A SURVEY ON CYCLES AND CHAOS (PART II)

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1 A SURVEY ON CYCLES AND CHAOS (PART II) Claude DIEBOLT & Caherine KYRTSOU Absrac: This paper is an exension of a previous publicaion in he journal Hisorical Social Research (Vol. 26, No. 4, 200). Our reamen begins wih a simple presenaion of he basic noions of chaos, and hen describe he relaed economeric ools.. Inroducion The erm complex economic dynamics is used o designae deerminisic economic models whose rajecories exhibi irregular (nonperiodic) flucuaions or endogenous phase swiching. The firs propery includes chaoic rajecories ha give bounded flucuaions which are sensiive o perurbaions. The second means ha he equaions governing change in sysem saes swich from ime o ime according o inrinsic rules. Or i means ha disinc ypes of qualiaive behavior, such as growh, oscillaion or decay, are exhibied in differen subses of he sae space; he sysem equaions resriced o a given subse hen appear o have a differen naure han heir resricion o oher subses, so ha each such resricion yields an idenifiable regime. LAMETA/CNRS, Universiy of Monpellier I, Deparmen of Economics, Espace Richer, Avenue de la Mer, Monpellier, Cedex, France. s: claude.diebol@lamea.univ-monp.fr kyrsou@lamea.univ-monp.fr

2 Chaoic processes have many very ineresing properies, only a few of which need o be menioned here. The firs is he exisence of aracors. Suppose ha many erms of he process have been generaed, so ha is large, and le x,m be a vecor of m adjacen values (x, x -,,x -m+ ). For a cerain value of m, called he embedding dimension, x,m will always lie on a paricular subse of he m-dimensional space, called he aracor of he process. A chaoic process is in a sense simple if is embedding dimension is low (say one o hree) and is complicaed if i is high. For example, he logisic map (x =µx - (-x - )) has a dimension of one whereas a whie noise process have very high dimension. The empirical esing in economics and finance finds pleny of evidence for nonlineariy bu none for low dimensional chaos. This suggess ha here are sochasic shocks occurring somewhere in he economy, so one has o ask how his fis in wih he chaos heory. Experimens have also shown ha adding a lile whie noise o a low dimensional chaoic signal, makes he deerminisic chaos exremely difficul o deec in shor series. Thus emerges he ineres of inroducing he approach of sochasic chaos. As i has been underlined by Chan and Tong (994), i is more realisic o model economic or financial dae wih a nonlinear deerminisic process perurbed by dynamical noise. The purpose of he paper is o presen he recen developed ess for chaos: he correlaion dimension, he Lyapunov exponens and he surrogae dae ess. 2. The correlaion dimension es The correlaion dimension was inroduced by Grassberger and Procaccia (983). The correlaion dimension is based on he idea ha if an aracor is chaoic, hen wo poins (X i, X j ) saring a differen posiions will be dynamically uncorrelaed as a resul of he propery of sensiive dependence on iniial condiions. However, since he poins are on an aracor, hey can approach each oher bu can never inersec. 2

3 The correlaion beween poins on an aracor can be defined in erm of spaial correlaion ha is formally measured by he Euclidean disance. Le {X }, =,2,T be a sample from a sricly saionary process. The ime series {X } can be "embedded" in a m-space by consrucing "m-hisories". The correlaion dimension can be calculaed from he correlaion inegral given by: C(ε, m, T m ) = T m ( T ) m T m i, j= ( H X i X j ) i j () as defined in he Par I (Diebol and Kyrsou, 200). The use of an Euclidean norm for compuing he correlaion dimension is considered no o be oo resricive. Brock (986, heorem 2.4) has proved ha he correlaion dimension is independen of he choice of norm. Le he correlaion inegral measure he fracion of oal number of pairs (x i, x i+,, x i+m- ), (x j, x j+,, x j+m- ), such ha he disance beween hem is no more han ε. The correlaion dimension can be defined as follows: ( ) ln C ε, m d c = lim (2) ε 0 ln ε For he small values of ε, Grassberger and Procaccia (983) esablish ha he spaial correlaion C(ε,m) grows according o he power law: ( ) ln C ε,m If d m = lim, hen lnc(ε,m ) d m lnε lnc(ε,m ) lnε d m C(ε,m ) ε d m, and ε 0 ln ε C(ε,m) grows exponenially. 3

4 I is necessary o noice ha when he embedding dimension m increases, he dimension d m is reached, such ha d* c is he esimae of he rue correlaion: d* c = lim d m m The mehod of he correlaion dimension represens a very imporan diagnosic procedure for disinguishing beween deerminism and sochasiciy. If d m ends o be a consan as m increases, hen d m yields an esimae of he correlaion dimension of he aracor, namely d* c. In his case, he ime series are consisen wih deerminisic behavior. If d m increases wihou bound as m increases, his suggess ha he underlying series are sochasic. 3. The Lyapunov exponen es The Lyapunov exponen mehod can be employed o deermine if a process is chaoic. The approach is based on he idea ha he disance beween wo poins is described by he larges Lyapunov exponen. The Lyapunov exponens measure he average rae of conracion (when negaive) or expansion (when posiive) of he rajecories on he enire aracor. They can be posiive or negaive, bu a leas one exponen mus be posiive for an aracor o be classified as chaoic. If he disance beween he rajecories grows exponenially, his is evidence of chaos since i shows ha he process exhibis sensiive dependence o iniial condiions. Ruelle (990) argues ha a chaoic series can only be disinguished if i has a correlaion dimension well below 2log 0 T, where T is he size of he daa se, suggesing ha wih economic ime series he correlaion dimension can only disinguish low dimensional chaos from high dimensional sochasic processes. 4

5 Thus, where λ is he larges Lyapunov exponen, he crierion is: Noisy chaos or sochasiciy if λ < 0, chaos if λ > 0 In he n-dimensional case, where y + = f(y ) (3) wih T, y R n, he Lyapunov exponen λ is defined (Lorenz, 989) by λ (T) =(/T)log 2 ( Λ (T) ), where Λ (T) are he eigenvalues of he n-dimensional Jacobian marix J (T). In general, all Lyapunov exponens can be calculaed according o he following equaion [see Wolf e al., (985)]: λ i = lim log2 ( Λ (T) i ) (4) T T When applying his mehod o financial-price series, many auhors confirme he difficuly of polluion from high frequency noise. The larges Lyapunov exponen λ ends o be greaer han he rue exponen and is convergence o a value appears difficul or even impossible. 3. Kanz algorihm (994) Kanz (994) has ried o solve his problem by consrucing a new algorihm for he esimaion of λ. Similar o Wolf e al. (985), he makes use of he fac ha he disance beween wo rajecories ypically increases wih a rae given by he maximal Lyapunov exponen. This divergence rae of rajecories naurally flucuaes along he rajecory, wih he flucuaions given by he specrum of effecive Lyapunov exponens. The maximal exponen λ τ is defined o be: λ τ () χ = lim ln ε 0 τ ( + τ) χ ( + τ) ε ε (5) 5

6 where χ() is he ime evoluion of some iniial condiion χ(0) in an appropriae sae space, is ime, and τ is relaive ime referring o he ime index of he saring poin, and ε = χ(0)-χ ε (0). χ()-χ ε () = εω u (), where ω u () is he local eigenvecor associaed wih he maximal Lyapunov exponen λ max. By definiion he average of λ τ () along he rajecory is he rue Lyapunov exponen. The mehod of Kanz requires consrucing he following equaion o provide he curve S(τ). The maximal Lyapunov exponen is he slope of his curve in he scaling region. S T ( τ ) = ln ( dis χ, χ i ; τ T = U i U ) (6) where: U is he neighborhood se and dis(χ,χ i ;τ) defines he disance beween a reference rajecory χ and a neighbor χ i afer he relaive ime τ. When noise is presen in he daa, he slope of he curve S(τ) changes as follows: s(τ) λ + σi, τ - dis(x, x i; τ) dis(x σi, τ (7), x i; τ ) λ is he esimae of he maximal Lyapunov exponen and σ i,τ is he sandard deviaion of he noise. S(τ) does no conain he embedding dimension explicily, bu neverheless i eners. This requires ha one fix a dimension m for he delay rajecories 2. 2 For more deails in he choice of embedding dimension, see Kanz (994). 6

7 3.2 Gençay and Decher algorihm (992) Gençay and Decher (992) ry o solve he problem in he Lyapunov exponen esimaion when a high level of noise is presen, by using an algorihm for he esimaion of λ, based on feedforward neural neworks. We presen briefly heir esimaion procedure below. We noice ha for he neural neworks esimaion we use he mehod of non-linear leas squares (Kuan and Liu, 995). In pracice i is very difficul o observe he sae of he sysem and know he acual funcional form, f, ha generaes he dynamics. The model ha i is principally used is he following: associaed wih he dynamical sysem in equaion (3) here is a viewer funcion h : R n R which generaes daa: x = h(y ) (8) We suppose ha all ha is available o he researcher is he sequence of he variables {x }. The well-known Takens heorem (98) saes ha, when m 2n+ we have: J m (y) = (h(y), h(f(y)),, h(f m- (y))) (9) which is generically an embedding, m he embedding dimension and n he dimension of he real sysem. For a funcion g : R m R m for which J m o f = g o J m on an indecomposable aracor, Decher and Gençay (990) show ha n larges Lyapunov exponens of g are he Lyapunov exponens of f. Thus, hey esimae he funcion g based on he daa sequence {J m (y )} and calculae he Lyapunov exponens of g. 7

8 The mapping g, which is o be esimaed may be given as follows: g : ( x, x,..., x ) x+ m u + m + m 2 x+ m 2 x+ m x y x+ and his reduces o esimaing x +m = u(x +m-, x +m-2,,x ) Finally, for a single-layer nework he leas-squares crierion for a daa se of lengh T is: T m = 0 L( β,w,b) = [x+m u N,m (x m ; β, w, b)] 2 (0) where: x m = (x +m-, x +m-2,,x ) is he inpu, u N,m (x m ; β,w, b) is he single-layer feed forward nework, ϕ ( u) = is he acivaion funcion, + exp( u) β,w,b: parameers o be esimaed, N is he number of hidden unis. 4. The surrogae daa es The surrogae daa es has been proposed by Theiler e al., (992) and vasly applied o real daa. Evidence of non-lineariy was ofen repored while in few works he null hypohesis could no be rejeced (Prichard and Price, 993). 8

9 The main idea of his es is o discriminae non-linear dynamics, if his can be deeced from he given series. Oherwise he null hypohesis canno be rejeced, which does no necessarily mean ha he examined process is sochasic linear. This is only one possible case. There are a number of oher possibiliies, such as he underlying dynamics is non-linear bu masked by noise, or he dimensionaliy is high and he daa size small, so ha deecion of non-lineariy canno be archived, or simply he daa record does no represen well he underlying sysem. To es he null hypohesis H 0 ha he original signal is generaed by a linear sochasic process undergoing a saic possibly non-linear ransform, an ensemble of M surrogae daa ses represening H 0 is generaed. To make his, he surrogae daa mus have he same auocorrelaion and he same empirical ampliude disribuion as he original signal. Then, a non-linear mehod is applied o he original and he surrogae daa giving he saisics q 0 for he original and q,,q M for he surrogaes. The H 0 is rejeced if q 0 is saisically differen from q,,q M. Typically, he confidence of rejecion is given in erms of he significance S: S= q 0 q σ q where q is he average and σ q he sandard deviaion of q i, i=,,m. Significance of abou 2σ suggess he rejecion of H 0 a he 95% level of confidence. The compuaion of S quanifies beer he difference beween original and surrogae daa han he simple ordering of he M+ q-quaniies followed in oher works (Schreiber, 999). For he generaion of he surrogae daa he algorihm of ampliude adjused Fourier ransform (AAFT) is usually applied. 9

10 The surrogae dae es can be also used as a validaion es. Afer obaining he surrogae series, we can applied he correlaion dimension and he Lyapunov exponens mehods. The comparison beween he resuling correlaion dimensions and Lyapunov exponens (original and surrogae daa) can allow us o deermine he robusness of he obained resuls. For some recen applicaions of he previous nonlinear ess o financial reurns series see Kyrsou (2002), Kyrsou and Terraza (2002a,b). REFERENCES Brock, W.A., (986): Disinguishing random and deerminisic sysems: Abridged version, Journal of Economic Theory, Vol. 40, pp Chan, K.S., and Tong, H., (994): A noe on noisy chaos, Journal of he Royal Saisical Sociey B, Vol. 56, No 2, pp Decher, W.D., and Gençay, R., (990): Esimaing Lyapunov exponens wih mulrilayer feedforward nework learning, Working paper, Deparmen of Economics, Universiy of Houson. Diebol, C., and Kyrsou, C., (200): A survey on cycles and chaos (Par I), Hisorical Social Research, Vol. 26, No. 4, pp Gençay, R., and Decher, W.D., (992): An algorihm for he n Lyapunov exponens of an n-dimensional unknown dynamical sysem, Physica D, Vol. 59, pp Grassberger, P., Procaccia, I., (983): Measuring he srangeness of srange aracors, Physica 9D, pp Kanz, H., (994): A robus mehod o esimae he maximal Lyapunov exponen of a ime series, Physics Leers A, 85, pp Kuan, C-M., and Liu, T., (995): Arificial neural neworks: an economeric perspecive, Journal of Applied Economerics, Vol. 0, pp

11 Kyrsou, C., (2002): Héérogénéié e chaos sochasique dans les marchés bourisers, Phd hesis, Deparmen of Economics, Universiy of Monpellier I, France. Kyrsou, C., and Terraza, M., (2002a): I is possibly o sudy chaoic and ARCH behaviour joinly? Applicaion of a noisy Mackey-Glass equaion wih heeroskedasic errors o he Paris Sock Exchange reurns series, forhcoming in Compuaional Economics. Kyrsou, C., and Terraza, M., (2002b): Sochasic chaos or ARCH effecs in sock series? A comparaive sudy, forhcoming in he Inernaional Review of Financial Analysis. Lorenz, H.W., (989): Nonlinear dynamical economics and chaoic moion, Springer Verlag. Prichard, D., and Price, C.P., (993): Is he AE index he resul of nonlinear dynamics?, Geophysical Research Leers, 20, pp Ruelle, D., (990): Deerminisic chaos: he science and he ficion, Proceedings of he Royal Sociey of London A, 427, pp Schreiber, T., (999): Is nonlineariy eviden in ime series of brain elecrical aciviy?, Wupperal preprin WUB Takens, F., (98): Deecing srange aracors in urbulence, in Dynamical Sysems and Turbulence, D. Rand and L.S. Young, eds., Lecure Noes in Mahemaics 89, Springer, Berlin. Theiler, J., Eubank, S., Longin, A., and Galdrikian, B., (992): Tesing for nonlineariy in ime series: he mehod of surrogae daa, Physica D, 58, pp Wolf, A. e al., (985): Deermining Lyapunov exponens from a ime series, Physica 6D, pp