A Portmanteau-type test for detecting serial correlation in locally stationary functional time series
AA PORTMANTEAU-TYPE TEST FOR DETECTING SERIALCORRELATION IN LOCALLY STATIONARY FUNCTIONAL TIMESERIES
AXEL BÜCHER, HOLGER DETTE, AND FLORIAN HEINRICHS
Abstract.
The Portmanteau test provides the vanilla method for detecting serialcorrelations in classical univariate time series analysis. The method is extended tothe case of observations from a locally stationary functional time series. Asymptoticcritical values are obtained by a suitable block multiplier bootstrap procedure. Thetest is shown to asymptotically hold its level and to be consistent against generalalternatives.
Key words:
Autocovariance operator, Block multiplier bootstrap, Functional whitenoise, Time domain test. Introduction
Over the last decades, technological progress allowed to store more and more data.In particular, many time series are recorded with a very high frequency, as for instanceintraday prices of stocks or temperature records. In the literature, data of this type isoften viewed as functional observations. Due to this development, the field of functionaldata analysis has been very active recently (see the monographs Bosq, 2000, Ferraty andVieu, 2006, Horváth and Kokoszka, 2012 and Hsing and Eubank, 2015, among others).The statistical analysis of functional data simplifies substantially if the observationsare serially uncorrelated (or even serially independent). In fact, a huge amount ofmethodology has been proposed solely for this scenario, whence it is important to vali-date or reject this assumption in applications. Moreover, in the context of (univariate)financial return data, the absence or insignificance of serial correlation is commonly in-terpreted as a sign for efficient market prices (Fama, 1970). Likewise, investors may beinterested in knowing whether functional counterparts like cumulative intraday returnsexhibit significant autocorrelation.If the observations are not only serially uncorrelated, but also centred and homoscedas-tic, then the time series is referred to as a functional white noise. Testing for thefunctional white noise hypothesis has found considerable interest in the recent litera-ture. For instance, inspired by classical portmanteau-type methodology in univariateor multivariate time series analysis (see Box and Pierce, 1970; Hosking, 1980; Hong,
Date : September 17, 2020.
Corresponding author:
Axel Bücher. a r X i v : . [ m a t h . S T ] S e p AXEL BÜCHER, HOLGER DETTE, AND FLORIAN HEINRICHS relevant serial correlations, see Section 3.2 for a rigorousdefinition.The paper is organized as follows: mathematical preliminaries, including a precisedescription of the hypotheses, are collected in Section 2. Suitable test statistics areintroduced in Section 3, where we also prove weak convergence and validate a bootstrapapproximation to obtain suitable critical values. Finite sample results are collected inSection 4, a case study is presented in Section 5 and all proofs are deferred to Section 6.2.
Mathematical Preliminaries
Throughout this document, we deal with objects in L p ([0 , d ), for different choicesof p ≥ d ∈ N . We denote the respective L p -norms by k · k p,d , with the specialcase k · k p, abbreviated by k · k p . Further, for functions f, g ∈ L p ([0 , f ⊗ g )( x, y ) = f ( x ) g ( y ).2.1. Locally stationary time series.
For t ∈ Z , let X t : [0 , × Ω → R denotea ( B| [0 , ⊗ A )-measurable function with X t ( · , ω ) ∈ L ([0 , ω ∈ Ω. Wecan regard [ X t ] as a random variable in L ([0 , X t as well.The expected value of [ X t ] in L ([0 , τ µ t ( τ ) = E [ X t ( τ )]. Similarly, the kernel of the (auto-)covariance operator of [ X t ] has ETECTING SERIAL CORRELATION IN LSFTS 3 a representation in L ([0 , ) with c X t ( τ, σ ) = Cov (cid:0) X t ( τ ) , X t ( σ ) (cid:1) and c X t ,X t + h ( τ, σ ) =Cov (cid:0) X t ( τ ) , X t + h ( σ ) (cid:1) . We refer to Section 2.1 in Bücher et al. (2020) for technical details.The sequence ( X t ) t ∈ Z is called a functional time series in L ([0 , stationary if, for all q ∈ N and h, t , . . . , t q ∈ Z ,( X t + h , . . . , X t q + h ) d = ( X t , . . . , X t q )in L ([0 , q . For the definition of a locally stationary functional time series we usea concept introduced by Vogt (2012) and van Delft and Eichler (2018) (see also vanDelft et al., 2020; van Delft and Dette, 2020; Bücher et al., 2020). To be precise wecall a sequence of functional time series ( X t,T ) t ∈ Z indexed by T ∈ N a locally stationaryfunctional time series of order ρ > u ∈ [0 , X ( u ) t ) t ∈ Z in L ([0 , { P ( u ) t,T : t = 1 , . . . , T, T ∈ N } with E | P ( u ) t,T | ρ < ∞ uniformly in t ∈ { , . . . , T } , T ∈ N and u ∈ [0 , k X t,T − X ( u ) t k = (cid:26) Z { X t,T ( τ ) − X ( u ) t ( τ ) } d τ (cid:27) / ≤ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) tT − u (cid:12)(cid:12)(cid:12)(cid:12) + 1 T (cid:19) P ( u ) t,T , for any t ∈ { , . . . , T } , T ∈ N and u ∈ [0 , ρ ≥ { ( X ( u ) t ) t ∈ Z : u ∈ [0 , } is L -Lipschitz continuous in the sense that E k X ( u ) t − X ( v ) t k ≤ C | u − v | , (2.1)for some constant C >
0, by local stationarity of X t,T . In the following discussionwe assume that X t,T (and hence X ( u ) t ) is centred, i. e. µ t,T = E [ X t,T ] = 0 for all t ∈ { , . . . , T } .2.2. Serial correlation in locally stationary time series.
In classical (functional)time series analysis, a time series is called uncorrelated if its autocovariances are zerofor any lag h >
0. In the locally stationary setup, a slightly more subtle version suggestsitself: we call a centred locally stationary functional time series of order ρ ≥ { ( X ( u ) t ) t ∈ Z : u ∈ [0 , } (i. e., E [ k X ( u ) k ] < ∞ for all u ) serially uncorrelated if the hypothesis¯ H := H (1)0 ∩ H (2)0 ∩ . . . (2.2)holds, where the individual hypothesis H ( h ) at lag h ∈ N is defined by H ( h )0 : k Cov( X ( u )0 , X ( u ) h ) k , = 0 for all u ∈ [0 , . (2.3)If, additionally, u Var( X ( u ) ) is constant, then the locally stationary time series will becalled functional white noise. As in Remark 1 in Bücher et al. (2020), it may be shownthat these definitions are independent of the choice of the approximating family.Throughout this paper, we will develop suitable tests for certain hypotheses relatedto ¯ H and H ( h ) in (2.2) and (2.3), respectively. Following the main principle of classical AXEL BÜCHER, HOLGER DETTE, AND FLORIAN HEINRICHS portmanteau-type tests for detecting serial correlations, we start by fixing a maximumlag H ∈ N and to test the hypotheses¯ H ( H )0 : k Cov( X ( u )0 , X ( u ) h ) k , = 0 for all h ∈ { , . . . , H } and u ∈ [0 , . (2.4)Note that ¯ H = T H ∈ N ¯ H ( H ) .2.3. Regularity conditions on the observation scheme.
In order to obtain mean-ingful asymptotic results, the following regularity conditions will be imposed.
Condition 2.1 (Assumptions on the observations) . (A1) Local Stationarity.
The observations X ,T , . . . X T,T are an excerpt from acentered locally stationary functional time series { ( X t,T ) t ∈ Z : T ∈ N } of order ρ =4 in L ([0 , , R ), with approximating family of stationary time series { ( X ( u ) t ) t ∈ Z : u ∈ [0 , } .(A2) Moment Condition.
For any k ∈ N , there exists a constant C k < ∞ such that E k X t,T k k ≤ C k and E k X ( u ) k k ≤ C k uniformly in t ∈ Z , T ∈ N and u ∈ [0 , Cumulant Condition.
For any j ∈ N there is a constant D j < ∞ such that ∞ X t ,...,t j − = −∞ (cid:13)(cid:13) cum( X t ,T , . . . , X t j ,T ) (cid:13)(cid:13) ,j ≤ D j < ∞ , for any t j ∈ Z (for j = 1 the condition is to be interpreted as k E X t ,T k ≤ D for all t ∈ Z ). Further, for k ∈ { , , } , there exist functions η k : Z k − → R satisfying ∞ X t ,...,t k − = −∞ (1 + | t | + · · · + | t k − | ) η k ( t , . . . , t k − ) < ∞ such that, for any T ∈ N , ≤ t , . . . , t k ≤ T, v, u , . . . , u k ∈ [0 , , h , h ∈ Z , Z ( u ) t,T ∈ { X t,T , X ( u ) t } , and any Y t,h,T ( τ , τ ) ∈ { X t,T ( τ ) , X t,T ( τ ) X t + h,T ( τ ) } , wehave(i) k cum( X t ,T − X ( t /T ) t , Z ( u ) t ,T , · · · , Z ( u k ) t k ,T ) k ,k ≤ T η k ( t − t , . . . , t k − t ) ,(ii) k cum( X ( u ) t − X ( v ) t , Z ( u ) t ,T , · · · , Z ( u k ) t k ,T ) k ,k ≤ | u − v | η k ( t − t , . . . , t k − t ) ,(iii) k cum( X t ,T , . . . , X t k ,T ) k ,k ≤ η k ( t − t , · · · , t k − t ) ,(iv) R [0 , | cum (cid:0) Y t ,h ,T ( τ ) , Y t ,h ,T ( τ ) (cid:1) | d τ ≤ η ( t − t ). Assumption (A1) restricts the non-stationary behaviour of the observations to smoothchanges, while the moment condition ensures existence of the cumulants. The cumulantcondition originates from classical multivariate time series analysis (see, e. g., Brillinger,1981). Similar assumptions were made by Lee and Rao (2017) and Aue and van Delft(2020) in the context of non-stationary functional data. Lemma 2 in Bücher et al. (2020)shows that (A3) follows from (A1), (A2) and an additional moment condition, providedthat a certain strong mixing condition is met.
ETECTING SERIAL CORRELATION IN LSFTS 5 Testing for serial correlation in locally stationary functional data
A test statistic for detecting serial correlation.
In this section, we propose atest statistic for detecting deviations from hypothesis (2.4) and prove a correspondingweak convergence result. A bootstrap device for deriving suitable critical values will bediscussed in the subsequent Section 3.3.The test statistic is based on the following observation: as X ( u ) t is centred we mayrewrite (observing (2.1)) hypotheses (2.3) and (2.4) as H ( h )0 : k M h k , = 0 and ¯ H ( H )0 : H max h =1 k M h k , = 0 , where M h ( u, τ , τ ) = Z u E [ X ( w )0 ( τ ) X ( w ) h ( τ )] d w. An empirical version of M h , based on the observations X ,T , . . . , X T,T , is provided bythe statistic ˆ M h,T ( u, τ , τ ) = 1 T b uT c∧ ( T − h ) X t =1 X t,T ( τ ) X t + h,T ( τ ) . The next theorem implies consistency of the empirical versions, which suggests to rejectthe null hypotheses in (2.3) and (2.4) for large values of the statistics S h,T = √ T k ˆ M h,T k , and ¯ S H,T = √ T H max h =1 k ˆ M h,T k , , respectively. Theorem 3.1.
Under Condition 2.1, we have, for any h ∈ N as T → ∞ √ T S h,T → k M h k , in probability. Moreover, for any H ∈ N , h ∈ { , . . . , H } as T → ∞S h,T (cid:32) k ˜ B h k , under H ( h )0 , + ∞ else , and ¯ S H,T (cid:32) H max h =1 k ˜ B h k , under ¯ H ( H )0 , + ∞ else , where ˜ B = ( ˜ B , . . . , ˜ B H ) denotes a centred Gaussian variable in L ([0 , ) H , whosecovariance operator C B : L ([0 , ) H → L ([0 , ) H is defined by C B f ... f H ( u , τ , τ ) ... ( u H , τ H , τ H ) = P Hh =1 h r ,h (( u , τ , τ ) , · ) , f h i ... P Hh =1 h r H,h (( u H , τ H , τ H ) , · ) , f h i . (3.1) Here, the kernel function r h,h is given by r h,h (( u, τ , τ ) , ( v, ϕ , ϕ )) = Cov (cid:0) ˜ B h ( u, τ , τ ) , ˜ B h ( v, ϕ , ϕ ) (cid:1) = ∞ X k = −∞ Z u ∧ v c k ( w ) d w, (3.2) AXEL BÜCHER, HOLGER DETTE, AND FLORIAN HEINRICHS with c k ( w ) = c k ( w, h, h , τ , τ , ϕ , ϕ ) = Cov (cid:0) X ( w )0 ( τ ) X ( w ) h ( τ ) , X ( w ) k ( ϕ ) X ( w ) k + h ( ϕ ) (cid:1) , for any ≤ h, h ≤ H . In particular, the infinite sum in (3.2) converges. It is worthwhile to mention that the distributions of the limiting variables in the previ-ous theorems are not pivotal under the null hypotheses. As a consequence, critical valuesfor respective statistical tests must be estimated, for instance by a plug-in approach orby a suitable bootstrap device. Throughout this paper, we propose a bootstrap approachwhich will be worked out in Section 3.3 below.3.2.
Detecting relevant serial correlations.
In the previous section, we considered“classical” hypotheses in the sense that we were testing whether the covariance operatorsup to lag H are exactly equal to zero. However, in concrete applications, hypotheses ofthis type might rarely be satisfied exactly and it might rather be reasonable to refor-mulate the null hypothesis in the form that “the norm of the autocovariance operatoris small”, but not exactly equal to 0. More precisely, given thresholds ∆ h > h ∈ { , . . . , H } , we propose to consider the following relevanthypotheses H ( h, ∆)0 : k M h k , ≤ ∆ h , ¯ H ( H, ∆)0 : k M h k , ≤ ∆ h for all h ∈ { , . . . , H } , (3.3)where H ∈ N is some fixed constant representing the maximal lag under consideration.The choice of the thresholds ∆ h depends on the specific application and has to be dis-cussed with the practitioner in concrete applications. Although this may be a dauntingtask, we strongly argue that one should carefully think about it as the classical im-plicit choice of ∆ h = 0 typically corresponds to an unrealistic null hypothesis in manyapplications.Consistency of ˆ M h,T for M h suggests to reject the above hypotheses for large valuesof ˆ M h,T . We propose to consider the “normalized” test statistics S h, ∆ h ,T = √ T ( k ˆ M h,T k , − ∆ h ) k ˆ M h,T k , , ¯ S H, ∆ ,T = H max h =1 √ T ( k ˆ M h,T k , − ∆ h ) k ˆ M h,T k , , whose asymptotic properties are described in the following result. It is worthwhile tomention that related test statistics like √ T ( k ˆ M h,T k , − ∆ h ) or √ T ( k ˆ M h,T k , − ∆ h )∆ h may be treated similarly, but that the respective tests exhibited a worse finite-sampleperformance in an unreported Monte-Carlo simulation study. Corollary 3.2.
Under Condition 2.1, we have, for any fixed H ∈ N and for T → ∞ , √ T (cid:0) ( k ˆ M h,T k , − k M h k , ) k ˆ M h,T k , (cid:1) h =1 ,...,H (cid:32) (cid:0) h M h , ˜ B h i (cid:1) h =1 ,...,H , ETECTING SERIAL CORRELATION IN LSFTS 7 where ˜ B , . . . , ˜ B H are defined in Theorem 3.1 and h f, g i = R [0 , f ( x ) g ( x ) d x . As aconsequence, S h, ∆ h ,T (cid:32) − ∆ h k ˜ B h k , if k M h k , = 0 , −∞ if k M h k , ∈ (0 , ∆ h ) , h M h , ˜ B h i if k M h k , = ∆ h , + ∞ if k M h k , > ∆ h . Moreover, ¯ S H, ∆ ,T (cid:32) ( max { max h ∈ N H h M h , ˜ B h i , max h ∈ O h − ∆ h k ˜ B h k , } if ¯ H ( H, ∆)0 is met , + ∞ else , where N H = { h ∈ { , . . . , H } : k M h k , = ∆ h } , O H = { h ∈ { , . . . , H } : k M h k , = 0 } and where the maximum over the empty set is interpreted as −∞ . As in Section 3.1, the limiting distributions under the null hypotheses are not pivotal,whence a bootstrap procedure will be introduced next.3.3.
Critical values based on bootstrap approximations.
The limiting distribu-tions of the test statistics derived in the previous sections depend in a complicatedway on the higher order serial dependence of the underlying approximating family { ( X ( u ) t ) t ∈ Z : u ∈ [0 , } and are rather difficult to estimate. To avoid the estimation,we propose a multiplier block bootstrap procedure.Following Bücher et al. (2020) the bootstrap scheme will be defined in terms of i.i.d.standard normally distributed random variables { R ( k ) i } i,k ∈ N which are independent of { ( X t,T ) t ∈ Z : T ∈ N } . Further, let m = m T and n = n T denote two block lengthsequences satisfying one of the following two conditions. Condition 3.3. (B1) The block length m = m T ∈ { , . . . , T } tends to infinity and satisfies m = o ( T )as T → ∞ .(B2) The block length n = n ( T ) ∈ { , . . . , T } satisfies m/n = o (1) and mn = o ( T )as T → ∞ .Next, let K ∈ N denote the number of bootstrap replications. For k ∈ { , . . . , K } and h ∈ { , . . . , H } , define multiplier bootstrap approximations for B h,T ( u, τ , τ ) = √ T { ˆ M h,T ( u, τ , τ ) − M h ( u, τ , τ ) } as ˆ B ( k ) h,n,T ( u, τ , τ ) = 1 √ T b uT c∧ ( T − h ) X i =1 R ( k ) i √ m ( i + m − ∧ ( T − h ) X t = i (cid:8) X t,T ( τ ) X t + h,T ( τ ) − ˆ µ t,h,n,T ( τ , τ ) (cid:9) , AXEL BÜCHER, HOLGER DETTE, AND FLORIAN HEINRICHS where ˆ µ t,h,n,T ( τ , τ ) = 1˜ n t,h ¯ n t,h X j =¯ n t X t + j,T ( τ ) X t + j + h,T ( τ )denotes the local empirical product moment of lag h with¯ n t,h = n ∧ ( T − t − h ) , ¯ n t = − n ∨ (1 − t ) , ˜ n t,h = ¯ n t,h − ¯ n t + 1 . Note that for n = T we obtain ˆ µ t,h,T,T = ˆ µ h,T for all t ∈ { , . . . , T } , whereˆ µ h,T ( τ , τ ) = 1 T − h T − h X t =1 X t,T ( τ ) X t + h,T ( τ )denotes the global empirical product moment. Let ˆ B ( k ) n,T = ( ˆ B ( k ) ,n,T , . . . , ˆ B ( k ) H,n,T ) andˆ B T = √ T (cid:0) B ,T , . . . , B H,T (cid:1) . The following result shows that this multiplier bootstrap isconsistent.
Theorem 3.4.
Suppose that Condition 2.1 is met and let ˜ B (1) , ˜ B (2) , . . . denote inde-pendent copies of ˜ B . Fix K, H ∈ N .(i) If Condition 3.3 (B1) and (B2) are met, then, as T → ∞ , (ˆ B T , ˆ B (1) n,T , . . . , ˆ B ( K ) n,T ) (cid:32) ( ˜ B, ˜ B (1) , . . . , ˜ B ( K ) ) . (ii) If Condition 3.3 (B1) is met and if Cov( X (0) , X (0) h ) = Cov( X ( w ) , X ( w ) h ) for any w ∈ [0 , and h ∈ Z , then, as T → ∞ , (ˆ B T , ˆ B (1) T,T , . . . , ˆ B ( K ) T,T ) (cid:32) ( ˜ B, ˜ B (1) , . . . , ˜ B ( K ) ) . It is worthwhile to mention that the assumption on Cov( X ( w ) , X ( w ) h ) in Theorem 3.4(ii)is met provided that X t,T = X t for some stationary time series ( X t ) t ∈ Z . In such asituation (for instance to be validated by a stationarity test in practice), using thebootstrap scheme with n = T over the one with n satisfying Condition 3.3 (B2) typicallyresults in better finite sample results, see Section 4 for more details.Subsequently, we reconsider the problem of testing for serial uncorrelation of a locallystationary time series using classical and relevant hypotheses. For the sake of brevity, weonly treat the hypotheses ¯ H ( H ) and ¯ H ( H, ∆) , which are defined in (2.4) and (3.3), respec-tively and involve multiple lags. For this purpose we consider the following bootstrapapproximations of the respective test statistics¯ S ( k ) H,n,T = H max h =1 k ˆ B ( k ) h,n,T k , for the classical hypotheses and¯ S ( k ) H,n,T, rel = H max h =1 h ˆ M h,T , ˆ B ( k ) h,n,T i ETECTING SERIAL CORRELATION IN LSFTS 9 for the relevant hypotheses. Finally, we propose to reject the classical hypothesis (2.4)whenever ¯ p H,n,K,T = 1 K K X k =1 (cid:16) ¯ S ( k ) H,n,T ≥ ¯ S H,T (cid:17) < α . (3.4)Similarly, the relevant hypothesis (3.3) is rejected whenever¯ p H,n,K,T, rel = 1 K K X k =1 (cid:16) ¯ S ( k ) H,n,T, rel ≥ ¯ S H, ∆ ,T (cid:17) < α. (3.5) Corollary 3.5.
Fix α ∈ (0 , , suppose that Condition 2.1 is met and let K = K T → ∞ .(i) If Condition 3.3 (B1) and (B2) hold, then the decision rule (3.4) defines a con-sistent asymptotic level α test for the classical hypotheses (2.4) , that is lim T →∞ P (¯ p H,n,K,T < α ) = α under ¯ H ( H ) , else.Similarly, for α < / , the decision rule (3.5) for the relevant hypotheses (3.3) satisfies lim T →∞ P (¯ p H,n,K,T, rel < α ) = 0 if k M h k , < ∆ h for all h ∈ { , . . . , H } , lim sup T →∞ P (¯ p H,n,K,T, rel < α ) ≤ α if ¯ H ( H, ∆)0 ∩ R is met, (3.6)lim T →∞ P (¯ p H,n,K,T, rel < α ) = 1 else , where R denotes the set of all models from the null hypothesis ¯ H ( H, ∆)0 for which k M h k , = ∆ h for some h ∈ { , . . . , H } and for which Var( h M h , ˜ B h i ) > for eachsuch h . In (3.6) , the value α is attained if k M h k , = ∆ h for all h ∈ { , . . . , H } .(ii) If Condition 3.3 (B1) is met and if Cov( X (0) , X (0) h ) = Cov( X ( w ) , X ( w ) h ) for any w ∈ [0 , and h ∈ N , then the same assertions as in (i) are met for n = T . The restriction to α < / h ∈ O H − ∆ h k ˜ B h k , in Corollary 3.2 is negligible (see Section 6for details). 4. Monte Carlo Simulations
A large scale Monte Carlo simulation study was performed to analyse the finite-sample properties of the proposed tests. The major goal of the study was to analysethe level approximation and the power of the tests for hypotheses of the form ¯ H ( H ) and¯ H ( H, ∆) , with H ∈ { , . . . , } . Moreover, we also provide a comparison with existing testsfor white noise / no serial correlation in the stationary setup, both for tests in the timedomain (Kokoszka et al., 2017) and in the frequency domain (Zhang, 2016; Bagchi et al.,2018; Characiejus and Rice, 2020). Models.
We start by employing the same (stationary) models as in Zhang (2016)and Bagchi et al. (2018). In particular, for the null hypothesis of serial uncorrelationfor any lag h , we consider: Model (N ), an i.i.d. sequence of Brownian motions; Model(N ), an i.i.d. sequence of Brownian bridges; and Model (N ), data from a FARCH(1)process defined by X t ( τ ) = B t ( τ ) s τ + Z c ψ exp (cid:16) τ + σ (cid:17) X t − ( σ ) d σ, where ( B t ) t ∈ Z denotes an i.i.d. sequence of Brownian motions and c ψ = 0 . X t = ρ ( X t − − µ ) + ε t , where ρ denotes an integral operator ρ ( f ) = R K ( · , σ ) f ( σ ) d σ, f ∈ L ([0 , K ∈ L ([0 , ) and a sequence of centred, i.i.d. innovations ( ε t ) t ∈ Z in L ([0 , K and ε t :(A ) K ( τ, σ ) = c g exp (cid:0) ( τ + σ ) / (cid:1) , ε t i.i.d. Brownian motions , (A ) K ( τ, σ ) = c g exp (cid:0) ( τ + σ ) / (cid:1) , ε t i.i.d. Brownian bridges , (A ) K ( τ, σ ) = c w min( τ, σ ) , ε t i.i.d. Brownian motions , (A ) K ( τ, σ ) = c w min( τ, σ ) , ε t i.i.d. Brownian bridges , where c g and c w are chosen such that the Hilbert-Schmidt norm of the ρ is 0.3.Note that the above models are stationary. Since our proposed methodology allowsfor smooth changes in the distribution of the underlying stochastic processes as well, weadditionally consider the following heteroscedastic locally stationary models:(N ) X t,T = σ ( t/T ) B t , (A ) X t,T = ρ ( X t − ,T ) + σ ( t/T ) B t , (A ) X t,T = σ ( t/T ) ρ ( X t − ,T ) + B t , where ( B t ) t ∈ Z denotes an i.i.d. sequence of Brownian motions, σ ( x ) = x + 1 / ρ isdefined as in model (A ). For model (N ), the null hypothesis holds true, whereas thealternative is true for models (A ) and (A ).4.2. Details on the implementation.
For the comparison with the tests by Zhang(2016) and Bagchi et al. (2018) (results in Table 1) and the evaluation of the finite-sampleproperties under non-stationarity (results in Tables 3 and 5), the data was simulated onan equidistant grid of size 1000 on the interval [0 , wwntests by Petoukhov (2020). ETECTING SERIAL CORRELATION IN LSFTS 11Model ¯ H (1)0 ¯ H (2)0 ¯ H (3)0 ¯ H (4)0 (B) (Z) Panel A: T = 128(N ) 7.3 6.3 5.9 5.6 1.8 4.2(N ) 4.9 4.4 4.2 4.2 1.1 5.4(N ) 5.4 4.5 4.2 3.8 4.7 5.9(A ) 99.8 99.5 99.3 99.1 66.5 83.7(A ) 98.4 97.9 97.1 96.5 51.7 83.1(A ) 99.8 99.7 99.7 99.7 84.9 68.3(A ) 91.8 88.5 85.7 82.7 37.0 65.8 Panel B: T = 256(N ) 5.3 6.1 5.9 5.1 1.9 4.2(N ) 4.4 5.3 4.8 4.2 1.4 4.8(N ) 5.0 4.4 4.0 4.0 6.0 5.2(A ) 100.0 100.0 100.0 100.0 91.5 99.2(A ) 99.9 99.9 99.9 99.9 84.4 99.5(A ) 100.0 100.0 100.0 100.0 99.1 98.2(A ) 99.6 99.4 99.3 99.2 65.9 99.1 ¯ H (1)0 ¯ H (2)0 ¯ H (3)0 ¯ H (4)0 (B) (Z) Panel C: T = 5126.1 5.4 6.0 5.1 2.8 4.75.9 4.8 4.6 4.8 1.9 5.94.3 5.0 4.7 4.0 6.3 4.9100.0 100.0 100.0 100.0 99.3 99.5100.0 100.0 100.0 100.0 98.3 99.8100.0 100.0 100.0 100.0 100.0 98.7100.0 100.0 100.0 100.0 90.4 100.0 Panel D: T = 10245.5 6.3 5.7 5.4 3.5 4.95.5 5.7 5.3 5.4 3.5 5.14.3 3.9 3.7 3.6 7.6 4.8100.0 100.0 100.0 100.0 100.0 100.0100.0 100.0 100.0 100.0 99.9 99.8100.0 100.0 100.0 100.0 100.0 100.0100.0 100.0 100.0 100.0 99.6 100.0 Table 1.
Empirical rejection rates of test (3.4) for the classical hypothe-ses (2.4) in the case of stationary models, for various values of the max-imal lag H in ¯ H ( H ) . The columns denoted by (B) and (Z) correspond tothe tests of Bagchi et al. (2018) and Zhang (2016), respectively. For computational reasons, we reduced the dimension by projecting the generateddata onto the subspace of L ([0 , D = 17 functions of the Fourierbasis { ψ n } n ∈ N , where, for n ∈ N , ψ ≡ , ψ n − ( τ ) = √ πnτ ) , ψ n ( τ ) = √ πnτ )to calculate the proposed test statistic.For the calculation of the bootstrap quantiles, we employed the data driven choiceof the block length m explained in Bücher et al. (2020). In the context of stationaryprocesses (models (N )–(N ) and (A )–(A )), it is natural to consider global estimatorsin the bootstrap procedure and we chose the bandwidth n = T . In fact, preliminarysimulations suggested that this choice of n leads to better finite sample behavior. For thenon-stationary models however, this choice is not reasonable and we used local estimatorsin order to avoid a possible bias. In this setting, we chose the bandwidth n = b T / c ,satisfying Condition 3.3 (B2). The number of bootstrap replicates was chosen as 200and each model was simulated 1000 times.4.3. Results for the classical hypotheses.
In the following, we denote by (B) and(Z) the tests proposed by Bagchi et al. (2018) and Zhang (2016), respectively. (M H ), H ∈ { , , } , denotes the multiple-lag test at lag H proposed by Kokoszka et al. (2017).Finally, (Spec s ) and (Spec a ) denote the spectral test as proposed by Characiejus andRice (2020), with static and adaptive bandwidth, respectively. The empirical rejectionrates of test (3.4) for the stationary models (N )–(N ) and (A )–(A ) are shown in Model ¯ H (1)0 ¯ H (2)0 ¯ H (3)0 ¯ H (4)0 (M ) (M ) (M ) (Spec s ) (Spec a ) Panel A: T = 100(N ) 6.7 6.0 6.2 5.7 5.4 4.8 6.7 5.1 5.7(N ) 5.1 4.5 4.8 4.4 3.0 3.5 3.7 4.3 5.0(N ) 5.3 5.8 4.9 5.2 4.3 4.4 5.4 18.8 21.8(A ) 97.5 96.3 95.4 94.6 96.6 92.3 88.0 100.0 99.7(A ) 95.3 92.7 91.2 89.3 89.9 78.8 69.1 99.8 98.9 Panel B: T = 200(N ) 4.5 4.3 4.6 4.3 5.4 5.4 5.4 4.6 5.5(N ) 4.6 5.1 4.3 4.0 3.1 3.4 3.8 4.7 4.6(N ) 3.2 3.2 3.2 3.2 5.1 5.3 6.3 26.2 28.8(A ) 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0(A ) 100.0 99.9 99.9 99.8 100.0 100.0 98.7 100.0 100.0 Panel C: T = 300(N ) 5.7 4.9 4.5 4.8 6.0 4.4 6.9 4.7 5.8(N ) 5.3 4.5 4.1 4.4 5.3 5.2 4.6 5.2 7.3(N ) 4.6 4.0 3.7 3.6 5.4 5.3 6.1 25.3 28.8(A ) 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0(A ) 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 Table 2.
Empirical rejection rates of test (3.4) for the classical hypothe-ses (2.4) in the case of stationary models, for various values of the max-imal lag H in ¯ H ( H ) . The columns denoted by (M i ) , i ∈ { , , } , and (Spec i ) , i ∈ { a, s } , correspond to the tests of Kokoszka et al. (2017) andCharaciejus and Rice (2020), respectively. Tables 1 and 2. We observe that the level approximation of the new test (3.4) is veryaccurate for all scenarios under consideration, and that the power is larger than for thecompetitors from the literature, in particular for small samples. A partial explanation forthis observation consists in the fact that tests based in frequency domain formulate thewhite noise hypothesis in terms of the spectral density operator and therefore implicitlyconsider the auto-covariance operators at any lag h . Although the power of test (3.4)slightly decreases with increasing H , it decreases slower than the power of the multiple-lag time domain test by Kokoszka et al. (2017). The type I errors of the tests (Spec s ) and(Spec a ) seem to exceed the level of 5% for model (N ). This difficulty might arise fromthe fact that the data is uncorrelated but dependent. In contrast, the level approximationof the proposed tests seems to be more accurate.The empirical rejection rates of test (3.4) for the locally stationary models (N ),(A ) and (A ) are shown in Table 3, for different sample sizes. We observe a reasonableapproximation of the nominal level and high power under the non-stationary alternatives.4.4. Results for relevant hypotheses.
We conclude this section with a brief discus-sion of the performance of the proposed test (3.5) for the relevant hypotheses (3.3). For
ETECTING SERIAL CORRELATION IN LSFTS 13
Model ¯ H (1)0 ¯ H (2)0 ¯ H (3)0 ¯ H (4)0 Panel A: T = 128(N ) 6.5 6.2 5.5 5.5(A ) 99.0 98.6 98.6 98.6(A ) 99.4 98.6 98.4 98.1 Panel B: T = 256(N ) 7.3 5.7 5.5 4.9(A ) 100.0 100.0 100.0 100.0(A ) 100.0 100.0 100.0 100.0 Model ¯ H (1)0 ¯ H (2)0 ¯ H (3)0 ¯ H (4)0 Panel C: T = 512(N ) 6.5 6.8 5.6 4.9(A ) 100.0 100.0 100.0 100.0(A ) 100.0 100.0 100.0 100.0 Panel D: T = 1024(N ) 6.9 6.5 6.5 5.0(A ) 100.0 100.0 100.0 100.0(A ) 100.0 100.0 100.0 100.0 Table 3.
Empirical rejection rates of test (3.4) for the classical hypothe-ses (2.4) in the case of locally stationary models, for various values forthe maximal lag H in ¯ H ( H ) . Model h = 1 h = 2 h = 3 h = 4(A ) 0.1419 ( . · − ) 0.0689 ( . · − ) 0.0336 ( . · − ) 0.0169 ( . · − )(A ) 0.0283 ( . · − ) 0.0138 ( . · − ) 0.0069 ( . · − ) 0.0037 ( . · − )(A ) 0.1996 ( . · − ) 0.1220 ( . · − ) 0.0755 ( . · − ) 0.0468 ( . · − )(A ) 0.0235 ( . · − ) 0.0117 ( . · − ) 0.0070 ( . · − ) 0.0048 ( . · − ) Table 4.
Theoretical values of k M h k , , obtained by simulation. Thenumbers in brackets correspond to the empirical variance of the simula-tion. this purpose we have calculated the quantities k M h k , for the models (A )–(A ) by a nu-merical simulation (specifically, we simulated 10 ,
000 time series of length T = 2 , D = 101, calculated for each time series thequantity k ˆ M h,T k , for h ∈ { , . . . , } , and used the respective means as an approximationfor k M h k ). The results can be found in Table 4. For the simulation experiment, we chosehypotheses corresponding to ∆ = ∆ h,w = w k M h k , with w ∈ { . i/
10 : i = 1 , . . . , } and h = 1 , . . . ,
4, such that the null hypotheses are met for w ≥ w <
1. The results can be found in Table 5, where we omit theresults for H ∈ { , } since they are qualitatively similar to the cases H ∈ { , } . Again,we observe convincing level approximations and good power properties.5. Case Study
Functional data arises naturally when time series are recorded with a very high fre-quency. To illustrate the proposed methodology, we consider intraday prices of variousstocks. More specifically, we consider prices over the time span from February 2016to January 2020, where each observation corresponds to the intraday price at a givenday. In particular, let P t ( x j ), t ∈ { , . . . , T } , j ∈ { , . . . , m } denote the price of a share, Model H \ w Panel A: T = 128(A ) 1 58.1 39.4 25.2 13.2 8.1 ) 1 72.7 50.3 31.0 16.7 8.8 ) 1 57.9 40.1 22.6 13.2 6.9 ) 1 61.0 42.3 27.2 15.7 7.7 Panel B: T = 256(A ) 1 87.4 66.4 42.6 21.8 10.7 ) 1 91.7 75.7 51.9 28.4 12.5 ) 1 87.1 65.5 41.1 20.8 8.9 ) 1 82.8 64.0 41.9 22.8 12.6 Panel C: T = 512(A ) 1 97.4 88.6 68.2 39.8 16.6 ) 1 99.7 95.2 73.5 40.2 16.3 ) 1 98.1 88.2 64.1 37.0 13.4 ) 1 97.3 87.1 60.1 32.9 14.2 Panel D: T = 1024(A ) 1 100.0 99.7 89.6 60.3 23.8 ) 1 100.0 99.8 93.9 64.8 25.9 ) 1 100.0 99.6 87.0 53.8 18.6 ) 1 100.0 98.2 85.5 52.5 21.2 Table 5.
Empirical rejection rates of the test (3.5) for the relevant hy-potheses (3.3) in the case of stationary models. observed at time points x j at day t . The lengths T of the considered time series dependon the different stocks as for some days observations are missing.Gabrys et al. (2010) define intradaily cumulative returns as R t ( x j ) = 100 { log P t ( x j ) − log P t ( x ) } , j ∈ { , . . . , m } , t ∈ { , . . . , T } . ETECTING SERIAL CORRELATION IN LSFTS 15 − − − − Boeing − Intradaily cumulative returns − Blackrock − Intradaily cumulative returns
Figure 1.
Intradaily cumulative returns of Boeing and Blackrock from8th to 12th of February 2016, where the x-axis corresponds to rescaledtime and the y-axis denotes returns.
Stock ¯ H (1)0 ¯ H (2)0 ¯ H (3)0 ¯ H (4)0 T Bank of America 80.6 80.1 79.7 79.4 982Blackrock 98.4 98.3 98.3 98.3 822Boeing 82.3 81.9 81.4 81.1 984Goldman Sachs 74.5 73.9 73.6 73.3 990JP Morgan 93.3 93.2 93.0 92.9 982
Table 6. p -values of the (combined) tests for the respective null hypothe-ses in percent. Throughout, we consider R t ( · ) as an L -function. Some exemplary intradaily cumulativereturn curves are displayed in Figure 1. The results of our testing procedure for detectingpossible serial correlations can be found in Table 6, where we employed K = 1000bootstrap replicates and considered up to H = 4 lags. The null hypotheses of serialcorrelation cannot be rejected at level α = 0 .
05, as the p -values clearly exceed α . Thus,our results match the common assumption of uncorrelatedness in the literature.6. Proofs
Proof of Theorem 3.1.
We prove that for any H ∈ N and as T → ∞ , √ T (cid:0) ˆ M ,T − M , . . . , ˆ M H,T − M H (cid:1) (cid:32) ˜ B := ( ˜ B , . . . , ˜ B H ) , (6.1)where ˜ B denotes a centred Gaussian variable in L ([0 , ) H , with covariance operatorgiven by (3.1). The statement is then a consequence of the continuous mapping theorem.By Theorem 1 of Bücher et al. (2020), the vector √ T ( ˆ M ,T − E ˆ M ,T , . . . , ˆ M H,T − E ˆ M H,T ) converges weakly to a vector of centred Gaussian variables ˜ B = ( ˜ B , . . . , ˜ B H )in L ([0 , ) H . Thus, (6.1) follows from Slutsky’s lemma, once we have shown thatlim T →∞ √ T k E ˆ M h,T − M h k , = 0 for any h ∈ N . For the latter purpose, invoke the triangle inequality to obtain √ T k E ˆ M h,T − M h k , = √ T (cid:18) Z (cid:13)(cid:13)(cid:13)(cid:13) T b uT c∧ ( T − h ) X t =1 E [ X t,T ⊗ X t + h,T ] − Z u E [ X ( w )0 ⊗ X ( w ) h ] d w (cid:13)(cid:13)(cid:13)(cid:13) , d u (cid:19) / = √ T (cid:18) Z (cid:13)(cid:13)(cid:13)(cid:13) b uT c∧ ( T − h ) X t =1 Z tTt − T E [ X t,T ⊗ X t + h,T ] − E [ X ( w ) t ⊗ X ( w ) t + h ] d w − Z uT − {b uT c∧ ( T − h ) } E [ X ( w )0 ⊗ X ( w ) h ] d w (cid:13)(cid:13)(cid:13)(cid:13) , d u (cid:19) / ≤ √ T (cid:18) Z (cid:26) b uT c∧ ( T − h ) X t =1 (cid:13)(cid:13)(cid:13)(cid:13) Z tTt − T E [ X t,T ⊗ X t + h,T − X ( w ) t ⊗ X ( w ) t + h ] d w (cid:13)(cid:13)(cid:13)(cid:13) , + (cid:13)(cid:13)(cid:13)(cid:13) Z uT − {b uT c∧ ( T − h ) } E [ X ( w )0 ⊗ X ( w ) h ] d w (cid:13)(cid:13)(cid:13)(cid:13) , (cid:27) d u (cid:19) / . The integral from T − {b uT c ∧ ( T − h ) } to u at the right-hand side is of order 1 /T .Further, by Jensen’s inequality and local stationarity, (cid:13)(cid:13)(cid:13)(cid:13) Z tTt − T E [ X t,T ⊗ X t + h,T − X ( w ) t ⊗ X ( w ) t + h ] d w (cid:13)(cid:13)(cid:13)(cid:13) , ≤ Z tTt − T k E [ X t,T ⊗ X t + h,T − X ( w ) t ⊗ X ( w ) t + h ] k , d w ≤ CT for some constant C >
0. Thus, it follows √ T k E ˆ M h,T − M h k , = O ( T − / ) , which completes the proof of the theorem. (cid:3) Proof of Corollary 3.2. If k M h k , = 0 for some h ∈ { , . . . H } , then √ T ( k ˆ M h,T k , −k M h k , ) k ˆ M h,T k , converges to zero in probability by Theorem 3.1 and Slutsky’s lemma.Hence, it is sufficient to assume that k M h k , = 0 for all h ∈ { , . . . H } . We then obtain √ T ( k ˆ M h,T k , − k M h k , ) h =1 ,...,H (cid:32) (cid:18) h M h , ˜ B h ik M h k , (cid:19) h =1 ,...,H , from the functional delta method (Theorem 3.9.4 in van der Vaart and Wellner, 1996),applied to the functional in Proposition 6.1 below. Apply Slutsky’s lemma to conclude. (cid:3) Proposition 6.1.
The function
Φ := k·k , from L ([0 , ) to R is Hadamard-differentia-ble in any M with k M k , > , with derivative Φ M ( h ) = h M,h ik M k , in direction h ∈ L ([0 , ) . ETECTING SERIAL CORRELATION IN LSFTS 17
Proof.
For any sequences h n → h with h n ∈ L ([0 , ) and t n → t n ∈ R \ { } , itholds k M + t n h n k , − k M k , t n = 1 t n Z [0 , M ( x ) t n h n ( x ) + t n h n ( x ) d x = Z [0 , M ( x ) h n ( x ) d x + t n Z [0 , h n ( x ) d x, which converges to 2 R [0 , M ( x ) h ( x ) d x = 2 h M, h i . The square root function in R isHadamard-differentiable at x > √ x ) = √ x . By the chain rule forHadamard-differentiable functions (Lemma 3.9.3 in van der Vaart and Wellner, 1996),the Hadamard-derivative of Φ is given by Φ M ( h ) = h M,h ik M k , . (cid:3) Proof of Theorem 3.4. (i) can be deduced directly from Theorem 2 of Bücher et al.(2020). For (ii) note that by Theorem C.3 of the supplementary material of the latter ar-ticle, it holds (ˆ B T , B (1) T , . . . , B ( K ) T ) (cid:32) ( ˜ B, ˜ B (1) , . . . , ˜ B ( K ) ), where B ( k ) T = ( ˜ B ( k ) T, , . . . , ˜ B ( k ) T,H )and˜ B ( k ) T,h ( u, τ , τ )= 1 √ T b uT c∧ ( T − h ) X i =1 R ( k ) i √ m ( i + m − ∧ ( T − h ) X t = i X t,T ( τ ) X t + h,T ( τ ) − E [ X t,T ( τ ) X t + h,T ( τ )] . Note that for u < b uT c + m − ≤ T − h , for any sufficiently large T ∈ N .Thus, rewriteˆ B ( k ) h,T,T ( u, τ , τ ) = ˜ B ( k ) T,h ( u, τ , τ )+ r mT b uT c X i =1 R ( k ) i (cid:18) T − h T − h X t =1 E [ X t,T ( τ ) X t + h,T ( τ )] − X t,T ( τ ) X t + h,T ( τ ) (cid:19) + O P (cid:16)q mT (cid:17) . For the second term on the right-hand side of the latter display, it holds by independenceof the random variables R ( k ) i , E (cid:13)(cid:13)(cid:13)(cid:13)r mT b· T c∧ ( T − h ) X i =1 R ( k ) i (cid:18) T − h T − h X t =1 E [ X t,T ⊗ X t + h,T ] − X t,T ⊗ X t + h,T (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) , ≤ Z [0 , m E (cid:20)(cid:18) T − h T − h X t =1 E [ X t,T ( τ ) X t + h,T ( τ )] − X t,T ( τ ) X t + h,T ( τ ) (cid:19) (cid:21) d( τ , τ )= m ( T − h ) T − h X t ,t =1 Z [0 , Cov (cid:0) X t ,T ( τ ) X t + h,T ( τ ) , X t ,T ( τ ) X t + h,T ( τ ) (cid:1) d( τ , τ ) , which is of order O ( m/T ) by the same arguments as in the proof of Theorem 2 of Bücheret al. (2020). Thus, ˆ B ( k ) T,T = B ( k ) T + O P ( p m/T ) and (ii) follows. (cid:3) Proof of Corollary 3.5.
The assertions for the null hypothesis H ( H ) follow from Theo-rem 3.4 and Corollary 4.3 in Bücher and Kojadinovic (2019). The null hypothesis H ( H, ∆)
08 AXEL BÜCHER, HOLGER DETTE, AND FLORIAN HEINRICHS may be treated by similar arguments as in the last-named corollary, observing that theweak limit of ¯ S H, ∆ ,T is stochastically bounded by max Hh =1 h M h , ˜ B h i on the positive realline. The assertions regarding the alternative hypotheses follow from divergence to in-finity of the test statistics and stochastic boundedness of the bootstrap statistics. (cid:3) Acknowledgements
Financial support by the German Research Foundation is gratefully acknowledged.Collaborative Research Center “Statistical modelling of nonlinear dynamic processes”(SFB 823, Teilprojekt A1, A7 and C1) of the German Research Foundation (DeutscheForschungsgemeinschaft, DFG) is gratefully acknowledged. Further, the research of H.Dette was partially supported by the German Research Foundation under Germany’sExcellence Strategy - EXC 2092 CASA - 390781972.The authors are grateful to V. Characiejus and G. Rice for providing their softwareand to N. Jumpertz for help with the data set.
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Heinrich-Heine-Universität Düsseldorf, Mathematisches Institut, Univer-sitätsstr. 1, 40225 Düsseldorf, Germany.
E-mail address : [email protected] (Holger Dette, Florian Heinrichs) Ruhr-Universität Bochum, Fakultät für Mathematik, Uni-versitätsstr. 150, 44780 Bochum, Germany.
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