Propagation of Memory Parameter from Durations to Counts 6 0 0 Rohit Deo∗ Clifford M. Hurvich∗ Philippe Soulier† Yi Wang∗ 2 n a J February 2, 2008 0 3 ] T S Abstract . h t a Weestablishsufficientconditionsondurationsthatarestationarywithfinitevarianceandmemory m parameter d ∈ [0,1/2) to ensure that the corresponding counting process N(t) satisfies VarN(t) ∼ [ Ct2d+1 (C > 0) as t→∞, with the same memory parameter d ∈[0,1/2) that was assumed for the 1 durations. Thus,theseconditionsensurethatthememoryindurationspropagatestothesamemem- v ory parameter in counts and therefore in realized volatility. We then show that any Autoregressive 2 Conditional DurationACD(1,1) modelwith asufficientnumberoffinitemomentsyieldsshortmem- 4 7 oryincounts,whileanyLongMemoryStochasticDurationmodelwithd>0andallfinitemoments 1 yields long memory in counts, with the same d. Finally, we present a result implying that the only 0 way for a series of counts aggregated over a long time period to have nontrivial autocorrelation is 6 for the short-term counts to have long memory. In other words, aggregation ultimately destroys all 0 / autocorrelation in counts, if and only if the countshaveshort memory. h t a KEYWORDS:Long Memory Stochastic Duration,Autoregressive Conditional Duration,Rosenthal- m typeInequality. : v i X r a ∗NewYorkUniversity,44W.4’thStreet, NewYorkNY10012USA †Universit´eParisX,200avenuedelaR´epublique,92001Nanterrecedex,France TheauthorsthankXiaohongChenandRaymondThomasforhelpfulcomments andsuggestions. 1 I Introduction There is a growing literature on long memory in volatility of financial time series. See, e.g., Robinson (1991), Bollerslev and Mikkelsen (1996), Robinson and Henry (1999), Deo and Hurvich (2001), Hurvich, MoulinesandSoulier(2005). Longmemoryinvolatility,whichhasbeenrepeatedlyfoundintheempirical literature, plays a key role in the forecasting of realized volatility (Andersen, Bollerslev, Diebold and Labys 2001, Deo, Hurvich and Lu 2005), and has important implications on option pricing (see Comte and Renault 1998). Given the increasing availability of transaction-level data it is of interest to explain phenomena ob- served at longer time scales from equally-spaced returns in terms of more fundamental properties at the transaction level. Engle and Russell (1998) proposed the Autoregressive Conditional Duration (ACD) model to describe the durations between trades, and briefly explored the implications of this model on volatility of returns in discrete time, though they did not determine the persistence of this volatility, as measured, say,by the decay rate of the autocorrelationsof the squaredreturns. Deo, Hsieh and Hurvich (2005)proposedtheLong-MemoryStochasticDuration(LMSD)model,andbegananempiricalandthe- oreticalexplorationof the questionas to which propertiesofdurations lead to long memory in volatility, though the theoretical results presented there were not definitive. Thecollectionoftime points ···t <t ≤0<t <t <··· atwhicha transaction(say,atradeofa 1 0 1 2 − particular stock on a specific market) takes place, comprises a point process, a fact which was exploited by Engle and Russell (1988). These event times {t } determine a counting process, k N(t)=Number of Events in (0,t]. For any fixed time spacing ∆t > 0, one can define the counts ∆N = N(t∆t)−N((t −1)∆t), the t′ ′ ′ number of events in the t’th time interval of width ∆t, where t = 1,2,···. The event times {t } ′ ′ k ∞k= −∞ also determine the durations, given by {τ } , τ =t −t . k ∞k= k k k 1 −∞ − Boththe ACDandLMSDmodels imply thatthe doubly infinite sequence ofdurations{τ } are k ∞k= −∞ a stationary time series, i.e., there exists a probability measure P0 under which the joint distribution of 1 any subcollection of the {τ } depends only on the lags between the entries. On the other hand, a point k process N on the real line is stationary under the measure P if P(N(A)) = P(N(A+c)) for all real c. A fundamental fact about point processes is that in general (a notable exception is the Poissonprocess) there is no single measure under which both the point process N and the durations {τ } are stationary, k i.e., in general P and P0 are not the same. Nevertheless, there is a one-to-one correspondence between the classofmeasuresP0 thatdetermine astationarydurationsequenceandthe classofmeasuresP that determineastationarypointprocess. ThemeasureP0 correspondingtoP iscalledthePalm distribution. The counts are stationary under P, while the durations are stationary under P0. Deo,HsiehandHurvich(2005)pointedout,usingatheoremofDaley,RolskiandVesilo(2000)thatif durations aregeneratedby anACD model andif the durations havetail indexκ∈(1,2) under P0, then theresultingcountingprocessN(t)haslongrangecountdependencewithmemoryparameterd≥1−κ/2, in the sense that VarN(t) ∼ Cn1+2d (C > 0) as t → ∞, under P. This, together with the model for returns at equally spaced intervals of time given in Deo, Hsieh and Hurvich (2005) implies that realized volatility has long memory in the sense that the n-term partial sum of realized volatility has a variance that scales as C n2d+1 as n → ∞, where C > 0. Deo, Hsieh and Hurvich (2005) also showed that if 2 2 durations are generated by an LMSD model with memory parameter d under P0 then counts have long memory with memory parameter dcounts ≥ d, but unfortunately this conclusion was established only under the duration-stationarymeasureP0,andnotunder the count-stationarymeasureP. This gapcan be bridged using methods described in this paper. Still, the results we have described above merely give lower bounds for the memory parameter in counts. Inthispaper,wewillestablishsufficientconditionsondurationsthatarestationarywithfinitevariance and memory parameter d ∈ [0,1/2) under P0 to ensure that the corresponding counting process N(t) satisfiesVarN(t)∼Ct2d+1(C >0)ast→∞underP,withthesamememoryparameterd∈[0,1/2)that was assumed for the durations. Thus, these conditions ensure that the memory in durations propagates to the same memory parameter in counts and therefore in realized volatility. Next, we will verify that the sufficient conditions of our Theorem 1 are satisfied for the ACD(1,1) 2 modelassumingfinite8+δmoment(δ >0)ofthedurationsunderP0,andfortheLMSDmodelwithany d ∈ [0,1/2) assuming that the multiplying shocks have all moments finite. Thus, any ACD(1,1) model with a sufficient number of finite moments yields short memory in counts, while any LMSD model with d > 0 and all finite moments yields long memory in counts. These results for the LMSD and ACD(1,1) models are given in Theorems 2 and 3, respectively. Lemma 1, which is used in proving Theorem 2, provides a Rosenthal-type inequality for moments of absolute standardized partial sums of durations under the LMSD model, and is of interest in its own right. Finally, we present a result (Theorem 4) implying that if counts have memory parameter d∈[0,1/2) then further aggregations of these counts to longer time intervals will have a lag-1 autocorrelation that tends to 22d −1 as the level of aggregation grows. Interestingly, this limit is zero if and only if d = 0. Thus, one of the important functions of long memory in counts is that it allows the counts to have a non-vanishingautocorrelationevenas∆tgrows,aswasfoundbyDeo,HsiehandHurvich(2005)tooccur inempiricaldata. Bycontrast,shortmemoryincountsimpliesthatcountsatlongtime scales(large∆t) are essentially uncorrelated, in contradiction to what is seen in actual data. To summarize, aggregation ultimately destroys all autocorrelation in counts, if and only if the counts have short memory. II Theorems on the propagation of the memory parameter LetE,E0,Var,Var0 denoteexpectationsandvariancesunderP andP0,respectively. Defineµ=E0(τ ) k and λ = 1. Our main theorem uses the assumption that P0 is {τ }-mixing, defined as follows. Let µ k N =σ({τ } ) and F =σ({τ } ). We say that P0 is {τ }-mixing if k ∞k= n k ∞k=n k −∞ lim sup |P0(A∩B)−P0(A)P0(B)|=0 n→∞B∈N∩Fn for all A∈N. Theorem 1 Let {τ } be a duration process such that the following conditions hold: k i) {τ } is stationary under P0. k 3 ii) P0 is {τ }-mixing. k iii) ∃ d∈[0,1) such that 2 Y (s)= ⌊kn=s1⌋(τk−µ), s∈[0,1] n n1/2+d P converges weakly to σB (·) under P0, where σ > 0 and B (·) is fractional Brownian motion if 1/2+d 1/2+d 0<d< 1 or standard Brownian motion B =B if d=0. 2 1/2 iv) n (τ −µ) p for all p>0, if d∈(0,1) supE0 k=1 k <∞ 2 n1/2+d  n (cid:12)P (cid:12)  for p=8+δ, δ >0, if d=0 · (cid:12) (cid:12) (cid:12) (cid:12)  Then the induced counting process N(t) satisfies VarN(t)∼Ct2d+1 under P as t→∞ where C >0. Remark: Inspection ofthe proofofTheorem1 revealsthatif d>0, only 4/(0.5−d)+δ finite moments are needed, where δ > 0 is arbitrarily small. The closer d is to 1/2, the larger the number of finite moments required. Remark: As pointed out by Nieuwenhuis (1989), if {τ } is strong mixing under P0 then P0 is {τ }- k k mixing. This weaker form of mixing is essential for our purposes since even Gaussian long-memory processes are not strong mixing. See Gu´egan and Ladoucette (2001). A LMSD Process Define the LMSD process {τ } for d∈[0,1) as k ∞k= 2 −∞ τk =ehkǫk where under P0 the ǫk ≥ 0 are i.i.d. with all moments finite, and hk = ∞j=0bjek j, the {ek} are i.i.d. − Gaussian with zero mean, independent of {ǫ }, and P k Cjd 1 if d∈(0,1) − 2 b ∼ j   Caj, |a|<1 if d=0  4 (C 6=0) as j →∞. Note that for convenience, we nest the short-memory case (d=0) within the LMSD model, so that the allowable values for d in this model are 0≤d<1/2. Theorem 2 If the durations {τ } are generated by the LMSD process with d∈[0,1/2), then the induced k counting process N(t) satisfies VarN(t)∼Ct2d+1 under P as t→∞ where C >0. To establish Theorem 2, we will use the following Rosenthal-type inequality. Lemma 1 For durations {τ } generated by the LMSD process with d ∈ [0,1), for any fixed positive k 2 integer p≥2, E0{|y −E0(y )|p} is bounded uniformly in n, where n n n τ y = k=1 k . n n1/2+d P B ACD(1,1) Process Define the ACD(1,1) process {τ } as k ∞k= −∞ τ = ψ ǫ k k k ψ = ω+ατ +βψ k k 1 k 1 − − with ω >0,α>0,β ≥0 and α+β <1, where under P0, ǫ ≥0 are i.i.d. with mean 1. We will assume k further that under P0, ǫ has a density g such that θg (x)dx>0,∀ θ >0 andE0(τ8+δ)<∞ for some k ǫ 0 ǫ k δ >0. R Nelson (1990) guarantees the existence of the doubly-infinite ACD(1,1) process {τ } , which in k ∞k= −∞ our terminology is stationary under P0. Theorem 3 Suppose that the durations {τ } are generated by the ACD(1,1) model, with the additional k assumptions stated above. Then the induced counting process N(t) satisfies VarN(t) ∼ Ct under P as t→∞ where C >0. 5 III Autocorrelation of Aggregated Counts Theorem 4 Let{X }beastationaryprocesssuchthatVar( n X )∼Cn1+2d asn→∞,whereC 6=0 t t=1 t and d∈[0,1/2). Then P n 2n lim Corr X , X =22d−1. t t n " # →∞ t=1 t=n+1 X X Proof: 2n n n 2n Var X =2Var X +2Cov X , X . t t t t " # " # " # t=1 t=1 t=1 t=n+1 X X X X Thus, n 2n 2n n Cov X , X =.5 Var X −2Var X . t t t t " # " # " #! t=1 t=n+1 t=1 t=1 X X X X The result follows by noting that lim n 2d 1Var( n X )=C, where C 6=0. (cid:3) n − − t=1 t →∞ P This theorem has an interesting practical interpretation. If we write X = N[k∆t]−N[(k−1)∆t] k where ∆t > 0 is fixed, then X represents the number of events (count) in a time interval of width ∆t, k e.g. one minute. Thus, n X is the number of events in a time interval of length n minutes, e.g. one k=1 k day. The theorem impliPes that as the level of aggregation (n) increases, the lag-1 autocorrelation of the aggregatedcounts will approach a nonzero constant if and only if the non-aggregatedcount series {X } k has long memory. In other words, the only way for a series of counts over a long time period to have nontrivial autocorrelation is for the short-term counts to have long memory. Since in practice long-term counts do have substantial autocorrelation (see Deo, Hsieh and Hurvich 2005), it is important to use onlythemodelsfordurationsthatimplylongmemoryinthecountingprocess(LRcD).Examplesofsuch models include the LMSD model (see Theorem 2), and ACD models with infinite variance (see Daley, Rolski and Vesilo (2000), and Theorem 2 of Deo, Hsieh and Hurvich, 2005). 6 IV Appendix: Proofs Let P denote the stationary distribution of the point process N on the real line, and let P0 denote the corresponding Palm distribution. P determines and is completely determined by the stationary distri- bution P0 of the doubly infinite sequence {τ } of durations. Note that the counting process N k ∞k= −∞ is stationary under P, the durations are stationary under P , but in general there is no single distribu- 0 tion under which both the counting process and the durations are stationary. For more details on the correspondence between P and P0, see Daley and Vere-Jones (2003), Baccelli and Br´emaud (2003), or Nieuwenhuis (1989). Following the standard notation for point processes on the real line (see, e.g., Nieuwenhuis 1989, p. 594), we assume that the event times {t } satisfy k ∞k= −∞ ...<t <t ≤0<t <t <.... 1 0 1 2 − Let t if k =1 1 u = k   τk if k ≥2 · Here, the random variable t1 > 0 is the time of occurrence of the first event following t = 0. For t > 0, define the count on the interval (0,t], N(t):=N(0,t], by s N(t) = max{s: u ≤t}, u ≤t i 1 i=1 X = 0, u >t. 1 Throughout the paper, the symbol =⇒ denotes weak convergence in the space D[0,1]. Proof of Theorem 1: By assumption iii), Y =⇒ σB under P0, where σ > 0. First, we will apply Theorem 6.3 of n 1/2+d Nieuwenhuis (1989) to the durations {τ } to conclude that Y =⇒ σB under P. Since the k ∞k= n 1/2+d −∞ {τ } are stationary under P0 and are generated by the shift to the first event following time zero k ∞k= −∞ 7 (see Nieuwenhuis 1989, p. 600), and since we have assumed that P0 is {τ }-mixing, his Theorem 6.3 k applies. It follows that Y =⇒ σB under P. We next show that the suitably normalized counting n 1/2+d process converges to the same limit under P. Define Y˜ (s)= ⌊kn=s1⌋(uk−µ) , s∈[0,1]. n n1/2+d P Note that for all s, Y˜ (s) = Y (s)+n (1/2+d)(u −τ ). From Baccelli and Br´emaud (2003, Equation n n − 1 1 1.4.2, page 33), for any measurable function h, E[h(τ )]=λE0[τ h(τ )] . (1) 1 1 1 Since u ≤ τ , and since assumption iv) implies that τ has finite variance under P0, using h(x) = x in 1 1 1 (1), it follows that n (1/2+d)(u −τ ) is o (1) under P. Thus, Y˜ =⇒σB under P. − 1 1 p n 1/2+d Let N(t)−t/µ Z(t)= . (2) t1/2+d By Iglehart and Whitt (1971, Theorem 1), it follows that Z(t) →d C˜B (1) under P as t → ∞, 1/2+d whereC˜ >0. Furthermore,byLemma2,Z2(t)isuniformlyintegrableunderP andhencelim Var[Z(t)]= t C˜2Var[B (1)]. The theorem is proved. (cid:3) 1/2+d Proof of Theorem 2: We simply verify that the conditions of Theorem 1 hold for this process. By definition {τ } is stationary under P0 and by Lemma 4, P0 is {τ } mixing. By Surgailis and k k Viano (2002), Yn =⇒σB1/2+d under P0, where σ >0 and by Lemma 1, supnE0 nkn=11/(2τ+kd−µ) p <∞ for P all p. Thus, the result is proved. (cid:3) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12) (cid:12) Proof of Theorem 3: We simply verify that the conditions of Theorem 1 hold for this process. 8 ByLemma4,{τ }isexponentialα-mixing,andhencestrongmixingandthusbyNieuwenhuis(1989), k P0 is {τ }-mixing. Furthermore, since all moments of τ exist up to order 8+δ,δ > 0, we can apply k k results from Doukhan (1994) to obtain Y ⇒CB, (3) n if 1var( n τ )→C2 >0, as n→∞. n k=1 k P It is well known that the GARCH(1,1) model can be represented as an ARMA(1,1) model, see Tsay (2002). Similarly, the ACD(1,1) model can also be re-formulated as an ARMA(1,1) model, τ =ω+(α+β)τ +(η −βη ) (4) k k 1 k k 1 − − whereη =τ −ψ iswhitenoisewithfinitevariancesinceE(τ8+δ)<∞. Theautoregressiveandmoving k k k k average parameters of the resulting ARMA(1,1) model are (α+β) and β, respectively. It is also known that for any stationary invertible ARMA model {z }, nvar(z¯) → 2πf (0), where k z f (0) is the spectral density of {z } at zero frequency. For an ARMA(1,1) process, f (0) > 0 if the z k z moving average coefficient is less than 1. Here, since 0 ≤β < 1, we obtain 1var( n τ ) =nvar(τ¯) → n k=1 k 2πf (0)>0, as n→∞. Therefore (3) follows. P τ Define y = 1 n τ . Since all moments of τ are bounded up to order 8+δ, (δ >0) under P0, n √n k=1 k k by Yokoyama (1980P), we obtain E0{|y −E0(y )|8+δ}≤K <∞, δ >0 (5) n n uniformly in n, provided that {τ } is exponential α-mixing, which is proved in Lemma 4. k Therefore, we can apply Theorem 1 to the ACD(1,1) model and the result follows. (cid:3) Proof of Lemma 1: We present the proof for the case 0<d< 1. The proof for the case d=0 follows along similar lines. 2 Also, we assume here that p is a positive even integer. The result for all positive odd integers follows by Ho¨lder’s inequality. 9