On the invertibility in periodic ARFIMA models
aa r X i v : . [ m a t h . S T ] A ug On the invertibility in periodic ARFIMA models
Amine AMIMOUR and Karima BELAIDE Department of MathematicsApplied Mathematics LaboratoryUniversity of Bejaia, Bejaia. [email protected] k [email protected] Abstract
The present paper, characterizes the invertibility and causality conditions of a periodicARFIMA (PARFIMA) models. We first, discuss the conditions in the multivariate case,by considering the corresponding p-variate stationary ARFIMA models. Second, we con-struct the conditions using the univariate case and we deduce a new infinite autoregressiverepresentation for the PARFIMA model, the results are investigated through a simulationstudy.
Keywords:
PARFIMA. Invertibility condition. Fractionally process. Simulation. Long rangedependence. Long memory.
A long memory processes have long been studied in the literature. (See, e.g. Granger andJoyeux [6] and Hosking [7] for early work, Baillie [3] for more background). Hosking [7] proposeda purely fractionally differenced autoregressive moving average process ( X t ) t ∈ Z denoted byARFIMA(0 , d,
0) or ARFIMA for short, given by the equation(1 − B ) d X t = ε t , (1.1)where d is a real number which denote the lag of the ARFIMA process, ε t is a sequenceof independent and identically distributed (i,i,d) random variables with mean zero and finitevariance σ , and B is the back shift operator, such that X t − j = B j X t . The invertibility concept1 INTRODUCTION AND NOTATIONS of time series models, loosely says that the ARFIMA model is invertible when we are able toexpress the noise process ε t as a convergent series of the observations X t . The characteristicequation for a ARFIMA process is ε t = ∞ X j =0 π j X t − j , (1.2)where π j = Γ( − d + j )Γ( − d )Γ( j +1) .Γ( . ) is a gamma function and it is given byΓ( z ) = ( R ∞ s z − e − s ds, si z > ∞ , si z = 0 , if z < , Γ( z ) is defined by the recurrence formula z Γ( z ) = Γ( z + 1) . In order for a ARFIMAprocess to be invertible, the infinite sum of the coefficients π j must be absolutely summable.However, when the lag d > − , the infinite sum of the coefficients π j is absolutely summable.Odaki[9] noticed that the ARFIMA process is invertible even when − < d . These condi-tions concerning the univariate case. Chung [4] used a Vector ARFIMA model to define theinvertibility condition of multivariate p-dimensional stationary process in the sense of Hosking[7]. Recently, the invertibility condition for the bivariate and multivariate case is achieved byKechagias and Pipiras [8], they extended the definition of the bivariate LRD of Robinson [10],to the multivariate case, concluding that the trigonometric power-law coefficients, can be usedto construct new univarite and multivariate causal LRD series.Since a several time series encountered in practice exhibit periodic autocorrelation struc-ture, a fact that cannot be explained, by the classical seasonal models or by the time seriesmodels with parameters invariants in time. In fact, Franses and Ooms [5] considered thePARFIMA models defined by X t = (1 − B ) − d s ε t , (1.3)where, d s is the fractional parameter which can vary with season s = 1 , , ,
4, to study quarterlyinflation in the UK, allowing the degree of fractional integration to vary with season, the seriesanalyzed in this study concern the quarterly inflation rate verified over a 33-year period, afterthe presentation of the correlogram, they concluded that the autocorrelations do not decreaserapidly, which indicates long-memory behavior in the series. Moreover, if the transformationof the series be performed by classical differentiation in the sense of Box Jenkins, the auto-correlations indicate that the differentiated series can be overdifferentiated, because the sumof the autocorrelations will be close to − .
5, then the observations suggest the utility possibleof ARFIMA models. Therefore, for a more appropriate modeling, which takes into accountseasonal variation they have proposed to merge the presence of long memory and the periodicdynamics, into a new model. The key result of these authors is that this model compared withthe periodic short-memory models and long-memory models with constant coefficients, not only2
INTRODUCTION AND NOTATIONS provides a plausible informative description of the periodic series, but also, can be useful forout-of-sample forecasts. Amimour and Belaide [2] studied recently, the probabilistics propertyof PARFIMA models, in the sense that the long memory parameter varies periodically in timesuch that d t + p = d t with period p ∈ N ∗ ; p >
1, denoted by PtvARFIMA and has the followingstochastic equation (1 − B ) d i X i + pm = ε i + pm , (1.4)where for all t ∈ Z , there exists i = { , ........, p } , m ∈ Z , such that t = i + pm and the varianceis periodic in t such that σ t + pm = σ t , the authors constructed also a local asymptotic normalityproperty for this model see [1]. Indeed, the PARFIMA models have received little attentionfrom time series analysts due to these complexities, one of the main difficulties is to achieving ainvertibility condition. Noting that the works cited above based only on the sufficient conditionof invertibility and causality, that iswhen d i >
0, the process (1.4) is invertible and has an infinite autoregressive representation isas follows ε i + pm = (1 − B ) d i X i + pm = ∞ X j =0 π ij X i + pm − j , (1.5)where π ij = Γ( j − d i )Γ( j +1)Γ( − d i ) , when d i < , the process (1.4) is causal and has an infinite moving-average representation is asfollows X i + pm = (1 − B ) − d i ε i + pm = ∞ X j =0 ψ ij ε i + pm − j , (1.6)where ψ ij = Γ( j + d i )Γ( j +1)Γ( d i ) .ψ ij ∼ v i j d i − , v i > , as j → ∞ . (1.7)See propositions (2.1) and (2.2) in Amimour and belaide [2]. For instance, precise informationon the invertibility conditions of a PtvARFIMA model is important to circumscribe a model’sparameter space. Moreover, verifying invertibility in the necessary and sufficient conditions3 PTVARFIMA AND P-VARIATE STATIONARY ARFIMA MODELS sense, is required for statistical inference, in this article we address the above topic. To find aninvertibility condition of PtvARFIMA.The work is organized as follows. We first discuss in section 2 some characteristics ofour model using his associated stationary multivariate process. In section 3, we focus on theconstructing of new infinite autoregressive representation of our model. We finish this work bythe simulation for illustrating the results in section 4.
The PtvARFIMA model is first introduced by Amimour and Belaide [2], this model extendsthe ARFIMA model from hosking [7]. PtvARFIMA time series exhibit long range dependence(LRD), and the periodic phenomenon, the periodic covariance function γ iX ( h ) echoes LRD anddecays periodically as the lag h increases, the asymptotic behavior of this function is given bythe following explicit form, as h → ∞ γ iX ( h ) ≃ γ X ( h ) ≃ σ
21 Γ(1 − d − d )Γ( d )Γ(1 − d ) ( h ) d + d − , if h ≡ p ] ,σ
21 Γ(1 − d − d )Γ( d )Γ(1 − d ) ( h ) d + d − , if h ≡ p ] ,..σ
21 Γ(1 − d )Γ( d )Γ(1 − d ) ( h ) d − , if h ≡ p ] .γ X ( h ) ≃ σ
22 Γ(1 − d − d )Γ( d )Γ(1 − d ) ( h ) d + d − , if h ≡ p ] ,σ
22 Γ(1 − d − d )Γ( d )Γ(1 − d ) ( h ) d + d − , if h ≡ p ] ,..σ
22 Γ(1 − d )Γ( d )Γ(1 − d ) ( h ) d − , if h ≡ p ] ....γ pX ( h ) ≃ σ p Γ(1 − d p − d )Γ( d )Γ(1 − d ) ( h ) d p + d − , if h ≡ p ] ,σ p Γ(1 − d p − d )Γ( d )Γ(1 − d ) ( h ) d + d p − , if h ≡ p ] ,..σ p Γ(1 − d p )Γ( d p )Γ(1 − d p ) ( h ) d p − , if h ≡ p ] . . In this section, we will focus on the PtvARFIMA model as a multivariate stationaryARFIMA time series, that is, R p − valued time series X m = ( X pm , ...., X p + pm ) T , where T denotes the transpose. 4 PTVARFIMA AND P-VARIATE STATIONARY ARFIMA MODELS
Proposition 2.1.
A PtvARFIMA model defined in (1.4) is a p -variate second order stationarylong memory time series.Proof. Suppose that for a > , a diagonal matrix N = diag ( n , ...., n p ) , we write a N = diag ( a n , ....., a n p ) . The periodic covariance function γ iX ( h ) previously given, can be expressedas a matrix function of a second order stationary series X = { X m } m ∈ Z , given by γ i,kX ( h ) ≃ σ i h D − I Rh D − I = (cid:0) σ i R i,k h d i + d k − (cid:1) , i, k = 1 , .., p, as h → ∞ , (2.1)where D = diag ( d , ...., d p ) , and R i,k > i, k = 1 , ...., p . Proposition 2.2.
The causal representation of the p − variate ARFIMA time series, associatedwith the PtvARFIMA model with period p is X m = ∞ X j =0 Υ j ε m − j , (2.2) and R i,k = Γ( d i )Γ( d k )Γ( d i + d k ) (cid:18) v i v k sin( πd k )sin( π ( d i + d k )) (cid:19) , (2.3) where Υ j is a diagonal matrix diag ( ψ j , ....., ψ pj ) and ε m is an R p − valued white noise ( ε pm , ...., ε p + pm ) T , satisfying E [ ε m ] = 0 and E [ ε m ε ′ m ] = σ i I .Proof. Let us calculate the autocovariance function γ i,kX ( h ) γ i,kX ( h ) ≃ σ i ∞ X j =0 v i j d i − v k ( j + h ) d k − ≃ σ i ∞ X j =0 v i v k j d i − ( j + h ) d k − ≃ σ i ∞ X j =0 v i v k j d i − ( j + h ) d k − ≃ σ i ∞ X j =0 v i v k j d i − ( j + h ) d k − = ∞ X j =0 v i v k h d i − (cid:18) jh (cid:19) d i − h d k − (cid:18) jh + 1 (cid:19) d k − , from (Appendix A, in Kechagias and Pipiras [8]) γ i,kX ( h ) ≃ σ i h d i + d k +1 v i v k Z ∞ z d i − ( z + 1) d k − dz ∼ R i,k h d i + d k − , where R i,k = v i v k Γ( d i )Γ(1 − d i − d k )Γ(1 − d k ) , and using the identity Γ( z )Γ(1 − z ) = π sin( πz ) , < z < , we deduce the expression of R i,k . A NEW INFINITE AUTOREGRESSIVE REPRESENTATION FOR PTVARFIMA
Theorem 3.1.
Let { X i + pm } m ∈ Z be the PARFIMA (0 , d i , process, given by the expression(1.4), then(i) When − < d i < , or | d i | < . { X i + pm } m ∈ Z is a invertible process with infiniteautoregressive representation given by ε i + pm = ∞ X j =0 Π j ( i ) X i + pm − j . (3.1) The coefficients Π j ( i ) j ≥ in (3.1) satisfy the following relation Π j ( i ) = − ( ψ j ( i ) + j − X l =1 Π l ( i ) ψ j − l ( i + k )) , (3.2) where the weights ψ j ( i ) j ≥ are given in (1.6) and l ≡ k [ p ] , Π ( i ) = 1 , with ∞ P j =0 | Π j ( i ) | < ∞ .(ii) When − < d i < , or | d i | < . { X i + pm } m ∈ Z is a causal process with infinite movingaverage representation given by X i + pm = ∞ X j =0 Ψ j ( i ) ε i + pm − j . (3.3) The coefficients Ψ j ( i ) j ≥ in (3.3) satisfy the following relation Ψ j ( i ) = − ( π j ( i ) + j − X l =1 Ψ l ( i ) π j − l ( i + k )) , (3.4) where the weights π j ( i ) j ≥ are given in (1.5) and l ≡ k [ p ] , Π ( i ) = 1 , with ∞ P j =0 | Ψ j ( i ) | < ∞ .Proof. For simplicity of notation, we use ψ j ( i ) = ψ ij and π j ( i ) = π ij .( i ) We have X i + pm = ∞ X j =0 ψ j ( i ) ε i + pm − j , (3.5)we can develop the series as follows X i + pm = ε i + pm + ψ ( i ) ε i + pm − + ψ ( i ) ε i + pm − + .............. = ε i + pm + ψ ( i )[ X i + pm − z }| { ε i + pm − + ψ ( i + 1) ε i + pm − + ψ ( i + 1) ε i + pm − ...... ] − ψ ( i )[ ψ ( i + 1) ε i + pm − + ψ ( i + 1) ε i + pm − ............ ]+ ψ ( i ) ε i + pm − + ψ ( i ) ε i + pm − ............... SIMULATION X i + pm − ψ ( i ) X i + pm − = ε i + pm +[ ψ ( i ) − ψ ( i ) ψ ( i + 1)][ X i + pm − z }| { ε i + pm − + ψ ( i + 2) ε i + pm − + ....... ] − [ ψ ( i ) − ψ ( i ) ψ ( i + 1)][ ψ ( i + 2) ε i + pm − ............ ]+[ ψ ( i ) − ψ ( i ) ψ ( i + 1)] ε i + pm − + ...............X i + pm − φ ( i ) z }| { ψ ( i ) X i + pm − = ε i + pm +[ φ ( i ) z }| { ψ ( i ) − ψ ( i ) ψ ( i + 1)][ X i + pm − z }| { ε i + pm − + ψ ( i + 2) ε i + pm − + ....... ] − [ ψ ( i ) − ψ ( i ) ψ ( i + 1)][ ψ ( i + 2) ε i + pm − ............ ]+[ ψ ( i ) − ψ ( i ) ψ ( i + 1)] ε i + pm − + ...............X i + pm − φ ( i ) X i + pm − − φ ( i ) X i + pm − ................ = ε i + pm , (3.6)such that φ j ( i ) = ψ j ( i ) − j − X l =1 φ l ( i ) ψ j − l ( i + k ) , (3.7)with l ≡ k [ p ] . Next it is easy to show that Π j ( i ) = − φ j ( i ) . This result is illustrated by thesimulation in the next section.( ii ) The proof is similar to (i) but d i must be replaced by − d i . In this section, we provide some simple examples, to illustrate the above theoretical results inthe invertible case, as well as the non-invertible case for p = 2. More precisely, we show thatthe absolute sum of coefficients Π j ( i ) is finite for − < d i < or | d i | < d < − and d < , d > − and d > or | d i | > N such that j = 0 , , ....., N . The sum of Π j ( i ) is summable absolutely if the differencebetween the previous and the present value of Π j ( i ), e.g. ( | Π N ( i ) − Π N +1 ( i ) | ), will be very lowor negligible. Numerical values of | Π N ( i ) − Π N +1 ( i ) | are presented in table (1) for the invertiblecase. Table (2) for non invertible case, we find that the results indicated in the tables reflectthe theoretical framework.For − < d i < or | d i | < | Π j ( i ) | converges,i.e the sum of | Π j ( i ) | stabilizes quickly in this case, since the quantity | Π N ( i ) − Π N +1 ( i ) | decreasewhen N increase and have inconsiderable values. Contrarily, if the values of d i are outside theintervals cited (see table 2), this demonstrates that the sum diverges even for large values of7 SIMULATION
N d d d d d d d d d d .
15 0 . . − .
49 0 .
75 0 . . . − . − .
710 1 . × − . . . × − . × − . × − . . . × − . × − . × − . . . × − . × − . × − . . . × − . × − . × − . . . × − . × − N d d d d d d d d d d − . − . − . − . − . − . − . − . .
49 0 . . . . . × − . . . . . × − . . . . . × − . . . . . × − . . . . . × − . | Π N ( i ) − Π N +1 ( i ) | ) for − < d i < or | d i | < NN d d d d d d − . . − . . − . − .
410 1 . . . . . . . . . . . . . . . | Π N ( i ) − Π N +1 ( i ) | ) for d < − and d < , d > − and d > or | d i | > N EFERENCES N , in this case the quantity | Π N ( i ) − Π N +1 ( i ) | is considerable and sometimes increases when N increase. In this article, we have discussed in one hand the p-variate stationary ARFIMA models asso-ciated with the PARFIMA model considered here. We have represented the model as a causalrepresentation, showing that the asymptotic behavior of the periodic autocovariance functioncan be obtained using the multivariate moving average representation. On the other hand,we have established a new infinite autoregressive representation for the PARFIMA model andillustrated the results via a simulation study.
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