The Dynamics of Squared Returns Under Contemporaneous Aggregation of GARCH Models

Transcription

1 The Dynmics of Squred Returns Under Contemporneous Aggregtion of GARCH Models Eric Jondeu (This version: June 2012) Abstrct The pper investigtes the properties of portfolio composed of lrge number of ssets driven by strong multivrite GARCH(1,1) process with heterogeneous prmeters. The ggregte return is shown to be wek GARCH process with (possibly lrge) number of lgs, which reflect the moments of the distribution of the individul persistence prmeters. The pper describes consistent estimtors of the ggregte return dynmic, bsed either on nonliner lest squres or on minimum distnce, when only ggregte dt is vilble. Monte-Crlo simultions demonstrte tht the proposed ggregtion-corrected estimtor (ACE) performs very well under relistic sets of prmeters drwn from U.S. equity dt. Finlly, the ACE is shown to outperform some competing estimtors in forecsting the dily vrince of U.S. portfolios. Keywords: Aggregtion, Heterogeneity, GARCH model, Voltility. JEL clssifiction: C13, C21, G17. Swiss Finnce Institute nd University of Lusnne, Fculty of Business nd Economics, CH 1015 Lusnne, Switzerlnd. E-mil: 1

2 1 Introduction Generlized AutoRegressive Conditionl Heteroskedsticity (or GARCH) models re commonly used to model the voltility of finncil returns, such s stock returns, interest rtes, or currency returns. Multivrite extensions llow to model the joint dynmic of severl processes. These models re now used s stndrd tools for sset nd risk mngement. In these res, one key issue is the level of ggregtion tht should be considered for modeling the dynmics of the portfolio return. Two lterntive pproches re redily vilble (see Andersen et l., 2005). On the one hnd, the sset-level pproch requires estimtion of the joint behvior of ll the ssets in the portfolio, thus llowing one to cpture the interctions between sset returns nd to evlute their implictions for portfolio return. In the cse of lrge portfolios, however, the computtionl burden renders this pproch brely fesible. On the other hnd, the portfolio-level pproch requires only modeling of the portfolio return, but it is often inpproprite for scenrio nlysis. This pproch is in generl preferred for computtionl resons. Another instnce in which the portfolio-level pproch is nturlly dopted is the modeling of sectorl indices, sset clsses, or risk fctors. In such cses, it is often impossible to identify the underlying individul ssets precisely, nd therefore the properties of the resulting portfolio cn be inferred from the ggregte process only. An importnt, yet often overlooked, issue rised by the portfolio-level pproch is tht stndrd strong GARCH models re not closed under ggregtion. This issue ws first ddressed by Nijmn nd Sentn (1996) for the ggregtion of two independent processes with the sme level of voltility persistence. They showed tht the ggregte portfolio does not shre the sme properties s the individul ssets. The properties of the resulting process, nmely the wek GARCH model, hve been studied by Drost nd Nijmn (1993), Nijmn nd Sentn (1996), nd Meddhi nd Renult (2004). Some issues rised by the estimtion of wek GARCH process hve been ddressed by Frncq nd Zkoin (2000) nd Komunjer (2001). One prticulrly striking result is tht, even in the simple cse of two independent ssets with the sme voltility persistence, stndrd 2

3 estimtion techniques fil to provide consistent estimtes of the prmeters driving the observed ggregte process (see Nijmn nd Sentn, 1996). The contemporneous ggregtion of lrge number of series hs been lredy investigted in closely relted field, nmely the ARMA processes. Erlier results hve been provided by Grnger nd Morris (1976) nd Grnger (1980). They hve estblished tht ggregting ARMA(1,1) processes results in n ARMA process whose number of lgs increses with the number of processes under ggregtion. Grnger (1980) shows tht in generl the ggregte process exhibits long-memory fetures. Since wek GARCH process cn be viewed s n ARMA process for squred returns, similr result holds for the contemporneous ggregtion of GARCH(1,1) processes. Ding nd Grnger (1996) nd Kzkevičius, Leipus, nd Vino (2004) hve investigted the ggregtion of n- component GARCH processes, in which the individul voltility processes re driven by common innovtion. Zffroni (2007) extends this reserch to the ggregtion of stndrd GARCH(1,1) processes in the context of n infinite number of series nd demonstrtes the symptotic behvior of the vrince of the ggregte process. He shows tht conditionl heteroskedsticity is preserved provided the degree of cross-sectionl dependence between the ssets is sufficiently strong. This pper provides some new results on the dynmic properties of portfolio composed of lrge number of ssets. The individul ssets re ssumed to be driven by multivrite GARCH(1, 1) vrince process with heterogenous prmeters. 1 As expected, ggregte squred returns follow n ARMA-type process, reflecting the wek GARCH nture of the ggregte returns. In generl, lrge number of lgs is required to fully cpture the effect of the ggregtion of heterogenous processes on the dynmic of the portfolio squred return. The first contribution of the pper is to show tht the prmeters driving the dynmic of the ggregte squred returns depend on the properties of the individul prmeters. They cn be interpreted in terms of moments of the cross-section distribution of the voltility persistence prmeters. This result hs importnt implictions from 1 The min difference with Zffroni s (2007) set-up relies on the form of the dependence cross the ssets. Zffroni considers two extreme cses, with purely idiosyncrtic innovtions or with common innovtions. This pper considers the cse of cross-correlted innovtions, in stndrd multivrite GARCH frmework. 3

4 modeling perspective. The fct tht the ggregte squred return does not shre the sme specifiction s the individul components is known for long time. However, there ws no cler description of how the ggregte specifiction should look like. This pper is the first one to clerly estblish the ggregte dynmic s function of the individul ones. Given the resulting dynmic of the ggregte process, the usul estimtor, i.e., the Qusi Mximum Likelihood Estimtor (QMLE) for strong GARCH(1, 1) process, is inconsistent. The pper describes two consistent estimtors for the prmeters of the ggregte process when only ggregte dt re vilble. These estimtors, bsed either on nonliner lest squre or on minimum distnce, tke dvntge of the reltionship between the prmeters of the ggregte voltility dynmics nd the moments of the cross-section distribution of the voltility persistence in order to reduce the estimtion burden. The proposed Aggregtion-Corrected Estimtors (ACE) do not rely on QMLE in order to circumvent the misspecifiction of the conditionl voltility. Using Monte-Crlo simultions, the new estimtors re shown to perform very well in finite smples for severl relistic prmeteriztions. They provide unbised estimtes of the prmeters driving the portfolio return dynmic. They lso llow inferring the properties of the cross-section distribution of voltility persistence prmeters, lthough they rely on ggregte dt only. Finlly, the forecsting performnces of competing estimtors of the ggregte squred returns re compred to the so-clled disggregted estimtor (DISE) bsed on the ggregtion of individul voltility forecsts. The ACE significntly outperforms the DISE nd the QMLE in forecsting the vrince of portfolio of U.S. equities. The reminder of the pper is orgnized s follows: Section 2 briefly describes the multivrite GARCH model used for the individul ssets nd provides conditions required for the model to be well behved. Section 3 considers the ggregtion of individul voltility processes nd provides the min results for the dynmics of ggregte squred returns. Section 4 describes the ACE, which correctly cptures the dynmic of the ggregte voltility process. Section 5 evlutes the performnces of the proposed estimtor using both simulted dt nd ctul dt on U.S equities. Section 6 concludes. 4

5 2 The Model for Individul Assets This section describes the set-up. The investment set is composed of lrge number of individul ssets, driven by correlted heterogenous GARCH processes. We ssume digonl vec multivrite model, introduced by Bollerslev, Engle, nd Wooldridge (1988), in which the dynmic of the unexpected returns, ε t = {ε i,t }, t Z, i = 1,, N, is given by: ε t = H 1/2 t z t, (1) X t = ε t ε t, (2) H t = Ω + A X t 1 + B H t 1, (3) where z t is the (N, 1) vector of idiosyncrtic innovtions with E[z t ] = 0 nd V [z t ] = I N, H t = {h ij,t } is the (N, N) conditionl covrince mtrix, nd X t is the (N, N) mtrix of cross-products of unexpected returns with elements X ij,t = ε i,t ε j,t. 2 Mtrices Ω = {ω ij }, A = {α ij }, nd B = {β ij } re (N, N) mtrices of prmeters, nd denotes the Hdmrd product, such tht {A B} ij = α ij β ij. In this specifiction, the individul vrinces nd the covrinces re ll described by GARCH(1,1) process with heterogenous prmeters. 3 Strightforwrd mnipultion of equtions (1) (3) gives multivrite ARMA(1,1) model for the cross-product of unexpected returns: X t = Ω + Γ X t 1 + v t B v t 1, (4) where v t = X t H t denotes the (N, N) mtrix of covrince innovtions nd Γ = {γ ij } = A+B is the (N, N) mtrix of persistence prmeters. The properties of X t re rther well known. They hve been studied in detil by Comte nd Liebermn (2003) nd Hfner 2 The squre root mtrix H 1/2 t in eqution (1) is defined s H t = H 1/2 t H 1/2 t. There re lterntive wys of defining the squre root of covrince mtrix, such s the Cholesky decomposition or the spectrl decomposition. We use the Cholesky decomposition in the empiricl prt of the pper. 3 Alterntive specifictions to the digonl vec model could be considered s well. The results of this pper could lso be esily dpted to the vech model (Bollerslev, Chou, nd Kroner, 1988) or BEKK model (Engle nd Kroner, 1995). 5

6 (2003) in similr frmework. In prticulr, the innovtion v t is wek white noise, with E[v t ] = 0, E[vech(v t )vech(v t ) ] = Σ v, nd E[vech(v t )vech(v s ) ] = 0, s t, lthough v t is not n i.i.d. sequence. In the sequel, we mke the following set of ssumptions, closely relted to those dopted in Hfner (2003). Assumption 1 (Positivity) The mtrices Ω, A, nd B re symmetric positive semidefinite lmost surely. Assumption 2 (Sttionrity) Ω < + nd ll eigenvlues of Γ = A + B re smller thn 1 in modulus. Assumption 3 (Innovtion process) The innovtion process {z t }, t Z, is i.i.d. nd its distribution hs finite fourth moments. Assumption 1 ensures tht the conditionl covrince mtrix H t is symmetric positive semi-definite lmost surely (Ding nd Engle, 1994). Assumption 2 ensures tht the multivrite GARCH process ε t is covrince sttionry (Engle nd Kroner, 1995). Under Assumptions 1 nd 2, the multivrite model is well behved, with positive semi-definite covrince mtrix t ech dte t. Assumption 3 implies tht the time dependency in the model is fully cptured by the covrince mtrix H t, so tht the innovtion z t is n i.i.d. vector. 3 Aggregtion This section describes the properties of portfolio composed of N risky ssets with strong GARCH(1,1) conditionl covrince mtrix. A risk-free sset my be included in the portfolio, provided ε i,t is defined s the unexpected excess return over the risk-free rte. Portfolio weights re denoted by w = (w 1,, w N ). The weights stisfy w i = O(N 1 ), with 0 < w i < 1, i, nd N i=1 w i = 1. 4 The ggregte unexpected return is defined s 4 Portfolio weights re ssumed to be constnt over time. This ssumption typiclly corresponds to strtegic lloction problem. We could lso consider the cse of n eqully-weighted portfolio, with w i = 1/N, i. The cse for time-vrying weights would rise new issues, in prticulr if their dynmics depend on the prmeters of the individul squred returns. The investigtion of this extension is beyond the scope of this pper nd left for future reserch. 6

7 ε (w) p,t = N i=1 w iε i,t, i.e., the cross-section men of ε i,t, where the exponent (w) indictes tht the men is computed over the N ssets with weights w. The ggregte squred return is then defined s: ( X (w) p,t = ε (w) p,t ( ) N ) 2 2 = w i ε i,t = i=1 N N w i w j X ij,t. i=1 j=1 In the following, we will omit the exponent (w) from the ggregte vribles to sve on nottions. For given weight vector, the cross-section moment of order k for prmeter ϑ driving the covrince mtrix dynmic is denoted by [ Ẽ(w) ϑ k] m (ϑ) k = N j=1 w iw j ϑ k ij, where the tilde is used to void confusion with the time series ver- N i=1 ge. Under Assumptions 1 to 3, the following proposition holds regrding the dynmic of the ggregte squred returns. Proposition 1 Let {ε t }, t Z, be n N-dimensionl strong GARCH(1, 1) process defined by equtions (1) (3), with X t = ε t ε t. Under Assumptions 1 to 3, the ggregte squred return X p,t stisfies in the limit (s T ): X p,t = h p,t + v p,t, (5) h p,t = Ω p + Ψ k X p,t k + Φ k h p,t k, (6) where h p,t is the liner projection of X p,t on F t = {1, X p,t 1, X p,t 2, }. The ggregte innovtion, v p,t = N i=1 N j=1 w iw j v ij,t, is wek white noise with zero men nd constnt vrince. Ω p is defined in Appendix 1. The ggregte squred return lso stisfies the infinite ARMA representtion: X p,t = Ω p + Λ k X p,t k + v p,t Φ k v p,t k. (7) 7

8 Prmeters Ψ k, Φ k, nd Λ k in equtions (6) nd (7) denote the following cross-section mens: Ψ k = Ẽ(w) [ψ k ], ψ 1 = α, ψ k = (λ k 1 Λ k 1 ) α, k = 1, 2, Φ k = Ẽ(w) [φ k ], φ 1 = β, φ k = (λ k 1 Λ k 1 ) β, Λ k = Ẽ(w) [λ k ], λ 1 = γ, λ k = (λ k 1 Λ k 1 ) γ = ψ k + φ k. Proof: See Appendix 1. There re severl comments regrding Proposition 1. First, the term h p,t in eqution (6) cnnot be interpreted s the conditionl vrince process of the ggregte return (Komunjer, 2001). Therefore, the min eqution of interest is the dynmic of the ggregte squred return (eqution (7)). As this eqution indictes, the ggregte return is driven by wek GARCH process with n infinite number of lgs. In fct, depending on the chrcteristics of the individul processes, the ggregte ARMA process my be of low order. For instnce, ssume tht prmeters {α ij } nd {β ij } re heterogeneous cross ssets, but tht ll of the {γ ij } re equl to single vlue, γ ij = γ, i, j, such tht ll the ssets shre the sme voltility persistence. In such model, the ggregte squred return simply writes s n ARMA(1,1) process: X p,t = Ω p + Λ 1 X p,t 1 + v p,t Φ 1 v p,t 1, (8) with Λ 1 = γ, Φ 1 = Ẽ(w) [β] nd Λ k = Φ k = 0, k > 1 in eqution (7) (see Appendix 1). A similr ARMA(1,1) process ws obtined by Nijmn nd Sentn (1996, eqution (10)) in the cse of two independent processes with the sme persistence prmeter. This exmple clerly illustrtes tht the dditionl lgs in eqution (7) rise from the heterogeneity in the persistence prmeter. Proposition 1 lso helps understnding why estimting wek GARCH(1, 1) model for the ggregte process cnnot correct for the heterogeneity cross the ssets. Indeed, it provides consistent estimtes of the ggregte prmeters (Ω p, Λ 1, Φ 1 ) in eqution (7) only if the dditionl lgs Λ k nd Φ k re ll equl to 0, k > 1, i.e., if ll of the {γ ij } re equl to single vlue. 8

9 Second, the dynmic of the ggregte squred return is driven by the properties of the sequences of prmeters {Λ k } nd {Φ k }. Contemplting eqution (7) revels tht the prmeters {Λ k } re relted to the moments of the cross-section distribution of the persistence prmeters {γ ij }. Indeed, strightforwrd computtion shows tht: k 1 Λ k = Ẽ(w) [λ k ] = Ẽ(w) [γ k ] Λ r Ẽ (w) [γ k r ], k 1. (9) As the {γ ij } re bounded between 0 nd 1 (under Assumption 3), the cross-section moments of order k, m (γ) k r=1 = Ẽ(w) [γ k ], re decresing to 0 s k, with 0 m (γ) s m (γ) s 1 m (γ) 1 < 1, s > 1. As eqution (9) clerly indictes, Λ k is liner combintion of terms of the form {m (γ) k, m(γ) k 1 m(γ) 1,, (m (γ) 1 ) k } with ll the terms involving sum of powers of γ equl to k. For instnce, the first Λ k re: Λ 1 = m (γ) 1, Λ 2 = m (γ) 2 (m (γ) 1 ) 2, Λ 3 = m (γ) 3 2m (γ) 1 m (γ) 2 + (m (γ) 1 ) 3, Λ 4 = m (γ) 4 2m (γ) 1 m (γ) 3 + 3(m (γ) 1 ) 2 m (γ) 2 (m (γ) 2 ) 2 (m (γ) 1 ) 4. Strightforwrd, but tedious computtion lso shows tht: (1) the {Λ k } re positive, given tht γ is bounded between 0 nd 1; (2) the {Λ k } re decresing to 0 s k increses to ; (3) the limit sum of the {Λ k } prmeters is given by: Λ k = [ ] k 1 Ẽ (w) [γ k ] Λ r Ẽ (w) [γ k r ] = r=1 Ẽ(w) [γ k ] 1 + Ẽ(w) [γ k ], (10) so tht 0 Λ k < 1. Therefore, there exist n ε > 0 nd K Λ N, such tht 0 Λ k ε, k > K Λ. Thus, we cn select the number of lgs in eqution (7) in such wy tht the contribution of the non-included terms Λ k, k > K Λ, is mde negligible reltive to tht of the included prt. It is worth noticing tht the estimtes of prmeters {Λ k } from ggregte eqution (7) cn be used to recover the cross-section moments of the individul persistence prmeters 9

10 {γ ij }. 5 The first moments of {γ ij } re: Ẽ (w) [γ] = Λ 1, (11) Ṽ (w) [γ] = Λ 2, (12) S (w) [γ] = (Λ 3 Λ 1 Λ 2 ) / (Λ 2 ) 3/2, (13) K (w) [γ] = ( Λ 4 2Λ 1 Λ 3 + Λ 2 1Λ 2 + Λ2) 2 / (Λ2 ) 2, (14) where Ṽ (w), S(w), nd K (w) denote the cross-section vrince, skewness, nd kurtosis, respectively. As the knowledge of {Λ k },,KΛ is equivlent to the knowledge of {m (γ) k },,K Λ, we cn deduce the cross-section properties of the individul persistence prmeter, even when only ggregte dt is vilble. Prmeters {Φ k } re defined in similr wy: k 1 Φ k = Ẽ(w) [φ k ] = Ẽ(w) [βγ k 1 ] Φ r Ẽ (w) [γ k r ], k 1. (15) r=1 Prmeter Φ 1 corresponds to the cross-section men of {β ij }. Subsequent prmeters Φ k pertining to lgs v p,t k correspond to cross-section co-moments between {β ij } nd powers of {γ ij }. For instnce, Φ 2 = Ẽ(w) [βγ] Ẽ(w) [β]ẽ(w) [γ] is the cross-section covrince between {β ij } nd {γ ij }. Consequently, estimtes of prmeters {Φ k } provide informtion bout the cross-section joint distribution of {β ij } nd {γ ij }. As bove, we cn compute the limit sum of the {Φ k } prmeters s: Φ k = [ ] k 1 Ẽ (w) [βγ k 1 ] Φ r Ẽ (w) [γ k r ] = r=1 Ẽ(w) [βγ k 1 ] 1 + Ẽ(w) [γ k ], (16) so tht 0 < Φ k < Λ k < 1, s ll the {β ij } nd {γ ij } lie in (0, 1), with β ij γ ij. Agin, number of lgs K Φ cn be selected in eqution (7) in such wy tht the terms Φ k, k > K Φ, re mde negligible. 5 A similr interprettion hs been proposed by Robinson (1978) nd Grnger (1980) for the ggregtion of utoregressive processes nd by Lewbel (1994) for the ggregtion of liner dynmic processes. 10

11 With similr computtion, we obtin tht in the limit the ggregte constnt term Ω p is defined s: Ω p = Ẽ(w) [ω] + Ẽ (w) [(λ k Λ k ) ω] = Ẽ(w) [ωγ k 1 ] 1 + Ẽ(w) [γ k ]. (17) Given these expressions, it is possible to investigte the conditions ensuring the sttionrity of the ggregte unexpected return process ε p,t s (K Λ, K Φ ). Covrince sttionrity requires tht V [ε p,t ] = E[X p,t ] <. The ggregte process (7) cn be rewritten s: Λ(L)X p,t = Ω p + Φ(L)v p,t, (18) where Λ(L) = 1 Λ 1 L Λ 2 L 2 nd Φ(L) = 1 Φ 1 L Φ 2 L 2, with L the lg opertor. The condition for invertibility of Λ(L) is tht the roots of Λ(z) = 0 lie outside the unit circle, which implies Λ(1) > 0 or, equivlently, Λ k < 1. This condition is gurnteed by eqution (10). Therefore, we cn rewrite eqution (18) s: X p,t = Λ(1) 1 Ω p + δ(l)v p,t, with δ(l) = Λ(L) 1 Φ(L) nd δ 0 = 1. We obtin tht the unconditionl vrince of the ggregte return is: E[X p,t ] = Ω p 1 Λ k = [ ] ω Ẽ (w) [ωγ k 1 ] = Ẽ(w) = 1 γ N N w i w j h ij, i=1 j=1 where h ij = ω ij /(1 γ ij ) denotes the unconditionl covrince between ssets i nd j. Under Assumptions 1 nd 2, the {h ij } re finite nd so is E[X p,t ]. Therefore, ε p,t is covrince sttionry process. 4 Estimtion of the Aggregte Model In this section, we describe two estimtion procedures for the ggregte process in the cse of contemporneous ggregtion, when no informtion is vilble bout the individul ssets. 11

12 Imposing strong GARCH(1,1) structure to the ggregte squred return: X p,t = h p,t + v p,t, (19) h p,t = Ω p + Ψ 1 X p,t 1 + Φ 1 h p,t 1, (20) will yield inconsistent estimtors in generl. Indeed the prmeters Ψ 1 nd Φ 1 will correspond to the theoreticl vlues Ẽ(w) [α] nd Ẽ(w) [β] only if the persistence prmeters re in fct homogeneous cross ssets, i.e. γ ij = γ, i, j. As consequence, except under very specil circumstnces, the ggregte squred return cnnot be expected to be strong GARCH(1,1) process, so tht the QMLE is n inconsistent estimtor. This result hs been lredy pointed out by Nijmn nd Sentn (1996) in the cse of two independent individul processes: the QML estimtor is pproximtely consistent in some cses nd clerly inconsistent in others. Komunjer (2001) hs estblished in their context the resons for the inconsistency of the QMLE, which lso pply to this pper The Lest-Squre Estimtor (LSE) A first pproch designed for the estimtion of wek GARCH(p,q) process hs been proposed by Frncq nd Zkoin (2000). This estimtor, clled Lest-Squre Estimtor (LSE), explicitly cknowledges the possible misspecifiction of the first two moments nd therefore does not rely on QMLE. It is bsed on the minimiztion of the sum of the squred residuls of the ggregte squred return process. The definition of the resulting Lest-Squre Aggregtion-Corrected Estimtor (LS-ACE) is given below. Definition 1 The Lest-Squre Aggregtion-Corrected Estimtor LS-ACE(K Λ,K Φ ), denoted by θ LS AC = (Ω p, Λ 1,, Λ KΛ, Φ 1,, Φ KΦ ), is defined s: θ LS AC rg min θ Q T (θ) = 1 T T (v p,t (θ)) 2, (21) t=1 6 Komunjer (2001) lso proposes new QMLE for wek GARCH processes, designed to ccount for the deficiencies of the stndrd estimtor in this context. Unfortuntely, the lrge-smple properties of this new estimtor re still quite disppointing, resulting in severe underestimtion of α, similr to tht obtined with the Nijmn nd Sentn (1996) pproch. 12

13 where K Λ K Φ v p,t (θ) = X p,t Ω p Λ k X p,t k + Φ k v p,t k (θ). (22) The expressions for Λ k nd Φ k re given in Proposition 1, the expression for Ω p is in Appendix 1. The consistency nd symptotic normlity of this estimtor hve been proved in Frncq nd Zkoin (2000). They lso describe how to construct the symptotic covrince mtrix of the estimtor, even under situtions where ε t is the innovtion of preliminry ARMA filter. In our context, the symptotic distribution is given by: T (ˆθLS AC θ 0 ) N(0, V LS ), with symptotic covrince mtrix V LS = J 1 IJ 1, where I = lim T V [ ] T θ Q T (θ 0 ) nd ( ) 2 J = lim Q T (θ 0 ) T θ i θ j re evluted t the true prmeter vlue θ The Minimum Distnce Estimtor (MDE) A second estimtor, which does not rely on QMLE, hs been proposed by Billie nd Chung (2001) in the context of GARCH(1,1) process. This minimum distnce estimtor (MDE) is motivted by the ide of replicting some properties of the dt. A typicl exmple of such properties is the utocorrelogrm of the ggregte squred returns. The objective of the estimtor is then to minimize the distnce between the theoreticl utocorreltions nd their empiricl counterprts. It is described more precisely in the following definition. Definition 2 The Minimum-Distnce Aggregtion-Corrected Estimtor MD-ACE(K Λ, K Φ,K ρ ), denoted by θ MD AC = (Ω p, Λ 1,, Λ KΛ, Φ 1,, Φ KΦ ), is defined s: θ MD AC rg min θ (ρ (θ) ˆρ) W (ρ (θ) ˆρ), (23) 13

14 where ρ (θ) = ( ρ 1 (θ),, ρ Kρ (θ) ) nd ˆρ = (ˆρ 1,, ˆρ Kρ ) denote the first K ρ theoreticl utocorreltions of n ARMA(K Λ, K Φ ) process nd their empiricl counterprts, respectively, nd W is weighting mtrix. The constnt term is estimted by Ω p = (1 K Λ Λ k)e [X p,t ]. A usul choice for the weighting mtrix W is consistent estimte of the inverse of the covrince mtrix of ˆρ. The theoreticl utocorreltions ρ (θ) re obtined using the pproch described in Brockwell nd Dvis (1991, section 3.3). Billie nd Chung (2001) give the symptotic distribution of the MDE: T (ˆθMD AC θ 0 ) N(0, V MD ), with V MD = (D C 1 D) 1, where D = ρ(θ)/ θ is evluted t the true prmeter vlue θ 0 nd C = {c ij }, c ij = (ρ k+i ρ k i 2ρ i ρ k )(ρ k+j ρ k j 2ρ j ρ k ), is the symptotic covrince mtrix of the smple utocorreltions (Brtlett, 1946). As the innovtion process v p,t is not i.i.d., we need to estimte the mtrix C in consistent wy. For this purpose, we follow the pproch described by Billie nd Chung (2001), bsed on the Newey nd West (1987) procedure. 4.3 Prmetriztion of the prmeter cross-section distribution As we discussed in Section 3, the prmeters {Λ k } nd {Φ k } decrese to 0 s k goes to infinity. Therefore, the numbers of lgs K Λ nd K Φ cn be selected to ensure tht the contribution of the non-included prt is mde negligible reltive to tht of the included prt, yielding consistent estimtors of θ AC. However, solving problems (21) nd (23) with lrge number of lgs nd unrestricted prmeters would be inefficient, s the sequences of prmeters {Λ k } nd {Φ k } re known to be restricted by their reltion to the moments nd co-moments of the cross-section distribution of the prmeters. 7 7 It turns out tht, in typicl finncil pplictions, the persistence prmeters re rther concentrted round vlues slightly below 1. This implies tht the convergence of the {Λ k } prmeters towrd 0 is fst s k increses. As the subsequent Monte-Crlo simultions will show, estimtion of eqution (7) provides essentilly unbised estimtes of the ggregte prmeters even for reltively smll number of lgs. 14

15 In principle, these reltions cn be explicitly incorported in the estimtion through flexible prmetric representtion of the cross-section distribution. 8 Assume tht the joint distribution of {β ij, γ ij } is known nd depends on new set of prmeters θ of smller dimension. Then the sequences of prmeters {Λ k } nd {Φ k } cn be obtined s functions of θ, so tht the innovtion in eqution (22) cn be rewritten s: K Λ K Φ v p,t ( θ) = X p,t Ω p Λ k ( θ)x p,t k + Φ k ( θ)v p,t k ( θ). (24) A similr frmework hs been described by Zffroni (2004b) for his goodness-of-fit test in n ARCH( ) process. We strt with the sequence of ggregte prmeters {Λ k }, which re relted to the moments of the persistence prmeters {γ ij } (eqution (9)). As in Robinson (1978), Grnger (1980), nd Gonçlves nd Gouriéroux (1988), we mke the following ssumption regrding the cross-section distribution of {γ ij }. Assumption 4 The cross-section distribution of the persistence prmeter {γ ij } is described by Bet distribution with prmeters p nd q: f γ (γ) = γp 1 (1 γ) q 1, B(p, q) where p, q (0, ) nd B(, ) is the Bet function. This prmetric distribution covers the rnge of vlues (0, 1), which represents the dmissible intervl for γ. The Bet distribution is ble to reproduce wide rnge of distribution shpes, like leftwrd symmetric bell shpe (for 1 < q < p) or continuously incresing distribution (for 0 < q < 1 < p). 9 The non-centrl moments of the Bet 8 Ω p cn be estimted directly s single free prmeter. 9 The cse q < 1 is known in the literture on the ggregtion of ARMA processes s the long memory cse becuse it genertes sequence of non-centrl moments tht is not bsolutely summble, therefore inducing long-memory pttern in the underlying ggregte series (see Grnger, 1980, Abdir nd Tlmin, 2002, Zffroni, 2004). In the GARCH frmework, long memory is ruled out (see Zffroni, 2004b, 2007). 15

16 distribution with prmeters p nd q re given by: Ẽ (w) [ γ k] = B (p + k, q) B(p, q) = Γ (p + q) Γ (p + k) Γ (p) Γ (p + q + k), (25) where Γ( ) is the Gmm function. For given vlues of p nd q, the non-centrl moments of {γ ij } re obtined by eqution (25) nd the prmeters {Λ 1,, Λ KΛ } re directly deduced from eqution (9), giving ll the utoregressive prmeters of eqution (7). As stted bove, the sequence of ggregte prmeters {Φ k } is relted to the comoments of the joint distribution of {γ ij } nd {β ij } (eqution (15)). Prmeterizing these co-moments is more problemtic, becuse it requires some ssumptions bout the kind of dependence between {γ ij } nd {β ij }. To our knowledge, there is no simple wy to define joint Bet distribution, whose dependence structure cn be described s free prmeter. As consequence, to void excessive restrictions on their joint distribution, we estimte the cross-section men Φ 0 = Ẽ(w) [β] nd the sequence of co-moments Φ k, k = 1,, K Φ, directly s free prmeters. The prmeter set for the LS-ACE nd MD-ACE estimtors therefore reduces to θ = (Ω p, p, q, Φ 1,, Φ KΦ ). 5 Performnce of the Estimtors This section ims t evluting the finite-smple properties of the estimtors. This evlution is bsed on Monte-Crlo simultions, which reproduce the properties of lrge smple of U.S. equities. We show tht the ACE provides unbised estimtes of the prmeters driving the ggregte squred returns. Finlly, we report some estimtes of the ggregte return dynmic using U.S. ggregte dt only. The comprison of the competing estimtors on rel dt confirms tht the ACE outperforms the usul estimtors. 5.1 Clibrtion Bsed on U.S. Equities The clibrtion of the individul prmeters for the Monte-Crlo simultions is bsed on smple of 75 U.S. compnies between Jnury 1988 nd December 2010 for totl of 16

17 T = 6, 000 dily observtions. 10 In the following, we use the generic nottion ϑ i ϑ ii for the prmeters. The individul vrince prmeters (ω i, α i, γ i ) of ech of the 75 individul stocks nd the covrince prmeters (ω ij, α ij, γ ij ) of ech of the 2,775 pirs of stocks re estimted using the flexible GARCH pproch of Ledoit, Snt-Clr, nd Wolf (2003), which ensures the positive semi-definiteness of the covrince mtrices. This subsection ims t describing the min properties of the prmeter estimtes, which will be used for the simultions. Tble 1 reports some summry sttistics on the prmeter estimtes of the individul conditionl vrinces nd covrinces, bsed on ML estimtion. The cross-section men estimtes of the vrince prmeters α i, β i, nd γ i re 0.049, 0.943, nd 0.992, respectively (Pnel A). The lrge vlue obtined for the persistence prmeters γ i is consistent with tht reported in the empiricl literture on stock returns. Regrding the conditionl covrince prmeters, the men estimtes of α ij, β ij, nd γ ij re 0.035, 0.942, nd 0.977, respectively (Pnel B). 11 To further investigte the chrcteristics of the individul prmeters, Bet distribution is djusted to ech set of prmeters (α i, α ij, β i, β ij, γ i, γ ij ). The tble reports the estimtes of prmeters p nd q for ech distribution. As it ppers clerly, there re some significnt differences between the chrcteristics of the vrince prmeters (α i, β i, γ i ) nd the covrince prmeters (α ij, β ij, γ ij ). Given the dominnt role plyed by the covrince terms in the ggregted vrince (97.3% of the terms involved), we mostly focus on in the sequel on the covrince prmeters. Figure 1 displys the histogrm of the prmeters {ϑ ij } nd the estimted Bet distribution f ϑ (ϑ), for ϑ = α, β, nd γ. As the figure illustrtes, the fit of the ctul prmeters is very good for ll the prmeter sets. We notice tht the rnge of vlues is in fct rther nrrow. All the estimtes of α ij rnge between 0.01 nd 0.07, ll the estimtes of γ ij re bove For simultion purpose, nother importnt property is the dependence between the individul prmeters. Clerly, we cnnot simulte α nd β independently from ech 10 The smple is composed of ll the compnies belonging to the S&P 100 t the end of 2010, for which prices were vilble on Dtstrem over the period. The estimtions strts in 1988 becuse the October 1987 crsh ws found to ffect the estimtion of the model. 11 These numbers re consistent with the restrictions α 2 ij < α iα j nd β 2 ij < β iβ j, required to ensure the positive semi-definiteness of A nd B (Assumption 1). This obviously results in the sme observtion for the persistence prmeter, i.e., γ 2 ij < γ iγ j. 17

18 other becuse it could imply vlues of γ lrger thn 1. Similrly, simulting ω nd γ independently from ech other could generte extremely errtic vlues for h = ω/(1 γ), when γ is close to 1. To ddress this issue, we consider the correltion between the individul prmeters estimted on U.S. equities (Pnel C). Regrding first the prmeters α, β, nd γ, the tble revels tht γ ij is positively nd strongly correlted with β ij but wekly correlted with α ij (0.88 nd 0.205, respectively). Therefore, we proceed s follows in the Monte-Crlo experiments: prmeters α i nd γ i re simulted from independent Bet distributions with the prmeters p nd q reported in Pnel A. Prmeters α ij nd γ ij re drwn from independent Bet distributions with the prmeters p nd q reported in Pnel B. Then, we define β i = γ i α i nd β ij = γ ij α ij. The clibrtion of the unconditionl vrinces nd covrinces lso requires dditionl informtion bout the constnt terms ω i nd ω ij, the unconditionl vrinces h i = ω i /(1 α i β i ) nd covrinces h ij = ω ij /(1 α ij β ij ). The men estimtes of the unconditionl vrinces h i nd covrinces h ij re nd 0.065, respectively. From Pnel C, we lso notice tht the correltion is highly negtive between γ ij nd ω ij ( 0.743) but close to 0 between γ ij nd h ij ( 0.191). Therefore, the simultions re performed s follows: The unconditionl vrinces h i re drwn from symmetric Bet distribution with p hv = q hv = 3 in the rnge of the estimted vrinces [ ] h v, h v. 12 The unconditionl covrinces h ij re drwn from Bet distribution with p hc = q hc = 3 in the rnge [h c, h c ]. We then define the constnt terms s ω i = (1 γ i )h i nd ω ij = (1 γ ij )h ij. 5.2 Simultion: Bseline Cse For ech simultion, smples of vrince prmeters (α i, γ i, h i ) nd covrince prmeters (α ij, γ ij, h ij ) for i, j = 1,, N, re drwn from their respective distribution, s described in the previous subsection. Then N time-series of individul innovtions {z i,t } t=1,,t re drwn from norml N(0, 1) distribution nd the unexpected returns {ε i,t } t=1,,t re constructed for i = 1,, N. The portfolio unexpected return {ε p,t } t=1,,t is obtined by 12 More precisely, if h i is drwn from stndrd Bet(p h,q h ) distribution, the unconditionl vrince is defined s from h i = h + (h h) h i, where h nd h denote the minimum nd mximum estimtes of the unconditionl vrinces, respectively. The choice of p h = q h = 3 ensures tht the resulting h i re rther dispersed in the intervl [ h, h ]. 18

19 ggregtion, with portfolio weights w = (1/N,, 1/N). Finlly, the prmeters driving the ggregte squred return X p,t = ε 2 p,t re estimted from ggregte dt only. We consider two lterntive estimtors in order to evlute the mgnitude of the bis induced by imposing prmeter homogeneity when deriving the ggregte squred return dynmic. The first one is bsed on the usul (implicit) ssumption of prmeter homogeneity, i.e., the QMLE of the strong GARCH(1,1) process: h p,t = Ω p + Ψ 1 X p,t 1 + Φ 1 h p,t 1. (26) The second estimtor, consistent with prmeter heterogeneity, is the Lest-Squre ACE of the wek GARCH(K Λ,K Φ ) process: 13 K Λ K Φ X p,t = Ω p + Λ k X p,t k + v p,t Φ k v p,t k. (27) In the bseline cse, the number of observtions per smple is T = 6, 000 nd the number of ssets vries from N = 20 to 40. Ech experiment is bsed on 1,000 replictions. It should be noticed tht these simultion experiments re not designed to exctly mtch ll the fetures observed on U.S. equity returns, but rther to mimic some of their min properties. 14 Tble 2 reports summry sttistics of prmeter estimtes for the QMLE nd ACE procedures. We begin with the cse N = 20, which is relistic number of sset clsses in strtegic lloction pproch. For the ACE, we report the estimtes of {Ω p, Ψ 1, Φ, Λ 1 }, for comprbility with the QMLE, s well s the estimtes of the Bet prmeters p nd q. As expected, the QMLE provides bised estimtes of the vrince prmeters. The most striking result is the severe downwrd bis in the γ-type prmeter (Λ 1 = Ψ 1 + Φ 1 ). The medin estimte is 0.835, while the expected vlue is This bis is not due to the estimtion of the α-type prmeter (Ψ 1 ), s its medin estimte is equl to 0.041, 13 The Lest-Squre ACE estimtor is bsed on K Λ = 20 nd 40 lgs nd K Φ = 5 lgs, so tht the first five terms Φ i, i = 1,, 5, re freely estimted. Results for the Minimum-Distnce ACE re very similr nd not reported to sve spce. 14 For instnce, ctul dt my be generted by symmetric GARCH processes nd/or ft-tiled innovtions, which re fetures not introduced in this experiment. 19

20 which is rther close to the expected vlue with nrrow confidence intervl. On the opposite, the medin estimte of the β-type prmeter (Φ 1 ) is fr from the expected vlue (0.794 insted of 0.937) with lrge uncertinty cross simultions. Incresing the number of ssets does not help in estimting the persistence prmeter, s the vlue of Λ 1 is still severely underestimted even with N = 40 (with medin estimte of 0.777). This result indictes tht the QMLE is not ble to generte the high persistence found in the simulted ggregte squred returns. With regrd to the ACE, the tble revels tht for N = 20 the persistence prmeter Λ 1 is correctly nd precisely estimted to be for both vlues of K Λ (20 nd 40). This result suggests tht moderte number of dditionl lgs is sufficient to correct for the ggregtion bis. The prmeter Ψ 1 is lso very well estimted, with medin estimtes of nd 0.039, respectively. Incresing the number of ssets in the portfolio does not lter the prmeter estimtes significntly. This result is importnt, becuse it suggests tht the ACE is ble to reproduce rther closely the properties of the ggregte process, even for reltively smll number of ssets. 5.3 Simultion: Robustness Check To evlute the robustness of the results presented bove, we performed dditionl simultion experiments bsed on lterntive ssumptions regrding the rnge of the unconditionl correltions, the distribution of the innovtion process, nd the choice of the portfolio weight vector. All simultion results, bsed on T = 6, 000 nd N = 40 ssets, re reported in Tble The first experiment relies on the effect of incresing the correltion between the ssets. As outlined by Zffroni (2007), dynmic conditionl heteroskedsticity of the ggregte process requires sufficiently strong cross-correltion. While the bseline cse ws clibrted with moderte positive correltion using the men vlue found on U.S. stocks 15 Other experiments essentilly left the ptterns lredy described unltered. In prticulr, there is no sizeble effect on the prmeter estimtes when the number of lgs in the ACE (K Φ nd K Λ ) is incresed or when the rnge of the unconditionl vrinces is widened. The results, not reported in order to sve spce, re vilble upon request. 20

21 (0.167), this experiment considers the cse of highly correlted ssets (ρ ij [0.75; 0.9]). As Pnel A revels, the medin estimte of the persistence prmeter Λ 1 obtined from the QMLE is in the sme rnge s in the bseline cse (0.759), nd therefore fr below the expected vlue. The estimte of Ψ 1 remins close to the expected vlue. ACEs turn out to be very robust to chnges in the rnge of correltions cross ssets. The estimte of Λ 1 only slightly decreses towrds its expected vlue. The second experiment considers non-norml distribution of the innovtion process. Although z i,t hs been ssumed to be normlly distributed in the simultions so fr, it is well known tht the empiricl distribution of sset returns is often symmetric nd/or ft-tiled. The interction between the vrince dynmics nd the distribution properties of returns hs been highlighted by Engle (1982) nd more recently by He nd Teräsvirt (1999). To illustrte the consequences of innovtions drwn from distributions with ft tils, Pnel B of the tble reports the results for t distribution with 5 degrees of freedom. As expected, the mgnitude of the bis in the QMLE is incresed. The medin estimtes of the prmeter Λ 1 produced by the QMLE is decresed from for the norml innovtions to for t(5) innovtions. Introducing symmetry into the innovtion distribution through skewed t distribution does not further ffect these prmeter estimtes with ny significnce. Agin, the properties of the ACEs re not ltered by the chnge in the conditionl distribution regrdless of the number of lgs K Λ. This result is consistent with the fct tht the ACEs, which re bsed on Lest-Squre estimtion, do not rely on ny prticulr distributionl ssumption (provided the innovtion s fourth moment is finite). The lst experiment evlutes the effect of the portfolio weights on the performnce of the estimtors. While the previous simultions were bsed on equl weights, we consider now portfolio with short sles: weights re rndomly drwn between 0.2 nd 0.2, with the sum of the weights equl to 1. Agin, the tble revels tht the QMLE underestimtes the persistence prmeter, lthough to lesser extent (Pnel C). On the opposite, the ACEs produce prmeter estimtes tht re very close to the expected vlues. These results suggest tht no positivity restrictions on portfolio weights re in fct required to 21

22 obtin consistent estimtors of the prmeters driving ggregte squred returns. The ACEs esily ccommodte portfolios with different weights or even with short sles. 5.4 Evidence from the U.S. Mrket Aggregte Dynmics The simultion results reported bove suggest tht, under contemporneous ggregtion, the dynmic of the ggregte squred returns cnnot be correctly estimted by the stndrd strong GARCH(1,1) model. Its min filure is tht it produces severely bised estimtes of the persistence prmeter. It is likely to hve drmtic implictions on vrince forecsting, n issue which we ddress in this section. We now consider the estimtion nd forecst of the ggregte squred returns dynmics, ssuming tht only ggregte dt is vilble, nd evlute the effect of the ggregtion bis. The ggregte return is here simply defined s the eqully-weighted verge of the individul returns on U.S. equities lredy discussed. Then, the dynmic of the ggregte squred return is modeled with the usul strong GARCH(1,1) process (eqution (26)) or with the wek GARCH(K Λ,K Φ ) process defined in Proposition 3 (eqution (27)). To void ny over-fitting in the forecst exercise, the smple is divided into two subperiods. The first 12 yers (from 1988 to 1999) re used for the estimtion of the model (3,131 observtions), nd the lst eleven yers (from Jnury 2000 to December 2010) re used for the out-of-smple forecsts (2,869 observtions) Estimtion of the Aggregte Dynmics Tble 4 reports the prmeter estimtes. For the QMLE, the usul GARCH(1,1) prmeters re reported. For the ACEs, we report the prmeters p nd q of the Bet distribution. We lso report the implied estimtes of the stndrd devition, skewness, nd kurtosis of the cross-section of {γ ij } deduced from equtions (11) (14). This llows comprison with the cross-section estimtes reported in Tble 1. As the tble revels, the estimtes of the persistence prmeters disply lrge differences between the QMLE nd ACEs, in line with the simultion evidence reported in Section 5.2. The QML estimte of Λ 1 is low (0.907), reflecting lrge downwrd bis. In contrst, both ACEs produce high persistence of ggregte squred returns (0.98 nd 22

23 0.979, respectively). These vlues cn be compred with the ctul verge of the persistence prmeters over the complete smple (0.977, s reported in Tble 1). It should be mentioned tht the prmeter Ψ 1 is over-estimted by the QMLE (0.127), wheres the vlue found with the ACEs (0.05) is lmost in line with the ctul verge (0.035). Although both ACEs provide very similr prmeter estimtes, the tble indictes tht the ACE(40, 5) lso more ccurtely reproduces the moments of the persistence prmeter distribution. Indeed, the estimtes of the cross-section vrince, skewness, nd kurtosis of {γ ij } implied by the ACE(40, 5) re closer to their empiricl counterprts (reported in Tble 1) thn the estimtes implied by the ACE(20, 5). To gin further insight on the reltive performnce of these estimtors in fitting ggregte squred returns, Figure 2 compres the empiricl ACF of the squred returns X p,t with the ACF implied by the QMLE nd the ACE(40,5). As is clerly evident from the figure, the empiricl ACF of the ggregte squred returns displys slow decy. The first-order utocorreltion of X p,t is bout 0.1. At order 50, it decreses to bout 0.04, nd t order 100 it is still bout The QMLE fils to produce slow decy of the ACF, lthough it correctly estimtes the first utocorreltions. The implied utocorreltion is indistinguishble from 0 t order 50. In contrst, this pttern is correctly reproduced by the ACE. The figure shows tht the ACE lso cptures the slow decy observed beyond Comprison of the Vrince Forecsts As the individul components re in fct vilble in our dtset, it is possible to define disggregte estimtor (henceforth lbeled DISE ), which uses ll the informtion vilble on individul returns. This estimtor consists in estimting the joint dynmics of ll the individul stocks (using the pproch of Ledoit, Snt-Clr, nd Wolf, 2003), forecsting ll individul vrinces nd covrinces, nd then ggregting these forecsts to produce forecst of the ggregte vrince. Although this pproch is rther time consuming, it is fesible in our context with moderte number of ssets. It is formlly defined s: ĥ DISE,t = N N w i w j ĥ ij,t, i=1 j=1 23

24 where ĥij,t denotes the forecsts of the conditionl (co)vrince t dte t deduced from the multivrite GARCH(1,1) model. The effect of the ggregtion bis cn be evluted sttisticlly by compring the reltive performnces of the disggregte nd ggregte estimtion techniques for forecsting the conditionl vrince of ggregte returns. One difficulty is tht the ggregte vrince t dte t is not observble, but mesured with noise by squred unexpected returns ˆX p,t. As shown by Ptton (2011), most stndrd loss functions used to rnk competing vrince forecsts re not robust to the use of noisy proxies nd therefore result in incorrect rnking of the competing forecsts. The vrince forecsts re thus compred using robust loss functions, which re selected mong the fmily described by Ptton (2011): L 1,t ( ˆX p,t, ĥt) = ( ˆX p,t ĥt) 2, L 2,t ( ˆX p,t, ĥt) = ˆX p,t /ĥt log( ˆX p,t /ĥt) 1, L 3,t ( ˆX p,t, ĥt) = ( ˆX 3 p,t ĥ3 t )/6 ĥ2 t ( ˆX p,t ĥt)/2, where ĥt denotes the vrince forecst. L 1 is the usul squred error, L 2 nd L 3 re symmetric loss mesures tht penlize under-predictions nd over-predictions, respectively. The test of the difference between two loss functions is bsed on the test developed by Diebold nd Mrino (1995) nd West (1996). For instnce, the loss difference between the forecsts bsed on the DISE nd the ACE is defined s: d k,t = L k,t ( ˆX p,t, ĥdise,t) L k,t ( ˆX p,t, ĥace,t), for k = 1, 2, 3, where ĥdise,t nd ĥace,t denote the vrince forecst bsed on the DISE nd the ACE, respectively. The Diebold-Mrino/West test is bsed on the t-stt ssocited with the loss difference DM k = d k / σ k, where d k is the smple men nd σ 2 k the smple vrince of the loss difference. Under the null hypothesis E[d k,t ] = 0, the t-stt is symptoticlly distributed s N(0, 1) Following Diebold nd Mrino (1995), the smple vrince is mesured s σ 2 k = 2π f d(0)/τ, where f d (0) is consistent estimte of the loss difference spectrl density t frequency 0, nd τ is the number of observtions over the out-of-smple period. 24

25 Tble 5 reports the Diebold-Mrino/West sttistics DM k for the test tht the estimtion techniques hve the sme forecsting bility over the out-of-smple period, for vrious subperiods. Severl results re of importnce. First, most test sttistics re positive, mening tht, in most cses, the ggregte techniques produce more ccurte vrince forecsts thn the disggregte estimtor. Although individul squred returns provide useful informtion for forecsting the ggregte vrince, the disggregte estimtor is plgued by the estimtion error surrounding the individul prmeter estimtes. The tble revels tht, t lest for the present dt, vrince forecsts bsed on ggregte dt re more ccurte thn those bsed on disggregte dt. The only exception occurs for the QMLE over the subsmple, which corresponds to period of very low voltility (except t the very end of the period). Under such circumstnces, the disggregte estimtor outperforms the QMLE, lthough insignificntly. Second, the ACE consistently domintes the disggregte estimtor for ll loss functions. Most of the test sttistics re positive nd significnt t the 1% significnce level, except over the subsmple. In prticulr, it performs very well for the loss function L 3,t focusing on over-predictions. In contrst, the QMLE(1, 1) fils in providing better forecsts thn the disggregte estimtor under the loss function L 3,t. To sum up, the ACE is ble to significntly reduce under- s well s over-predictions reltive to the disggregte estimtor, while the usul QMLE brely out-performs the disggregte estimtor. Presumbly, this is due to the bility of the ACE to correctly reproduce the persistence of the ctul squred returns. 6 Conclusion This pper describes the dynmics of ggregte squred returns in the presence of lrge number of heterogeneous ssets, when the covrince mtrix is driven by strong multivrite GARCH(1, 1) process. In the limit, the ggregte squred return is n ARMA process with n infinite number of lgs. This result estblishes reltionship between the prmeters of the ARMA process nd the moments of the cross-section distribution of the persistence prmeter. The proposed estimtion procedures explicitly cknowledge 25

26 this reltionship to prmeterize the sequence of prmeters of the ARMA process, nd therefore to reduce significntly the computtionl burden in the estimtion of the ggregte process. In contrst to the usul QMLE, the estimtion procedure provides unbised estimtes of the prmeters driving ggregte squred returns nd performs very well in finite smples. The pper lso evlutes the effect of the ggregtion bis on the estimtion of ggregte squred returns over lrge smple of U.S. equities. Once the ggregtion bis is dequtely corrected, the ggregte squred return process is more persistent thn suggested by the usul QMLE. In ddition, the ACE produces more ccurte vrince forecsts thn disggregted or QML estimtion techniques. Some issues currently remin unresolved. In prticulr, it is not cler how to hndle the ggregtion of symmetric GARCH processes. The min difficulty tht rises in this cse is tht the individul vrince symmetries re lost when only ggregte returns re considered. Another interesting extension is the cse of time-vrying weights. We hve investigted the cse of constnt weights, which includes the eqully-weighted portfolio. For sector indices or sset clsses, however, the weights of the vrious components my vry over time, for instnce in correltion with the reltive mrket cpitliztion of the individul ssets. In this cse, the prmeters of the ggregte squred return process re themselves time-vrying s they reflect the cross-section verge of the individul prmeters bsed on time-dependent weights. In such instnces, the composition effects must be explicitly tken into ccount. These extensions re left for further reserch. 26