A bootstrap test to detect prominent Granger-causalities across frequencies
aa r X i v : . [ q -f i n . S T ] O c t A bootstrap test to detect prominentGranger-causalities across frequencies
Matteo Farn´e Department of Statistical Sciences,University of Bologna, Italy
Angela Montanari
Department of Statistical Sciences,University of Bologna, Italy
October 23, 2018 Electronic address: [email protected] ; Corresponding author bstract
Granger-causality in the frequency domain is an emerging tool to analyze the causal relation-ship between two time series. We propose a bootstrap test on unconditional and conditionalGranger-causality spectra, as well as on their difference, to catch particularly prominentcausality cycles in relative terms. In particular, we consider a stochastic process derived ap-plying independently the stationary bootstrap to the original series. Our null hypothesis isthat each causality or causality difference is equal to the median across frequencies computedon that process. In this way, we are able to disambiguate causalities which depart signifi-cantly from the median one obtained ignoring the causality structure. Our test shows powerone as the process tends to non-stationarity, thus being more conservative than parametricalternatives. As an example, we infer about the relationship between money stock and GDPin the Euro Area via our approach, considering inflation, unemployment and interest rates asconditioning variables. We point out that during the period 1999-2017 the money stock ag-gregate M1 had a significant impact on economic output at all frequencies, while the oppositerelationship is significant only at high frequencies.
Keywords : Bootstrapping, Causality, Spectral analysis, Statistical tests, Monetary Policy,Euro Area
Introduction
As a statistical concept, causality has a central role both from a theoretical and a practicalpoint of view (see Berzuini et al. (2012)). In time series analysis, the concept that was tobe called Granger-causality (GC) was first introduced by Wiener in the context of predictiontheory (Wiener, 1956) and then formalized by Granger in the context of linear regression mod-elling of stochastic processes (Granger, 1969). Causality measures in the frequency domainwere first proposed in Pierce (1979) as R measures for time series. In Geweke (1982) andGeweke (1984) the fundamental concepts of unconditional and conditional Granger-causalityin the frequency domain were introduced (and extended in Hosoya (1991) and Hosoya (2001)respectively).While the use of GC in the time-domain dates back to the sixties, GC in the frequency do-main has become increasingly popular in recent years. In Lemmens et al. (2008) the causalitystructure of European production expectation surveys is analyzed by the methods of Pierce(1979) and Geweke (1982) comparatively, which require to study appropriate frequency-wisecoefficients of coherence. The same approach is used in Tiwari (2014) for exploring the re-lationship between energy consumption and income in the United States. The advantage offrequency-domain GC lies in the disentanglement of the causality structure across a rangeof frequencies, while traditional time-domain GC only provides an overall indication on thepresence of a causality relationship. The aim of our paper is to provide an inferential toolto mark up the strongest causalities in the frequency domain, in order to draw meaningfulremarks about the causality structure.In Ding et al. (2006), bootstrap thresholds are computed to make inference about Geweke’sunconditional and conditional GC measures in the context of neurological data, via the ran-domization approach of Blair and Karniski (1993). A further extension of that approachcan be found in Wen et al. (2013), and relevant applications in the neurophysiological con-text include Brovelli et al. (2004), Roebroeck et al. (2005) and Dhamala et al. (2008), whereexplicit VAR estimation is avoided by a nonparametric approach. More recently, a compre-hensive computational and inferential strategy for time-domain and frequency-domain GC1pectra has been proposed in Barnett and Seth (2014).In Breitung and Candelon (2006), a parametric test for Granger-causality in the frequencydomain is proposed. Its convergence rate is O ( T − / ) (where T is the time length) and itspower is decreasing as the distance of the frequency of interest from π increases (even ifYamada and Yanfeng (2014) show that the same test is still useful at extreme frequencies).The test is based upon a set of linear restrictions on the parameters of the (possibly coin-tegrated) VAR model best representing the series (we refer to L¨utkepohl (2005) for VARselection and estimation).Applying such a test to time series with a rich causality structure, like macroeconomicones, most of causalities are often flagged as significant. In addition, test precision may sufferat extreme frequencies when T is not large. Nonetheless, such procedure is widely used in theliterature. For example, a relevant application for studying the relationship between real andfinancial business cycles can be found in Gomez-Gonzalez et al. (2015).Some nonparametric testing approaches were also proposed in the literature. Hidalgo(2000) estimates VAR filters via generalized least squares and then accordingly derives atest statistics for GC. Hidalgo (2005) extends the framework of Hidalgo (2000) to the mul-tivariate case computing relevant quantiles under the null via resampling bootstrap. InAssenmacher-Wesche and Gerlach (2008), the Philipps spectral estimator (see Phillips (1988))is exploited to estimate causality both at frequency 0 and at the rest of frequencies in a cointe-grated setting. Such a method is used in Berger and Osterholm (2011) to test the relationshipbetween money growth and inflation in the Euro Area.The present work proposes a complementary approach to the classical testing frameworkof the no-causality hypothesis. Our aim is to detect prominent cycles, i.e. cycles which aredominant compared to others for explaining the causality relationship. We would like toanswer the following question: ’Which causalities are larger than the median causality thatwould hold in case of stochastic independence?’ The need for such a tool rises to distinguish themost relevant causalities for the causality structure of the process among significant causalitiesin the classical sense. 2n order to reach this goal, we approximate the data generating process under the nullapplying the stationary bootstrap of Politis and Romano (1994) independently to each se-ries. We derive the desired bootstrap quantile of the median causality and we compare eachcausality to it. The median is chosen because we need a unique comparison ground for eachcausality, and under the null there is no reason to suppose that causalities are stochasticallydifferent. Our test is adaptive with respect to the true spectral shape and can be used as acomplementary tool to classical tests, provided that T is large enough to ensure T → ∞ .We exploit the described tool for studying the mutual relationship between economicoutput and money stock in the Euro Area, as, in so doing, we can identify characteristicfrequencies, i.e. characteristic time periods of the causality structure. The problem has beenwidely addressed as far as the US economy is concerned. In Belongia and Ireland (2016),for instance, the methodology of Friedman and Schwartz (1975), based on structural VARmodels, is revisited and applied to U.S. data across the period 1967 − Let us suppose that the past values of a time series Y t , i.e. Y t − , Y t − , . . . , help predicting thevalue at time t of another time series X t , that is, Y t − , Y t − , . . . add significant informationto the past values of X t ( X t − , X t − , . . . ) for predicting X t . In that case we say that Y t X t .We now briefly recall the bases of Granger-causality spectral theory. We follow the ap-proach in Ding et al. (2006), which we refer to for the details. Suppose that X t and Y t , jointlycovariance-stationary, follow a non-singular V AR ( k ) model. Defining Z t = [ X t , Y t ] ′ , we have Z t = A Z t − + . . . + A k Z t − k + ǫ t , (2.1)where ǫ t ∼ N (0 , Σ ), Σ is a 2 × A , . . . , A k are 2 × ω we define the transfer function P ( ω ) of Z t in (2.1) as P ( ω ) = I − k X j =1 A j e − ijω − , − π ≤ ω ≤ π, (2.2)which is invertible if and only if the roots of the equation det( I p − P kj =1 A j L j ) = 0 (where L is the lag operator) lie within the unit circle. Setting P ( ω ) = P XX ( ω ) P XY ( ω ) P Y X ( ω ) P Y Y ( ω ) , the definition (2.2) allows to define in a compact way the model-basedspectrum h ( ω ) as follows: h ( ω ) = P ( ω ) Σ P ( ω ) ∗ , − π ≤ ω ≤ π, where ∗ denotes the complex conjugate.Setting Σ = σ υ υ γ , we need for computational reasons to define the transformmatrix S = − υ γ , from which we derive the transformed transfer function matrix ˜P ( ω ) = S × P ( ω ). The process Z t = [ X t , Y t ] ′ is normalized accordingly as Z ∗ t = ˜P ( L )[ X t , Y t ] ′ and becomes Z ∗ t = [ X ∗ t , Y ∗ t ] ′ . 4he unconditional Granger-causality spectrum of X t (effect-variable) respect to Y t (cause-variable) is then defined as (Geweke, 1982) h Y → X ( ω ) = ln (cid:18) h XX ( ω )˜ P XX ( ω ) σ ˜ P XX ( ω ) ∗ (cid:19) . (2.3)In the empirical analysis, the theoretical values of coefficient and covariance matrices will bereplaced by the corresponding SURE estimates (Zellner, 1962).Moreover, we can define the conditional Granger causality spectrum of X t respect to Y t given an exogenous variable W t (conditioning variable). Suppose we estimate a VAR on[ X t , W t ] ′ with covariance matrix of the noise terms Σ ′ = σ ′ υ ′ υ ′ γ ′ . and transfer function G ( ω ) (defined as in (2.2)). The corresponding normalized process of [ X t , W t ] ′ (according tothe procedure described above) is denoted by [ X ∗ t , W ∗ t ] ′ .We then estimate a VAR on [ X t , Y t , W t ] ′ with covariance matrix of the noise terms Σ = σ XX σ XY σ XW σ Y X σ Y Y σ Y W σ W X σ W Y σ W W and transfer function P ′ ( ω ). Building the matrix C ( ω ) = G XX ( ω ) 0 G XW ( ω )0 1 0 G W X ( ω ) 0 G W W ( ω ) , we can define Q ( ω ) = C − ( ω ) P ′ ( ω ), which is a sort of “conditional” transfer function matrix.The theoretical spectrum of X ∗ can thus be written as h X ∗ X ∗ ( ω ) = Q XX ( ω ) σ XX Q XX ( ω ) ∗ ++ Q XY ( ω ) σ Y Y Q XY ( ω ) ∗ + Q XW ( ω ) σ W W Q XW ( ω ) ∗ . X t (effect-variable) respect to Y t (cause-variable) given W t (con-ditioning variable) is (Geweke, 1984) h Y → X | W ( ω ) = ln (cid:18) h X ∗ X ∗ ( ω ) Q XX ( ω ) σ XX Q XX ( ω ) ∗ (cid:19) . (2.4)Both h Y → X ( ω ) and h Y → X | W ( ω ) range from 0 to ∞ , with − π ≤ ω ≤ π . h Y → X ( ω )expresses the power of the relationship from Y to X at frequency ω , h Y → X | W ( ω ) expresses thestrength of the relationship from Y to X at frequency ω given W . Therefore, the unconditionalspectrum accounts for the whole effect of the past values of Y t onto X t , while the conditionalspectrum accounts for the direct effect of the past values of Y t onto X t excluding the effectmediated by the past values of W t . The same measures are more easily defined in the time-domain. In that case, they are defined for the process as a whole (not frequency-wise as inthe frequency domain).Granger-causality spectra h Y → X ( ω ) and h Y → X | W ( ω ) can be interpreted as follows. If h Y → X ( ω ) >
0, it means that past values of Y t help predicting X t , and ω is a relevant cycle.If h Y → X | W ( ω ) >
0, it means that past values of Y t in addition to those of W t help predicting X t , and ω is a relevant cycle. Significant frequencies give us some hints on the relevant delaystructure of the cause variable with respect to the effect variable.We remark that these measures do not give any information on the sign of the relationship,which is given by time-domain measures like the correlation coefficient. It rather describesthe strength, i.e. the intensity, of the causal relationship. The inference on Granger-causality spectra in the frequency domain is still an open problem.In fact, differently from the corresponding time-domain quantities, the limiting distributionfor unconditional and conditional spectra is unknown (see Barnett and Seth (2014), section2.5). In spite of that,Breitung and Candelon (2006) test the nullity of unconditional and conditional GC at eachfrequency ω , imposing a necessary and sufficient set of linear restrictions to the (possibly6ointegrated) VAR model best fitting the series. The resulting test statistics is distributedunder the null as a Fisher distribution with 2 and T − k degrees of freedom (except for ω = { , π } at which the distribution is F (1 ,T − k ) ), where k is the VAR delay and T is the timeseries length.As said in the Introduction, that test applied on macroeconomic series often flags most ofcausalities as significant, due to the rich causality structure. For this reason, in order to disam-biguate among significant causalities the most prominent ones, we propose a complementarybootstrap testing approach. At each frequency ω , we test the null hypothesis H : t ( ω ) = t med ,against the alternative H : t ( ω ) > t med , where the functional t ( ω ) may be the unconditionalGC h Y → X ( ω ), the conditional GC h Y → X | W ( ω ) or their difference h Y → X ( ω ) − h Y → X | W ( ω ), and t med is the median of t ( ω ) across frequencies under the assumption of stochastic independence.Since the distributions of h Y → X ( ω ) and h Y → X | W ( ω ) are unknown, we approximate thedistribution of each t ( ω ) under the null by the stationary bootstrap of Politis and Romano(1994). A similar approach was originally proposed by Ding et al. (2006), which tests thesame null hypothesis of Breitung and Candelon (2006) by the randomization procedure ofBlair and Karniski (1993), retaining the maximum causality across frequencies. Differently,our procedure tests by bootstrap the equality between each unconditional or conditionalcausality and the median causality under the assumption of stochastic independence. Apply-ing the stationary bootstrap to each time series independently of the other ones approximatesthe no-causality situation, because it approximates the Markov chain best representing inde-pendently each series. We stress that unconditional and conditional spectra must be assessedseparately, because their distributions are in general different.In Ding et al. (2006), the comparison between unconditional and conditional Grangercausalities is performed using the randomized t-test of Blair and Karniski (1993) on the boot-strapped series of the causality peak across frequencies. This approach is suitable for theircase, where they perform psychological/neurological experiments, which allow to have mul-tiple trials data. On the contrary, in the economic context, we can not perform such a testbecause we only have a single realization. This is the reason why we take the difference7etween unconditional and conditional GC, which allows to determine if the conditioningvariable has a significant impact (amplification or annihilation) on the causal relationship inour time-dependent data context.Our idea derives from Politis and Romano (1994), according to which each Fr´echet-differentiable functional may be successfully approximated by the stationary bootstrap, and the resultingbias depends on the sum of the
Fr´echet-differential h F evaluated at each observation, giventhat the distance between the empirical and the true distribution function of X t is small.The bootstrap series obtained via the stationary bootstrap ofPolitis and Romano (1994) are stationary Markov chains conditionally on the data. It meansthat each bootstrap series X ∗ , . . . , X ∗ T is a Markov chain conditionally on X , . . . , X T . Supposethat we apply the same procedure to X t , Y t and W t , obtaining the stationary bootstrap series X ∗ t , Y ∗ t and W ∗ t . Computing unconditional and conditional Granger causality spectra on thoseseries equals to assess causalities under the assumption of stochastic independence, because theentire stochastic behaviours of X ∗ t , Y ∗ t and W ∗ t are explained by the conditional distributionsof X ∗ t | X ∗ t − , Y ∗ t | Y ∗ t − , W ∗ t | W ∗ t − . Therefore, testing each Granger-causality computed on theoriginal series X t , Y t and W t against the median causality computed across frequencies on X ∗ t , Y ∗ t and W ∗ t is effective as a test for causality strength in relative terms.Let us consider ˆ r ( ω ) = ˆ h Y → X ( ω ), which is defined as (2.3) where the coefficient matrices A j , j = 1 , . . . , k , and the error covariance matrix Σ are replaced by the corresponding SUREestimates (Zellner, 1962). We know that SURE estimates ˆA j , j = 1 , . . . , k , are rationalfunctions of the data, thus being Fr´echet-differentiable . ˆ˜P ( ω ) and ˆΣ are functions of the ˆA j ,thus being rational in turn; the same holds as a consequence for ˆ h ( ω ). Therefore, ˆ h Y → X ( ω ),the natural logarithm of a rational function of the data, is Fr´echet-differentiable . At thispoint, as pointed out in Politis et al. (2012), page 30, even if the median is not
Fr´echet-differentiable , the bootstrap for ˆ h Y → X ( ω ) is still valid, provided that the density function of median ( h Y → X ( ω )) is positive. As a consequence, according to Politis and Romano (1994),paragraph 4.3, we can estimate consistently any quantile of the distribution of the median ofˆ h Y → X ( ω ) under the null hypothesis. 8onsidering ˆ r ( ω ) = ˆ h Y → X | W ( ω ), which is defined as (2.4) where the coefficient matricesand the error covariance matrix are replaced by the corresponding SURE estimates, a similarreasoning can be carried out. The same applies to the estimated difference of ˆ h Y → X ( ω ) − ˆ h Y → X | W ( ω ), that is Fr´echet-differentiable apart from the case ˆ h Y → X ( ω ) = ˆ h Y → X | W ( ω ), whichholds with null probability.We stress that our aim is not to represent the common multivariate distribution function F of the process Z t = [ X t , Y t , W t ]. That problem is an estimation one, which would beeffectively solved by parametric or residual bootstrap. Our aim is to exploit the randomprocess Z ∗ t = [ X ∗ t , Y ∗ t , W ∗ t ] to derive the bootstrap quantile q ∗ − α which satisfies P ( r ∗ med ≤ q ∗ − α ) = 1 − α, where α is the significance level and r ∗ med is the bootstrap median across frequencies ofunconditional, conditional GC or their difference under the assumption of stochastic inde-pendence. Since r med is Fr´echet-differentiable , P ( r ∗ med ≤ q ∗ − α ) approximates consistently P ( r med ≤ q r, − α ) as T → ∞ under the null hypothesis of stochastic independence.In more detail, suppose that r is a Fr´echet-differentiable functional, that is, there existssome influence function h F such that r ( G ) = r ( F ) + Z h F d ( G − F ) + o ( || G − F || )with R h F dF = 0 ( || . || is the supremum norm). We define the mixing coefficient α X ( k ) = sup A,B | P ( A, B ) − P ( A ) P ( B ) | where A and B vary over events in the σ -fields generated by { X t , t ≤ } { X t , t ≥ k } . Thefollowing Theorem holds. Theorem 2.1
Suppose that X t , Y t and W t are strictly stationary random variables with dis-tribution functions F X , F Y , F W . Assume that, for some d ≥ , E ( h F X ( X ) d ) < ∞ , k α X ( k ) d d < ∞ and P k k α X ( k ) / − τ < ∞ . Further assume that these assumptionsalso hold for Y t and W t . Then, if the distribution function F Z of the random vector Z t =[ X t , Y t , W t ] can be factorized as F X F Y F W , it holds P ( r ( ˆ F ∗ Z ) − r ( F Z ) ≤ q r (1 − α )) = 1 − α for any Fr´echet-differentiable functional r under the assumption T → ∞ . We refer to Appendix 4 for the proof.Due to the nature of our test, we need to exclude any stochastic process with a constantGranger-causality spectrum different from a white noise. Suppose that Z t is a stochasticprocess with auto-covariance matrices R j = cov ( Z t , Z t − j ), j ∈ Z + . Throughout the paper,we need to assume that, in case there is at least one non-zero causality coefficient at one delay j ∈ Z + , the resulting covariance matrix is not diagonal, i.e. the effect and the cause variableare not uncorrelated. At the same time, we need to assume that each auto-covariance matrix R j , j ≥
1, is positive definite.We clarify the expressed constraints by an example. Consider the case of a
V AR (1) withthe following parameters: Σ = diag (1 , A = .
50 0 . According to Wei et al. (2006),p. 392, the vectorized covariance matrix of a vector AR (1) process is vec ( R ) = ( I − A ⊗ A ) − vec ( Σ ) and the vectorized autocovariance matrix at lag k is vec ( R k ) = vec ( R ) A k .Therefore, we have R = diag (1 . , R , R , . . . are singular matrices. The sameholds, for instance, if we suppose A = . .
50 0 . In that case, the covariance matrixresults R = diag ( , .3 Testing procedure We now report in detail the testing procedures relative to the three functionals. • For the functional ˆ h Y → X ( ω ), our bootstrap procedure is – Simulate N stationary bootstrap series ( X ∗ t , Y ∗ t ) given the observed series ( X t , Y t ). – On each simulated series ( X ∗ t , Y ∗ t ):1. estimate a VAR model on ( X ∗ t , Y ∗ t ) via SURE using BIC for model selection.2. at Fourier frequencies f i = iT , i = 1 , . . . , [ T ], compute h Y ∗ → X ∗ (2 πf i ).3. compute median { f i ,i =1 ,...,T/ } h Y ∗ → X ∗ (2 πf i ). – Then, compute q uncond, − α , the (1 − α )-quantile of the bootstrap distribution atStep 3 across the N bootstrap series, where α is the significance level. – Finally, at each f i , flag ˆ h Y → X (2 πf i ) as significant if larger than q uncond,α . • For the functional ˆ h Y → X | W ( ω ), the procedure becomes – Simulate N stationary bootstrap series ( X ∗ t , Y ∗ t , W ∗ t ) given the observed series( X t , Y t , W t ). – On each simulated series ( X ∗ t , Y ∗ t , W ∗ t ):1. estimate a VAR model on ( X ∗ t , W ∗ t ) and ( X ∗ t , Y ∗ t , W ∗ t ) via SURE using BICfor model selection.2. at Fourier frequencies f i = iT , i = 1 , . . . , [ T ], compute h Y ∗ → X ∗ | W ∗ (2 πf i ).3. compute median { f i ,i =1 ,...,T/ } h Y ∗ → X ∗ | W ∗ (2 πf i ). – Then, compute q cond, − α , the (1 − α )-quantile of the bootstrap distribution at Step3 across the N bootstrap series. – Finally, at each f i , flag ˆ h Y → X | W (2 πf i ) as significant if larger than q cond,α . • For the functional ˆ h Y → X ( ω ) − ˆ h Y → X | W ( ω ), the procedure is – Simulate N stationary bootstrap series ( X ∗ t , Y ∗ t , W ∗ t ) given the observed series( X t , Y t , W t ). 11 On each simulated series ( X ∗ t , Y ∗ t , W ∗ t ):1. estimate a VAR model on ( X ∗ t , Y ∗ t ), ( X ∗ t , W ∗ t ) and ( X ∗ t , Y ∗ t , W ∗ t ) via SUREusing BIC for model selection.2. at Fourier frequencies f i = iT , i = 1 , . . . , [ T ], compute h Y ∗ → X ∗ (2 πf i ) − h Y ∗ → X ∗ | W ∗ (2 πf i ).3. compute median { f i ,i =1 ,...,T/ } h Y ∗ → X ∗ (2 πf i ) − h Y ∗ → X ∗ | W ∗ (2 πf i ). – Then, compute q diff, α and q diff, − α , the ( α )- and (1 − α )-quantiles of the bootstrapdistribution at Step 3 across the N bootstrap series. – Finally, at each f i , flag ˆ h Y → X (2 πf i ) − ˆ h Y → X | W (2 πf i ) as significant if smaller than q diff, α or larger than q diff, − α .We provide an R package, called “grangers”, which performs these routines.In addition, we extend our framework to test the nullity of r (2 πf i ), i = 1 , . . . , [ T ], acrossthe frequency range. In order to do that, we apply Bonferroni correction, that is, we applythe test procedure to each frequency with significance level αT . In this way, we ensure thatthe overall level is not larger than α under the null. This approach is conservative: anyway,the test still has a power of 100% as the VAR process tends to non-stationarity. In order to clarify the interpretation of our results, we need to define the concept of “promi-nence” in a formal way. At a significance level α , given a random time series sampled by theunderlying data generating process, any functional r ( ω ) is said to be maximally prominentat frequency ω if P { r ( ω ) > r med } > − α , where r med is the median of r ( ω ) across frequen-cies. As a consequence, the power of our test procedure approaches 1 as r ( ω ) is maximallyprominent .We define the prominence rate at frequency ω as the expected probability of r ( ω ) to bemaximally prominent: prom ( ω ): P ( r ( ω ) > q − α ). The prominence rate answers the question“Which is the probability that r ( ω ) is maximally prominent?” The degree of prominence atfrequency ω is then defined as dp ( ω ): P ( r ( ω ) > r med ). Instead, the power at frequency ω isdefined as power ( ω ) = P (ˆ r ( ω ) > q − α ). Denoting the solutions of the characteristic equation12et( I p − P kj =1 A j L j ) = 0 in decreasing order by λ , . . . , λ q , the maximal power at frequency ω is defined as mp ( ω ) = lim | λ |→ power ( ω ). For our test we observe max ω ∈ ]0 , π ] mp ( ω ) = 1.In general, ˆ r ( ω ) is significant if larger than r ∗ med at a significance level α . As explained inSection 2.2, the distribution of r ∗ med consistently resembles the one of r med by the stationarybootstrap. The level of our test, as expected, is approximately equal to the chosen significancelevel α under the null, i.e. in case of zero-causality at all frequencies (white noise process).We now describe the performance of our test in a number of situations. First of all,suppose that we simulate 100 replicates from a VAR process in the form (2.1) with k = 1, Σ = diag (1 ,
1) and no causality coefficients. The VAR delay is selected for each bootstrapsetting by BIC criterion. Our tested coefficient matrix A is A , ( jj ) = 0 , . , . , . , j = 1 , A , ( jj ) is distant from 1. If A , ( jj ) = 1 (double random walk), the rejection rate increases at lowfrequencies, according to the shape of prominence rate and degree of prominence, until 0 . k = 1 and A , ( j = 0 . , j = 1 ,
2. This processhas an unconditional causality which decreases as the frequency increases. For A , ( j = 0 . . . . . A , ( j = 1 (Figure 2), the powerat the lowest frequency is one, reflecting the prominence rate and the degree of prominence.The same case is tested for the conditional causality, with very similar results.We now compare an unconditional and a conditional causality which are zero at all fre-quencies. For both cases, the rejection rates stand below 5% at all frequencies. If we comparetwo decreasing causalities having the shape above described ( A , ( j = 0 . , j = 1 , . . . .1 0.2 0.3 0.4 0.5 . . . . . . Granger−causality y to x frequency U n c ond i t i ona l G C . . . . . . Rejection rate frequency R e j e c t i on r a t e NEWBC0.1 0.2 0.3 0.4 0.5 . . . . . . Prominence rate frequency P r o m i nen c e r a t e . . . . . . Degree of prominence frequency D eg r ee o f p r o m i nen c e Figure 1: Case with k = 1, A , ( j = 0 . j = 1 ,
2. In dotted the significance level α = 0 . . .
95. In dashed the rejection rate of BC test. . . . . Granger−causality y to x frequency U n c ond i t i ona l G C . . . . . . Rejection rate frequency R e j e c t i on r a t e NEWBC0.1 0.2 0.3 0.4 0.5 . . . . . . Prominence rate frequency P r o m i nen c e r a t e . . . . . . Degree of prominence frequency D eg r ee o f p r o m i nen c e Figure 2: Case with k = 1, A , ( j = 1, j = 1 , .1 0.2 0.3 0.4 0.5 − Difference frequency D i ff e r en c e differenceuncond. causalitycond. causality 0.1 0.2 0.3 0.4 0.5 . . . . . . Rejection rate frequency R e j e c t i on r a t e . . . . . . Prominence rate frequency P r o m i nen c e r a t e . . . . . . Degree of prominence frequency D eg r ee o f p r o m i nen c e Figure 3: Comparing an unconditional and a conditional zero causality. − Difference frequency D i ff e r en c e differenceuncond. causalitycond. causality 0.1 0.2 0.3 0.4 0.5 . . . . . . Rejection rate frequency R e j e c t i on r a t e . . . . . . Prominence rate frequency P r o m i nen c e r a t e . . . . . . Degree of prominence frequency D eg r ee o f p r o m i nen c e Figure 4: Comparing an unconditional and a conditional decreasing causality A , ( j = 1.15rominent causality difference, as the degree of prominence and the prominence rate confirm.Moreover, consider the VAR models described in Breitung and Candelon (2006), para-graph 4. Those models have k = 3, A k, ( j = 1, k = 1 , j = 1 and A k, ( j = − ω ∗ ), k = 2, j = 1. Such coefficient structure results in a null causality at frequency ω ∗ . On these settings,we can compare the results of our test to the results of “BC test” by Breitung and Candelon(2006), which appear in dashed line. In addition, we test the sensitivity of the results tothe condition number of the covariance matrix, setting Σ = diag (1 , Σ = diag (0 . , Σ = diag (5 , ω ∗ = π and Σ = diag (1 , . . ω ∗ , resembling the shape of the degree of prominence (Figure 5). On the contrary, BC testshows a rejection rate of 0 . ω ∗ , and 1 at extreme frequencies. Setting Σ = diag (0 . , . .
2, while ours is approximately constant around0 . X is much smaller than the one of Y ,such that X is close to a null process, and the underlying causality is small and detected asconstant across frequencies. Setting Σ = diag (5 ,
1) (Figure 7), the rejection rate of both testsstands around 1, except from a value of 0 . ω ∗ . This occurs because the magnitude of Y is much smaller than the one of X , such that any non-null causality is detected as maximallyprominent.Setting A k, (22) , k = 1 ,
3, to 0 .
25 and 0 . ω ∗ = 0, ω ∗ = π , ω ∗ = π and ω ∗ = π ,we note that our competitor is less precise, as described therein, particularly for the first twocases, because the rejection rate is considerably above 5%. Its rejection rate for non-zerocausalities is 100%, while ours resembles the shape of the degree of prominence, which tendsto 0 for null causalities with particular intensity for the cases ω ∗ = 0, ω ∗ = 1.16 .1 0.2 0.3 0.4 0.5 . . . . Granger−causality y to x frequency U n c ond i t i ona l G C . . . . . . Rejection rate frequency R e j e c t i on r a t e NEWBC0.1 0.2 0.3 0.4 0.5 . . . . . . Prominence rate frequency P r o m i nen c e r a t e . . . . . . Degree of prominence frequency D eg r ee o f p r o m i nen c e Figure 5: Case with ω ∗ = π , A k, (22) = 0, Σ = diag (1 , k = 1 ,
3. In dashed the rejectionrate of BC test. . . . . Granger−causality y to x frequency U n c ond i t i ona l G C . . . . . . Rejection rate frequency R e j e c t i on r a t e NEWBC0.1 0.2 0.3 0.4 0.5 . . . . . . Prominence rate frequency P r o m i nen c e r a t e . . . . . . Degree of prominence frequency D eg r ee o f p r o m i nen c e Figure 6: Case with ω ∗ = π , A k, (22) = 0, Σ = diag (0 . , k = 1 , .1 0.2 0.3 0.4 0.5 . . . . Granger−causality y to x frequency U n c ond i t i ona l G C . . . . . . Rejection rate frequency R e j e c t i on r a t e NEWBC0.1 0.2 0.3 0.4 0.5 . . . . . . Prominence rate frequency P r o m i nen c e r a t e . . . . . . Degree of prominence frequency D eg r ee o f p r o m i nen c e Figure 7: Case with ω ∗ = π , A k, (22) = 0, Σ = diag (5 , k = 1 , • the magnitude of VAR roots, which has the effect to extend the range. In general, therejection rate is perturbed at low frequencies as the process is closer to non-stationarity; • the true underlying spectral variability, which in turn depends on the relationship be-tween the magnitude of causality and non-causality coefficients; • the condition number of the autocovariance matrices R j , j ≥
0, which masks the un-derlying spectral variability.In Table 1 we report the rejection rates of the test on all causalities jointly consideredobtained by Bonferroni correction. We note that the test has power approximately 0 .
05 incase of no-causality (Case 3), and approximately 1 in case of non-stationarity (Cases 2 and7). 18ase Rejection rate1 0.482 0.983 0.054 0.625 0.676 0.167 0.99Table 1: Test on all causalities jointly considered obtained by Bonferroni correction.
While remembering Friedman and Schwartz’s general statement (seeFriedman and Schwartz (2008)) that “In monetary matters appearances are deceiving: theimportant relationships are often precisely the reverse of those that strike the eye”, in thissection we study the co-movements of gross domestic product (GDP) and money stock (M3and M1 aggregate) in the Euro Area. We test in the frequency domain both the direct linkfrom one variable to the other one and the indirect link with respect to further explanatoryvariables like the inflation rate (HICP), the unemployment rate (UN), or the long-term interestrate (LTN).Published works on this research topic make use of time-domain methods: some of themuse factor modelling (Cendejas et al., 2014), some others use likelihood methods (Andr´es et al.(2006), Canova and Menz (2011)), or large-dimensional VAR models (Giannone et al., 2013),or VAR models with time-varying parameters (Psaradakis et al., 2005). A good review forthe pre-Euro period may be found in Hayo (1999), which explored the relationship betweenbusiness cycle and money stock in EU countries via a Granger-causality analysis in the timedomain, exactly as Tsukuda and Miyakoshi (1998) did for the Japanese economy.On the contrary, we apply the inferential framework for GC in the frequency domain de-veloped in Section 2. Differently from Breitung and Candelon (2006), which tests the nullityof Granger-causalities at each frequency, our test is able to discern prominent causalities incomparison to others. In this way, we provide explicit inference on unconditional and condi-19ional GC. HICP, UN and LTN are used as conditioning variables, with the aim to discountfor the mediating power of each of the three variables with respect to the relationship betweenoutput and money supply. The same approach also allows us to compare unconditional andconditional GC relative to the same directional link.
We have considered the time series of GDP at market price in the Euro Area (chain linkedvolumes in Euro) and the monetary aggregate M3 and M1 (outstanding amount of loans tothe whole economy excluded the monetary and financial sector, all currencies combined). M3is also called “broad money”, M1 “narrow money”.There is not a general consensus on which measure of money supply is the most appro-priate. While the Federal Reserve has officially ceased to publish M3 series since 2006, theM3 index of notional stocks, i.e. the annual growth rate of the outstanding amount (alsocalled “base money”), is still used by the ECB as the official measure of short-term circulat-ing money. For a nice discussion on the role of M3 as a policy target for central bankers seefor example Alves et al. (2007).Since our goal is to focus on the effect of monetary policy on output, we restrict our analysisto the period 1999-2017, when the ECB has taken actual decisions on the Euro Area. Monthlyseries (all but GDP) are made quarterly by averaging. We can thus denote our series by
GDP t , M t , M t , HICP t , U N t , LT N t , where t = 1 , . . . ,
56 (there are 56 quarters from Winter 2001to Autumn 2014). The data are drawn from the ECB Real Time Research database wherenational figures are aggregated according to a changing composition of the Euro Area acrosstime (see Giannone et al. (2012)). We refer to and ECB(2012) for technical and computational details.According to Dickey-Fuller test, the logarithmic transform of
GDP t , M t , M t are non-stationary, as well as the three conditioning variables HICP t , U N t and LT N t . There-fore, following Friedman and Schwartz (1975), we pass all series by Hodrick-Prescott filter(Hodrick and Prescott, 1997), with the canonical value of λ = 1600, in order to remove any20 uro Area log−GDP Time l og − G D P . . HP−filtered cycle: log−GDP
Time2000 2005 2010 2015 − . . . Euro Area log−M3
Time l og − M . . . HP−filtered cycle: log−M3
Time2000 2005 2010 2015 − . . . Euro Area log−M1
Time l og − M . . . . HP−filtered cycle: log−M1
Time2000 2005 2010 2015 − . . . Figure 8: GDP, M3 and M1 in logs - Euro Area. In dashed the extracted trend.
Euro Area HICP rate
Time H I C P . . HP−filtered cycle: HICP rate
Time2000 2005 2010 2015 − . . Euro Area UN rate
Time UN . . . HP−filtered cycle: UN rate
Time2000 2005 2010 2015 − . . Euro Area LTN rate
Time L T N . . . HP−filtered cycle: LTN rate
Time2000 2005 2010 2015 − . . Figure 9: HICP, UN, LTN rates - Euro Area. In dashed the extracted trend.21rend and to extract cyclical components. We do not use Baxter-King filter (Baxter and King,1999), as suggested in Belongia and Ireland (2016), because we have not enough end of sampledata. Cycle extraction is performed via the R package “mFilter”.Figures 8 and 9 contain the plots of
GDP t , M t , M t and HICP t , U N t , LT N t respectively.Left figures contain the original series and the estimated trend, while right figures contain theestimated cycles. Figures 10 and 11 show the ACF of the extracted cycles. The patterns arevery similar across series: positive for the first 4-5 quarters, negative for all quarters around2 years and non-significant elsewhere. U N t shows a rebound for the quarters around 5 years. LT N t is no longer significant after 2 quarters. Figure 12 shows the CCF for the couplesGDP-M3 and GDP-M1. Their pattern is similar: we have positive correlation around 0 andnegative correlation at sides around the lag of 2 years.Since our ultimate goal is to infer about the cause-effect relationship of money stock andeconomic output, we test at each frequency the equality between Granger-causality spectraand the median GC across frequencies, both unconditional and conditional on the inflationrate, the unemployment rate and the long-term interest rate. In this way, we can displaythe relevant cycles in the causality structure of the relationship from GDP to M3 (M1) andviceversa. Due to the use of Fast Fourier Transform, the frequencies used are the following: f i = i , i = 1 , . . . ,
40, because T = 76. The frequency range is re-scaled to [0 ,
2] for thequarterly frequency of our series.Relevant VAR models, estimated including an intercept by the R package “vars” (Pfaff et al.,2008), are selected by the Bayesian Information Criterion (BIC), imposing a maximum of fourlags. BIC is used because we know that BIC is correctly estimating the unknown numberof delays, while AIC may overestimate it, thus increasing the probability to estimate non-stationary VAR models. In any case, all roots of estimated characteristic polynomials arestrictly smaller than one. In the end, the resulting number of delays is then fixed across thebootstrap inference procedure for each VAR estimation. The number of bootstrap samples is1000.Note that for computational reasons BC test cannot be computed for k = 1. Besides, its22 − . . . . Lag A C F ACF Euro Area log−GDP − . . . . Lag A C F ACF Euro Area log−M3 − . − . . . . . . . Lag A C F ACF Euro Area log−M1
Figure 10: ACF of GDP, M3 and M1 in logs - Euro Area. − . − . . . . . . . Lag A C F ACF Euro Area HICP rate − . . . . Lag A C F ACF Euro Area UN rate − . . . . . . . Lag A C F ACF Euro Area LTN rate
Figure 11: ACF of HICP, UN, LTN rates - Euro Area.23
15 −5 0 5 10 − . − . . . . . . Lag A C F CCF GDP−M3 −15 −5 0 5 10 − . − . − . . . . . . Lag A C F CCF GDP−M1
Figure 12: CCF GDP-M3 and GDP-M1 - Euro Area.p-value is constant across frequencies (except the last one) for k = 2. BC test requires a largenumber of delays, while ours works for all values, given that the resulting VAR is stationaryand non-singular. Therefore, we can not compare directly our test to BC test on real data,because BC is not useful for all cases with k ≤ We start describing VAR estimates on the couple GDP-M3. Our lag selection procedurechooses 2 lags. In the
GDP t equation, GDP t − and GDP t − are heavily significant, while M t − and M t − slightly are (at 5% and 10% respectively). This results in a GC spectralshape which is approximately constant across frequencies. In the M t equation, M t − isheavily significant, while GDP t − is at 10%. The corresponding GC shape is prominent atlow frequencies.Concerning the couple GDP-M1, our VAR lag selection procedure chooses 2 lags. In the GDP t equation, GDP t − , GDP t − and M t − are heavily significant. The related uncondi-tional GC shape is prominent at low frequencies only. In the M t equation, only M t − isheavily significant, while GDP t − has a p-value of 12%. The resulting GC spectral shape is24hus prominent only at very low frequencies.In Figures 13 and 14, unconditional and conditional GC spectra from M3 to GDP andviceversa are reported. The same spectra from M1 to GDP and viceversa are reported inFigures 15 and 16 respectively. In dashed the bootstrap threshold at 5% is outlined. Indotted, the same threshold for the overall test obtained by Bonferroni correction is depicted. . . . . . . M3 to GDP frequency G r ange r − c au s a li t y s pe c t r u m . . . . M3 to GDP on HICP frequency G r ange r − c au s a li t y s pe c t r u m . . . M3 to GDP on UN frequency G r ange r − c au s a li t y s pe c t r u m . . . . M3 to GDP on LTN frequency G r ange r − c au s a li t y s pe c t r u m Figure 13: GC spectra M3 to GDP . . . . . . . . GDP to M3 frequency G r ange r − c au s a li t y s pe c t r u m . . . . GDP to M3 on HICP frequency G r ange r − c au s a li t y s pe c t r u m . . . . . GDP to M3 on UN frequency G r ange r − c au s a li t y s pe c t r u m . . . . GDP to M3 on LTN frequency G r ange r − c au s a li t y s pe c t r u m Figure 14: GC spectra GDP to M3We first comment conditional GC spectra for the couple GDP-M3. Conditioning on HICP,the level of significance of M t − and M t − is increased in the GDP t equation. This resultsin a GC decreasing across frequencies and prominent across the entire frequency range. Inthe M t equation, the level of significance of GDP t − increases to 5%. As a result, GC isprominent until the period of 1 year. Conditioning on UN, in the GDP t equation the levelof significance of M t − is 5% while M t − is no longer significant. This results in a GC25 .0 0.5 1.0 1.5 2.0 . . . . M1 to GDP frequency G r ange r − c au s a li t y s pe c t r u m . . . . . . . . M1 to GDP on HICP frequency G r ange r − c au s a li t y s pe c t r u m . . . . . . . M1 to GDP on UN frequency G r ange r − c au s a li t y s pe c t r u m . . . . . . . M1 to GDP on LTN frequency G r ange r − c au s a li t y s pe c t r u m Figure 15: GC spectra M1 to GDP . . . . . . GDP to M1 frequency G r ange r − c au s a li t y s pe c t r u m . . . . . GDP to M1 on HICP frequency G r ange r − c au s a li t y s pe c t r u m . . . . . . GDP to M1 on UN frequency G r ange r − c au s a li t y s pe c t r u m . . . . GDP to M1 on LTN frequency G r ange r − c au s a li t y s pe c t r u m Figure 16: GC spectra GDP to M126rominent only across the left half of the frequency range. In the M t equation, GDP t − is no longer significant, resulting in a non-prominent GC everywhere. Conditioning on LTN,the level of significance is 5% for M t − and 10% for M t − in the GDP t equation. Thecorresponding GC is prominent across the entire frequency range. In the M t equation, GDP t − is significant at 5%, causing again GC to be prominent everywhere.We now comment conditional GC spectra for the couple GDP-M1. Conditioning on HICP,in the GDP t equation M t − is still heavily significant. The spectral shape is almost the sameas the unconditional one. In the M t equation, the level of significance is quite smaller, so thatthe only prominent causality is at the lowest frequency. Conditioning on UN, M t − is stillheavily significant in the GDP t equation. The spectral shape is very close to the unconditionalone (even if slightly weaker). In the M t equation, GDP t − has a p-value of 20% and therelated GC shape is close to the unconditional one. Conditioning on LTN, M t − is stillsignificant at 1% in the GDP t equation, causing GC shape to be almost the same as the oneconditioning on UN. In the M t equation, GDP t − has a p-value of 26%. As a consequence,we observe prominence only at the lowest frequency.Concerning the overall test on all causalities, we observe the absence of any significancein four cases out of sixteen: the GC spectra from M3 to GDP, unconditional and conditionalboth on UN and LTN, and the GC spectrum from GDP to M3 conditional on UN. We remarkthat this test is conservative in nature: however, it allows to adequately contextualize thesignificance of individual tests.Finally, GC spectral differences are reported in Figures 17, 18 for the couple M − GDP ,in Figures 19, 20 for the couple M − GDP . From M3 to GDP, we only observe a remarkableamplification power of HICP at low frequencies. UN and LTN show no-causality influenceeven according to the overall test. From GDP to M3, HICP and LTN show amplificationpower at the lowest frequency and annihilation power around the period of 2 years. Onthe contrary, UN amplifies the causal relationship across the left quarter of the frequencyrange. From M1 to GDP, the three conditioning variables show annihilation power at verylow frequencies. From GDP to M1, UN is observed to amplify the link at low frequencies,27 .0 0.5 1.0 1.5 2.0 − . − . . . M3 to GDP difference HICP frequency G r ange r − c au s a li t y s pe c t r u m − . − . . . M3 to GDP difference UN frequency G r ange r − c au s a li t y s pe c t r u m − . − . − . . . . . . M3 to GDP difference LTN frequency G r ange r − c au s a li t y s pe c t r u m Figure 17: GC spectral differences M3 to GDP − . − . − . . . . GDP to M3 difference HICP frequency G r ange r − c au s a li t y s pe c t r u m . . . GDP to M3 difference UN frequency G r ange r − c au s a li t y s pe c t r u m − . − . . . GDP to M3 difference LTN frequency G r ange r − c au s a li t y s pe c t r u m Figure 18: GC spectral differences GDP to M328 .0 0.5 1.0 1.5 2.0 − . − . . . . . . . M1 to GDP difference HICP frequency G r ange r − c au s a li t y s pe c t r u m − . . . . . M1 to GDP difference UN frequency G r ange r − c au s a li t y s pe c t r u m . . . . M1 to GDP difference LTN frequency G r ange r − c au s a li t y s pe c t r u m Figure 19: GC spectral differences M1 to GDP − . − . . . . . . GDP to M1 difference HICP frequency G r ange r − c au s a li t y s pe c t r u m − . − . − . . . GDP to M1 difference UN frequency G r ange r − c au s a li t y s pe c t r u m − . − . − . . . . . GDP to M1 difference LTN frequency G r ange r − c au s a li t y s pe c t r u m Figure 20: GC spectral differences GDP to M129hile the impact of HICP and LTN is not remarkable even according to the overall test.
To sum up, the causal relationship from M3 to GDP is prominent only conditionally on HICP,which appears to be an amplifier, at low frequencies. We can say that conditionally on HICPthe low frequency components of M3 appear to be good predictors of the same componentsof GDP one step ahead. The causal relationship from GDP to M3 is also present at lowfrequencies, except if we condition on UN, which shows a strong annihilation power. On thecontrary, the causality from M1 to GDP is prominent, both unconditionally and conditionallyon HICP, UN and LTN, at all frequencies. The three explanatory variables show a remarkableannihilation power at low frequencies. In the end, the causality from GDP to M1 appearsstrong at low frequencies only. The impact of HICP and LTN on the link can be assumed tobe non-remarkable, while UN shows a remarkable amplification power at very low frequencies.
The motivating application of this paper was the study of the time relationships between M3(M1) aggregate and GDP in the Euro Area. Our ultimate goal was to determine how M3(M1) affects (or is affected by) economic output, both tout court and taking into accounttheir relationship with monetary (inflation rate), economic (unemployment rate) or financial(interest rate) variables.Granger-causality unconditional spectrum analysis turned out to be a very effective tool tofind out the most relevant time delays in the reciprocal dynamics of two variables. This is dueto the fact that, by this frequency-domain tool, we can capture all time delays simultaneously(synthesis power). We can also take into account the latent relationship with some othervariables, computing Granger-causality conditional spectrum.In this context, we have developed a testing procedure which is able to mark up prominentfrequencies, which are frequencies at which the (unconditional or conditional) causalities aresystematically larger than the median causality. A simulation study has shown that our test30an be used as a complementary tool to Breitung and Candelon (2006), since we do not marksignificant causalities but causalities particularly prominent with respect to others. In thisway, we can disambiguate among significant causalities the most prominent ones. In the sameway, we are also able to compare unconditional and conditional spectra detecting prominentcausality differences.Our test has a general validity, as it only requires the stationary bootstrap of Politis and Romano(1994) to be consistent on the data generating process of interest under the hypothesis ofno-causality. Therefore, our procedure may find application in different fields than macroe-conomics, like neuroscience, meteorology, seismology and finance, among others. However,monetary economics is a very suitable application field, as the time series of interest oftenpresent a rich causality structure, and the need rises to disambiguate among significant cyclesthe most prominent ones.From an empirical point of view, we have been able to say that the relationship betweenmoney supply and output is present in the Euro Area across the period 1999-2017. We haveprovided evidence that M3 (M1) in some cases reacts to economic shocks, in some othersit acts as a policy shock with respect to economic output. We have observed that the linkbetween GDP and M1 is much stronger in both directions than the link between GDP andM3. In particular, the causal relationship from M1 to GDP appears to be prominent at allfrequencies, while the opposite one is prominent at low frequencies only.In conclusion, we can say that in the Euro Area money stock cannot be considered anexogenous variable tout-court , since its interrelation with economic output is complex andalso depends on further explanatory variables in a nontrivial way. Nonetheless, the intensityof the causal link from money to output appears to be stronger than the reverse one.31 ppendix
Proof of Theorem 2.1
Let us define the random vector Z t = [ X t , Y t , W t ]. We assume that X t , Y t and Z t are stochas-tically independent, which causes the null hypothesis of no-causality to hold. This is likeassuming that the distribution function F Z can be factorized as F X F Y F W . In addition, weassume that X t , Y t and Z t are strictly stationary.By Politis and Romano (1994) (paragraph 4.3) we know that √ T ( r ( ˆ F X ) − r ( F X )) = 1 √ T T X i =1 h F ( X i ) + o ( √ T || ˆ F X − F X || ) , (4.1)where ˆ F X is the empirical density function of X t and F X is the corresponding true distributionfunction. The same equation holds for Y and Z .If, for some d ≥ E ( h F X ( X )) d < ∞ , and if it holds P k α X ( k ) d d < ∞ , then √ T P Ti =1 h F X ( X i ) is asymptotically normal with mean 0 and variance E ( h F X ( X i ) ) + 2 ∞ X k =1 cov ( h F X ( X ) , h F X ( X k )) . (4.2)The same equation holds for Y and W if E ( h F Y ( Y )) d , α Y ( k ) d d < ∞ , and E ( h F W ( W )) d , P k α W ( k ) d d < ∞ respectively.At the same time, if P k k α X ( k ) / − τ < ∞ , P k k α Y ( k ) / − τ < ∞ , P k k α Z ( k ) / − τ < ∞ for some 0 < τ < /
2, the stochastic processes ˆ F X − F X , ˆ F Y − F Y ,ˆ F W − F W converge in supremum norm to a Gaussian process having continuous paths andmean 0. Therefore, √ T ( r ( ˆ F ) − r ( F )) is asymptotically normal with mean 0 and variance(4.2).Moreover, for each random variable the distribution of √ T ( r ( ˆ F ) − r ( F )) is approximatedvia the distribution of √ T ( r ( ˆ F ∗ ) − r ( ˆ F )), where ˆ F ∗ is the empirical density function obtainedvia stationary bootstrap. This holds because the two distributions converge to the sameGaussian process under previous weak dependence assumptions, provided that T → ∞ .32t this point, since we assumed the stochastic independence of X t , Y t and Z t , the weakdependence assumptions on E ( h F ), α X and P k α ( k ) d d are transmitted to the whole process Z t . Therefore, for any Fr´echet-differentiable functional r we can write P ( r ( ˆ F ∗ Z ) − r ( F ) ≤ q r (1 − α )) = 1 − α under the assumption T → ∞ . R package “grangers”
Our paper is complemented by an R package, called “grangers”, with five functions performingthe calculation of unconditional and conditional Granger-causality spectra, bootstrap infer-ence on both, and inference on the difference between them (see https://github.com/MatFar88/grangers ).The package also contains the data used for the analysis, and two functions performing thetests of Breitung and Candelon (2006) on unconditional and conditional Granger-causalityrespectively.
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