On uniqueness of moving average representations of heavy-tailed stationary processes

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1 MPRA Munich Personal RePEc Archive On uniqueness of moving average representations of heavy-tailed stationary processes Christian Gouriéroux and Jean-Michel Zakoian University of Toronto, CREST 3 March 204 Online at MPRA Paper No , posted April :47 UTC

2 On uniqueness of moving average representations of heavy-tailed stationary processes BY CHRISTIAN GOURIÉROUX CREST and University of Toronto. AND JEAN-MICHEL ZAKOÏAN EQUIPPE University Lille 3 and CREST, 5 boulevard Gabriel Péri, Malakoff Cedex, France. zakoian@ensae.fr SUMMARY We prove the uniqueness of linear i.i.d. representations of heavy-tailed processes whose distribution belongs to the domain of attraction of an α-stable law, with α < 2. This shows the possibility to identify nonparametrically both the sequence of two-sided moving average coefficients and the distribution of the heavy-tailed i.i.d. process. Some key words: α-stable distribution; Domain of attraction; Infinite moving average; Linear process; Mixed causal/noncausal process; Nonparametric identification; Unobserved component model.. INTRODUCTION By definition, a real linear process X t can be written as a two-sided moving average of a strong white noise: X t = a j ɛ t j, t Z, where ɛ t is a sequence of independent and identically distributed i.i.d. real random variables, and a j is a sequence of Moving-Average MA coefficients. Provided that such a process exists, an important issue is the uniqueness of the MA representation in. Uniqueness of the linear representation of non-gaussian i.i.d. processes plays an important role in time series, for instance in the analysis of time reversibility see Hallin, Lefèvre and Puri 988, Breidt and Davis 992. From a statistical point of view, this problem is equivalent to the nonparametric identification of the sequence a i and of the common distribution of the variables ɛ t. See Rosenblatt 2000 for a review of statistical applications of non-gaussian linear processes. While Gaussian linear processes generally admit several moving average representations, non- Gaussian linear processes have been shown to admit an essentially unique MA representation, under different regularity conditions on the MA coefficients and the moments of the i.i.d. process. The literature dealing with uniqueness of MA representations of non-gaussian processes includes Lii and Rosenblatt 982, Findley 986, 990, Cheng 992, Breidt and Davis 992. The sufficient uniqueness conditions obtained in these articles are summarized in Table. They require at least finite variance of the independent process ɛ t, except in Breidt and Davis 992 see their Remark 2, p In the present article, we provide a direct proof for the identifi-

3 2 cation of the MA representation when the errors belong to the domain of attraction of a stable law, with index α < 2. This case is especially important in view of the increased interest, in the recent time series literature, in heavy-tailed distributions. In particular, it has been noted that mixed causal/noncausal processes of the form were adequate for modeling the speculative bubbles observed on fat-tailed financial series, for instance when the noise and the observed process do not have a finite expectation see Gouriéroux and Zakoïan 203. Finally, we extend the identification result to the multivariate framework, especially to unobserved component models. 2. UNIQUENESS RESULTS FOR UNIVARIATE PROCESSES We start by recalling conditions ensuring the existence of a MA representation of a strictly stationary process. 2. Existence The existence of infinite MA processes, as in, is generally established under the assumption that the variables ɛ t belong to L or L 2. An extension was established by Cline Theorem 2., 983, who gave conditions for the existence of infinite MA with i.i.d. random variables admitting regularly varying tails. The following is a straightforward consequence of Proposition 3.3. in Brockwell and Davis 99. PROPOSITION. Consider the two-sided moving average X t = written on a process Z t such that, for some s 0, : α h Z t h, 2 sup E Z t s < t and h= α h s <. Then, the series X t converges absolutely with probability one. If, in addition, Z t is a strictly stationary process, then X t is also strictly stationary. We now turn to the uniqueness problem, first in the case where X t = ɛ t is an i.i.d. process Uniqueness of the MA representation of an heavy-tailed i.i.d. process Let ɛ t and ɛ t denote two i.i.d. processes. Suppose that and, for some s 0,, ɛ t = a j ɛ t j, t Z, 3 E ɛ t s < and a j s <. 4 By Proposition, the latter conditions entail the existence of the infinite MA appearing in the right-hand side of 3. Note also that this entails E ɛ t s <, using the inequality x + y s x s + y s for x, y 0. Let Ψ, Ψ denote the characteristic functions of ɛ and ɛ, respectively.

4 The characteristic function of ɛ t, ɛ t h with h 0 is, for u, v R, E e iuɛ t +vɛ t h = Ψ uψ v = E e iua j+va j+h ɛ t j = Ψua j + va j+h. The last equality holds by the dominated convergence theorem, which applies because the products e iua j+va j+h ɛ t j n have unit norm. j= n Hence, for u, v R, log Ψ u + log Ψ v = For expository purpose, we first consider stable variables. a Stable variables log Ψua j + va j+h. 5 Let us recall that the distribution of a stable variable X with parameters α, β, σ, µ, denoted X Sα, β, σ, µ, has the characteristic function exp[ σ α s α { iβ sign s tan } πα 2 + iµs], if α, Ψs = exp[ σ s { + iβ sign s 2 π log s } + iµs], if α =. For a stable distribution with index of stability α 0, 2, the moment condition in 4 is satisfied for s 0, min{α, }. See Samorodnitsky and Taqqu 994 for further details on stable variables. PROPOSITION 2. If ɛ t and ɛ t have stable laws, ɛ t Sα, β, σ, µ and ɛ t Sα, β, σ, µ with α, α < 2, if a j s < for s 0, min{α, }, then all the MA coefficients a j s except one are equal to zero in the MA representation 3. b Variables in the domain of attraction of a stable law By definition, a variable X belongs to the domain of attraction of an α stable law, denoted X DAα, with α < 2, if for any i.i.d. sequence X i, with X d = X, there exist sequences of constants A n R and B n > 0 such that X + + X n A n B n d Z, 6 where d denotes the convergence in distribution, and the distribution of Z is α-stable. The tails of variables belonging to DAα have the following characterization: X DAα if and only if its cumulative distribution function is such that: F x c Lx x α, F x c 2Lx x α, when x, where L is a slowly varying function and c, c 2 0, c + c 2 0 see for instance Embrechts, Klüppelberg, Mikosch 997, Theorem Hence, if X DAα, E X s < for s 0, α. Note that we can have E X α < which is not possible for α stable variables. Let us start with a simple extension of Proposition 2. that is, L is measurable positive, defined on 0,, such that lim x Ltx/Lx = for all t > 0. 3

5 4 PROPOSITION 3. If ɛ t DAα and ɛ t has a stable distribution with index α, with α, α < 2, if a j s < for s < α, then all the MA coefficients a j s except one are equal to zero in 3. Let us now extend the identification result, when both i.i.d. processes ɛ t and ɛ t are in the attraction domain of a stable law. PROPOSITION 4. If ɛ t DAα and ɛ t DAα are such that E e iuɛt = e α L /u, E e iuɛ t = e α L /u, 7 with α, α < 2, where L, L are slowly varying functions, if for any τ > 0 there exist constants M, K > 0 such that for all t > 0, L tx L x K + tτ, 8 sup x >M and if a j s < for s < α, then all the MA coefficients a j s except one are equal to zero in Uniqueness of the MA representation of an heavy-tailed linear process We now study the uniqueness of representation. Assume that X t = a j ɛ t j = a jɛ t j, t Z 9 where ɛ t and ɛ t are two sequences of i.i.d. real random variables, and a j, a j are two sequences of MA coefficients with, for some s 0, : a h s < and a h s <. 0 h= h= The random series in 9 converging a.s., we may define abɛ t := a jɛ t j and a Bɛ t := a j ɛ t j where B stands for the backward shift operator. The next proposition shows that under appropriate assumptions on the i.i.d. sequence and the MA coefficients, the representation of X t is essentially unique. PROPOSITION 5. Assume that ɛ t DAα and ɛ t DAα, satisfying conditions 7-8, that a B is invertible, with {a B} = c jb j such that c j s <, and that a j s <, for s < α. Then, if 9-0 hold, we have for some constants c R and l Z. ɛ t = cɛ t l and a j = ca j+l, j Z From Proposition 5, we see that the MA representation is identifiable up to a change of scale and a drift of the time index on ɛ t. 3. UNIQUENESS RESULTS FOR MULTIVARIATE HEAVY-TAILED LINEAR PROCESSES The literature on the identification of multivariate MA processes is rather limited. Sets of conditions for identification have been derived in Chan, Ho and Tong 2006, when the p- dimensional observed process and the q-dimensional noise processes are square integrable, when

6 p q i.e. an order condition, and when the q components of the noise are i.i.d. see their Condition 3. The order condition is required in the proof to be able to apply the approach by Cheng 992, 999. This section will complete such results by considering processes with fat tails. We start by an extension of Proposition Uniqueness of the decomposition of an heavy-tailed i.i.d. process as a sum of independent stable MA The next result extends Proposition 3 by considering the decomposition of a non-gaussian stable one-dimensional noise as the sum of independent stable MA processes: ɛ t = k= a k,j ɛ k,t j, t Z where ɛ k,t are K independent sequences of stable i.i.d. real random variables, and the {a k,j ; j Z} are K sequences of real numbers such that, for some s 0,, 0 < a k,j s <. PROPOSITION 6. If ɛ t DAα with α < 2, the variables ɛ k,t have stable laws, ɛ k,t Sα k, β k, σ k, µ k with α k s, 2 for k =,..., K, then in each sequence {a k,j ; j Z} all the a k,j s except one are equal to zero in Uniqueness of the MA representation of a vector process with fat tails We now show that, under appropriate conditions, a vector process X t has at most one representation of the form X t = ABɛ t, 2 where is ɛ t is a vector i.i.d. process, with independent components following stable distributions with distinct tail indices. More precisely, consider a p-dimensional process X t = X t,..., X pt admitting two MA representations given by X t = ABɛ t = A Bɛ t, t Z, 3 where ɛ t, ɛ t are two K-dimensional i.i.d. processes with stable and independent components, and AB = a i,k B i=,...,p;k=,...,k, A B = a i,k B i=,...,p;k=,...,k are infiniteorder lag polynomials with matrix coefficients. Let a i,k B = a i,k,jb j and a i,k B = a i,k,j Bj. Assume that, for some s 0,, 0 < a i,k,j s <, 0 < a i,k,j s <. PROPOSITION 7. Assume that ɛ t and ɛ t are two K-dimensional i.i.d. processes satisfying 3, with stable and independent components ɛ k,t and ɛ k,t having stable laws, ɛ k,t Sα k, β k, σ k, µ k and ɛ k,t Sα k, β k, σ k, µ k, with α k, αk s, 2 for k =,..., K and α <... < α K. Suppose that, for any i, k, the lag polynomial a ik B is invertible, with {a i,k B} = c i,k,jb j such that c i,k,j s <. Then, for i =,..., p, and 5

7 6 k =,..., K for some constants c i,k R and l i,k Z. a i,k,j = c i,k a i,k,j+l i,k, j Z, 4. CONCLUDING REMARKS The identification results of this paper can be used in both parametric and nonparametric analyses of linear processes. The proofs also suggest a nonparametric estimation method, for a linear process X t based on an i.i.d. sequence of heavy-tailed variables ɛ t, as in, by the following steps. i Estimation of α. Since X t DAα when ɛ t DAα see for instance Embrechts, Klüppelberg, Mikosch 997, Theorem A3.26, the tail index α can be consistently estimated by a standard approach. For instance, the Hill estimator can be used for its main properties under various assumptions see Embrechts, Klüppelberg, Mikosch 997 Theorem ii Estimation of the MA coefficients. The analysis of the joint characteristic function of X t, X t h for h 0 and v = tu in a neighborhood of u = 0, provides estimation of a j + ta j+h α when ɛ t Sα, β, σ, µ. Estimates of a finite number of MA coefficients can be deduced by truncating the infinite sum and solving the moment restrictions. iii Estimation of the distribution of ɛ t. Having estimated the lag polynomial ab, approximations of the variables ɛ t can be deduced by computing the residuals ˆɛ t = âb X t, where the X u for u < 0 are replaced by 0. Then, the common distribution of the ɛ t s will be estimated from the residuals. APPENDIX Uniqueness of non-gaussian MA representations in the literature Table gives a summary of the main conditions obtained in the literature for the uniqueness of the MA representation in. Apart from such conditions, it is generally also assumed that the spectral density of X t is positive almost everywhere. Note that the condition given by Breidt and Davis Remark 2, p.386, 992, as an alternative proof to their Proposition 3., is based on the following result: THEOREM KAGAN ET AL. 973, P.94. Let ɛ j be a sequence of independent random variables, and {a j }, {b j } be two sequences of real constants such that i The sequences {a j /b j : a j b j 0} and {b j /a j : a j b j 0} are both bounded; aj ɛ j and b j ɛ j converge a.s. to random variables U and V respectively; ii U and V are independent. Then, for every j such that a j b j 0, ɛ j is normally distributed. In the context of MA processes, this result can be used as follows. Suppose that 3-4 hold, where ɛ t and ɛ t are two i.i.d. processes. Then the variables ɛ t = are independent. It follows that if the sequences a j ɛ t j and ɛ t h = a j+h ɛ t j {a j /a j+h : a j a j+h 0} and {a j+h /a j : a j a j+h 0} are both bounded, A

8 Table. Uniqueness conditions for the non-gaussian MA representation Papers Condition on ɛ t Condition on MA coefficients Lii and Rosenblatt 982 All moments finite j aj < Findley 986 All moments finite a 2 j < Findley 990 Finite non-zero rth cumulant a 2 j < Cheng 992 Finite variance a 2 j < Breidt and Davis Prop. 3., 992 Finite variance Fractionally Integrated ARMA Breidt and Davis Rem. 2 p. 386, 992 E log + ɛ t < Fractionally Integrated ARMA 7 then either the variables ɛ t and ɛ t are normally distributed, or all the a j s except one are equal to zero. The conclusion thus coincides with that of our Proposition 4, but is obtained under a less restrictive distributional assumptions on the i.i.d. processes, but b more restrictive assumptions on the sequence of coefficients a j. Breidt and Davis Remark 2, p. 386, 992 showed that A is satisfied for a MA polynomial of the form Az = Ψz/Φz = j=0 a jz j, where Ψz and Φz are finite-order polynomials under standard assumptions. For more general processes, the boundedness assumption can be very restrictive. For instance, it precludes sequences recursively defined by a j = 0 for j 0 and a j = λ j a j for j, where λ j is a sequence converging to zero or to infinity. APPENDIX 2 Proof of Proposition 2 In view of 5 with v = 0, we have, for all u R, σ α α = σ α a j α α. Thus α = α and, without loss of generality, we can take σ = σ and a j α =. We then have, by taking v = tu in 5, a j + ta j+h α = + t α, t R. A2 The right-hand side of this equality is a differentiable function of t except at t = 0. Let us study the differentiability of the left-hand side. Suppose that there exists j 0 such that a j0 0 and a j0+h 0. The function t a j0 + ta j0+h α is everywhere differentiable except at t 0 = a j0 /a j0+h, with t 0 0. The contribution of all terms of the infinite sum displaying non differentiability at t 0 is a j + ta j+h α = t t 0 α a j+h α, j J j J where J = {j Z a j + t 0 a j+h = 0}. This contribution must vanish; otherwise, we would have in the right-hand side of A2 a function which is differentiable at t 0 while the left-hand side is not. Hence j J a j+h α = 0, in contradiction with the fact that this sum is bounded below by a j0+h α 0. Therefore, there exists no integer j 0 such that a j0 0 and a j0+h 0, which establishes the property. Proof of Proposition 3 By Ibragimov and Linnik Theorem 2.6.5, 97, for a variable ɛ DAα, where the limiting stable distribution has location parameter µ = 0, we have E e iuɛ = e α L /u, A3

9 8 in the neighborhood of the origin, for some slowly varying function L. This equality also holds if µ 0, the limit law in 6 being uniquely determined up to positive affine transformations. By 5 with v = 0, we thus have α L = σ α a j α α, A4 in the neighborhood of 0. Thus α = α and L is constant in the neighborhood of 0. It is not restrictive to assume σ α = L / in the neighborhood of 0 and a j α =. We thus are lead to the proof of Proposition 2. Proof of Proposition 4 By 5 with v = 0 and A3, we have in the neighborhood of 0, α L = a j 0 i Let us first show that α = α. If α < α, the left-hand side term in α α L = a j 0 a j α α L. A5 a j u a j α L, a j u tends to 0 when u 0, while the right-hand side term does not. We also get a contradiction if α < α, hence α = α. ii Now we will show that L a j α L, when u 0. A6 We will use the next property see for instance Embrechts, Klüppelberg, Mikosch 997, Theorem A3.2. LEMMA. If L is a slowly varying function, for any a, b > 0, the convergence Ltx/Lx when x is uniform on the segment [a, b]. Using Lemma and A5 with α = α we have, in the neighborhood of 0, for some constants a, b > 0 to be chosen later, L a j α L L a j α a ju L a j 0 n j= n a j α sup t [a,b] L L := S,n u + S 2,n u + S 3,n. t + L a j α L j >n a j 0 a ju + a j α j >n

10 We have, in the neighborhood of 0, for τ = α s, S 2,n u = L a j s a j τ a ju L a j s a j τ K + a j τ, j >n a j 0 j >n a j 0 which is smaller than an arbitrarily small ς > 0 for n sufficiently large, n > N say. The same property holds for S 3,n. Now let [a, b] = [ min j N { a j, a j 0}, max j N { a j, a j 0} ] in S,n u. We have t L S,n u a j α sup t [a,b] L, which tends to zero using the uniform convergence of L. Thus we have established A6 and, without generality loss, we can make a scale change on the a j s to get, for u 0, L = a j α L. A7 iii Using A7, Equation 5 can now be written, for u, v in the neighborhood of 0, uv 0, as α L + v α L = a j u + a j+h v α L v a j u + a j+h v a j+ta j+h 0 = a ju+a j+h v 0 a j u α L + a j+h v α L. v Hence, for v = tu, t > 0, we get, for u in the neighborhood of 0, a j + ta j+h α L a j + ta j+h = a j α L + ta j+h α L. t By the same argument as before, using the uniform convergence of the slowly varying function L see Lemma, the asymptotic behavior of these sums in the neighborhood of u = 0 is the same if we replace terms of the form L, with x > 0, by L. We thus have, for any t > 0, x a j + ta j+h α = and we are led to the discussion of Proposition 2. a j α + ta j+h α, Proof of Proposition 5 We use Theorem 2. in Kokoszka 996, which establishes sufficient conditions for two infinite-order lag polynomials whose coefficients may not be absolutely summable to commute. From the assumptions made on the coefficients of the polynomials ab and {a B}, we have ɛ t = {a B} abɛ t, where the random series {a B} abɛ t converges absolutely a.s. By Proposition 4, it follows that 9

11 0 {a B} ab = cb l for some constant c and some integer l Z. Thus ab = ca BB l and the proof is complete. Proof of Proposition 6 Similarly to 5, we have for u, v R, log Ψ u + log Ψ v = log Ψua k,j + va k,j+h, k= A8 by the independence between ɛ k,t and ɛ l,t for any k, t l, t. By the arguments leading to A4 we thus have α L = σ α k k a k,j α k α k, A9 k= in the neighborhood of 0. Thus α k = α =: α for all k, and L is constant in the neighborhood of 0, L = k= σ α k a k,j α. Now by taking v = tu in the neighborhood of u = 0, we find that, similarly to A2, k= σ α k a k,j + ta k,j+h α = + t α Now we can use the following elementary lemma. k= σ α k a k,j α, t R. A0 LEMMA 2. For α > 0, let f γ : t R f γ t = + γt α and let f : t R f t = t α. Let Γ = {γ i } i I a family of distinct real numbers, with I Z. The family of functions {f, f γ ; γ Γ} is linearly independent. It follows from A0 and Lemma 2 that for any k, j either a k,j or a k,j+h is equal to zero. Thus, for any k, the set {a k,j, j Z} reduces to a singleton. The conclusion follows. Proof of Proposition 7 By the independence between ɛ k,t and ɛ l,t in the one hand, and ɛ k,t and ɛ l,t in the other hand, for any k, t l, t, we get from the i-th equation of 3 k= Thus, by arguments already used, log Ψ ua i,k,j + va i,k,j+h, = k= σ k α k a i,k,j α k α k = K k= k= log Ψua i,k,j + va i,k,j+h. σ α k k a i,k,j α k α k, in the neighborhood of 0. It follows that {α k ; k =,..., K} = {αk ; k =,..., K} and thus α = α <... < α K = αk and σ k α k a i,k,j α k α k = σα k k a i,k,j α k α k for k =,..., K. It is thus not restrictive to assume σ k = σk and a i,k,j α k α k = a i,k,j α k α k. Now,

12 similarly to A0, k= σ k α k a i,k,j + ta i,k,j+h α k = k= σ α k k a i,k,j + ta i,k,j+h α k, t R, which, using the fact that the powers α k are different, entails, for k =,..., K, a i,k,j + ta i,k,j+h α k = a i,k,j + ta i,k,j+h α k, t R. A Let η t be an i.i.d. sequence of symmetric α k - stable distributed random variables, η t Sα k, 0,, 0. The left hand side of A characterizes the distribution of the process a i,k Bη t. It follows that a i,k Bη d t = σ k a i,k Bɛ k,t. Therefore, we have d η t = σ k {a i,k B} a i,k Bɛ k,t and we can conclude as in the proof of Proposition 5. REFERENCES Breidt, F.J. and R.A., Davis 992 Time-reversibility, identifiability and independence of innovations for stationary time-series. Journal of Time Series Analysis 3, Brockwell, P., and R., Davis 99 Time Series : Theory and Methods. Springer Verlag, New-York. Chan, K.-S., Ho, L.-H. and H. Tong A note on time-reversibility of multivariate linear processes. Biometrika 93, Cheng, Q. 992 On the unique representation of non-gaussian linear processes. Annals of Statistics 20, Cheng, Q. 999 On time-reversibility of linear processes. Biometrika 86, Cline, D.B.H. 983 Infinite series of random variables with regularly varying tails. Unpublished document, The University of British Columbia. Embrechts, P., Klüppelberg, C. and T. Mikosch 997 Modelling extremal events. Springer. Findley, D. 986 The uniqueness of moving average representation with independent and identically distributed random variables for non-gaussian stationary time series. Biometrika 73, Findley, D. 990 Amendments and corrections: The uniqueness of moving average representation with independent and identically distributed random variables for non-gaussian stationary time series. Biometrika 77, 235. Gouriéroux, G. and J-M. Zakoian 203 Explosive Bubble Modelling by Noncausal Process. CREST Discussion Paper Hallin, M., Lefèvre, C. and M. Puri 988 On time-reversibility and the uniqueness of moving average representations for non-gaussian stationary time series. Biometrika 75, Ibragimov, I.A. and Yu. V. Linnik 97 Independent and stationary sequences of random variables. Wolters-Noordoff, Groningen. Kagan, A., Linnik, Yu. V. and C. Rao 973 Characterization problems in mathematical statistics, Wiley: New-York. Kokoszka, P. S. 996 Prediction of infinite variance fractional ARIMA. Probability and Mathematical statistics 6, Lii, K.S. and M. Rosenblatt 982 Deconvolution and estimation of transfer function phase and coefficients for non-gaussian linear processes. Annals of Statistics 0, Rosenblatt, M Gaussian and non-gaussian linear time series and random fields. Springer Verlag, New York. Samorodnitsky G. and M. S. Taqqu 994 Stable Non-Gaussian Random Processes, Chapman & Hall, London.

Strictly Stationary Solutions of Autoregressive Moving Average Equations

Strictly Stationary Solutions of Autoregressive Moving Average Equations Peter J. Brockwell Alexander Lindner Abstract Necessary and sufficient conditions for the existence of a strictly stationary solution