V j, t = Σ gl, –j X ~ l =0 j –j (27b) ( Lj (=L(j L=–(2L)(l–=102–)(21 – )2) ) )V j, t = Σ gl, j X t – j = V j, t + i V j, t ρˆ ( λj ) ( ( [ ( ( ) ( ( ) ] [ ) ) ( ( ] [ ) ( ) ( ) ) ( ) ( ] ( ( ) ) V = g~l, j X t – j = V j0,t + i V j1,t N – τ – 1N –~τ –X1 ~~XYj, t ~ YlΣ ~ ~ =0 τ = 0τ, = N 1, 0… , 1,,advocated fortransform, for …j ,–N1j – by Whitcher and Craigmile W Σj, tWWj, jt,wavelet ~ ΣHilbert + τ j, t + τ tW The maximal overlapN~discrete L j –t1= L j – j –Nτ j –t = τ (2004), makes use of a recently introduced class of filters based on the Hilbert wavelet pairs (24)(24) γˆ ( λγˆτj )( = λ j ) = 1 N –1 N~–1X ~~XY ~ Y ~ ~ (HWP) and τutilizes (maximal version W Wj, jt, t W – 2,, –… Σ Σ ~the~ non-decimated =, ––12,, … (,N–j (–N1j) –of 1) the dual-tree complex τ = –τ1overlap) + τ j, t + τfor for j, t W Nj of –Nτj –t =τL j –t1=–Lτ j – 1– τ wavelet transformation Kingsbury (2001).6 The filters in a Hilbert wavelet pair are approximate otherwise otherwise and, as in usual Hilbert transforms of each other discrete wavelet transformation, form a basis for a collection of orthogonal filters This time, however, the approximate analyticity of the λj ) γˆτ ( λγˆjτ )(bandpass ρˆτ ( ρˆjτ )(to (25) like the wavelet = j )compute = filters enables quantities with direct analogy to the Fourier(25) analysis σˆ ( λσˆj1)(σλˆ 2j () λσˆ j2)( λ j ) coherence and the wavelet phase angle {l0g}l0 {}g l0 } be conjugate quadrature mirror filters, i.e Let {hl0 }{hand 0 0 0 =h0 hlh+l2 nh=l + 20n;= 0n; ≠ 0n;≠ 0g;l0 =g(l0–=1)(l +–11h)Ll0+11hlL0 l Σl hlΣ Σl; (hΣll0 )(2hl=0 )12; = 1Σl; hlΣ l ;= l l – – – – an introduction to the Kingsbury’s transform see Selesnick, Baraniuk, Kingsbury (2005) 1 1 {hl }{h{l g}l }{g l } ~ ~~ ~ ~ ~ h j, l h=j,hl j0,=l h+j0,il h+j1, li h j1, l g~ j, l g~=j,g~l 0j=, l g~+0j,il g~+1j, il g~ 1j, l ) ( =0 Maximal overlap discrete lHilbert wavelet transformation L –1 ~ ~ ~ 6 For ) (26)(26) (27b) j ˆ Nˆ j –– otherwise 00.5.5 j otherwise otherwise 11 –1 ˆρˆ((λλj ))]]±jj±–ςςαα tanh tanhNˆ–1[ρ[= γˆτ (γλˆτj () λ j ) j = – ˆ N L j j ˆ 2j N γˆτ ( λ j ) j ρˆτ ( ρˆτj )( =j ) = (25) Nj j ––33 (25) (25) σˆ ( λσˆρˆ1jτ()(λσˆ j2j)()σλˆ=2j ()λ j ) J B ruzda 14 σˆ ( λ j ) σˆ ( λ j ) Lj j j NˆNˆj j ==NN 22 j––LLj j {hl0 }{h{l0g}l0{}g l0 } 0 –j {hl } {g l } ( Ljj = = (L –– 2)(1 –– – j ) ) L 0 20 0Lj 00 0 hl 0()hl =) 1;= 1;Σ hΣ ; 0;n0≠n00≠; 0;g l g=l (=–1≠()–l1+1) lh+1L0 h–10L0––l –l l +1 Σl hlΣl =hl0 =; 0Σ;l Σ(Σ 2jl + l0 hh + n0= l 2hnl = Σ hl hl +2n = 0; n 0; NN ––gττ l––11 = ( –1) hL –1–l lhl = ; Σl (hl ) = 1; l l ~ ~ ~X ~ ~Y ((LLj ==((LL––l 22)()(11––22––j j)))) ~ 11 = 0, 1, …, Njj –– Σ j W jj,,Xtt W jjY,, tt ++ ττ for τ = ~ 1 1 Nin –such and {hl }{h and couple of such filters chosenN that their wavelet function {l g}l {}g l } be1 another τ tt == LLajj ––way 11 jj – {hl } {g l1 } – –ττ– –11 N N is an~approximate transform of the mother for couple of filters.~~Figure ~ ~ ~ ~ 1Hilbert ~ ~~XXthe ~NN~Y––11Yfirst = γˆˆττ (( λ1jj1))wavelet = Σ 1W ~ ~ h +~ il h j1, l ~ ~ ~ ,,NN–j j –––11 ~ X for ~ YY ττ==00,,11,,… h j, l h=examples j,h l j= W W +t +ττ X for j,jseen t, t W j,jt, as , l +j, li h j,of Σ presents can be localized versions of1, cosine ~ for … 2, …, –– ( Njj –– 1) W W(26) τ= h j, l =such h j, l +Hilbert i h j1, l wavelet pairs.N~N~They , j t jj,, tt ++ ττ – – Σ = τ – t =t =~ LLj ––11 j, t(26) τ – j j ~ ~= Nj j –– τ tt == LL jj –– 11–– ττ ~0 ~ ~ (26) g~ j, l g=waves il g+~j,ilg j, l~the and sine As thej– Hilbert transform is built into a couple j,g l j, l g+j,forming (24) g j, l = g 0j, l +Fourier i g~ 1j, l transformation NN 1–1 (24 γˆγˆττ((λλj j))== 11 ~~Xwavelet ~0 otherwise ~~ X W~YY of orthogonal wavelet filters, it automatically adapts toΣthe scales This feature makes W – – – – ~ for = … N , , , ( ) L –1 L –1 ~ τ W W + , , j t j t τ – – – – Σ ~ for = … N , , , ( ) ~ ~ ~ jj τ j, t j, t + τ ~ h X L~=–10W +~i1W ~ – – (27a)methods producing τ tγ=ˆtother =LLj j –11–ττ time-frequency ~ NNj j –– τto (27a) the approach attractive W j, t W=j, Σ t =hΣ l,~ t– W j j, t~+ j, ti W j, t j, t~ 0as compared l,particularly j XW t –j j = τ ( λ jj ) (27a) = h X = W + i W τ – l = Σ j , t l , j t j j , t j , t l =0 ρˆττ ( j ) = (2 otherwise 00ˆ classic instantaneous amplitudes, demodulation method (see, e.g., l = phases and frequencies j = like otherwise σˆ 11 ( λthe j )σ ( λ jj ) – L j L –1 ~ ~modern ~1 Priestley the empirical mode decomposition (see Huang, ~ ~1981, –01V ~ §11.2.2) L= (27b) Shen 2005) g~ ,~–j X=t –Vj~or ~1 γˆγˆττ((λλ ) t =gΣ (27b) V j, t V=j, Σ +j,it V+j1,itV j, t ~ ~ l, j XlV t j = j, t g j ) consists in simultaneously applying } {g 00j} (27b) a pair (25) X = V + i V ˆ l =0 ρ The maximal overlap Hilbert wavelet transform { h – ( ) = , , , t j , t j t l j t j j ˆ Σ ρ l=0 (25) ττ ( j j ) = ll l=0 ˆσˆ1 ((λλj ))σˆllσˆ2 ((λλj )) σ j j of wavelet (and scaling) filters in their non-decimated (maximal overlap) forms As a result, two 0 0 l + hll0 = ; Σ (hll0 ) = 1; Σ hll0 hll0++ 22 nn = n≠ ≠ 0; g ll0 = = the = = 0of; the = ( ––1) l +1 hLL0 ––11 –– ll 00 which 00Σ sequences of coefficients are obtained, are real and imaginary parts final wavelet l l l l l {{hhl l }} {{ggl l}}l 0 j j coefficients In other words, the following filters are used: }Σ{((ghhlll1100}))22 ==11;; Σ hhl0l0hhl0+l0+22nn==00;; nn≠≠00;; ggl0l0 ==((––11))l +l +11hhL0L0–1–1––l l ΣlΣhhl0l0==00{;h; ll11Σ Σ l l l l l ~ 00 ~ 11 l h~ = = + h i h + , , j l j l jj,, ll j, l {{hhl1l1}} {{ggl1l1}}j, l (26) ~ ~ ~ g jj,, ll = g 0jj,, ll + i g 11jj,, ll = + ~ ~ ~ hh~j,jl, l ==hh~j0,j0l, l ++ iihh~j1,j1l,Ll –1 L –1 ~ ~ ~0 ~1 ~ X = W~ ~ ~ 11 h ~~ = g ~~00W ~= ~ W = hscaling W jj0,, tt + + ii W W jj1,, ttin the form: j , t g + i g j , t ll,, jj X tt –– jj = These filters produce the complex coefficients gj,jl, l =wavelet gj,jl, l + i gand j,Σ jl, l (26) (26) ll == 00 (2 (2 ~~ LL–1–1~~~ LL ––11 ~~00 ~~11 ~1 ~j,00t i V~ ~ W h~hl,l,j jX=Xt –t –j j = W ++i=iW ~ (27a) Wj,jt, t ==Σ = W W j , t (27a) (27a) V g X Σ j , t + jj1,, tt j, t = Σ ll,, jj tt –– jj = V jj,,j,tt t + l =l =00 j, t l = l=0 LL––11 ~ ~ ~ VV~j,jt, t ==Σ g~g~l,l,j jXXt t––j j ==VV~j0,jt0,t ++iiVV~j1,j1t, t (27b) (27b) (27b) Σ l =l =00 (2 In the empirical part of the paper we utilize the HWP filters introduced by Selesnick (2002) The real and imaginary parts of the Selesnick’s filters are of the same length and have the same squared gain function Under a specified degree of approximation (L) to a Hilbert pair (approximation to analyticity) the Selesnick’s procedure produces short filters with a given number of vanishing moments (K) The length of each HWP(K, L) filter is equal 2(K + L) In our study we ‘kK‘kK lL’.lL’ apply mid-phase solutions for HWP(K, L), and denote them kKlL ~ ~ X~ Y ~ Y Let W j,XtW and W j,t j ,tW j ,t be complex-valued wavelet coefficients obtained through filtering Xt and Yt Assuming that the wavelet have moments to eliminate deterministic trend ~ X ~lL’ ~Xfilters ~ X 0enough ~ ~ X 1~vanishing Y ~Y X~ Y ~_Y ~ _Y ~ Y = ‘kK =j, tE =W S XY S( λof tλ) j=, tseries, E) =W Ej, tWW Ej, tW+j,Xti0W+wavelet ‘kK lL’ XYj ,(the j, tj, tW j,itW j,W t j, tW j, it W j,itW j,= tof (X , Y ) for scale λ is defined as (see components the time-varying spectrum j (28)(28) ~~XX 0~ ~~X Y0Y ~ Y 0~ X 1~~XY11 ~ _Y _~ X 0~ ~X Y0 ~_Y 1~_X 1~~X 1Y 0t~ Y 0t ~ ~ _, ti)Q_ i Q X Y Whitcher, W + ( ( ) ( , ( ) , W Ej,j,t tWWj, tjj,,ttW+j, W W W W i W i W W W W W W W = C = C , t t t λ λ λ λ lL’ = E =2004): W ‘kK W Craigmile t j, t j, tj, t j, t j, t j, tj, t j ) j, t j, t j, tj, t j, t XY XYj j XY XYj ( ( ) )[( [( )( )( )] )] ) )( ( )] )] [( [( ~ (~λ ,(tλ) , t ) ~ ~ ~ ~ ~ ~ WC C W ~ ~ , t ) =~ E (W ~ S (= E [(W +~iWW ~ )() =W~E [~(W_ iW~~+ iW)] = )(~W ~_ iW ~)] = ~ ( λ , t ) = E (W W ) λ (28) _ ~ ~ ~ _ ~ _ _ (28) ( λt ),(=tλ)E , t )~ ~ S Q( λQ =, E [(W~ (W~W~W ~+)W=~E= [~E(WW~[(W~)+_iWWi (W~)(+~WWW~ ~Wi_WW)~ ~)]i =W(~W~ )]W= C W( λ ,Wt) _ i)]Q= C( λ (,λt), t)(28)i Q ( λ , t) C~YI (W (~dλI )(, dt + )W X ~XI (~d=IE (),d[Y(W), ) W ) _ i (W W _ W W )] = C ( λ , t ) _ i Q ( λ , t ) ( λ , t) j,t S XY j ,t j X j,tXY XY C XY j XY t Y XY jj,t j jX j, t j, t X XXY t j, t Y0 Yj Y j, t Y Xj,1t j, t X0 X0 j,Yt j, t X1 j, t j, t Xt X j, t X0 j, t j j, t j X X Y 0Y j, t j, t XYj t Y XY YX01 jj,,tt X j, t X1 j, t X0 j, t Y0 j, t j, t X1 j, t Y1 Y1 YX01 jj,, tt Y j, t j, t X j, t X0 j, t Y1 j, t Y1 j, t j, t Y1 j, t X1 j, t Y0 j, t X0 Yj,0t j, t Y0 j, t Y1 j, t Y1 j, t XY XY X1 j, t j j Y0 j, t XY XY XY j j XY XYj j XY XY j j [ [ XY XYj j XY XYj ] ] j j j f are ) ( f )the time-varying wavelet cospectrum and quadrature spectrum where CSXY ( λS( jf,)t )(=fand QXY (SλWZj , St()WZ (29)(29) Q ( λ j ,XYt ) XY ()1=– e(1 –i 2πef )i 2dπX f )(1d–X e(1 –i 2πef )i 2dπY f ) dY XY (quad-spectrum), respectively If the wavelet cospectrum and quadrature spectrum not depend X t ~ I (d X ), Yt ~ I (d Y ) QXY ( λ j , t ) X t ~ IW (d X=W),ΔdY=Xt Δ ~d XIX(d Y )Z =ZΔd=Y YΔdY Y X t t t t t t t t SWZ ( f ) X t ~ I (d X ), Yt ~SI (d(Yf)) = XY( f ) 21 i π f d1 S WZ ≈ ≈ ∈ ∈ S S ( f ) ( f ) S S ( ) ( ) f [ f [ , 1X ](,11– ]e i 2π f ) dY λ λ λ λ ( e ) – (30)(30) j j = XY i 2jπ f XYdj XY S XY ( f ) XY (29) j+1 22j+j1 j X SWZ ( f i)2 π f d Y ( e ) ( e ) – – S XY ( f ) = (29) dY i π f dX d X f 2d Y 12 12 2 –Xte)tC=i22 πC A A ( λ , (t(λ)1=,–teS) =WS(t λ)= ,Δ(tλ()1,= ()λ Z,(ttλ)=+,ΔtQ ) +YtQ ( λ ,(tλ) , t ) (31)(31) XY (28) (29) ~ ~ (~ ) (~ ~ ~ Pj, t = W jX, t , W jY, t , ℜ W j,Xt W jY, t , ℑ W j,Xt W jY, t 18 ) T (36) J B ruzda c2 + d g [ a, b, c, d ] = N –1 ~ ~ N –1 ~ ~(37) ~ ~ ab Qˆ XY ( λ j ) = _ ~ Σ ℑ W j,Xt W jY, t = _ ~ Σ W j,Xt W jY, t1 _ W j,Xt1 W jY, t0 Nj t = L j –1 Nj t = L j –1 ~ _ (38) (0, Rabc, j (0)) at the last stage of the analysis N j Kˆ XY (the K XY ( λ j ) ~ AN λ j ) significance and the Netherlands However, of relationships is hard to confirm as there are only 20 nonboundary wavelet coefficients available at this level Due T Rabcwavelet (P j, t ) Tangle S abcd(wavelet (Pj,2t ) ,delay) examination should , j (0) = ∇g ,~ j (0) ∇g to this our results of the phase be treated ~ time ~ ~ ~ ~ Pj, t = W jX, t , W jY, t , ℜ W j,Xt W jY, t , ℑ W j,Xt W jY, t with caution Figure Spresents running wavelet coherences and time delays, which generally point abcd , j ( ) at an increasing correlation at all leads and lags, while at 2the 2same time showing a decrease of the c +d P j, t months g g [ a, b,changes c, d ]T =take place in the case of Portugal and Spain lag parameter often below Opposite ab To confirm our results wavelet correlation0.5analysis was also performed This time the short ~ _ to execute a level wavelet decomposition (0) itKˆ possible Rˆ abcd , jN Daubechies d4 filter was made K XY ( λ j ) ~ AN (0, Rabc, (39) j XY ( λ j ) j (0)) ( λ j ) ± ς what Kˆ XYapplied, α ~ N j filter which has similar squared gain function to the Besides, we used also the la12 Daubechies Rabc, j (0) =wavelet ) T S abcd , j (0) with ∇g(P j, t correlations ∇g (Pd4 j, t ) ,were computed (see complex k4l2 filter Firstly, global and running ς α (1– α/2) Figures and 7) Except2 for some countries the Netherlands or Ireland the results point at an S abcd ,like j() increasing instantaneous C XY (linear λ j ) ≠ 0dependence between the Polish and euro zone business cycles, g especially in the case of the short cycles.P j,They confirm also that the most significant relationship t d takes place in the caseg of Besides, there is an evident jump in the level of instantaneous 0.5 (40) ([aGermany , b, c, d ]T ) = tan ) Rˆ abcd, j (0what c correlations for scale for the most recent data investigated, can be attributed to the recent (39) Kˆ XY ( λ j ) ± ς α ~ N j This is observed, between others, for financial crisis (the so-called ~ ˆ Great Recession of 2007–2009) (41) Nj θXY ( λ j ) – θXY ( λ j ) ~ AN (0, Rabc, j (0)) correlations with the French and Spanish short cycles Further, we computed time delay estimates ς α (1– α/2) via de following formula: Rabc, j (0) ( T ) ( ) ( ( ) ( ) ( ) ) ( ( ( ) k k T d ( ) k =– τˆ WPA j θˆ XY ( λ j ) 2π fj j j k (43) where WPA denotes an estimation via the wavelet phase angle (see also Figure 4) A lthough the results obtained with the three wavelets and the two estimation methods differ substantially, they are usually in agreement as for the sign of the delay parameter, even if different decomposition levels are considered In particular, the wavelet cross-covariance based delay estimates point at a lead time of length 1–2 months between the Polish and the euro zone (EA16) short business cycles, while the WPA-based method indicates a lead of about half a year In interpreting the results one should take into consideration several sources of estimation errors and how they influence the estimates Firstly, different wavelets estimate cyclical components differently In particular, the d4 wavelet filter is much more imprecise in isolating frequency 9 When (36) (37) (38) (42) (42) θˆ XY ( λ j ) g ([a, b, c, d ] ) = tan c where WCC stands forτˆ wavelet (43) = – cross-correlator 2π fj with the biased version of (24), which produced slightly The estimation was performed ~ Nj θˆXY ( λsimulation ~ AN (0, R 0)) unbiased wavelet j ) – θXY ( λ j ) analysis abc, j (the better results in short samples in our companion than cross-covariance estimator (see Bruzda R 2011).(0Only nonboundary wavelet coefficients were used ) abc, j The results are given in Table 1, where for comparison purposes we included also estimates of the ~ ~ = arg maxCovW~ X W~ Y (k ) = arg max Cov (W j,Xt , W jY, t + k ) wavelet time delay in the form: τˆ WCC j (35d ) C ( λj ) ≠ ~ X ~Y = arg maxCovW~ XXY ~ Y (k ) = arg max Cov (W j, t , W j, t + k ) τˆ WCC j j Wj WPA j ) testing for significance of the wavelet coherence for the last scale with the block bootstrap method, we were not able to reject the nonsignificance of the wavelet coherence at the nominal level of 10% for each of the series examined We resampled blocks of length from differenced observations and then reintegrated the series The overlapping block bootstrap method was applied (40) (41) (42) (43) Business cycle synchronization according to wavelets … 19 components of time series as compared to la12 (see the squared gain functions of the two wavelet filters in Figure 2), while at the same time it is times shorter than la12, what results in a much larger number of wavelet coefficients unaffected by the extrapolation method at the ends of the samples Secondly, simulation experiments show that in the case of low signal-to-noise ratios the WPA delay estimator is biased in small samples towards 0, while the WCC method, although asymptotically optimal, is characterised by a much larger small sample variance, what leads also to larger MSE (see simulation analysis in our companion paper) Generally, when a more precise estimation of frequency components is needed, the wavelet time delay (43) seems to be a better choice for time delay estimation in small and medium size samples It is worth mentioning that, to a large extent, our results are in line with the majority of other studies on business and growth cycle synchronization that include Poland and the euro area, especially those in Hughes Hallett, Richter (2007), Woźniak, Paczyński (2007), Skrzypczyński (2008), Adamowicz et al (2009) and Konopczak (2009) In particular, Hughes Hallett, Richter (2007) and Woźniak, Paczyński (2007), using the short-term Fourier transform, document a high coherence of growth cycles for Poland and the euro zone in the post-transition period for fluctuations of length coinciding with the typical horizon of monetary policy Skrzypczyński (2008) with the help of spectral and correlation analysis finds that the Polish short and long deviation cycles lead the corresponding euro zone cycles and that the degree of synchronization is higher for the short cycle (up to years) that for the long one (6–7 years in length) Besides, he documents that other countries in our region seem to show a slightly faster convergence to the European cycle than the Polish economy In turn, Adamowicz et al (2009) in a voluminous investigation find a moderately high level of synchronization with the euro zone as a whole, while the highest level of dependence has been affirmed in the case of Germany In addition, they notice that the Polish cycle usually leads the cycling components of the euro zone member countries Besides, they provide a tabular listing of empirical studies on business cycle synchronization for Poland and the EU countries together with their major finding (p 91, Table 5.1) Finally, Konopczak (2009) points at a generally high level of business cycle synchronization between Poland and the euro area measured with concordance indices and correlations of recession probabilities, especially in comparison with other CEE countries investigated (the Czech Republic, Estonia, Latvia, Lithuania and Slovakia), although the levels of demand and supply shocks correlations are rather low In the second part of the empirical study we concentrate exclusively on the euro zone countries and analyse the same production indices in the longer period from January 1991 till June 2010, what gives a total of 234 observations, except for Greece, for which data starting from 1995 were only available The longer data set made it possible to include the sixth decomposition level into our examination In this part of the study we used wavelet analysis of variance and correlation as well as wavelet coherence and phase angle examination Firstly, we computed the running wavelet variances for scales 8, 16 and 32 corresponding to decomposition levels 4–6 (see Figure 8) The d4 Daubechies wavelet filter was applied Looking at the wavelet variances it is possible to find certain general patterns of volatility changes in business cycle components of the series under scrutiny, at least for the short- and medium-term business cycles: for the majority of the economies in our sample the short business cycle fluctuations show fairly stable level of volatility, except for the very beginning of the series as well as the most recent period of the Great Recession of 2007–2009, while at the same time the fluctuations up to years in length are decreasing in their amplitudes Among 20 J B ruzda exceptions there are Greece and the Netherlands, for which we observe an increasing wavelet variance for both the short- and medium-term fluctuations For certain countries in the sample the jump in volatility of business cycles due to the Great Recession is substantial (for example, Germany as well as the euro area as a whole), although there are also countries with business cycles that seem no to be affected by the crises (Portugal and Ireland) The business cycle convergence within the euro zone was examined with the help of running wavelet correlations as well as running wavelet coherences and time delays (see Figures 9–10) For each country in the sample the changing levels of business cycle synchronization with the euro zone (EA16) were examined The results of wavelet correlation analysis point at a quite stable synchronization in the case of the short cycles, with a slight increasing tendency in the most recent period that can be attributed to the contagion effects and the global financial crisis, while at the same time we may observe a decreasing level of correlations for cycles above years in length The lack of a more systematic changes in the instantaneous dependencies makes the endogeneity hypothesis of the OCA criteria hard to verify To this end longer data series will be necessary The local wavelet analysis of coherence and time delay, performed for the shortest cycles only, points additionally at approximately constant phase relationships between each country cycle and the European business cycle, except for Ireland and Portugal, in the case of which a lagged relationship turned to a leading one, i.e for the most recent data the euro zone cycle is lagging behind these countries cycles Also the strength of linear dependences at all leads and lags measured with the wavelet coherence not changes much Conclusions The bivariate continuous discrete wavelet analysis provides a summary of evolutionary cross-spectral characteristics of the processes under scrutiny with good localization properties, high computational efficiency and without an excessive redundancy of information that takes place, e.g., in the continuous wavelet analysis These features make the approach particularly attractive in processing economic time series which are often subject to structural breaks, local trends and changing cyclical characteristics In the paper we made use of these properties in order to describe changing patterns of business cycle synchronization between Poland and euro zone member countries Besides, we addressed also questions concerning the endogeneity hypothesis of the optimum currency area criteria as well as the recent changes in business cycle variability In our study wavelet coherence and phase angle examination in the global and local (short-term) versions was applied The study was supplemented with wavelet analysis of variance and wavelet correlation and cross-correlation examination Our empirical results point 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 However, to obtain more firm results much longer data sets are needed, as the wavelet analysis of bivariate spectra is rather data demanding Business cycle synchronization according to wavelets … 21 References Adamowicz E., Dudek S., Pachucki D., Walczyk K (2009), Synchronizacja cyklu koniunkturalnego polskiej gospodarki z krajami strefy euro w kontekście struktur tych gospodarek, in: Raport na temat pełnego uczestnictwa Rzeczypospolitej Polskiej w trzecim etapie Unii Gospodarczej i Walutowej, 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the MODWT case Level Scaling L=4 L = 12 Level Wavelet Level Wavelet Level Wavelet Level Wavelet 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 10 25 Business cycle synchronization according to wavelets … Figure Example Hilbert wavelet pairs k4l2 k4l4 2 1 0 -1 -1 -2 -2 10 10 15 Note: see the description in the text Figure Wavelet coherences and wavelet time delays Poland–EA16 3 3 Poland–Netherlands 10 4 Coherence 3 4 Poland–Italy -10 10 -5 -20 Poland–Greece Poland–Ireland 10 0.5 -10 1 Poland–France 10 0.5 -5 Poland–Portugal -10 -20 1 0.5 Poland–Germany Poland–Greece Coherence Coherence Coherence -5 Poland–Portugal -5 -10 Poland–EU27 0.5 -10 1 Coherence -5 Time delay -5 -10 -15 Poland–Spain 0 0.5 Time delay -5 Poland–Netherlands 0.5 Time delay Poland–Italy Time delay Poland–France Time delay -10 Coherence Coherence 0.5 Time delay Poland–Spain Time delay 0.5 Time delay 0.5 Time delay Poland–Germany Coherence Coherence Coherence 0.5 Poland–EU27 Time delay Poland–EA16 Poland–Ireland -10 Note: see formulas (33) and (34) for decomposition levels 1–4 corresponding to oscillations with period lengths 2–4 (up to a quarter), 4–8 (about half a year), 8–16 (about year) and 16–32 (below years) together with large sample 90% confidence intervals; numbers on horizontal axes are the decomposition levels j 26 J B ruzda Figure 92 94 96 98 Poland–Italy 0.5 88 90 92 94 96 98 Poland–Netherlands 0.5 88 90 92 94 96 98 88 90 92 94 96 98 96 10 Poland–Germany 88 90 92 94 96 88 90 92 94 96 98 Poland–Netherlands 88 90 92 94 96 98 Poland–Greece -10 98 Poland–Italy 10 98 88 90 92 94 96 98 Time delay 88 90 92 94 96 98 Poland–France 0.5 88 90 92 94 96 98 0.5 88 90 92 94 96 98 0.5 88 90 92 94 96 98 0.5 88 90 92 94 96 98 90 92 94 96 98 Poland–France 88 90 92 94 96 98 Poland–Spain 10 88 90 92 94 96 98 Poland–Portugal 10 -10 Poland–Ireland 88 10 -10 Poland–Portugal -10 Poland–Spain Poland–EU27 10 -10 Time delay Coherence 94 -10 Poland–Greece 0.5 10 -10 Time delay 92 -10 Time delay 88 90 10 88 90 Time delay Poland–Germany 0.5 -10 0.5 Time delay 98 Poland–EU27 Time delay 96 Coherence 92 94 Coherence 88 90 Time delay Poland–EA16 Coherence 0.5 10 Coherence Poland–EA16 Time delay Time delay Coherence Coherence Coherence Coherence Coherence Running wavelet coherences and wavelet time delays 88 90 92 94 96 98 96 98 Poland–Ireland 10 -10 88 90 92 94 Note: see formulas (33) and (34) for level in windows consisting of 10 nonboundary wavelet coefficients after circular shifting in order to align them to the original data; numbers on horizontal axes are the mid-points of the subsamples of circularly shifted wavelet coefficients 27 Business cycle synchronization according to wavelets … Figure Wavelet correlations Poland–EA16 Poland–EU27 Poland–Germany Poland–France Poland–Italy 1 1 0 0 -1 -1 Poland–Spain -1 Poland–Netherlands -1 Poland–Portugal -1 Poland–Greece 1 1 0 0 -1 -1 -1 4 Poland–Ireland -1 -1 Note: see formula (13) for decomposition levels 1–5 corresponding to oscillations with period lengths 2–4 (up to a quarter), 4–8 (about half a year), 8–16 (about year), 16–32 (below years) and 32–64 (up to years) together with 90% confidence intervals computed with the inverse hyperbolic tangent transformation – see formula (23); numbers on horizontal axes are the decomposition levels j 28 J B ruzda Figure Running wavelet correlations 120 -1 40 60 80 100 120 140 -1 80 100 120 40 60 80 100 120 140 Poland–Spain 80 100 120 40 60 80 100 120 140 -1 Poland–France 80 100 120 40 60 80 100 120 140 Poland–Greece 80 100 120 80 100 -1 Poland–Italy Poland–Netherlands 80 100 Poland–Ireland 40 60 80 100 120 140 Poland–Italy 80 100 120 Poland–Portugal -1 120 40 60 80 100 120 140 -1 40 60 80 100 120 140 -1 0.5 120 -1 Poland–Germany -0.5 -1 Poland–Netherlands -1 Scale Scale 16 40 60 80 100 120 140 -1 Poland–Greece -0.5 -1 Scale 16 Poland–France -1 -1 0.5 Scale 120 -1 Poland–Portugal 100 -1 Poland–Spain 80 Scale 100 -0.5 Poland–EU27 Scale 16 80 Scale 16 Poland–EU27 0.5 Scale Scale -1 Poland–Germany -1 Scale 16 40 60 80 100 120 140 Scale 16 Scale 16 -0.5 Scale -1 Scale 16 -0.5 0.5 Scale 16 Poland–EA16 Scale Scale 16 0.5 Scale Poland–EA16 Scale Scale 40 60 80 100 120 140 Poland–Ireland -1 80 100 120 Note: see formula (13) for scales and 16 corresponding to oscillations with period lengths 16–32 (below years) and 32–64 (up to years); computation in windows of 30 nonboundary wavelet coefficients after aligning them to the original data; numbers on horizontal axes are the mid-points of the subsamples of circularly shifted wavelet coefficients 29 Business cycle synchronization according to wavelets … Figure Running wavelet variance -3 x 10 Ireland 50 100 150 200 0.5 50 100150 200 -3 x 10 Ireland 50 100 150 200 50 100 150 50 100 150 200 -4 x 10 Italy Scale 32 50 100 150 200 -4 x 10 Netherlands Scale 32 -4 x 10 Netherlands 50 100 150 200 50 100 150 200 50 100 150 200 -3 x 10 Greece -4 x 10 EU27 Scale 32 Scale 16 Scale 16 Scale 16 0.5 50 100150 200 -3 x 10 Greece 50 100 150 200 0.5 50 100 150 200 Scale 16 Scale Scale 1.5 -4 x 10 Italy 50 100 150 200 -3 x 10 Netherlands -4 x 10 Germany 0.5 50 100 150 200 0.5 -3 x 10 Portugal 0.5 50 100150 200 50 100 150 200 -3 x 10 Ireland Scale Scale 32 Scale 32 50 100 150 200 0.5 0.5 -4 x 10 Spain -3 x 10 Italy 0.5 50 100150200 -3 x 10 Portugal 50 100 150 200 50 100 150 -4 x 10 EU27 Scale 32 50 100150 200 -4 x 10 France -4 x 10 Germany Scale 16 -4 x 10 Portugal 0.5 50 100 150 200 -3 x 10 Spain Scale -4 x 10 Germany Scale 0 50 100150200 0.5 50 100 150 200 Scale 32 Scale 32 Scale 16 -4 x 10 Spain -4 x 10 France -4 x 10 EA16 0.5 50 100 150 200 50 100 150 200 Scale 32 Scale 4 Scale 16 Scale -4 x 10 France -4 x 10 EA16 50 100 150 200 Scale 16 Scale 4 Scale 16 Scale -4 x 10 EA16 Scale 16 Scale 50 100 150 200 -4 x 10 EU27 50 100 150 200 Note: see formula (9) for scales 8, 16 and 32 corresponding to oscillations with period lengths 16–32 (below years), 32–64 (up to years) and 64–128 (up to 10 years); results obtained with d4 Daubechies wavelet filter of length and windows of 50 wavelet coefficients for levels 4–5 (15 coefficients for level 6) unaffected by circularity after aligning them to the observations in the sample; numbers on horizontal axes are the mid-points of the subsamples of circularly shifted wavelet coefficients 30 J B ruzda Figure Running wavelet correlations 0.5 0.5 -0.5 50 100 -1 150 EA16–Greece Scale 16 0.5 EA16–Greece 0.5 -0.5 -0.5 50 100 150 -1 80 100 120140 160 -0.5 EA16–Spain Scale 50 100 EA16–Portugal EA16–Greece 0.5 -0.5 50 100 EA16–Ireland 0.5 -1 100 150 Scale 32 110 115 120 EA16–Portugal 0.5 -1 80 100 120140 160 EA16–Ireland 0.5 -1 EA16–Spain -0.5 -0.5 50 110 115 120 0.5 -1 EA16–Portugal 1 80 100 120140 160 0.5 -1 150 -0.5 -0.5 -0.5 110 115 120 EA16–Spain 0.5 -1 80 100 120140 160 0.5 -1 150 0.5 -1 EA16–France -0.5 -0.5 -0.5 110 115 120 -1 150 Scale 32 100 0.5 -1 EA16–Netherlands -1 50 110 115 120 0.5 -0.5 -0.5 0.5 Scale Scale -0.5 -1 80 100 120140 160 Scale 32 -0.5 EA16–Italy -1 110 115 120 Scale 32 EA16–Netherlands EA16–Netherlands -0.5 0.5 -1 80 100 120140 160 EA16–France Scale 32 -1 150 0.5 Scale 100 -0.5 Scale 32 50 Scale 16 Scale -0.5 -0.5 Scale EA16–Italy 0.5 EA16–France Scale -1 0.5 -1 80 100 120140 160 EA16–Germany Scale EA16–Italy 0.5 -1 -1 150 Scale 16 Scale 100 Scale 32 50 1 -0.5 -0.5 -1 0.5 Scale EA16–Germany Scale 0.5 -1 Scale 32 EA16–Germany Scale 16 Scale 110 115 120 EA16–Ireland 0.5 -0.5 80 100 120140 160 -1 110 115 120 Note: see formula (13) for scales 8, 16 and 32 corresponding to oscillations with period lengths 16–32 (below years), 32–64 (up to years) and 64–128 (up to 10 years); results obtained with d4 Daubechies wavelet filter of length and windows of 50 wavelet coefficients unaffected by circularity after aligning them to the observations in the sample; numbers on horizontal axes are the mid-points of the subsamples of circularly shifted wavelet coefficients 31 Business cycle synchronization according to wavelets … Figure 10 100 120 140 EA16–France 0.5 100 120 140 EA16–Portugal 0.5 100 120 140 EA16–Ireland 10 -10 100 120 140 EA16–France 10 -10 100 120 140 EA16–Portugal 10 -10 100 120 140 100 120 140 EA16–Spain 0.5 100 120 140 EA16–Italy 0.5 100 120 140 EA16–Netherlands 0.5 100 120 140 Scale 8: Time delay Scale 8: Coherence 140 0.5 Scale 8: Time delay 120 EA16–Germany Scale 8: Time delay 0.5 100 EA16–Germany 10 -10 100 120 140 EA16–Spain 10 -10 100 120 140 EA16–Italy 10 -10 Scale 8: Time delay EA16–Ireland -10 Scale 8: Coherence 140 Scale 8: Coherence 120 EA16–Belgium 10 Scale 8: Coherence 100 Scale 8: Time delay Scale 8: Time delay 0.5 Scale 8: Time delay EA16–Belgium Scale 8: Time delay Scale 8: Coherence Scale 8: Coherence Scale 8: Coherence Scale 8: Coherence Running wavelet coherences and wavelet time delays 100 120 140 EA16–Netherlands 10 -10 100 120 140 Note: see formulas (33) and (34) at level in windows consisting of 10 nonboundary wavelet coefficients after circular shifting in order to align them to the original data; numbers on the horizontal axis are the mid-points of the subsamples; numbers on horizontal axes are the mid-points of the subsamples of circularly shifted wavelet coefficients ... “modification of the DWT”’ Business cycle synchronization according to wavelets … There is also a growing interest in applying wavelet methodology to examine business cycles and their synchronization and... empirical results point 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... changes in business cycle components of the series under scrutiny, at least for the short- and medium-term business cycles: for the majority of the economies in our sample the short business cycle