1.Revisiting stock market index correlations Mehmet Dalkir University of New Brunswick,Department of EconomicsP.O. Box 4400Fredericton, NB E3B 5A3Phone: (506) 453-4858 Fax: (506) 453-4514 e-mail: mdalkir@unb.ca

2. AbstractComovement of stock market indices increases during crashes, and does not come down when the turmoil settles down. This paper explains upgrade of comovements during turmoil periods with theories from Bayesian learning and dynamical systems involving synchronization. Our main conclusion is that the correlation does not go down because it is learned during the turmoil. This learned level of correlation has high precision, so there is little doubt that it is at a higher level because of a numerical discrepancy. The belief that market movements are loyal to each other turns into a self-fullling prophecy. Traders follow other markets closely before making trading decisions. So the belief that interdependence between markets are high during the crash turns into reality by correlated actions of traders in dierent markets avoiding correlation to fall to its previous level after the crash. Keywords: stock market index correlations, Bayesian learning, synchronization, stochas- tic systems. JEL classication codes: C11, C16, D83, D84, D85, G15. 3. Revisiting stock market index correlations 11 IntroductionLee and Kim (1993) nd that average weekly cross market correlations between 12 majorstock markets increase from 0.23 before the October 1987 crash in the U.S. markets to 0.39afterwards. Even one and a half year after the crash, for the period of June 1989 to December1990 average weekly cross market correlations were 0.41. Forbes and Rigobon (2002) reportthat during times of high volatility in the series under consideration, correlation coecientbetween these series is biased upwards. Through a detailed common factors analysis, Lee andKim (1993) conclude that the phenomenon of closer co-movements among stock marketsafter the crash existed regardless of the extent of the world stock markets volatility. Forbesand Rigobon (2002) explain why the correlations get high during the crash, but they do notexplain the phenomena observed by Lee and Kim (1993). Why the correlations stay highafter the crash, regardless of the volatility? This paper is an attempt to explain the stickinessin correlation coecients between stock markets by referring to theory of Bayesian learning. Our main conclusion is that the correlation does not go down because it is learned duringthe turmoil. This learned level of correlation has high precision, so there is little doubt that itis at a higher level because of a numerical discrepancy. The belief that market movements areloyal to each other turns into a self-fullling prophecy. Traders follow other markets closelybefore making trading decisions. So the belief that interdependence between markets arehigh during the crash turns into reality by correlated actions of traders in dierent marketsavoiding correlation to fall to its previous level after the crash. Is there a draw back in high level of synchronization between markets over the world? Oneconcern may be diminishing opportunities of cross-country hedging when markets start to 4. Revisiting stock market index correlations 2move up and down altogether. Cross-country hedging may be of secondary importance aftercross-industry hedging when the world economies are synchronizing through trade. Also, thecorrelation coecient measuring comovements at higher frequencies may fail to measure thesecular growth in the indices. So hedging opportunities still may exist for investors targetinglong term growth rather than short term speculations.2. Bayesian Learning One renovation brought by this paper is the utilization of the phase, rather than themagnitude of the stock market indices. This approach is convenient in terms of qualitativeclassication of the signal relative to the prior distribution of beliefs. By phase or inno-vation in phase it is meant the direction of change in the market index. For example, ifwe arbitrarily take zero as the most recent peak point of the market index, (in radians),or 180o (in degrees) will be the point when it is making the rst dip after that. And thenthe phase of the index will keep on increasing (sometimes decreasing) and when it reaches2 (or 360o ) we will call it a complete cycle. In converting the index magnitude to phase,we are losing the amplitude of the index, and gaining the ability to attach a direction tothe perceived index values and the signals about the indices. To our understanding, gainsdominate losses for this case: in practice, the initial concern on the index is usually notits absolute value, but its direction of change. Phase denition of the index still has thiscapability. What the phase series lacking is the amplitude of the index (which is assumedto be constant through this paper), which has secondary importance beside the phase. In the literature, Bayesian learning systems usually posed within the framework of thenormal (Gaussian) distribution, or distributions with a support on the line (or on a plane). 5. Revisiting stock market index correlations 3In this paper, a distribution with support on a circle (or on a torus) will be considered. Thecounterpart of the normal distribution on the circle is the von Mises distribution which hassome desirable properties similar to the normal distribution. These properties are given byMardia and Jupp (2000, pp.41-43). In this study, one additional property of the von Misesdistribution is crucial: like the normal distribution, it is in the conjugate family of itself.That is, if the prior and the signal xt are von Mises, then the posterior is von Mises as well. 2.1 The circular case The setup of the model is similar to the Gaussian case as expressed in Chamley (2004): 1. The value of the natures parameter is chosen randomly before the rst period ac- cording to a von Mises distribution M (, ), where is the mean direction and is the precision. 2. There is a countable number n of individuals that receive a private signal xi , where the stochastic process {xi }i=1,...,n is dened on a probability space (, F, P ) where F is a -algebra on the set , and P is a probability measure on the measurable space (, F). Specically, xi = + i . Corr( i , j ) = 0 for i = j, and M (0, ) which implies xi M (, ). All individuals have the same payo function from their actions a: U (a) = E[(a )2 ]. Every individual i is assigned to a time t exogenously. Individual t chooses his action at [0, 2) once and for all in period t. Each individual chooses an action only once, and at each time period, there is only one individual acting. 3. The public information at the beginning of period t is made of the prior distribution M (, ) and the set of previous actions Ht = {at1 , . . . , a1 }. 6. Revisiting stock market index correlations4 2.2 Perfect observation of actions The common payo function induces the individuals to choose their beliefs t Et [] astheir action: at = t for all t. Since all individuals know the payo function is common, theyare aware that the actions reect beliefs perfectly. So the public belief on one period later,t+1 is equal to the individual ts personal belief: t+1 = t . The public belief t M (t , t ) is updated according to the Bayesian rule below (Bagchi, 1994) to become the public belieffor the next period: t+1 M (t+1 , t+1 ). sin xt + t sin tt+1 = tan1,(2.1) cos xt + t cos t222 t+1 = t + + 2t cos(t xt ).(2.2)The updating rule in the von Mises case is particularly interesting: unlike the normal case,the updating rule for precision depends on the signal xt and the prior mean t . Threeparticular cases are listed below:1. Assertive signalWhen xt = t for a certain time interval t, then t grows by the precision of the signal:t+1 = t + . This case is identical with the Gaussian case.2. Orthogonal signalWhen |t xt | = /2 at a certain time interval t, then 2 grows by 2 : 2 = 2 +2 tt+1t t+1 < t + .3. Negative signalWhen |t xt | = at a certain time interval t, then t may increase or decrease: 7. Revisiting stock market index correlations5 t+1 = |t |.In general, limt P (A()t < t+1 < t) = 1.1 If the incoming signal has any vectorcomponent in the direction of the common belief, denitely there will be gains in terms ofprecision. Before continuing with the divergence property in the circular case, the linear case willbe summarized for comparison purposes. The updating of mean and variance of the public belief is referred to as Gaussian learningrule in the linear case, and is a consequence of the normal distribution being the conjugate2 family of itself. The Gaussian updating rule is particularly interesting since the updatingof the precision is independent of the mean . In the linear case, the recursive equation for precision of belief t could be identied asa function of time:t+1 = + t .(2.3) where is the precision of the signal st . So precision grows indenitely in time, and public2 belief becomes almost certain as t . The variance t = 1/t converges to zero like1/t. One diculty with the linear case is that every signal contributes to the precision of thepublic belief. A reduction in the precision of beliefs is impossible for this case by signalscontradicting the learned value of a parameter. By adopting the circular support for thedistribution of the signal, we have the exibility of qualitatively classifying the signals as1see proof of Lemma ??. 0 < A() I1 ()/Io () < 1, for 0 < < where Iq () is the modied Bessel function of the rst kind of order q. A() is increasing.2That is, if the signal st is normal, then the set of possible common distributions for the prior t and the posterior t+1 form the conjugate family of the normal distribution. 8. Revisiting stock market index correlations6assertive, orthogonal, and negative. Specically, the following proposition applies for the circular case: Proposition 2.1 t+1 > t when cos(t xt ) > 2t . If t is growing, history is gainingmore importance over the private signal through formula (??).Next, we will propose and prove a lemma showing that t+1 / converges to t in distri-bution. Lemma 2.1 limt P (t+1 / > t) = 1.Proof. From (??) and (??) we have the recursive relationtzt+1 = zt + ut zt+1 = zo + uj . (2.4) j=1for zt and ut complex where zt t cos t + it sin t , ut cos xt + i sin xt ,and zo cos + i sin . Then we obtain the real recursive equations ttt+1t+1 cos t+1 = cos + cos xj Ctcos xj =cos t+1 cos ,j=1j=1 ttt+1 t+1 sin t+1 = sin + sin xj Stsin xj =sin t+1 sin .j=1j=1 Ct and St are the sucient statistics that represent the history Ht similar to the sucientt statistic Tt j=1 sj in the linear case. Mardia and Jupp (2000) give the distributions forRt Ct2 + St2 , and t tan1 (St /Ct ). We will establish divergence of t+1 / without 9. Revisiting stock market index correlations 7using the distribution function explicitly. We use (??) and (??) and Rt = Ct2 + St2 to derive 2an expression for 2 Rt : 2 2 Rt = 2 + 2 2t+1 cos(t+1 ). 2t+1 (2.5)From Mardia and Jupp (2000, p.75) E[2 Rt ] = 2 t2 2 + 2 t(1 2 ) = 2 + E[2 ] 2 E[t+1 cos(t+1 )]2t+1 (2.6)where is the mean resultant length of the distribution under consideration. Specically forthe von Mises case, A(). Collecting terms and dividing by t2 , we get 22 1E[2 /t2 ] = 2 2 + t+1[1 2 ] 2 + 2 E[ 2 t+1 cos(t+1 )].(2.7)t t t All the terms except the rst vanishes as t . The last term vanishes since from the caseof an assertive signal we know sup t+1 () = + t, and thus limt t+1 /t2 = 0with certainty. Since the cosine function is bounded, it is also true thatlimt t+1 cos(t+1 )/t2 = 0 with certainty. We have established thatlimt E[2 /t2 ] = 2 2 , or t+12t+1 lim E2 t2 2 = 0(2.8)t 10. Revisiting stock market index correlations 8 3 which implies 2t+122 L1 2 t+1 p 2 t+1 d 2 . (2.9) 2 t2 2 t2 2 t2Since the square root function is measurable and continuous over the domain [0, ), we alsot+1 t+1 havet d . Sincetconverges to a constant in distribution, we may conclude thatt+1 t+1 limt Pt > = 1 limt P > t = 1. Proposition 2.2 In a circular social belief system with updating rules (??) and (??), indetermining agents actions, the probability that the weight of public belief (relative to privatesignals) tending to innity becomes equal to one as t .Proof. This is a direct indication of Lemma ??. Remark 2.1 Notice that proposition ?? is for a belief distributed with some circular distri-bution updated by (??) and (??). It is valid for, but not limited to beliefs distributed with avon Mises distribution. Remark 2.2 Mardia and Jupp (2000) also show that the conditional limiting distribution oft+1 when the belief has a von Mises distribution is also von Mises: t+1 |t+1 M (, t+1 )so that E[t+1 |t+1 ] = as t . The expected value of the public belief, given the valueof precision of the belief, converges to the true value as time goes to innity, and with t+1tending to innity in the sense of Lemma ??.2.3 Learning a leading indicator In this section, a two dimensional circular learning system will be investigated. For thesake of simplicity, two stock markets will be considered. The intrinsic values of markets3 The converge concepts here are L1 : convergence in the rst moment, p : convergence in probability, and d : convergence in distribution. 11. Revisiting stock market index correlations 9are given by the rm prots, which uctuate with frequencies 1 and 2 for market 1 andmarket 2 respectively. These frequencies are not known by traders. Traders in these twomarkets play a guessing game: their returns from trade are maximized if they can forecast themovement of the market value with minimum squared deviation from the actual frequencyj , j = 1, 2. In other words, beating the market maximizes prots. As explained in theprevious section, the decision rule implies perfect revelation of beliefs, and the forecast ofeach trader becomes public belief one period later. The support of the distribution in this case, similar to a circle, will be a manifold, namelya torus. It will be assumed that the expected cosine of the dierence between signal x1t andx2t is the cosine of a certain angle . For this case, Mardia and Jupp (2000) propose adistribution on a torus with density proportional to: exp{1 cos(x1t 1 ) + 2 cos(x2t 2 ) + (cos x1t , sin x1t )D(cos x2t , sin x2t ) } (2.10) cos sin where D = a which is a multiple of a rotation matrix. sin cos Also, we used the notation (, ) to denote transpose of a vector. This setting is similar to the one dimensional setting discussed before. Nature picks twovariables 1 and 2 , the constant change (frequency) in rm prots in two stock markets ateach time interval, from a distribution proportional to exp{1 cos(1 1 ) + 2 cos(2 2 ) + (cos 1 , sin 1 )D(cos 2 , sin 2 ) } (2.11)where D is as dened above. 12. Revisiting stock market index correlations 10 The parameters of the distribution, 1 , 1 , 2 , 2 and D are known to all investors inboth markets. Each time interval has two sub-intervals, t and t+ . Signal x1t arrives atthe beginning of t (or t) and signal x2t arrives at the beginning of t+ . Since the signalsare correlated through matrix D, agents will use x1t as a leading indicator to form theirexpectations on x2t . The Bayesian updating rules for this case will be jt cos jt + j cos xjtjt+1 = tan1 ,(2.12) jt sin jt + j sin xjt2 2 2 jt+1 = jt + j + 2j jt cos(xjt jt ) for j = 1, 2. Association between the signals will be learned according to the updating rulesbelow:Rt cost + a cos(x1t x2t ) t+1 = tan1, (2.13)Rt sint + a sin(x1t x2t )Rt+1 = Rt + a2 + 2aRt cos(x1t x2t 22 t )where the belief on cos has mean cos t and precision Rt at the end of period t. Noticethat by Remark ??, Proposition ?? applies to the belief on . The agents can use theidentities Et cos(x1t x2t ) = t cos t and Et sin(x1t x2t ) = t sint to deriveEt cos(x2t 2t ) = t cos(x1t 2t t ).(2.14)The unknown parameter t could be approximated as t A(Rt + A1 (A(1t )A(2t ))) underthe assumption that = 1 2 . Note that t is increasing in Rt , 1t , and 2t . 13. Revisiting stock market index correlations11 The crash scenario is as follows: at time t the index of market 1 makes a big jump downas a result of an outlier local shock higher in magnitude. The investor t in market 2 forecaststheir market index taking market 1 index as a leading indicator using (??). Their forecastindicates a fall in their index as there is a learned association between two indices. Investort in market 2 starts to sell in order not to get hit by the forecasted downfall. However, sinceit is known to everyone that actions reect signals perfectly, investors in markets 1 and 2regard this as a private signal to sell received by investor t independent from the downturn inmarket 1. The investor t + 1 in market 1 take this as a leading indicator, and the downturnin market 2, initiated by market 1 will bounce back, further pulling the index of 1 down. The accuracy of the association Rt may fall at the beginning, but as both indices startfalling fast, Rt increase with repetitive observations arriving conrming similar changes (as-sertive signals) in the phases of the indices. This is a direct consequence of proposition??. With increasing Rt investors start to ignore the new signals about the protability ofthe rms with stocks in their own markets, but go with their belief that the indices have astrong association between them. The strong association between the indices turns into aself-fullling prophecy. Even when small local shocks start to take eect again, the learned strong associationbetween the indices prevails. It takes a long time before the investors de-learn the strongassociation between the indices through negative signals, and for the correlation between theindices to go down to the level before the shock.3Dynamical Systems ApproachIn this section, an alternative approach will be used to explain the downward stickiness of 14. Revisiting stock market index correlations 12the correlation coecient after crashes. The setup is a continuous dynamical system whereXt and Yt are the indices of the stock markets 1 and 2 respectively. The system presentedbelow is not based on learning theories, so it is discussed only heuristically as a supportingargument for the learning approach from a dierent point of view. 3.1 Synchronization of indices It is known that two systems linked with coupling parameters 1 and 2 , and naturalfrequencies 1 and 2 will synchronize, and there will be a steady-state equilibrium wherethe phase dierence between the series is constant over time. An unstable state is alsopresent where the phase dierence will be uctuating, generating a quasiperiodic state intime domain (Stratonovich, 1967, chapter 9, sec.2). Denition 3.1 Two coupled systems are dened, in general, asXt = f1 (Xt ) 1 P1 (Xt , Yt ), (3.1)Yt = f2 (Yt ) + 2 P2 (Xt , Yt ), 1 > 0, 2 > 0.where P1 and P2 are deterministic functions. When the functions f1 and f2 are assumed tobe generating series with constant amplitudes, and with noisy frequencies averaging to 1and 2 respectively (i.e., the processes {Xt } and {Yt } go through a steady state cycle whenindependent) the analysis could be carried out in terms of phase series only. 1t = 1 1 Q1 (1t , 2t ) + 1t ,(3.2) 2t = 2 + 2 Q2 (1t , 2t ) + 2t , 15. Revisiting stock market index correlations 13where 1t and 2t are additive noise terms on observed frequencies of the series. The functionsQ1 and Q2 are the phase space counterparts of functions P1 and P2 respectively, derivedunder the assumption of constant amplitudes. For details on phase space representation andapproximation of functions, see Aronson, Ermentrout, Kopell (1990) and Pikovsky et. al.(2001, pp.222-229).Assuming that synchronization occurs in only one frequency of the series, and omittingall the resonance terms, the system can be represented in terms of the phase dierencet 1t 2t between the series.t = (2 + 1 ) sin t + 1t 2t ,(3.3)with a rst degree Fourier expansion of functions Q1 and Q2 that are assumed to be identical.Also 1 2 .Stratonovich (1967) gives the solution for the stationary distribution of under boundednoise4 , and for 2 + 1 > 0. When = 0, this distribution is von Mises, with M (0, ), (2 + 1 )/ 2 where 222 2 V ar(1 2 ) 1 + 2 212 , jV arj , j = 1, 2,12Cov(1 , 2 ). The correlation coecient r in this case will be an increasing function of as derived by Koopmans (1974): r A()(3.4) where A() I1 ()/I0 () and Ip is the modied Bessel function of the rst kind of order p .Since A() is monotonically increasing, this approach implies increasing correlation betweenthe series when covariance between the noise terms is increasing. Also when 2 + 1 > 0, 4 Namely, if |1 2 | < . 16. Revisiting stock market index correlations 14the correlation between the series will be increasing. This feature is similar to features of themodels introduced by Barnett and Dalkir (2007, equation 3.34) and Dalkir (2004, equation8).The case where = 0 is slightly more complicated. In this situation the mean of thestationary distribution is not zero. Under bounded noise the phase dierence will be stablearound the steady state solution = 0.5 If we dene , then the time average of will be 1 2 |Ii ()|2 sinh() (3.5) where / 2 and is as dened above (Stratonovich, 1967). Also Ii () is the modied Bessel function of the rst kind of complex order.The dynamical system approach is also in parallel with the Bayesian learning approachof section 2. When the index of market 1 starts falling as a result of a local shock, andif the investors in market 2 are aware that 12 is nonzero, they would know that the localshock in market 1 will aect the index of market 2 as well. During times of turmoil, andwith uncertainty in the model (??) investors in market 2 may believe that it is not onlythe shock being transmitted by the correlated shocks, but the parameter 2 is increasingas well. That belief will further decrease the average change in the phase dierence asindicated in Stratonovich (1967, pg.242) (or increase the correlation coecient r). Likewise,if the markets have comparable sizes, investors in market 1 will observe that both indicesare moving together, so they may as well believe that 1 is increasing. Following their beliefthat two markets are now structurally close with a higher 1 and a higher 2 , investors in 5 If |1 2 | < 2 where = sin11 +2 ,