WAVELET VARIANCE RATIO TEST AND WAVESTRAPPING FOR THE DETERMINATION OF THE COINTEGRATION RANK. Burak Eroğlu, Istanbul Bilgi University

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1 CEFIS WORKING PAPER SERIES First Version: November 27 WAVEET VARIANCE RATIO TEST AND WAVESTRAPPING FOR THE DETERMINATION OF THE COINTEGRATION RANK Burak Eroğlu, Istanbul Bilgi University

2 Wavelet Variance Ratio Test an Wavestrapping for the Determination of the Cointegration Rank September 26, 26 Abstract In this paper, I propose a wavelet base cointegration test for fractionally integrate time series. This propose test is non-parametric an asymptotically invariant to ifferent forms of short run ynamics. The use of wavelets allows one to take avantage of the wavelet base bootstrapping metho particularly known as wavestrapping. In this regar, I introuce a new wavestrapping algorithm for multivariate time series processes, specifically for cointegration tests. The Monte Carlo simulations inicate that this new wavestrapping proceure can alleviate the severe size istortions which are generally observe in cointegration tests with time series containing innovations that possess highly negative MA parameters. Aitionally, I apply the propose methoology to analyse the long run co-movements in the creit efault swap market of European Union countries. Keywors: Fractional integration, Cointegration, Wavelet, Wavestrapping Introuction After the seminal work of Granger (98), researchers were extensively concerne with the cointegration tests to fin the number of long run relations among integrate variables. Barring a few exceptions, most of these tests are evise in the time omain. However, the frequency omain properties of integrate variables also have consequential implications in statistical analysis. For instance, Granger (966) ha observe that nonstationary variables have typical power spectra in which the low frequency components have ominating importance. Following the Granger s (966) observation, a few scholars use the Fourier transform to construct spectral unit root tests (e.g. Choi an Phillips (993), Robinson (994) an Chambers et al. (24)) an frequency omain cointegration tests (e.g. evy (22), Morana (24) an Nielsen (24)). Nonetheless, Fan an Gencay (2) claim that the Fourier transform is not very appropriate when ealing with nonstationary variables since it lacks time resolution an only provies frequency omain information. Moreover, the authors point out that most of time series in Economics an Finance have complicate behaviors which cannot be analyze by the Fourier transformation (Fan an Gencay, 2). To resolve this issue, Fan an Gencay (2) an Trokić (26) exploit the wavelet filters an evelop new spectral methos for testing unit roots. Similarly, I employ these spectral techniques in a multivariate setting an propose a new wavelet base nonparametric metho for testing cointegration in fractionally integrate systems. The metho I propose combines the Nielsen s (2) variance ratio cointegration test an the wavelet theory. This new testing esign has several avantages over the existing cointegration tests. First, it is a fully nonparametric test an oes not necessitate the estimation of a parametric moel as in Johansen (988), or requires the selection of an optimal banwith as in Phillips an Ouliaris (988). Secon, the asymptotic istribution of the test is nuisance an tuning parameter free, an is invariant to the short run ynamics of the ata. Thir, the evelope metho can be applicable to a wie range of fractionally integrate series as well as stanar I() variables. Fourth, the frequency omain nature of the test give us the opportunity to focus on low frequency fluctuations an approximately remove any the problematic short run ynamics. Since the long run (co)movements appear in low frequency fluctuations, my metho can easily capture the cointegration relations. Finally, by utilizing wavelets in cointegration framework, I propose a multivariate version of the wavelet base bootsrapping metho particularly known as wavestrapping. This metho is also nonparametric an very successful in removing size istortions appearing in cointegration tests. The test propose in this paper has goo size an power features in small samples. Nevertheless, in general, cointegration tests are negatively affecte by complicate short run ynamics in the ata. An important cause of such ynamics is the highly negative MA roots in the innovation structure of stochastic trens. This, in fact, has been a wiely recognize problem in the unit root testing literature. Since testing for cointegration is a multivariate generalization of unit root testing, it is not surprising that the problems such as these appear in the cointegration literature as well. For instance, Mallory an ence (22) claims that if a negative MA

3 error structure is present in the ata, using the stanar asymptotic critical values for Johansen s (988) cointegration test cause severe overrejection of the true null hypothesis. Moreover, from the simulation exercises we observe Nielsen s VR test (2) also suffers severe size istortions in the aforementione case. The wavelet base test can partially remove these size istortions. This feature can be attribute to the ability of wavelets to approximately filter out the short run ynamics in the observe ata an the stochastic trens as well. In orer to remove the remaining size istortions, I esign a wavelet base bootstrapping routine for my test without forsaking its non-parametric nature. In particular, Trokić (26) showe how the wavelet base bootstrapping can be use to effectively reuce the size istortions in the case of unit root tests. Here, I evelop a new metho in a multivariate setting by using a similar wavelet base bootstrapping algorithm. This metho is conucte via a wavelet filtering technique, namely the Discrete Wavelet Transform (DWT). To the best of my knowlege, this is the first paper which implements wavestrapping in a multivariate cointegration setup. Furthermore, I generalize the wavelet base bootstrapping approach to general fractionally integrate processes. The rest of the article is organize as it follows: Section 2 reviews the DWT. In Section 3 I propose the new wavelet base variance ratio cointegration test. Section 4 is evote to wavestrapping. Section 5 illustrates the small sample properties of the propose test uner ifferent scenarios. In Section 6 I emonstrate the results of the empirical application of the propose methos. The proof of the theorems an the tables for the simulation results can be foun in the appenix. Throughout the paper, the following notation is use: D p enotes weak convergence in istribution, expresses convergence in probability, x inicates the closest integer to x. Further, all the limits in this paper are taken as the sample size goes to. 2 Wavelet Transform In relate recent stuies, the use of wavelets in unit root an cointegration analysis has attracte some attention. In particular, Fan an Gencay (2) evelop a wavelet base unit root test. In their paper, the authors iscuss that wavelets operate in both time an the frequency omain an aapt themselves to capture the nonstationarity features of variables across a wie range of frequencies (Fan an Gencay, 2). This makes the wavelet transform a proper instrument for nonstationary time series an also cointegration analysis. Accoringly, for the construction of the new cointegration test, I utilize these tools, which are introuce below. A wavelet, ψ(t), is efine as a real-value wave-like function fluctuating in a finite omain with the following properties: ψ(t)t = an ψ(t) 2 t = These two properties implies, a wavelet function must take non-zero value in a finite time perio, but all the epartures from zero shoul be cancelle out (Gencay, et al., 2). By using a wavelet function, one can construct the continuous time wavelet transform (CWT) of a time series x t as follows: W (u, s) = x t ψ u,s (t)t where ψ u,s (t) = s ψ ( ) t u s is translate by u an ilate by s. Note that u is the location an s is the scale parameter an W (u, s) is calle as wavelet coefficients. The parameter s, which take values in the R, allows wavelets to work uner ifferent resolutions or in other wors ifferent frequencies. However, it is almost impossible to analyse all wavelet coefficients for ifferent scales. Furthermore, the CWT creates a reunant transformation for time series ata. Hence, the CWT is not an appropriate tool in my analysis. Nevertheless, the Discrete wavelet transformation (DWT), which shares the funamental properties of the CWT, provies a non-reunant ecomposition with finite number of scales. Consequently, the DWT is more suitable instrument for my stuy. We can efine the DWT with two separate filters. First, let h = (h, h,..., h ) enote a iscrete wavelet (or high pass) filter with the finite length. h l correspons to a filter coefficient for all l =,...,. The high pass filter h l sums to, l= h l = an it has unit energy, l= h2 l = like the CWT filters. These properties imply that the iscrete wavelet filters fluctuate for a short perio an fae out. As a result, wavelet filters oes not treat all ata points in the same way, unlike the Fourier transform. This feature is very useful to analyse nonstationary time series processes since the properties of nonstationary processes may alter through time. 2

4 In the DWT, we also have an aitional filter g (low pass filter) which complements the high pass filter. The low pass filter g can be obtaine by the quarature mirror relationship. Unlike the high pass filter, g sums to 2, l= g l = 2, but it has unit energy l= g2 l =. Convolving the ata with these two filters, we ecompose the time series process into its high frequency an low frequency components. et {x t } T be the observe time series process with yaic length T = 2J for some integer J. Then, we efine the matrix of the DWT coefficients W = [ W, W2,..., WJ, ], V J where for j =, 2,..., J Wj is the vector of j-th level wavelet coefficients an V J is the vector of J-th level scaling (approximation) coefficients 2. In this representation, the vector of the approximation coefficients VJ is associate with the fluctuations of x t on the scale 2 J an the vector of the wavelet coefficients Wj is associate with the changes on the scale 2 j. Remark that, the scale is inversely proportional with frequency. As a result, VJ correspons to the lowest frequency an W correspons to the highest frequency components of the transforme series. Moreover, the approximation coefficient VJ has length of J an Wj has length of j for each j =, 2..., J. In practice, the wavelet an the approximation coefficients can be obtaine by the pyrami algorithm which is firstly propose by Mallat (989). This algorithm starts by filtering the observe series with the high pass (h) an the low pass filters (g) to obtain the first level wavelet an scaling coefficients: V,t = g l x 2t l mo T an W,t = l= h l x 2t l mo T for all t =, 2,..., T () where the filtering is carrie out by the convolution of the observe series an the filters. et the vectors W = { } W,t an V = { } V,t enote the first level wavelet an scaling coefficients respectively. Then, the matrix [W, V ] constitutes the first level DWT of the time series x t. For the levels j = 2,..., J, we can obtain the j-th level coefficients by simply filtering the scaling coefficient of the level j : Vj,t = l= l= g l Vj,2t l mo T an Wj,t = l= h l V j,2t l mo T In the construction of my test statistic, I only use the first level DWT of the observe time series processes 3. Although Trokić (26) consiers higher level transformations, it is clear from his finings that the first level DWT generates best results by means of power while the higher level DWT has slight size improvements in testing. This result is in accorance with the fact that in each application of the high pass an low pass filters, the half of the sample size of the previous level coefficients is lost. As I mentione earlier, I propose a wavelet base bootstrapping routine. In this routine, I utilize the higher level transformations than the level. The higher level wavelet coefficients provies more components to resample in bootstrapping. Furthermore, I use a multisignal version of the DWT in wavestrapping. In this version, I transform more than one series simultaneously. Although it seems this is not ifferent than applying the DWT to iniviual series an stacking them in matrix form, the multisignal DWT help us preserving the cross-correlation structure between the resample series an the original series. Further etails will be given in subsequent sections. The next section will combine the Variance Ratio cointegration testing with the wavelet theory to buil up a powerful nonparametric cointegration test. 3 Wavelet Variance Ratio Cointegration Test To construct my cointegration test, I first efine the concepts of fractional integration an cointegration. For this part, I mostly follow Nielsen (2). The quarature mirror relationship can be characterize by: g l = ( ) l+ h l for l =,..., (Fan an Gencay, 2). 2 Throughout the paper z notation inicates that z is an object associate with the wavelet ecomposition with filter length. oes not refer to power operator. 3 In the lieterature, there are other variants of wavelet transformation such as the maximum overlap iscrete wavelet transform an the iscrete wavelet packet transform. However, in our analysis, the DWT gives the best results accoring to the simulation exercises. Consequently, we skip the explanation of these techniques in this paper. 3

5 Definition. The p vector time series Y t is fractionally integrate of orer, enote by Y t I() if Y t = + v t, t =, 2, 3,... (2) where v t has a continuous spectral ensity matrix that is boune positive semi-efinite, an boune away from the zero matrix at all frequencies, an the fractional integration operator + can be efine as: + y t = t π j ()y t j (3) j= where π j () = Γ(j+) Γ(j+)Γ() is the fractional binomial coefficient (Sowell, 99) an only the past values of y t with a positive inex enter the integration. With this specification, the right han sie of (3) is always well efine for all values of (Nielsen, 2). Note that although the series in (2) are never stationary for any, they are asymptotically stationary for < /2. As in Nielsen (2), I also use the term stationary for this case. In aition, if =, Y t becomes a stanar unit root process, but the negative inexe history oes not have any impact on the process. Definition 2. The p vector Y t I() is sai to be cointegrate with the rank r if there exist a full rank p r matrix β such that β Y t I( b) for b > where r p. Further, I assume b < /2 <. This efinition generalizes the concept of cointegration to the fractional systems. Note that if we choose = b =, the above characterization collapses to the stanar I()-I() cointegration framework. The representation in Definition 2, therefore, gives us a chance to eal with many ifferent types of integrate time series which we encounter in the economics an finance literature. Furthermore, remark that there is an extra assumption on the fractional orer of the cointegrating resiuals which is restricte to be less than /2. This implies that the cointegrating resiuals are stationary for any value of. This technical assumption is neee for the proofs of the asymptotic theory establishe in the paper. Still, it is not too restrictive to impose such conition which says that the equilibrium error is stationary. Nonetheless, in simulations I show that when b > /2, my test still generates power against the null hypothesis. Aitionally, the above assumption implies that we can have at most p cointegrating relations among p time series. When r = p, we have all variables in Y t are stationary. To connect the above efinitions of the fractional integration an the cointegration with testing an asymptotic theory, Nielsen (2) suggests the following assumption: Assumption. (Assumption of Cointegration) Y t is the p vector of the observable variables which are I(). There exists a full rank orthonormal p p matrix R = [R p r, R r ], where R p r has (p r) columns an R r has r columns with r p such that [ ] R + I p r Y t = (p r) r (p r) (p r) ( b) u t (4) + I r where I m is an m-imensional ientity matrix an u t is generate by a linear stationary process u t = Ψ()ɛ t = Ψ k ɛ t k t =, 2,... The p p coefficient matrices Ψ k satisfies k /2 Ψ k <, rank(ψ()) = p if k= r =, rank(ψ()) p r + an rank(ψ ()) = p r if r, where Ψ () is the upper left (p r) (p r) block of Ψ() = Ψ k ; the remaining blocks being Ψ 2 () an Ψ 2 (), an Ψ 22 (). Finally ɛ t are i.i. with k= E[ɛ t ] =, E[ɛ t ɛ t] = I p, E ɛ t q < for some q > max(4, 2/(2 )). This is the riving assumption of the Variance Ratio (VR) test of Nielsen (2) an also of my test. Essentially, this assumption states that when r, Y t is cointegrate. On the other han, when r =, all variables in Y t follow istinct I() processes which o not constitute any long run equilibrium. More importantly, equation (4) epicts a factorization for the cointegrate system. The orthonormal matrix R factorizes the observe times series Y t into two components which asymptotically behave very ifferently from each other. The first component can be interprete as a (p r)-imensional non-stationary factors. These factors are not cointegrate with each other but they construct the base of the long run equilibrium among the observe variables in Y t. As a result, these factors can be linke to the common stochastic tren representations of cointegrate variables (see Stock an Watson (988)). On the other han, the secon component of the factorization represents the stationary component of the equilibrium. These ynamics are associate with the cointegrating resiuals or the equilibrium error which is illustrate in Definition 2. With these two separate set of processes, the characterization of the cointegration system is complete. Moreover, notice that when p > r >, the (p r)-imensional k= 4

6 non-stationary factors become the common stochastic trens. However, when r =, we cannot claim that they are common across the observable variables. The other important outcome is that when r =, there is no stationary components in the system efine above. Using this assumption, I construct a wavelet base cointegration test for fractionally integrate time series. In my analysis, I consier the type II fractionally integrate processes with the eterministic components. These components are restricte to mean an time tren for simplicity. The following assumption epicts the characteristics of the observe time series processes with these eterministic terms: Assumption 2. The observe time series X t is generate by: X t = α δ t + Y t where for j = δ t =, for j = δ t = an for j = 2 δ t = [, t]. Further, Y t I() is the pure stochastic component. In orer to remove the eterministic components from the observe time series, we apply a OS emeaning or etrening proceure. Hence, I use the resiuals from the regression of the observe time series X t on δ t. The resiuals are given by Ŷt = Y t (ˆα α) δ t. Now applying the low pass filter (with length ) to each element of the time series Ŷt, we obtain ˆV,t. For analytic purposes, we use the compactly supporte Daubechies class wavelet filters an their variants (see Daubechies (988) for the further iscussion). Here, we first efine V,t = l= g ly 2t l mo T which is the first level wavelet coefficient of the stochastic component Y t. Aitionally, if we use the emeane or the etrene series in my analysis, we can set ˆV,t = l= g lŷ2t l mo T. Accoringly, this object has the following form: ˆV,t = g l Y 2t l mo T (ˆα α) g l δ 2t l mo T Further, we apply the fractional integration operator to ˆV,t to obtain: l= ˆV,t = + ˆV,t is the variance- Now we can efine the objects A T = ˆV,t ˆV,t an BT = ˆV,t ˆV,t. Note that A T covariance matrix of ˆV,t an BT is the variance covariance matrix of ˆV,t. l= We are intereste in testing the number of cointegrating relations for the observe series. For this purpose, we construct the null hypothesis as H : r = r. Now, We can efine the test statistic for this null hypothesis as Λ p,r ( ) = T 2 p r i= λ i where λ i s are orere eigenvalues of the ratio of two variance covariance matrices The following theorem summarizes the main result of the paper. A T (B T ) (5) Theorem. Suppose that Assumptions an 2 hols. For > an r =,, 2,..., p Λ p,r ( ) D U p r (, ) { Up r (, ) = tr B p r j, ( ) } (s)b p r j, (s) s B p r j,, (s)b p r j,, (s) s as T where B p r j, (s) is p r imensional vector of stanar (j = ) or emeane (j = ) or etrene (j = 2) inepenent Fractional Brownian motions with fractional orer an B p r j,, (s) is p r vector of inepenent Fractional Brownian motions with fractional orer + for all j =,, 2. These Fractional 5

7 Brownian motions can be efine as: B p r, (s) = W p r (s) [ B p r j, (s) = B p r W p r (s) ( ) W p r (s)d j (s) s ( ] D j (s)d j (s) s) D j (s),, (s) = W p r [ B p r j,, (s) = + (s) W p r + (s) ( ( ) s D j (s)d j (s) s ) W p r (s)d j (s) s for j =, 2 (6) ] (s r) D j (s)s Γ( ) for j =, 2 (7) where D j (s) = if j =, an D j (s) = [, s] if j = 2 for s. These processes have same form as in Nielsen (2). Theorem inicates that the new wavelet base test statistic has the same asymptotic istribution as Nielsen s VR (2) test. This new test also carries over the most important features of Nielsen s (2) test such as being nuisance an tuning parameter free. The test statistic only epens on an an so there are no short run ynamics in the limiting istribution 4. Moreover, the filter length is asymptotically irrelevant, since it oes not appear in U p r (, ). Finally,an aitional theorem for the asymptotic power properties of the propose test is presente below: Theorem 2. Uner the assumptions of Theorem, the test that rejects the null H : r = r when Λ p,r ( ) > CV ξ,p r (, ), where CV ξ,p r (, ) is foun from P (U p r (, ) > CV ξ,p r (, )) = ξ This implies that the propose test has the asymptotic size ξ an is consistent against the alternative H : r > r In Theorem 2, CV ξ,p r (, ) is the asymptotic critical value for the null hypothesis H : r = r. This theorem is important for constructing a testing strategy to etermine the cointegration rank as inicate in Nielsen (2). Since the asymptotic power is against the alternative H : r > r, we can aopt a recursive testing scheme. Starting from the null H : r =, if H is rejecte, we continue with testing the null H : r =. This continues until we cannot reject the new null hypothesis or r = p. The value of r where we stop testing is the estimate for the cointegration rank. 4 Wavestrapping for Cointegration Test The wavelet variance ratio test propose in the previous section performs better than the stanar variance ratio test in the presence of the negative MA innovations. However, my test still suffers size istortion, particularly for highly negative MA roots. In orer to aress this issue, I exploit the wavelet nature of the propose test to further eliminate any remaining size istortions. A common practice in the literature of unit roots is to apply a sieve parametric bootstrap metho, such as in Nielsen (29). Nevertheless, such bootstrap methos rely on parametric techniques such as specifying the parameter governing the sieve length. Even though Nielsen (29) proposes a non-parametric unit root test, he relies on a parametric bootstrap for fixing size istortion issues. This, of course, implies that the bootstrappe test no longer enjoys the avantages of being fully nonparametric, so this coul be somewhat of a isavantage. My purpose in this paper, is to construct a wavelet base non-parametric bootstrap metho which oes not rely on any parametric estimation of the cointegration system. Further, apart from reucing size istortions, this nonparametric wavestrappping metho will be an alternative to parametric bootstrap for cointegration rank etermination (see Cavaliere et al. (22)). The metho propose here is similar to Trokić (26) an Percival an Walen (26)) with significant extensions. As its name inicates, wavestrapping is a bootstrap like proceure in which resampling is one via the wavelet transform of the observe series. The main iea behin this proceure stems from the fact that the iscrete 4 can be etermine a proiri to the testing or it can be obtaine by pretesting as suggeste in Nielsen (29) an Nielsen (2). 6

8 wavelet transform approximately ecorrelates long memory time series (Percival an Walen, 26). After obtaining an resampling the ecorrelate coefficients, one can obtain the resample process via the reconstruction filter. Moreover, my metho is structurally ifferent from the existing bootstrapping routines for cointegration testing. While other techniques use regression base resampling algorithms, I apply the wavestrapping routine to the common stochastic factors. These factors can easily be obtaine nonparametrically from the eigenvalue problem of equation (5). Accoring to Nielsen (2), uner the null hypothesis H r = r if we efine η(p r ) as the p (p r ) matrix in which each column is the eigen vector corresponing to the (p r ) smallest eigenvalues, these eigen vectors are in the irection of the non-cointegrate component (Nielsen, 2). Then, if we post-multiply η(p r ) with Y we obtaine the stochastic trens upto a linear transformation, say Z = Y η(p r ). The number of the stochastic trens may be more than one. In this case, my routine becomes multivariate wavestrapping algortihm. The DWT base wavestrapping can be summarize below:. Fix the Monte Carlo replications as MC, the significance level as α an the number of the bootstrap replication as B. 2. Given the observe time series Y T p matrix with length T = 2 M, set the level of the DWT as J = M Consier the null hypothesis H : r = r. Compute the WVR test statistic Λ p,r ( ) 5 with the compactly supporte wavelet with the filter length. 4. Calculate the T (p r ) imensional stochastic trens Z as Z = Y η(p r ) where η(p r ) is the matrix of the eigen vectors efine above. 5. Similar to the classical bootstrapping methos, we nee to resample the time series uner the null hypothesis in which Z consists of p r inepenent unit root processes 6. First, obtain the T (p r ) matrix e = Z as the first ifference Z. 6. Apply a J level multisignal imensional DWT transformation to e columnwise to retrieve the wavelet ecomposition w = {V J, W, W 2,..., W J }. 7 Note that V J is a J (p r ) matrix, of which each column is corresponing to the J th level approximation coefficients of the column vectors of e. Similarly, for j =,..., J W j is a j (p r ) matrix, of which columns correspon to j th level wavelet coefficient of the column vectors of e. 7. Apply a parallel resampling scheme to only high pass filter part of the ecomposition 8. For bootstrap iteration b =,..., B, obtain the resample wavelet ecomposition { } w (b) = V J, W (b), W (b) 2,..., W (b) J by resampling each W j with replacement for j =, 2,..., J. More clearly, let W j,t be the t th row of j th level wavelet coefficient an U (b) t be an i.i. sequence of iscrete uniform istribution on {, 2,..., j} 9. Then W (b) j,t = W (b) j,u. t 8. Finally employing the multisignal reconstruction filter to w (b), obtain e (b) an Z (b) t 9. Compute the test statistic for the new wavestrappe sample Z (b), say Λ (b) p,r ( ). = t i= e (b) i.. Repeat the steps 7-9 B times to generate the wavestrappe empirical istribution for the test statistic.. Calculate the wavestrap p value from this empirical istribution: p m = B B b= { } for m =,..., MC Λ p,r ( )>Λp,r (b) ( ) 2. The wavestrap rejection probability for the Monte Carlo simulation can be compute as: RP = MC MC {p m <α} m= 5 I rop the notation, because of not complicating the presentation. 6 In case of Fractional Cointegration, my metho also works but it is slightly ifferent. Instea of assuming we have p r unit roots, we assume we have (p r) I() series uner the null. Consequently instea of wavestrapping the first ifference series, we nee to fractionally ifferentiate the observe series an apply the wavestrapping routine. 7 This is equivalent to applying DWT each column of e. 8 As state in Tang et al. (28), resampling the low pass filtere coefficients may istort the wavestrapping algorithm, since this component may not be uncorrelate 9 The efinition of U (b) t is very frequent in the bootstrapping literature (see Kilian (998) an Cavaliere et al. (22)) 7

9 The steps - with MC = escribes the wavestrapping proceure for a particular sample. Step 2 is for the size istortion an the power evaluation exercises an it requires to compute M C(B + ) test statistic (Trokić, 26). Trokić (26) states that steps - of the DWT base resampling is a ouble bootstrap exercise which is relatively expansive to compute. However, Trokić (26) inicates that one can use the Fast Double Bootstrap (wavestrap) proceure in which we only nee to calculate 2M C number of the test statistic to inexpensively compute the rejection probability in step 2. In orer to achieve this, first set B = an estimate the RP as: RP RP F DW = MC MC m= {Λp,r ( )>Q ( α)} where Q ( α) is the ( α) th quantile of the wavestrap test statistic Λ p,r ( ). Remark. In step 2, it is state that the matrix of observations Y T p shoul have yaic length. This may be restrictive for most cases in Economics an Finance. However, applying the signal extension techniques such as zero paing, symmetrization, smooth paing,... etc, one can still use the DWT transform for the variables with non-yaic sample sizes (Strang an Nguyen, 996). Moreover, software programs such as Matlab an R provies functions which computes the maximum number of possible ecomposition level given the sample size T. Remark 2. The so-calle parallel bootstrap is frequently applie in VAR literature (see Kilian (998) an Cavaliere et al. (22)). However, these methos require i.i. innovations for resampling. In my metho, practitioners o not nee further action for the ecorrelation of the innovations since wavelet transformation oes this automatically without any parameter estimation. Aitionally, the parallel bootstrap preserves the cross variable correlation. This feature save us from bothering about cross correlation between common stochastic components. Remark 3. When (p r ) =, my wavestrapping routine is equivalent to Trokić s (26) algorithm. 5 Small Sample Properties of the Wavelet VR Test an Wavestrapping In this section, I evaluate the size an size-ajuste power performances of the propose test, its wavestrappe version an Nielsen s (2) variance ratio test via Monte Carlo simulations. Here, I restrict myself to the use of the compactly supporte wavelets, namely the Daubechies wavelets with the length 2 an 4, an the east Asymmetric wavelets (Symlets) with length 4, 8 an 6. The notation use for these filters are given in Table. Furthermore, I consier the tests results for the null hypotheses r =, r = 2. I initially set (p r) = 2 an generate two inepenent unit roots as the stochastic trens. Using these trens, I obtain the observe time series. On the other han, I both consier stanar I()-I() cointegration an I()-I(-b) fractional cointegration. However, in the fractional cointegration case, I set = for simplicity, but let the other crucial parameter b take non-integer values. Table : The Wavelet Filters Use in the Analysis Filter Filter length Haar 2 D 2 4 S 2 4 S 4 8 S 8 6 et ɛ,t, e 2,t an v 2,t be the inepenent an ientically istribute stanar normal raws. We obtain the stochastic tren z,t as AR() unit root process. Applying the wavelet VR testing proceure efine in Section 3, we can fin the value of the test statistic in each simulation. Also note that we reject the null when the test statistic is larger than the corresponing critical value. Consier now the DGPs for each test of interest: The Daubechies wavelet with filter length 2 is also known as Haar I also implement my metho with other wavelets from Daubechies an Symlets, but by means of size an power, the best results are obtaine by using those reporte. 2 I run the simulations for the null H : r = 2, results are similar an can be provie upon request from the author 8

10 (a) H : r = z,t = z,t + ɛ,t ɛ 2,t = e 2,t + θe 2,t z 2,t = ρ c z 2,t + ɛ 2,t y,t = z,t y 2,t = z,t + z 2,t where ρ c = ( c/t ). Note that if c = then y,t an y 2,t are not cointegrate an so r = hols with common stochastic trens z,t an z 2,t. However, if c > then z 2,t is a stationary process an these two variables are cointegrate, thereby satisfying the alternative hypothesis. Moreover, MA() parameter of ɛ 2,t, theta is crucial for checking the size istortions. (b) H : r = z,t = z,t + ɛ,t ɛ 2,t = e 2,t + θe 2,t z 2,t = ρ c z 2,t + ɛ 2,t y,t = z,t y 2,t = z,t + v 2,t y 3,t = z,t + v 2,t + z 2,t Similar to the case above, if c =, z 2,t will be nonstationary an we have only one cointegrating relation which is between y,t an y 2,t. However, if c > then z 2,t will be stationary an we have further another cointegrating relation between y,t an y 3,t. Consequently, r = 2. Further, θ has a simular role as in the DGP (a). (c) H : r = z,t = z,t + ɛ,t z 2,t = ρ c z 2,t + e 2,t y,t = z,t u 2,t = v 2,t + θv 2,t y 2,t = z,t + u 2,t y 3,t = z,t + u 2,t + z 2,t where ɛ,t, e 2,t an v 2,t are stanar normal raw which is inepenent of each other. This DGP is very similar to (b) by means of cointegration structure: if c =, z 2,t will be nonstationary an we have only one cointegrating relation which is between y,t an y 2,t. However, if c > then z 2,t will be stationary an we have further another cointegrating relation between y,t an y 3,t. Consequently, r = 2. Note, that the MA() coefficient θ only appears in the cointegrating resiuals uner both the null an the alternative. () H : r = z,t = z,t + ɛ,t ɛ 2,t = e 2,t + θe 2,t z 2,t = b + ɛ 2,t y,t = z,t y 2,t = z,t + z 2,t where b controls the fractional cointegration such that iff b = then y,t an y 2,t are not cointegrate an so r = hols with the stochastic trens z,t an z 2,t. However, if b > /2 3 then z 2,t is a stationary process an these two variables are cointegrate, so we are uner the alternative hypothesis. 3 I also consier the cases b < /2. Results are reporte in the tables. 9

11 (e) H : r = z,t = z,t + ɛ,t ɛ 2,t = e 2,t + θe 2,t z 2,t = b + ɛ 2,t y,t = z,t y 2,t = z,t + v 2,t y 3,t = z,t + v 2,t + z 2,t As in the DGP (), if b =, z 2,t will be nonstationary an we have only one cointegrating relation which is between y,t an y 2,t. However, if b > /2 then z 2,t will be stationary an we have another cointegrating relation between y,t an y 3,t with fractionally integrate resiuals. Consequently, r = 2. Remark that we use the same critical values in all simulations since in all simulations we have p r stochastic trens uner the null hypothesis. Critical values are also obtaine by simulation 4. After we fin the critical values, we perform the power an size exercise for the cointegration rank r =, an also consier = {., }. Consier the size performance of the cointegration tests uner ifferent scenarios. Table 3 summarizes the results for the null H : r = in stanar I()-I() cointegration setup. From this table, we can observe that Nielsen s (2) VR an the propose wavelet VR tests with ifferent wavelets are generating similar size performances, which is close to nominal size, when MA() coefficient is positive uner ifferent eterministic ajustment an also for =. an. However, the presence of the negative MA roots severely istorts the size performance in all of these cases, especially when we have eterministic component ajustment. When θ =.9, the least size istortion is obtaine by the S 2 an S 4 uner any scenario. A similar pattern in Table 4 is also apparent for H : r = case. These finings imply that Symlet filters woul be goo choice if we want to reuce the size istortion in cointegration tests. To further remove remaining size istortions an obtain better small sample inference, we also conuct wavestrapping exercise for my tests. With the help of wavestrapping, we can achieve approximately 75% size reuction in Haar an D 2 filters base tests an 85% size reuction in the tests base symlet filters compare to FWVR test. Another important outcome from Tables 3 an 4 is that parameter rastically affect size performance uner the negative MA root scenarios. Although = causes slight unersizing in wavestrapping case, still the tests generate by this value have much more better size performance than the tests with =.. On the other han, a natural question to ask is whether the presence of negative MA innovations in cointegrating resiuals impact the size performance. In orer to unerstan this, I consier the setup when r = uner the null hypothesis. Notice that we nee at least one cointegrating relation uner the null hypothesis. Table 5 summarizes the results for this cases where the ata is generate accoring to the DGP (c). From this table, it is clear that the negative MA root presence oes not affect the tests uner the null hypothesis. However, this is an expecte results because only stochastic trens are asymptotically important for Nielsen s (2) an my test statistics. I, next, consier the size-ajuste power features of the cointegration tests. For this exercise, I utilize the DGPs (a), (b), () an (e). While DGPs (a) an (b) correspons to classical I()-I() cointegration framework (Tables 6-9), the DGPs () an (e) correspons to Fractional cointegration case (Tables -3) 5. First, take Classical I()-I() with =. case in to account. Table 6 isplays that wavelet base tests can prouce power uner the alternative hypothesis with small sample size (T=28). Although Nielsen s (2) tests seems to be more powerful when MA coefficient θ is positive, the negative values of θ engener more power for the wavelet base tests relative to Nielsen s (2). Especially, D 2 an S 2 filters prouce generally more power than other filters when they are use in the wavelet base test. However, the best results for wavestrapping are obtaine by using Haar filter in almost any case. Another substantial result from Table 6 is that power rops for all test when we inclue eterministic component ajustment into the moel. Furthermore, using = leas to power loss in most cases except the scenario where θ =.9. This power loss is eliminate by increasing the sample size. Table 7 clearly emonstrates that all wavelet base tests an Nielsen s (2) have full power, but wavestrapping versions of my tests suffer slight power loss except the one with Haar filter. 4 I simply simulate the functional of the fractional Brownian motions appeare in asymptotic istribution many times to obtain the critical values 5 I ont inclue the size exercise for the fractional cointegration, since If the stochastic trens are still I(), than the null hypothesis for fractional cointegration an stanar I()-I() cointegration are ientical.

12 Very similar results can be observe in Tables 8 an 9 for the null hypothesis H : r =. Nonetheless, the wavelet tests with D 2, S 2, S 4 an S 8 gain power quickly with.9 MA() coefficient. This pattern is not persistent for other values of θ. Consier Tables to 3 for the size-ajuste power properties of the propose methos in the fractional cointegration framework. The wavestrappe version of FWVR test with Haar filter shines in almost all cases. From other filters, D 2 an S 2 are oing fine, but the power results wavestrappe tests with S 4 an S 8 is not satisfactory. On the other han, when θ =.9 or.5, FWVR tests with =. an = are proucing similar power performance. However, this pattern oes not last in positive values of θ an the tests with =. have the lea by means of the size-ajuste power performances. For the null hypothesis r =, results are similar for =., whereas the filters except Haar are performing poorly in wavestrapping exercise when we consier emeaning or etrening. Finally, in the large samples, FWVR an its wavelet version is performing quite well by means of size ajuste power. To sum up, it seems using Haar, D 2 or S 2 in the wavelet base cointegration tests an the wavestrapping routine yiel satisfactory results by means size an size-ajuste power. These filters can be also mixe in testing an wavestrapping steps. For instance, one can use S 8 in FWVR test an Haar in wavestrapping algorithm. For brevity of the paper, I just provie results with the case where same filter is utilize in the test statistic an wavestrapping metho. Furthermore, =. seems to be optimal choice even though = is oing better in a few case. However, = usually fails to beat =. when we have positive MA() innovations. One can sacrifice the power gain obtaine by the test constructe with = to get a metho working balance in all situations. 6 Empirical Application I apply my cointegration testing methoology to the European creit efault swap (CDS) Rates ata with ifferent maturities. The creit efault swaps are one of the most common type of creit erivatives, which can hege the buyer against loses arising from a efault (ongstaff et al., 25). Thus, we can say that the CDS rate of a particular country inicates the efault risk perception of the investors against that country. Accoringly, The main iea in using CDS rates in cointegration analysis is to check whether a particular maturity CDS rates of ifferent countries within European Union co-move in the long run. I consier 8 countries, namely Austria, Belgium, France, Germany, Italy, Netherlans, Portugal an Spain. 6 For these countries, I collect one, two, three, five, seven an ten year CDS rates from Thomson Reuters. I use weekly ata from 23 July 28 to 23 November This interval is the longest perio for all countries an all maturities simultaneously. Furthermore, in the unreporte unit root tests for these variables, I fin all the CDS rates in my ataset follow a unit root. In orer to conuct cointegration test, I group the variables into 5 subsets of the ata accoring to the maturities. That is, each subset contains 8 countries CDS rates for a fix maturity. I separately employ Johansen s (988) trace test(jt ), FWVR test an its wavestrappe version to 5 atasets. I report the results in Table 2. In this table, we observe that accoring to JT test there is 5 cointegrating relations for one year CDS rate at.5 significance level. However, FWVR test fins 4 cointegrating relations at.5 significance level. Further, the wavestrappe FWVR test can only etect cointegrating relation among the one year CDS rates of the ifferent countries. One interesting result is that the wavestrappe FWVR test conclues no cointegration relation for the CDS rates with higher maturity than one year, while JT an F W V R fins at least two long run relations among these variables. For two year CDS rates JT an F W V R tests reveal that cointegrating rank is 4. Whereas, the wavestrappe test can only fin the presence of cointegraton at. significance level. On the other han, when the maturity of the CDS rate increases, both JT an F W V R test etect smaller number of cointegrating relations. For instance, for the cases with five, seven an year CDS rates, JT test points the rank as 4, 3 an 2, respectively, but F W V R test fins 3, 3, an 2 cointegrating relations for these cases, respectively. To sum up, the empirical results emonstrate that F W V R test tens to fin less cointegrating relations than JT test in general for the CDS rate exercises. Nonetheless, both tests inicate the presence of the long run comovement among CDS rates with all maturities. On the other han, from the wavestrappe test results for 6 The broaest analysis can be one with these countries. Incluing the other EU countries ecreases the length of the ata set. 7 The original ata is aily, but, I convert it to weekly ata as in ongstaff et al. (25). The conversion is one by taking the mi of the week (Wenesays) value of The CDS rates.

13 shorter maturity CDS rates, we can conclue that these variables co-move in the long run at least with one cointegrating relation. For the longer maturities, this result cannot be supporte statistically. As a result, the wavestrappe version of my test raws quite ifferent results than JT an F W V R tests in CDS rate exercises. Nevertheless, this result is consistent with the finings of Beirne an Fratzscher (23). The authors (Beirne an Fratzscher, 23) claim that after the sovereign ebt crisis in Europe, there is sharp rise in the sprea of sovereign yiels across countries. Moreover some countries become more risky, thus CDS rates are mostly etermine by country specific factors (Beirne an Fratzscher, 23). 7 Conclusion In this paper, I have propose a non-parametric wavelet base cointegration testing metho for fractionally integrate time series processes as well as the usual unit root variables. This metho can be regare as an alternative version of Nielsen s (2) variance ratio test. In fact, these two tests share many common features. First, both tests are fully nonparametric an o not require the estimation of a regression moel to remove short run ynamics. They also possess the same nuisance an tuning parameter free limiting istributions if one has a consistent estimate of the fractional integration parameter of the observe variables. Although these two tests enjoy same properties asymptotically, they may exhibit ifferent small sample characteristics in particular scenarios. Monte Carlo experiments inicate that when we compare power performances, my metho can compete with the stanar VR test in most cases an also beats it when we have the MA innovations with a negative root. On the other han, if we stuy size istortions, the wavelet base test has superiority over the stanar VR test uner a wie range of specifications. Even so, the size istortions in the extreme cases, such as in the presence of the negative MA roots in the stochastic factors cannot be eliminate effectively by either of the tests. To hanle such scenarios I propose a multivariate version of the wavestrapping proceure. Instea of wavestrapping the observe time series vector however, I wavestrap the stochastic factors which in fact carries all statistical features of the cointegration test in the Nielsen (2) setup. The simulations emonstrate tremenous improvement in size istortion even in the extreme negative MA root instances. Further, The DWT base proceure exhibits local power uner the ifferent innovation structures. Removing the size istortions in the particular cases is not the only benefit of the wavestrapping proceure. It also provies an alternative bootstrapping metho for the etermination of cointegration rank. To my knowlege this is this first attempt to implement a wavelet base bootstrapping algorithm in multivariate time series, particularly in a cointegration framework. Furthermore, the parametric bootstrapping methos which require the estimation of VECM moel an the selection of the lag length parameter(cavaliere et al. (22) an Swensen (26)), this new metho remains fully nonparametric an tuning parameter free. In orer to apply wavestrapping in cointegration framework, we only nee to estimate the stochastic trens. These trens can be nonparametrically estimate from the eigenvalue problem of Nielsen s (2) VR test explaine in Section 3. Finally, this paper has expane the wavelet framework for the unit root testing to the multivariate moels by proviing the erivations for not only multiple time series processes but also for fractionally integrate systems. I believe that these finings may she light on the properties of the wavelets transform of fractionally integrate variables in a multivariate setup. References Beirne, J. an Fratzscher, M. (23). The pricing of sovereign risk an contagion uring the european sovereign ebt crisis. Journal of International Money an Finance, 34:6 82. Cavaliere, G., Rahbek, A., an Taylor, A. (22). Bootstrap etermination of the co-integration rank in vector autoregressive moels. Econometrica, 8(4): Chambers, M. J., Ercolani, J. S., an Taylor, A. R. (24). Testing for seasonal unit roots by frequency omain regression. Journal of Econometrics, 78: Choi, I. an Phillips, P. C. (993). Testing for a unit root by frequency omain regression. Journal of Econometrics, 59(3): Daubechies, I. (988). Orthonormal bases of compactly supporte wavelets. Communications on pure an applie mathematics, 4(7): Fan, Y. an Gencay, R. (2). Unit root tests with wavelets. Econometric Theory, 5(26):

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