An alternative correlation dimension statistic

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(2) Introduction. I. n recent years the application of the Gra&<;:berger and Procaccia (U&P) correlation dimension (CD) (1983) in economics has heen of inlerest. This is a statistical. procedurc applied to time &eries which originated with the purposc of measuring the. fractal dlmension of strange attractors and was then used to distinguis,h between stochas:üc and deterministic (e.g, chaotic) proccsses. fn a 1atcr deve!opment, Brock et al. introduced a related statistic (DBS statistic). tcsting for ITD (identically and indepcndently distríbuted) series whose maín applkations include testing for nonlinearity in stochastic processes. (See for ex:amplc Sheinkman and LeBaron [1987], and Brock and Ma!liaris [1989].) One of the problems plaguing thcse applications has bcen the large data rcquirc· ments of tllese statistical procedures. Time series numbering 20,()()() are common in physical sdence appJications:, whilc results of applications tu shorter time series with lengths bctween 500 and 2,000 -it being difficult enough to obtain such economic data- have suffered from a bias towards low dimensions in t:he statistical results, The purpose of this paper is tú prosent a modified statistic which not only calculates dl.mension and simultaneously tests for the liD null, but is also not biased towards Iow dimensions and more scnsili ve to the presencc of stochao;tic structure in the shorter time series mentioned above. Moreover, we give a method to estimate the significance of tests conductcd by means of these stati~>iics. based on the re.shuffling test and not involving appro);:imation to the normal dlstribution and the calculation of variance. \Ve conduct several tests to compare the two statist.ics:, namely thc correlation dimension ln its usual form (co for short) and in it<:: modified fonn (SCD for Statistical Correlation Dimension). The test'i consist in comparing. for the CD and seo, 1) Thc mean of 2,000 realizations of liD series ofleng!h 500 calculated for several. stochastic processes. 2) The power of tl1e reshuffling test (described below) applied tu distinguish the presence of scveral proce.sses. 3} Results obtainnd from applying the reshuffiing tests to 103 series of residues obtained from:M.Hton Friedman's Monetary and W, Cox's Treasury Sccurity long-tenn. time series. 4) The dimensions ofBrock and Sayers' (1988) original time series. 5) The dímensíons ofthe CRSP daily long~term time series.. The paper contaíns two parts. Frrst, in sections 1-3, we define the statistica! correlation dimension (SCD) and deveJop sorne of its propertics, as well ao; obtain significance lntervals for the random reshuffling techníque. Second,. in sections 4-8. we compare results obtained in applications ofboth statistics,.

(3) Mayer 1 An Altenuuivc Correlation lJimeMiotl Statistic. 1. The Grassberger and Procaccia Corretatwn Dimension Let us first briefly give a one.-paragraph intuitive introduction to thc O&P en and íts applicarion. for the sake of readers less familiar with thls: specialized topic. Consíder any time series {a,: t= 1, , , . , N}, Theessential idea behind the concept of currelation dimension is to ~ystematically consider not just single elemcnts of the time series but the sets of m~vet.~tors whose m cntries are regularily-lagged members of the time series. Tilis set of vectors, called m~hi;;torics, have a graph of points in Euclidea.n space Rm, which for a large enough m rcveals the nature ofthe time series. Thus, if for sufficlently large m the m-th member of an m-hlstory is n determínistic function (or almost everywhere a locaHy ímplicit function) of the p:rcceding elements belonging to the m~vcctor, the set of m-histories will He in Rm on sorne ''surt'ace," while íf the m-th member is a stocha"'tic function of the previous mernbers. the m-histories will onJy cluster about sorne "'surface," tending to fill a regíon of Rm. Now, the dimension of these grapbs of points may be measured by the exponent oí how fast the number of "close" points grows as lhe distance allowcd between them is increased. In the case of chaotic dctenninistic processcs this number may turn out lo be a fraction corresponding to polnts dusterlng on "strange" (fractal) subsets of the surfaces rneniioned a.bove. In general, determinislic processes yield (with enough data) a finite. dimemüon even as m grows, indefinitely. while sufficiently regular stochastic processc.s will yield (again, with enough data) the largest possíble embedded dlmcnslon, m, since becam~e of their random nature thcy tend to fill space, Thus stochastic processe:s (high dlmension) may be distinguished from delerminisitic processes (low dimension). Let us add that in the- presence of stochastic structure and finite data we shall get in-between resultb (e.g. a dimension growing wlth m but less than m) which may be useful for detecting such structure. Clettrly a functíonal transfonnation of the data should not increase the dimension in the deterministic case (gfven sufficíent data) but in lhe stochastic case, if the transformation whitens the series, the dimension shouhl rise. Thls is the basL.:; for tlle Brock res.idue test, described below. Now Jet us write sorne of thesc notions precisely.. 1.1. Definition. Let sil= { a1 : f ~ 1, ... , N} be a finite series of real numbers. Forea.ch m= 1, 2,, .. , N, 'te !\¡ ('t = 1 in the applications in thís paper unless otherwise stated) let H(~, m, 1)={ (ar ... ,a 1 •(m-l~): t=. 1, ... ,N-(m-1)<);:: IR m. be the set of -r-Jagged m-histories of :A., • For tite sake of oompletenes:s, wc mention the following theorem, Consider the following type of deterministic process. 2.

(4) Maycr 1 1\n Alternaave Correlarion Dimenvioll Statistic. ----------------. dx -=f(x). dt. XE. d. H. (!). for whkh we have a set of m.ea¡;:urements a,=g(x,)E~,. rc:.\1. (2). (We could instcad treat a difference equation.) Herc d is the dimension of state space. Takens (1980) proves that generically, for m ;?: 2d + 1. the set of linút JX!Ínts of H(<ll., m, t) as Ntends to infinity (for any fixed ~) forms a diffeomorphic image ofthe corresponding set of limit points (attractor) of {x1} which l.herefore preserves such properties as the dimension mcntioned above. Ht~re "generically" m.eans that the measurement functlon g is non-singular, i,e., that it is sensitive to the state space variables. Thus: any time sequence A of measurements of a proce.ss descrihable by ( 1), even when the state variables x andJor the evolution function f may be unknown, can, ín general, be used to detect its stochastic or determlnistic nature by the methods beíng. described.. 1,2. Definltion, Por any finítc set H ¡;;Ir with cardinality #H let CH(€): (0, =)->. ro. !] given by Cn(e; il{(f,y)E HxH:x;<yJ¡x-yll <t) ). illf(#H- l). be Lhe relatíve frequency with which the distance betwee-n pairs of vectors in H ís less than E. Hcre 1i l1 is: some norm of RIÓl, • lJsing this functiou G&P ( 1983) define the CD.. 1.3. Definitlon. Thc correlation dimension of asequence 81 ;{a,: t= I, ... , N)is. where. -¡·. D(m) - m\:-4u. log cl{ít!,m, ,¡(€) • ,. Joge. An application of Takcns' thoorem ( 1980) shows that for sequences originated deterministicaJly as in ( l}. D ís: thc au.ractor's "llmit capadty". a <:lose lower bound of.

(5) Mayer 1 An A!tcmative Correlation Dimension Swtistir. the Ha.usdor:ff dimension and is indcpendent of 't, as N tends to infinity, On the other hand, for sufficiently regular stochastic sequences, it may be shown that undcr the same oonilitions D(m) tends tu m (see for example Brock (1986)). D is a1so independenl of the nonn used in Rm, '\Vñen s! ís generated by (1) illld (2), it may he shown thot D,; d, the dimension of state ~pace. When s4 is the re-alization of a stochastíc process, i t may be shown that plim,y~,D(m)=m,. 2. 1"he Statistical Correlaiion Dimension We instead define: 2.1. DefinitJon. The staristical correlation dimension (SCD) of a sequence dl:={a,:r~l. , ... ,N}ís. whcre logC"(",m,<¡(e) • logC¡¡r"- 1, ~(E) Thus D(m) = D' (m)D(l). For stochasúc processes D(l) =l. For deterministk processes D(l) = l if D;, l whilc D' (m)= l if D,; l. Thus D' (m) dístinguishes stocha.stic from deternúnistic processes ¡ust as D(m), but will yíeld D' (m)= 1 for D ,; 1, The theorems which establish the general propcrties of D(rn) also establish those of D' (m). For liD stocha~tic processes and the maximum norm. This relationship between the tcrms in the quotient yiclds theorems on the dí5trlbution Qf D' (m) (over realizations :il of a stochastic process). The relationship above shows that, for UD processes, CncJJ., m, 1;¡(E) behaves as a power of Cn(d, 1, r¡(t) rather than as a power ofe. On tbe other hand. for finite samples of more general processes. the growlh of both functions must decay as e gets larger nnrl forthese reaoons, especlallyfor smallersample sizcs, the growth of CH{sl, m, tJ{E) is better compared witb CH{sl-.I, t;(e) than witb e. 4.

(6) Mayer J A11 Altemative Correlation Dimenxirm Smtistic. Let us mentíon bcfore proceeding that in practicc, the G&P CD of a time series, D, is detemrined by means of the follow.ing proccdure. For each m, D(m) is calculated as the coefficient in a linear regression oflogCH(st,m, 1:¡(E) with respe-ct to loge, in a region of srnall values- of e: where the relation between both variables is approx.imately lineaL 1 Similatíiy, we ca1culate D (m) as the coefficient Jn a linear regression of logCH{.<A, m, t¡(E) with respect to logCH{st, J,>t¡(E). Wc now prove sorne nsymptotic properties of D'(m). These follow the appHcation of the tcchnique of U-~tatisücs {see Serfling, 1980), to CH(si, 111, t)C e) and related statistics by .Brock et al. T o give a synthetic exposition we begin by quoting two theorems, the fi.rst of whlch sununarizes re<ults by .Brock and Baek ( 1991 ). 2.2. Theorem. Let \Z¡} be a strictly stationary proces.o; which is aboolutely regular. (A definition is ommitted for brevity. There are alternative condítíons on the rate of dccay of dependence overtime yielding fue same result. See Denker and Keller, 1983, p. 507.) Define for each m, n, f..'. the random variables: C(m, n, 8) =eH( (Z,....,z,,. }, ·~ 1) (E) using the maximum norm. Generically (1fthe asymptotic variance does not tend to zero) there exist numbers JL(tn, E), o(m, t) for which, as n - t =, C(m, n, E) is AN(JL{m, e),~ a (m, e)). (asymptotically nol'IlllÚ). If {ZJ are ITD then. 2.3. Theorem. (Serfling, 1980, pagc 124.) Suppose that !he .equcnce ofvector-valued random variables X.= (X!, ... _x!¡ isAN(¡.t, n"1 :E) (asymptotically nol'IlllÚiy distributed) wlth I:= (<.Tg) a covariance matrix. Let g: ~ R be a real-valued function having a. ur:. nonzero differcntial at.r,.,. ~-E IM:k. Thcn. 5.

(7) Mayer 1AnAfternallve Correlati<Jn Dimensio!l Statistic. --------. The following propenies of D'(m) are direct applications of thcse theore!1'4;;, 2.4. Theorem. Let {Z¡} be a strictly stationary proccss which is absolutely regular-. 1 Define for cach n, m, E for which C(1, n, E)< l the r..mdom variables:. , . ) Jog C(m, n, e) ·-D (m, n. e = Jog C(l, n. F) using the maximum norm. Generically, D' (m, n, e) is asymptotkally normally distributed as n ~ =. If {ZJ are UD then D' (m, e)= lim, _, ~ E(D' (m, n, e))= m. The asymptotic mean is m. The asymptotic variance cr2 (m, e) depends on thc asymptotic means and covariance¡.; of C(m, n, e) and C(l, n, E). The analogous statement holds if we replace D' (m, n, e) wíth the coefficíent for the :slope in a linear regression of logC(m, n, e) with respect to logC(I, n, e) (as functions of E). Thcn tbe asymptotíc variance a2(m) is a function of the a~rnptotic means and covariances of { C(m, n, e1), C(l, n, €¡) }, where {e} is the set over whicb thc regression is defined. •. Por sequences originated stochastlcally theorem 2.4 provides, as does theorem 2.2 (DBS statistic), a hypothesJs test for the UD null. Actua!ly calculating the asymptotic standard deviation of the. (asymptotically nonnally distributed) CD, as obtained from the linear regression, would involve an exceedingly large number of computation& (uf the order of N~tt CD computations). An altemative method for obtaining signifi<:ance íntervals is preslmted below. Sorne further conslderations follow from theorem 2.4.1n practice it turns out that the comparison of logC11(d, ""'~ 11(E) or logCu(d, 1, r;(E) witb log(E) requires vcry large samples. Ramsey, Sayers & Rothm.an (1990) show empirícally, in thelr criticism of applications of the CD, that in a regression of logCH(sll, m. t¡(E) against log(e) bia..o;; !s introdu¡,.~ed by m and N (state space dimcnsion and sample slze). By contrast the seo avoids this problem from it"> definition, c-ancelling out thc comparison with log(e). Moreover, since D' (m, E)= m independently of e, we have one less limit proces.s requiring 1arge amounts of data. The SCD, besides testing the nn null, ls a numerical method for the calculation of the CD for <limension-; greater than 1,. 1 TI1erc. are severa! det1nitiuns which yield thc result. which we omit fur brevity. 6.

(8) Mayer 1 AnAfter!Udlw: Corrciatiotl Dimensirm Statistic. 3. The Residue and Reshuffling Tests:. Signijicance lntervals A few PQint::; rema1n to be made regardlng thc application of thc CD. As wa'\ already mentioned, ín lhe case úf stochastic time ~;eríes, if these are whitened by a funct1onaJ transformaííon before applying the CD, a ::;maller data sct will be needed to oblain the correct high dimensional results. On the othcr hand. it can be shown that such a functional transformation will not raise the dimension in the deterministic case -if in practice there is enough data and the transformation does not in vol ve many variables. This procedure, known as thc residue test, was proposed by Brock (1986), An additional procedure is to randomly reshufflc the series (with rcplacement). If there is structure, whethcr stochastic or dcterministic, it wiH be eliminated. so the value of the en shtluld cise. Thus, gíven the CD of a time series. one may apply the following significance test for low dimension. Obtain the CD of k independent random reshufflings of thc original series and see wbether if the original i~ lower than all of these. We can find significance intcrvals for this test which are val id índcpendently of the dístribution and which do not require a variance calcu1ation. (1ndeed this ho1d~:> for the same test appHed to any statistica1 proccdurc w.ith a continuous distribution function.) Suppose that for a given data set the CD of random reshufflings has a dlstribution well-approximated hy an integral P(CD E f). ~. f F(x)d.x 1. (as we may for data sets for which the NN resultiug reshuffled CD' s are approximate1y continuously distributcd). The probabil.ity that a given C.Oo (corresponding to the original series) is smaller than k indcpendent random reshuf.flings CD, l :;: i::; k is tben welJ.approxímated by P(CD0 <mín 15 ,"'(CD1))=J. R. = [ - ___!_. (J. (J. k. F(y)dyJ F(x)dx. (x.-J. F(y)dy))k+'. k+ll(.t,""} These oonf1dcnce inrenrals are iudependent of thc density function F, In particular, this ímp1ies that when applying the random reshuffling test ít is unneccssary for F' to be closc to the normal dislribution, orto estímate its vnriance. Abo, we may calculate the CD hy mean:) of a regression on a con~iderable discrete set of € ( which would makc tbe calculation of variance prohibitive) obtaining a much more robust statistic.. 7.

(9) Mayer 1 An Altenlátivc Correlatirm Dimenslon Stuiútü:. 4. Compari/ion of the seo with the Cl> To obtaín an infonnal qualitative compati!:>on of the two stalistics first observe figure.<:; l. which report D(m) and V'(m)D(l) for the following series of 823 elemenK Stochastic series: 1) A uniformly ctis.tributedrandom variable. 2) Autoregre.s!:>ive series y1 r:- 0.9y1 1 + w, with W; uniform on an línterval Chaotic dcterministic series: J) Thc logistic: xl+ 1 = 3.57x, (1 - x,),. 2jTheHenonscries: x:+l""" 1 + y1 - L4x;,y,+ 1 ::;:;;Q3y2• 3j The Lorentz series: X¡~= lO(x3 -x2).x2'::;:;; 28.0x2 -x;: X¡X3, x·/ =- 2.66x~ +x 1x2 4) The Rossler series: x 1' = - x2 - x3, x{ =x1 + 0.2't2, XJ' = 0.2 + (Xt -S.?).x3. As can be observed in the iigures, in the case of the chaotic series both statistics gave similar results. Howe-ver, the results were different ln the stochastic case, The proposed rncthod of calculation reduced the bias towards Iow dimension pre.-.ent ú1 the melhod of G&P. W e now report comparison of the seD and CD in the- five different ways rnentioned in section l. Let it be stated that both statistics were calculated together by the .same computer program. dumging only the tables against which the linear regrcssion::; of log CH(.1l. m, t)(e) were calcu1atcd from log CH{d, 1, ,.l(e) to logc. In the stocha.<;tic case D(l) is very closc tú l in practicc, so !he graphs show D(m) and D'(m). The Monte Cario tests uscd the fotlowing chaotic and stochastic processes. Realizations of the chaotic processcs differed in that a uniformly raodotn nwnber of iterations was first discarded, producing realizations with random initlal points dis~ tributed according to the proCé8SC$' inherent ergodic distributions. Chaotic processes (run in doubleprc-cisíon and verified to have a chaotic attactor): 1) The Iogistic x,. 1 = 3.51x, (l -x,) withx0 =0.255 ' 2) The Henon series: x • ::;::¡¡ 1 + y - 1.4x;,y;+t ~0.3y with Xo =0.5, Yo =0.5, 1. 1. 1. 2. (simílarily), Stochastic processes (let Ut be a normal random variable ohtained as the normal-. ized average of 12 uniform random variabJes): 3) The autoregressíve process Yt+ 1 = .6y, +u,. 4) The absolute value ARCH proccss y,= lu,_ ti u,, 5) The same process with autoregre5sion: Yr+ 1 = .6y1 + 1u1 _ 1 l u1•. R.

(10) Mayer 1 A11 Afrenu;tive Correlatitm llimen.sion Statistir. 6) A nrumal random varíablc wilh variante depcndiug on .(;tates following a. Mark:ov Process: Yr=U;U1,. '1 1 , P tar·fr=Ia. "¡ A ii• ~];;;;. 1 ;;:;:,' t,J::. . . ~ 2 ' A ;;;; (·_2 .S\ 6. .4)'. 7) The same process with autoregression: Yr+ 1 ;;;; .6y1 + a 1u1• The first test obtained the mean of the SCD and CD of 2,000 reshuffles of realizations of length 500 uf the.~ stochastic procer;.ses. Thus, the dimensions of llD processes with different distributions are ohtalned. Figures JT, in ditTerent scales, show thnt the SCD is slightly upward rather than very downwar<.l biased, for !ID processes. Also the SCD resulte, for diíierentiiD disrribulíons are relativcly much less variable than theCD results, which are more dependenton thc distribution. In praclice, the samercsult holds for non-liD proces.ses.. 5. Power of the Reshuftling Test for lhe SCD and the CD How powerful is the random reshuffiing test for wh.ich significance intcrvab; were ohtained in section 3? That is, what ís the probability that it will reject the presencc of structure when it is indeed present? To test this we applied the randomreshuffling test (with k= 20) to the Séven processes dcscribed in the previous section. We carried out tbís tesr for thc sen and CD defmcd for dífferent values of t (see definition 1.1), that í.s, for differently lagged m-histories. The values for -t were 1, 2,. 3, 4, JL The resuits are presented in figur~ In. In the case of the logistic, both statistics obtained a 100% score for cadt value of t up to 16 dimensions. For the Henon series. the result{i wcre very similar (onc must recaii that w.ilh finite precision arithmetk, a chaotic process will pre.sent a s:tochastic oomponent afterenough iterations). But in the stocha'\tic case, the SCD performed cons1stently belter than the CD. obtaining pmvcr ratings of 95% or above ovet a much wider range of dimensions and values of '!. Thc failure of both tests in detectiug variance depending on states following a Markov Proe,-csl:f point"' to the large data requircments necessary to delect such fuzzy structure,. 6. Summary Statistics of 1/J3 Reshuffling Tests on Residues of M, Friedman 's Monetary and W. Cox's Treasury Security Long. .term Time Series Milton Pdcdman's monetary and W. Cox's Treasury Security long-term time series wcre examined for the presence of e haos and for nonlinearity. Thc residue tests cons.istcd in extracting oorrelation in thc mean and variance, as well as in sorne cases; íntegrated corrola.tion in variance. Evidence was found for these non-linear stochastic structures. The estimaüon proceduros are presented in Appendix A. Altogether, the.

(11) Maycr 1 An Altematiw Correlation Dimension Stmislic. numbcr of residue series subjected to thc random reshuffliog test wa~:; 103, enough to compare the sen and CD in the context of practtcal appiü:ation. \Ve report summary statístics for the rcshuffling test with k= 20 as appHcd to the macroeconomic time series resldue:s mentioned ahove (the CD and the SCD of 103 residue series and of 20 re._llhufflings of each). Lct us describe thc surnmary statistics. For each test and each rnethod we ha ve- the following picces of data: the mean, Jowest and highest values of the dlmcnsions obtained in the 20 random reshufllings, and the rlimension of the original macroeconomic rcsidue (or norma1ized) series. Figures; IV .1 and IV .2 contain the means o ver the 103 teS,t\1 of each of these pieccs of data, for each method respectively. Thus we graph the mean of the unshuffled, Iowest, highest and mean rumeusiuns. (The mean mean dimension is a mean ovcr 20 random reshufflings of 103 sets of data.) The downward bias of the G&P method is apparent, with mean dimcnsion not rising above 8 and mean highest dimension not rising above 12, ín 64 dimensional phase space. On the other hand, the SCD mean is very dosc to the theoretical1y expectcd mean. It presents a srnall upward bias due mainly to the asymmetric distributíon of D'(m) aboutm. whlchis strccbed upwards (lhe mode is probably at m). Funhermore , tbc unshuffled dimensions are much closcr ro the Jowest dimensions in the ca're of D(m) than in the case of D'(m). F1gure V shows that thi.s last fact, which is a measure of the sensitivity of the statistic to structure, is uue not only absolutely but also relativeJy. He.re we norma1ize each of the pieces of data by dividing by the corresponding mean dimension and comparing the normalízed mean lowest dimcnsions with the norma1ized mean unshuft1ed dimensions (whích contain non-random and therefore very improbable structure). We i1nd that the unshuffied series can be much more rcadlly distingnished from the shuffled series in the case of the seo, which is therefore that much more ~ensítivc lo structure in the series. We summarize by observing that not oniy are meaningful results obtained when rcmoving structure progressively, but these are obtained for a wide range of phase spacc dimension.s. Figure V would suggest that in the reshufflíng tests carried out for macroeconomic time series in this paper, thc ~t detection of structure is obtaíned aboul m= 24. Section 5 above also shows that the power of the reshuffling te$it may maximize at quite high dimension~. Thu~, the SCD seems to precJude the need for data sets oftheoolerofN =!()m suggesred in Ramsey, Sayers &Rothman (1990) fortheCD.. 7. The SCD ami CD of Broclr ami Sayer's Time Series To ohtain results directly comparable to others in the lite-rature. we applied the sen and CD to the time series t.hat Brock aod Sayers used in their 1988 articJe {figure VI). Here we obtained dimcnsions for the f1l"St differences, for autoregresslvc residues of urder 1, and for absolute-value ARCH resíducs of order 1. Thcse were identical in 10.

(12) Mayer 1 An Alternntiw Cotrelation Dimensi.on StaúsJi.c. form to thosc applied to the monetary series, cxcept that because some of the time series are short (between une and two hundred) we only used one lag (see Appendix A, part 1 for the cstimation procedures). A more carcful anaJysis would have to considera wider array of stl'U.\.:tures, although in fact most of these series are too short for a good applícation of dimension e.stimates. We reproduce the ave-rage of so me of the results obtaincd for the economic serie."' in the following tables.. Tuble U Average Corrdadon Din1ension Obtained for Residues of Brock and Sayers 1988 Econorrúc Time-Series Correlarion Dimension. m. First Difs AR! ARCH1. 1. 2. 5. 10. 20. 30. 3.52 3.65 4.05. 4.42 4.35 5.20. 4.80. 0.84. 1.63. 2.83. 0.99 0.99. 1.73. B6. 1.80. 3.14. 4.99 6.17. Tuble !.2 Average Statistica1 Corrcla!ion Dimension Obtained for Rcsidues ofBrock and Sayers 1988 Econornic Time-Series Statistical Correlarion Dimensi'on. ---.. m. First Dlfs ARI ARCH1. j. 2. S. lO. 20. 3n. 1.85. 3.78 3.88 4.21. 5.54. 9.04 9.74. !2.0. 1.87 1.93. 5.95 6.94. 11.9. 13.5 17.6. The maJo conclusion is that the apparent convergence toa low dimension of 5 or 6, as originally nhtaincd, is spuri:ousiy introduced by thc CD. The dimensions of the first differonces as obtaincd by the SCD already show consistent growth wíth phase dimension, poinüng to structure of l>tQChastic naturc. Moroover, most of the series' ARCH resídues show a higher dimension (including Pig Iron, whlch was not found to rise by Brock aud Sayers), while the AR resldues do not dsc very much. Tire case is thus strengthened for thcir nonlinear, stochastic nature, (If the structurc were linear or if it were s.puriously indru:ed by the regress.ions, the AR residue dimensioos would rise compar.ibly). Sharp gradient change:s in any of thc:. 11.

(13) Muyer 1 An Al;emative Correlarton Dimensivn Suuistit. graphs are best inlcrpreted as pointíng to the uppcr Umit of phasc space dimensions yielding meaningful results. It is worth commenting that the determínistic sunspot time series used a'i control by Brock and Sayers, which yíelds consistently ristng tlimensions, may not in fact currespond to a low dimensional process, since it is govemed by a nonlinear partial diíf\.~rential system, to which Takcns' theorem does not apply, and which may have infinite or at least very high dimensional states. ~1oreúver. ARCH processes have been iinked to diffusion (Ne-lson, T990) so that the extraclion of ARCH Js not meaningless ín this context. 8. The SCD and CD ofthe CRSP Daily Long..tJJrm Serie• The ncxt point of compariwn is a single re.sbuffling test applied to residues from a CRSP serie-: with lcngth 6409. Three pairs of graphs show the dimensions of thc first dlffcrences, autoregressive residucs, and EGARCH residues a.;; provided by Craig. Hiemstra (sce Craig Hiemstra. 1992) (figures Vll). It is interesting to note that l.he CD provJdcs a downward biased and much lcss stablc &tatistíc even for thís length of series, yielding a spurious dimension of 5, while the SCD yields smnothiy rising dimensions, which are unbiased in the nD case. The graphs also confirm that the SCD provide..;; a mon:" sensitive test fur stochastic structurc. sinc,c lhe dimensions of the original unshuffled series. are much further from the mean than the shuffie.d series. The presence ofEGARCH is strongly confinned by OOth statistics, though the SCD detects additíonal structure for m-histories with m> 24 wíth a confidence of 5%, which is. only vaguely índicated by thc CD.. 9. Conclusions Regarding thc sen. the conclusions of this article are the foHowin:g: The SCD is a correlation dimension statistic improving the numerical computation of the CD in the stochastic case and its performance for distinguishing deterministic from stochastic and linear from nonlinear processes, In most of the cases, we see meaningful results up to high dimensional phase space. for series with lengths of five hundred or more. The SCD is a more powerful statistic for the det;e{:tion of stochastic structure than. the CD. Applied to these relatively small-si?.ed data sets. it is unhiased in the stochastic IID ca~, provides a test of thc liD null. reports low dimen~íons in the t::haotic cases, and is vcry responsive to succesive removal of structure by the residue test method (see appendíx A). These results are confirmed by each of the succcssive test-, applied. Combined. 12.

(14) Mayer 1 An Alternartve Corn:latinn Dimcnsfon Súitistir ~------------------~----~--. with the significance intervals obtained for the reshuffling tc:,t, the SCD proviúes an improved statistic for detecting stochastic structure. The downward bias of the CD and its low .sensitivity to structure when applied to relative1y short time series, accounl for many of thc ambiguous resu!ts in the Econouú.,; literature applying the CD, including spuriously low~dimensionai results.. Regarding the macroeconomic time series examined, the condusions are the followíng; Evidc-nce for deterministic chaos was not found in the long-term monetary and Treasury Securitics time ~;eries cxamined. Strong evideru::e was found for nonlinear stochastic be-haviour, especially in the forro of structurc in lhe variance. Evidence was found for conditionally autorregrcsive linear hcteroskeda.;;ticity in the standard dcvíation but also cvidem:e for othcr forros of structure. In the case of the price index of tbe Treasury Bills il was found that integrated autoregressive conditionnl betcro!:)keda.\tkity may be pre~;ent in the price variations.. Appemlix A. Dlmew;íon Tests App/ied to M. Frú3dman 's Monetary amllf. Cox's Treasury Security lAng-term Time Serie!!i A. l. Applicatirm to :Milton Friedman 's ~.Yonetary Time Series. Monthly variations between thc dates of May 1907 and December 1968 for the following monetary quantities as published by M. Friedman were used: Currency Held by tbe Public (MI), Total Dcposits at Commcrcial Banks, Deposits al Mutual Savings Banks, Total (sum of previous tbrec) (M3). (The length of the original series is 740.) In cach case the autocorrelation statistic Q detects the presence of autocorrelation, for the series itself and more !Strongly for its <lbsolute valU(\ therefore detccting conditional heteroskedasticity, To apply tbe residue test rhe series where transfol1:lled estitnating sever:al models. These rnodels extracted structure resulting from linear autocorrelation in mean, autocorrelation in variance. and change of regime. Modcls extractlng linear autococrelation in the mean were estimated by ordinary least squares (OLS): l.. R,= Po+ 1: lliRH +~, i""' 1. where L. was set to l or 12. and ~are thc least square residues. The residues of these regressions are not autocorrelated, as detected by t1:te- Q statistic, Howcver, conditional hereroskedaslicity was detectcd by means of the follow~ ing OLS regression:. 13.

(15) Maya 1 An Altern.ativc Corrt:lation Dimension Statistic. L. ¡¡;,¡~Ele+. I. s,¡~,. i=I. l~ 1 l1s a coru;istem estimator of the standard deviation of R1 (Schwert 1989; Pagan and Schwert 1990).ln tum, this condition is sufficient to apply tbe rcsidue test (Nuissance Parameter Theorem, W. A. Brock and W. D. Dechert [1989]). Models extracting condilional autoregressive heteroskcdasticity werc generated hy cmnbining tbe prcvious estimatc of the standard deviation with a new autoregrcssion of the mean t~stimated by wcighted 1east squares. The dimcnsion of the n~idue of the standard deviation was a1so calculated. Models extracting -struc'ture in the formof a change in re gime replac.cd the constant in cach regression with two dununies adding to one, representing the periods up to and after World War IT respeclively. The resulting SCD may be seen in figures Vffi.l, VTII.2, VIIT.3. corresponding respectivcly to the residues of modeJs e-xtracdng linear structure. ARCH structure and finally to the residues of tbe variance estimates, (Abbreviations used in the graphs are explained in Table II.) Each figure includes cases in whkh lags are chosen ato. 1 or 12, and lhe change of regime dummies are included or not. (These are grapbs of rl (m); D( 1) was consistently very el ose to 1 for all macroeconom.lc residues considered in this articlc.) Let us first examine the SCD' s of the first differcnces ofMl, Deposits at Commercial Banks arui M3. In figure Vill.l, the.se series pre<ent c<>nsi>tently low SCD'•, reflecting lhe presence of snucture. Thc SCD for the original first dlfferences yielded results almost identical to the seo for the linear rcsidues, This means both that the linear autoregression introduced no t>lJUrious suucture in the data, and tbat not much of the structure in the data was removed by ít. Figure VIII.2 is quite different. We see the sen consis.tr,ntly rising according to the progressive removal of structure, aJmost to the expected vaJue for llD series, However, figure vm.3, which exatn.ines the estimated variance residues, shows the presence of structure which has not been taken into llCCount by ARCH. Indecd. the method of extracting the structure attributed to the change of regime (which is not very. sophi.:;ticated) seems to introduce sorne spurious structure in the varíance, since it lowers thc SCD. although it does remo ve struct.ure ftom the original series. Thus1 we wouJdexpcct additional or different structure in the variance than that hc:re oonsidered. Let us now examine the seo' s of Deposits at Mutual Savings Banks. In figure VITI.l a somewhat-structured stochastic nature i:>i reflected; the change of regime has not much effect upon the conditional means and a certain amount of structure is removed by the linear autoregression. In figure VII1.2, change of rcgime does not make mucb difference in any ofthe ca.~es considered, while a certain amount of ARCH seems to be presem in the series. In t1gure VITI.3lt is apparent that sorne structure other thrul thosc here consid.ereü is present in the variance, which howe\-er does show sorne autoregressive structure.. 14.

(16) Maye.r 1AnAltertUJ.live Correlation DimensWn Sra:istic. A.2. Application to W Cox's 'treasury Secu.rities Time-Series Monthly variations between the dates of Jan 1942 and Dec 1984 for the follúwing quantitics were used: Market V alue of Federal Deht; Prlcc Index of Trea.sury Debt (any annuitie~ to maturity); Prtce lndBx ofTreasury Bilis (any annuities to maturity); Price Indexes of Treasury Dcbt (0-l, 1-5, 5-10. lO+ annuities to maturily}, as published by W. M. Cox. (The length of the original scri"" is 516.) The first diffcrences of thesc series showed low autocorreJation, but their absolute values were highly autocorrelated, thc-refore detecting conditionaf hete.roskeda..,.ticity. To apply the residue test lhe series whcre transformed cslimating the variance in severa! ways. Thesc models extracted struc-ture in the variance resulting frnm autocorrelation. The original series were then nonnalized according to the estimatcd variances. Except for two cases mentloned below, the modcl estimated was x,e N(O,v,) L. V1 """Bo+. L 0 v +T); 1 1. ¡,¡. where x, was the f1rst díffercnce of the series being considered and L was set to 2 or 6. Then thc Uimension was calculated for x,lvr and Tlt· Autoregressive structure in the means was only detccted and e.xtracted in the case of the Market Value of Privately Held Markelable Federal Debt, as for the monetary series above, but with 6lag,. In the case of the Price lndex of the Treasury Bills, it was nccessary to conslder an autoregressive strucrttre in the first dífferences of variance hefore the stochastic nature of thc series yielded high CD. The model considered was. cr. ~,=o,+. r li,t;, +11, i"' 1. Again X 1 was the first difference of the original series, and the dimensin was ca1cuJated for x/v1 and llt· In fue case of thc Mllfket Valuc of Privately Held Marketable Federal Debt the .series was found to he autocorrelated in the me.an., so ARCH was extracted as for the monctary series, but with 6lags. The rc~ulling SCD's rnay be seen in figure IX, whích shows in each graphresults for a single series in tum. (Again sce Table TT for .abreviations in thc graphs.) Thesc are. !5.

(17) lr!ayer 1 An Altematit'e Corre.larfi:m Dimensivn Suuistic. thc SCD •s for: the tirst difference of thc time series; the first differeocc nonnaHzed according to auwcorrelated variances at Iags 2 and 6; the residucs of these variance estimatcí5. In the second graph for thc Price Index of thc- Trcasury Bilis the sequence is similar, but the quantiries estímated are the first differences of the variance. The Graph for the ~arket V aluc of Príva~ely Hcld Marketable Federal Debt shows instead dirne-ns-ions for the residue!ii of thc wcighted linear regression; for the weighted residues, and for the varíance residues, a" welJ as for the first differences normalized wtthout taking into acC(JUnt the alltoregressive component of the rucan. This last procedure is lhe only onc to introduce spurious structurc. as can he seen from the docreased dimension. The other two procedures raise the dimension, with the residues of variance showing similar structure to the residues of the ftrst difterences.. In evecy oilier case. it may be seen that the successive removal of strucU1re r.úses the cstimated SCD, in sorne cases frorn what appeared to he a oonsistently 1ow dimensional rcsult In the case of the Price Index of the Treasury Bms tlüs proce..l:is yíelded high SCD only when thc conditionai autoregressive heteroskeda:;;ticity is considercd to be an integraled process{conditionalautocorrelationin tbe firstdifference of variance), though then the rcsidues of the first differences of variance continued lO prescnt much structurc. Comparison ofthe SCD ofthe Prk:e Index ofthc Treasury Debt at different maturitJes shows that there Js sorne structure in the shorterterm wbich fades as the length of maturity is incroased, different from the structurcs hcre considered, It ís interesting that in each case the series and variance re~idues gave simUar SCD's, supporting the hypothesis that most of the structure of thcse series ís in the variance.. Appendix B. Numerieal lmplementation ami Choice ofe To minimize the computation requiremenls in the cak:ulation of the CD an algorithm was developed which: a) calculates the di~tribution of distances recursívely in the phase spacc dimension; h) mainly uses short integer arithmetic; e) has.low mcmory require~ ments. This is hascd on the following, If. ís themaximomnonn distance between the m-histories (ai.···•aí +m-I and a1,••• ,aJ+m-l)., (Wc apply here only the case t = 1) then. 16.

(18) Mtlyer 1 An Altemative Correlatfon Dimension Statistic. so that, for cach dífferencej- i,. b;+l is obtainéd tttursively from blj, requiring. in. memory only a diagonal vector of norms of the differcnces. A set of values of E is represented as a .sequence of intcgers. Once bb has been cakulated, thc relevant information of its magnitude is therefure an integer, repre~ sentable in one byte if the number of E is less than 2.56. Sine e the linear interval~ on which the regressions are calculated shift considerably when m is. íncremented, these were det1ncd, after much expcrimentation, in temu;. of the values of Cs(:A.m, ¡¡(e:) as a foUows:. Thesc interval< of 8 correspond typically (lo a factor of 2) to [0,002,0JJ05] for m= 1 and [0,02,0,045] for m= 30. Observe lhat tbe lower bound represents a systemalic eHmination of noise. The optimaf boundii must vary with .sample size.. The program for tbis algorithm, implcmented in Turbo Pascal for 286 or 4t:S6lntel processors (a mathcrnatical ooproce-.ssor proves very u&eful), is avai1able, at the División de Economfa, CIDl:L. Biblwgraphy Baek, E. G. and W. A Brock (1992), ''A General Test for Nonlinear Granger Causality: Bivariate Moder', preprint. Bameu, William A. and PJng Chen {1988), "The Aggregatton Theoretic Monetary Aggregates .are Chaotic and Ha ve Strange Attractors: An Econometric Application of Mathematical Chaos", William Barne.lt, Ernst R. Berndl and Halben Wbite (cds.}, Dynamic fumomic Mnde.lling, pp. 199~245. Bamett, WH1iamA. andSeungmook S. Choi ( 1989), "'A Comparison Between tbeConventinnal &onometrk Approach to Strucrural C'--0nference and the Non Each Metric Cbuotk Attraclor Approach", in William Brunett, John Geweke ami Karl Shell (eds.), Ecmwmic Comple.xity, pp. 141-212. Boidrin, Michele, and M. Woodford (1990), "EquilíbriumModels Displaying Endoge- nous Fluctuations and Chao.'ii. A Survey", loumal ofM(metary Economics, vol. 25, pp. 182~222. Brock, W.A. (1986). "DistinguishingRandomand Determín'istic Systems: AbridgcdVerston", Joumal ofEcoiWmic Theory. vol 40, pp, 168-195. Brock, W. A. and W. D. Dechert (1987), ''Theorem'l on DiQ:tinguishing Deterministic from Random Systcmt~", Social Syslems Research 8702, University of Wisconsin-Madison. Rrock, W. A nnd C. Sayers (1988), "Is the Business Cyele Characterizerl by Determinisüc ChaQ.sT', Joumal of Monetary Ecunomics, vol. 22, pp. 71-90. Rrock, \V. A . .and A. G. Malllatis (1989). Differential Equations, Stahílity and Chaos in Dynamlc Economics, Advanced Te~tbook;; in Econmnics, North-Holland. Brock, W. A. and W. D. Dechert (1989), ''Statistical lnference 'Theory for :Vfcasures of Complexity in Chaos Theory rutd Nonlinear Science.. , in N. B. Abraharn et al. (eds.). Mensures of Meavures of Complexity and Chaos, PlenurnPress• .Kew Ynrk.. 17.

(19) :::ito~hastic. ttnd Chuotic Suries. '•'. --·---. ' "'. ,.--~·/'~. 8. '. //. /. e. //. ". /. '. •' '' ". ". ''. m. ". "•. ~. 4. ,.--//·,/~~~~~4-'---+-~~,~~ ~-. 1. :~~- ~ ::_:j:-~=j 2. e. •. 3. 5. 4. 6. 9. :. j. 8~. " •. '. •'. or. •. 4. '. ". '. m. • •". • ". 2. o. JO. --·-,. iO. '. e. --+-. /. c-~. ~ ··_J~_. ------"---··. j. 5. 4. G. 7. 8. f·h<~u ;)¡:><><:<1 \J;Ifi<!n~w::. Uniform -8-----. l<'IP~iw. -. Autoregrfe~~;.,,.. .. LIT'f'lllt. Figure L?. ~. Henon. -4- Ro!:sler.

(20) DhnPnswu <.>f \lome IJD l'rocesse.s Mean fDr !'lamplf' af 2000 Rcshuffles. '' "\ '\. ¡. •' l. '. ,,' l. '' '' ' '". ,. m. •''. $i;.,tlslicto! C.;rnr!i>tí<Jn. Oilúé>~l<i"'". 72¡ ~-. 64. '. "f" 4C :---. ". ". lG :-. ' ". ". '. ". o. Dhw:mdon oí some Tifl. P!'Q<.'-N~ses. M<'!"rm for Sllmpl<' of ZOOO reshufl'les C<>ne1aUnn Dirncnaicm. 14 ,-. '' ' '. '. '\. ¡. ". ". ''. 1. ···'. 1. 1. '". "¡ '. •'. +. •'. ... ,..

(21) POWOilt'IMJ. -1-q.l\~ u~. n-u •••loo .. -""m. ~. UnO<O A=>•il-• ia>l•. 11>lllft! v~¡-. ,,.....-------·-.. OQO. ·~. 0?0. ~ ~-. ,,a,. :~e. ". P~;. Qn. ;7'. 04b~. •' '. ~-~3 -,V_.._. í. í. .. "' w{. o '05$,. 1. .. r~-----. ~B~r. M. 'l C!ii,. !. o•¡. e u,. " "·' ""' lll. .~. " '"0' ~~ L. tl\•~"\o""' "'"~. m. ti"""' _Tal,, ~11.1. "'' \ R~:l •'. o~¡¡~ J. ~e. \tHL. '. ''. e•e. ~,. •. .;:L ' \'Jr. '. •. '• • • • -scp. ". :H. co~r. -+-'-4. w. "'"'"'t '"c.t~-. . ... ·'--1-. o~" Qb-. og~f. '""--. -----. --. "•. ..,__,.. i!h(U. " ''""'"' J7!. <n~f. ~o~. '. co. ~•. -~e. , __ , __ WJ!,4< '"'"~ '"""'"''. M M!'\W'"' "'''". "il""' '"'. V~>la:ll"" 1>11 ... 1"9111"...... l'owwr 'ltil 1. "H~. v..,t:."!':.~= ~~~::!',_.. P"'""'". ~i~f' (:~~~" ~+. }1•. :~r~. i. "•" '. .''.. .. 1 ~¡;~. ~·"''. -,a~:~"". l'ti-'!'Mt 6 11.,,..,.,¡ Raú"' -1-o:lll~. •. __ ,. ~""-'""" ""~ '""" ~· r~~,,.,,. .¡¡¡. Q,!t. OH•". .,'. _j. .T. '. n~~". OM. ---·. .,,..,...,_,.,. -lo• ,.,,• .~,,..,.¡,.~ ""'"" Al!ell. Al>tú--..oAii!l:$!. wlll ttn-nv,...lu ,._. " .,.. ... ' • • • • " " ". '... /. ~ ~ '. /. ~ "~-~~. cg:. J. .. 1. OH'. OA 1. U]o ¡. ols. 1. 1. ;;-;. il!#. :'". ---J-.1--~ ~. '-4-----'----.--"'-'_____,___,__; li). ll. 1<. to. búof. v~----"--. 1. ·~--:-J ,Q. '~. 1<. ""'""" IP4'~ "'-4<'C<>. m-'"""'t"' "''". flQ~,_p¡¡. ,>¡¡•. ~. ro-h""'""'-. ·~<t~ L~Q• ~~. ~!QU"' E '-'. 1. '".

(22) ""--:w¡. ¡o....,~-IM .. ll!l. """u"''~",.''""',... '-t'hM.'I. u.....,".., _.._"""". oo~r,---~. :. ~". ,,_,~,. "'~ 01~~e~~. o ..S e ~<>>-. ·' ,, óó 1. •.;-. • '. ~= ~ 1. 0.2' Ul~'. nd-. ll05'-. o--. '. "'- '"""'""'>;;-· '"'11'<-12 ?10-"'" ). -Tl·d·. o:>•>IO"m"" '-'"" "'<:>'# 7 f(¡¡~"'. ¡¡¡¡. -~· V"""t A•eH. ---""~ll!:n Ylt)K...,....._III...,.. .. ~ ......- d.............,. ,_.¡, -"1/\N?. "'"hlrurl•• '<'1111 ''-'~~' <11 l. m•lll•*<>'loo. ,._,,,,. fj;¡U"' HU. "''""'.,¡ ...- '""'-~ .....••.... w...... ,.,. - - lillk"'l!:t'. :ú-hl••m ... .,_,¡~. ,..,~,. ""'-'''" ~. .. 1!!.2. AmOI~IO. nv~i'>Wl. ~t '!'k!~. A,;,,.._,, jooloo wllll v.. ., ... ~,. -.ll""lna .,.,,_ '"'"-. m-Mol'''"¡,..,.-q4~<ri~. l'(JUMH1.

(23) ._,t. POW•• 3 ti"""' Jwl<lt6¡¡1.Wn S.rl ... ~1':'$1~ ~. U•un .,.,,., "Uh R~ttd""'. '"-"'"-' .,.,,.,..,. ·~J. "'. m ól""""' ""h 10¡¡1 <!( ~. '"'""'"'''"' ,.., h '"'"". "'"~"" !ltb. n~u"'-'''' ,..,.,., -·'l'al4 _ _ Mm!!. •--.....,hmomlll. ..u••-•""" '. A!MIU .. 'lt<lo;:.. J\1!e1f. -·~. 'l';c'\.. ' 1' '. .:T "'"'. 1. o.~,. ". CH. ,::,,. ·. ...... ' o~(···· '•" OH-·· 1) ~b. " "' '·'. 01$. ~''. ~,.;¡¡. ''. '. -~. '. "--.,,/·--.!. ' j. 0-JS~. ~. '. '" '. '. lkbb:WI.. '""' ""il"" f<Q>,¡WHPI. ""'-. " ' &·t'•>"><'~ ..,,,~ 1"(1> "!~><<! ,H;. ;,. . _,_.....,. d ). ;.,w.r lltll Y. ~tllt\11 ~. "' '. Aftk> .. CJ1'1M'Ift. ,..,........., '"11""''"1". '"'"""el> io(h>WIIIIJ !,(......... fn> ..... ,.,IQ wl!l> .,~,-. ,..,..,.... {\ /\. J\¡ \. '. ,,..,._.,~~··~ ' '+.J ~. ~h""a. '"""". lO. 1~. 11. 1>. .,,,.~·•~,;~. r;;-o.•""'"" ·~nh I"C' "' ro;r••• 111.~. :<. m· ni""""''"'"· :~J• ~ll ''0'"" j.

(24) 1owwt '!'1m 3. 1'<1- '1111# l. llu<>• ,......,...,....._ """'"". ll•non S••l.. "ltb. ~""""'" :~mM Y"h". .... ~. m"nl><~"~' "'''" i'<J' '" •. ,n•"''"""'"' "'"m ~ ct ~ p, ... ,,. ;;;<. -·~·. A ' l b _ . ¡ f l ' - vHb ;u...,¡,. Allell. ftO~"'. HiA. -·""'f> vm,.. JU..lll<"o Va!<U Al!Ca. Yrt• Wo..-oapu! ... ~~-. ' Mh. ·-. .~1. "'. ,, "~~y'. Jj~:r~~. "' '". "J;t ~\. 01.;(. ~."'\ ~. Q O~( o~'. ' o~vle,,. ''' "-·~r ' ;n;; ". ~'". ¡,,,. 1). 16,(JH-. 06.~~.. c.cgr. •. • "'. ,_.,.t,\thlLn wl'lo. "<041'0. :. ''. '. ". ~~!X. 1<. ~'. ' "g!r '" o•t~". ::J. '. '. (·". O o~~. ;, ~ ¡¡;;~. l/,l6~. ozL. 01~'. ''f '*. (>•··. '. ''. {. ){., .. ,.¡. t<l~ll-. *>1-l••. '. •. \1!•41<41""". ". .. ". ,..,t,::: t. ''V''"~. ,,..,.,~. u~. ~'i'MlT. 'l\i;>Wl!ltla&<:~. V..fi~~••. 1. Var!atll5 V!ffl. ""'"'.." ....,_ 1-.l.. 'l't!b. "a"""' 'N>o..,. ""''"" . . ,!!_i~g ~.~ ......... ,....,.... ~. '*. " ""' "' r-M ~ "~. N. '•' '. '" " "' :" " " '"'' "''•• ~~. ;:;,u. ,,..¡¡¡''"''"" "'""· ''V'"'~ fl\.lo•• 1 11 '. m-~'""'~.>' '1'1'~. '"'"'o! •. :<::.""' lll~. '' ".

(25) -tllnt2. n...n. '""'"" 1\oot. ~. Ll~O<II A111M•q-•~• '•"-. "''""' .. tw.. ~"'Od'*'. (OUl<l\ Vlll!l<ll. 'r------~-. :;¡t'" t. '. 7~. W!. '. --. QH. . ". ~~. •'. <¡~. f. \l.i>t. " '"' '. o. ••. QH¡. .....•. :¡¡~· '" .......... :. "". '~. ül. l. ". '. 6. ~. ltl. .'. ~~. ". ~..,,.,.- d!"'ff'l"~·-. m-tm=~«>. '. ''. '. ". ....~w"' -•• "'~'~ t<>;ll "' ,. wllt !<¡;p <1! ¡¡. ~!~A<>. PJ>tVtHO. ~~. ~~~·. .... --. rn-M»w\u Aw. ilH'ti-1... "dh l - " " ~. ~~ n;;~:.,m:. "'1 ,_.,., flqciO. ........... lil>lt_....,"'_'_ . ~~· """'-"". • ''. •• • '. j. ''. '. "'"""''kl"'"'"' ·-". w. W->4~. •. l~. '"'<UI>n.¡;,,..,¡..,. _. ... •Ub y .........,.. ''. V. «. ~110Ml7. 1. • '. 1!11. 'lln!6. A11b,.,_olu kr.._ Wllb 4l>l<>l~!t Vul~• 1\ICII. MM>I<IIo-U~ ~~ ..,.w'""'~. ,..,... J. 1. '. '. '. '. -seo. '. '. .J. " ". ---<--co. ·"-'' ~' C1. m-~1'"'"'"' '''"~ "WJ'e"'''~. rr.-h:.,.,.;,.. "''" '"'"' "' i'l¡¡~R> l!(!i. •J.

(26) MeAn DimenBi~n~ of Retl:.u.Jfl"d S"d"~ 11r4""be!-ge•·-p,""'""d" Mdhnrl. M~an Jl!m(!"n~ions of. f'.ehhUfflnrl. S"ri~s. Slnli~ticnl M~lh(VI. ¡·::tr ". \!V. "o. ". '. ''. m. o. o. o. oL-~-~-. o' ". '·~··~. 'l60Sllf~W11\~a...,_. Pt:mm SpMe Dlm.,!ll!wn. N. <ro ••u••i l ·. ~-. ili><JY:•<. w...... ~---. i'\¡¡\. Comparison nf Methods Oimension Compan:d wilh LowRFlllim.t:>nsion in Te"'t of Twenty Raudom Reshufflings. ". ' m ~. J. '. '. '. d. e 0. '' '1 '1 ". n. V. '. m. 'n. '. MD Mer.:n Duncuswn. M:.: Meen Loweo;l Dnne:t:hO:t. F'igur>? V. (101 ~uícl'l] (103 t.r~·-~).

(27) Urocl< on<l ~oyel'O' Serie• Dimon•Mn• M l'lr.t [liff<r<n«o. fuocJ.; on<l ~ayero' Ser'~" Dimenoion• ot ni .. t Olff~r.nc~•. ••. • '. •. ••. '" '" " " " .."Sp«c"" Oimonoion Ph~. -. ..._.._... Oun.opol ~. -;<-- GHP. ......... ·-~. .•'. .. .. ... • • '" " " " Phaoe Space D!rnen•lon. '. .__ ·-· -·· ....... '"'. "'•.....,. u~m"'"""""t"*-. """" ..... YIJU>e VI O Dl'<'lrlt ond Sayoro• Seneo of Autorreg>eolvo Do.M"'". urock attd Snrer•' Serlo• Dlmrnm<>n• of Auto...,greooive ,..,;dueo. .. ... IJim"""i""". . ... .... :: r. .. •. •• "". ." .•. "". ". •. /. /. . . . . .. .. ........... - ........ .•. ...•...,..,,. lod >rod. 3w~pol ~. """""""""''....-. """. 1IOW'O. ,_. u....,,,"""...--. ..... ¿f. ."[ ..."1. /"/. .. 1. •. ,.. .. .. .. P~a.., Sp""~. '"'. "'. ". ~. ~. "i. .. """''". w. ,.. /. ~. . ·-· .,,,.......... .. .. "''"". /. ". ... Brock and Snye:ra' Ser! u Dimen>iono of ARCH R•oldueo. / . /. ... •. PhMe Space Dlm•lloton. vt.a. Broci< and Say"""' Sor1e• Dim<noio"" at AHCH reBiduo•. .·¡ ... .. •.. /'. • ". "l'h.,.o SJ.>«ee Dlmen•lon. •'. ¡. ,.. lo•••<w•"'. ~. •• ". JJ!meu•ton. ......... ""'"'''",............ "•''=. FLou,... VI~. .. ,. •. ". .. .. .. " '" ". f'h.., Sp""" Dm1en••uu. ·-· "'. ''"~·~·"' Uh<"'PloJ""•"....... f¡¡uro VI O. '"· '"' ·····~·.

(28) ~f\SI'. UGI'. Dimen•lon ol Flrot. Dimenoion or Firot D!lforon ...... n1ffer~nceo. .. "! " ". o. o. ". m. o. ' o'-~~~-:~.. .~. o a '""'""'o'""""' Pila... Sp~~•. '". "". ~·. H~ol~~-. ''". •o. •a. so. e•. ........ n,u,.. Vll.a. Yll'"" VIl.! Auto>reg¡-e>o>ve. ,,.. Phaoe Space D!menolon. Dimension. Aulore~•.,••iv•. Dhnenoiorul. CRSP Reo.idue DimenM<>ruo. .."f. "'. " " ". o. '. o. o. ". "" •. ". m. o. ". 2< ,. .o •e Phaoe Spaoe Uunen"m. 19N0240<8006<. '". Ph""e Space Dlmeno1on. '"""'"''""' ....... ,.... ""'" '""""'. so. ••. """"""'"•' F>JU" V51. <. CM o E.gucll R..;due Dinl$R•iullo. CR!lP. F-t••ch Jleoid"e Dimen•io"". .". " ". ". ". •. ". o. '. o' o. o. '. •. 1B243B.048. ... M. " ". . ". 1024324048MR4. Phaoe Spaco Dlmen•ion o-..oo.. ""'''"""'. ... ""' ....,,,...

(29) 0<"f>O!ltts at Cormwu·dul Hanks. Curr<m<'>Y in Cit'('UJ.;tlí.ou Linear R¿sidue Di.;:nemdons. 'e'. •'. '. ' ' '. ' ". '. '. e. )~. '. '. ,,. 'l e. l. '. J. '. e e. e. l. •'. '. '' '. '. o. n. ". ''. D. m. m. ' '. s.. >. o. e e. Linear Rosidue 0\mt;nsions. S. ," e e. n. e. '. ". n. o n. ?hmw Space Oirnenslon. -~r. h•"'-W. --+-. A!l • :. ffi. --. e. ,._-e. 41'··!. --1--. AA<-12. - - >IR-!ii. ;;.~. <J. F'¡gure Vlll 1 1. nrpo»ib <~.t Mutual SavingE 81trtl<<> Linear Residu" Dimensions. ' '' ' " •'. ' '" '. e. ' '. Tüt.nl Money Supply. ''. '. e l. /. :: f. n,. .,,. "[. .'. '1. e. 4. ¡. i. 1 ¡. ,.. n. ''. 32 1. 1. ' Nr. "D "'''. "D. m. m. n. ". '. c.. Dim*"nsion~. ,,. '1. '. ' ,,'. Linear Rtntdue. e. '. ,L.. -~~-~-----~~-. ". ?4 Phe~e. 02. 4l!. 4fl. 56. Space Dlmensior. Al'l--l. - - M-12. -+· e ~. 64. ' ' ' 'e. '. ,r Ro~. AT<-l~. Cll. e. hc•t V\(. '. AFi-J. o·~. ~! R~~. "" !3,. F¡gwre vuu 4. ~Il_.

(30) Dcpoifits ul Corrun<;'l'Ci!il Uanks. Cut·t·cnt.t,V in CircuLation ARCU Reúdu"' Dimronl>:n:.s. e. •'. '. o. ''. ''. • •'1. •. "r". '' ' ' '. ~. /. '. ' '. '. "'" f. ''. ,r. •' • •. r. ' ". •. ' "'. •. ". lE. ' "'. '. " ' •. __ ___¡_____ __. '. '. '. " ". ". 41!. ·-. G4. 56. ~. e ~~. Ali.CI"-1~-. ~l!C::iw[.\i'. Fi~Ur<O. ·~. ~<•~. m. " ". ~. 10. "f. • • oe '. •"'. ". /. ",,. ". MlCH-1.. -~--. ' -~. " ". ~. '1 e. ••. • '1. ,. D. ~. ~~R...,. " ". ARCil- !_. ,,. ". Total Money Supply. Dlm<:>n~ion~. ARCH t'.esldu'l' Dimenri«n.s. ' l. '. 54. i. 'l • '1. 55j -'16. i. e. ' ' '1. "r 32. r. Z4. r. , l ,. 1. 1. .,. 40. Figure vm 2 2. 'l '. :;::. M'«..ti-12.. ". Depoaits 1H Mutual Sa'>'ings A;mks. '. ". ~2. Ph.ase Space DinHm.,wn. •.ru:·u-J ~. ,./"''. __. ,,____......--~··_____........--. Vl!l.2. 1. ARCU Re,.idue. .. /. 24-. /. /;/ //" .. 40. '. ?lnnm Sput:e JJímension. AllCH-1 ~·. JZ. '. ARCIT Retrhh!f' mmenHi<>n". '. 1. m n. 8. •.. '" "Spa,;,;:" D;mension 40. 40. '". 'íH. ?ht~n: ~{Y.-. ~. •. ::~. "t. "' ' 'm 1:1'. 115:-. • 'r , ' D' ,. :: r. .\l:{I.:H-!2. ~. ;;; o! f<'"t!; ~. J\RCH-iZ.. ". }'1¡;uro; VE!2 3. ARCH-1.. ,' ,'. --·~·--~·. e. '. ". MiCH-1 ·~. ··'. ---~-'. ~·. 40 re " " " Ph&sti Sp!lc"' Thme::tllwn l'l. Ali::'.JJ- :J<. ~. e"' Rttg M!(}l~;.\i'". F'igurf' VIII. ~. " :~. 4. AflC!I-l,. C{. ::2:.

(31) Dt:'po:;;it" ul Commer<'i0.l DankP Variance R"'~idw• Dn;wmüons. Cutrency :n Circutntwn Vud.uw'! Re-slduc Dlmension8. ;:;. ''. ''. <. t ''. • ~.. '. '' '•. •f. ' '. ''. n. ''. i G~. 5~. ' ' 4~ '. '. • ''. •'. •'. ,1. ". ''. n. '. ' •'. o. o. ' 'n o. ' ' •o ,1 ' e. m. ~. •. :¡. o. o. •'. '. n. -. e of ll&g AIKh-12. Cll. Deposit<l M M•.lt!J.o.l Varlanee Rél';idllP-. S. Ss.~•i.nj'!'S. _,_ e ~r. Total Money Supply Varianee Residuc Oimn~ot.if)u". IJs.nkr<. Dirr:wn~ion.s. '. S. • ' •'. ' '. e ?. •. ' " e' " ' '. '. • ". "f.. .. "D. o'. m. '. ". :<'. '. "'. ~. e. o. o. --~~-~-···. ,,. .•·.--..I.........--:.u !H. o' oi. o. ,.. n. R•t. A'<f'B ¡. f vi. dlCII-!2. !J(t"!';!-li(. AWll-1, VI!. ¡;¡¡. n.-~. J.m1l-!? ¡m. ~~--. e.

(32) Mucko.'l Vulue of l'rivutely Ileld Marketulole F<:nleral Debt Reddue Dhnensiotul. '. ''. ' ' •'. ~. e. '. <. "l. ,_, 58'. /. 'l. '" "f " "" ''. 'e. o. ''. ". D. '. ~. ///. /. '. '. ' '1. ::l. '. /. o. ' ' ' ''. /. e. '. 1. "D 1 m ~. " "'. 9. J2. 4P. '". "'. "~. Phase Spuce Uunen!!:lon.. '. ". --. 32,. •. ':,,f.. u. '" f. '. / /. 04 ;-. ::t ' "l "'. '. /. '. Prk'-' l11d11x of Treasury Debt Annuítit!S lQ Mv:tur:ity·,. All Residue DimemtiMl>'~. fír•! :_:¡¡¡ ARCH-~. -. ~. 7[.S-l). -+-. • "•1. Phase 5pace Dnnen1'um. n'. !{lit. ARM'-. VIl All.W-(1. '. F;gur« :XJ. F1gure IX 2. Prive lndex of Trl'inury Elonds Annuittes to Maturity: AU Re>'lidn>:< Dhneu'!i<:.n:>-. '. j~ F. ' wr ' 1. eL. '. 1'. '. ' ' '. 1 '. e. • '. "'. '1. 1. ,. '. •. ". n. ''. 0 m. ~. '. ,, t. ,,eo. ""'.o t ". J:O. A. '. llonds. ''. •'. ''. Tt·0a~ury. Annuities to ~aturily: AU Re'llidw: Dimetudons. '. "'. '. Pl'IOP fndex uf. e '. ". Phas<- 8pm:e. 56 ~mension. •. " ". ". "' ' o '. ". '. " Nrr• Al'i~AV-l-+- N.i'¡; AtitAV~e Vf!. rwur-e IX 8. ~RMV·. f. Figure lXA.

(33) ,e. Price Index of Treasur:y Debt. Price Index of Treasury Debt. Annuities to Maturity: 0-1. Annuilies lo Malurity: 1-5 Residue Dimensions. Resi<lne Dino<>nsion.«. ". '. e" o. ". '. '". '' " 'e'. ". e. 6. ' " "' ' '". m. 64. r. 1. e. 56. ''. 56. '". " 10 r-. [J. 10. :J2. ''. ~2. '. e. -. '" 'e. ~·1. o. '" ". m. '. ,,. " '. ". Phase Space Dimenswn. Va ARAV. , . -_ _c __ _c ___ ,_ _~---"---"---"--~. 32. ' ' '. Phase Space Dtmenswn. VIl ARAY-2. 2. Figm·, JX 5. Price Index of T!"A"m·y nebt. Price Index of Treasury Dcbt Annuitie;; to Mnt.lwily: 10 or more Residue Dimensions. Annuities to Maturity: 5-10 Residue. 'e. DiOI!:'ll~iou~. 6. '. '. '. ' '. '. e'. :: t. '. '. ". "e' o. '. '. 'e' o. " 6. "'•. ". ' e. ". " ' ''. "r. ". 56. ". " '' " ' ' ". 32. ". ". o. o. o. ---·-. -------------~. o. '. o. Plwse ):;pace Dimensions. VR ARIIY-2. --+--. NFJ) Af~V-2 VR ARAV-6. F1gure 1:\.7. o. m. o. Flrst Dtf. ". -+--. NFD AllAV-G. ' ' ". o o. 56. Phase Spacf! DimPnsion Firot Uil VIl. I.RAV-~. -+-. NFD ARAV-2 YH ARAV-6. F1gure JX.8. -+- Nrn ARAV-B.

(34)