Asymmetric Impact of Financial Integration to International Nonsynchronous Trading Effects in Developed and Emerging Equity Markets

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1 Theoretcal Economcs Letters, 04, 4, Publshed Onlne August 04 n ScRes. Asymmetrc Impact o Fnancal Integraton to Internatonal Nonsynchronous Tradng Eects n Developed and Emergng Equty Markets KHoon Jmmy Hong Unversty Technology o Sydne, Sydney, Australa Emal: hong.jmmy@gmal.com Receved May 04; revsed 6 June 04; accepted 4 July 04 Copyrght 04 by author and Scentc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY). Abstract Ths paper nvestgates the mpact o the nternatonal equty market ntegraton to the nternatonal nonsynchronous tradng eects (INTE). The paper nds that the nancal market ntegraton would ncrease INTE, n general, and the mpact monotoncally decreases over the lag length. However emprcal evdence suggests that the ncrease s asymmetrc among developed and emergng markets. Further theoretcal nvestgaton reveals that the level o volatlty and autocorrelaton are postvely related to the ncrease n INTE. The paper concludes that the relatvely hgher level o volatlty and autocorrelaton n emergng markets could mtgate the ncrease n INTE rom nancal market ntegraton. Keywords Nonsynchronous Tradng, Equty Market Integraton, Prce Autocorrelaton. Introducton Ths paper nvestgates the mpact o nancal equty market ntegraton to nternatonal nonsynchronous tradng eects (INTE). It should be emphaszed that ths paper looks nto the cross market nonsynchronous tradng. It s derent rom the nonsynchronous tradng wthn one market, whch most exstng studes nvestgate. The paper employs a stochastc process to provde a theoretcal ramework. The contrbutons are clear: Frst t provdes a theoretcal ramework to nvestgate the nonsynchronous tradng eect. Second, t theoretcally and emprcally nvestgates the eect o equty market ntegraton to INTE. And thrd, the paper provdes prce level How to cte ths paper: Hong, K.J. (04) Asymmetrc Impact o Fnancal Integraton to Internatonal Nonsynchronous Tradng Eects n Developed and Emergng Equty Markets. Theoretcal Economcs Letters, 4,

2 evdence nstead o return level evdence. The paper nds the theoretcal and emprcal evdences that more ntegrated equty markets ncreases the INTE. However the emprcal ndngs show that the ncrease s asymmetrc between developed and emergng markets. Further nvestgaton reveals that the hgher volatlty and autocorrelaton n emergng equty markets mtgate the ncrease n INTE ncurred by the equty market ntegraton. The paper also nds that the sze o the eect o equty market ntegraton to nonsynchronous eects decreases over tme. It s well known that nonsynchronous tradng causes auto correlaton n equty prce and return. There are plenty o lterature that addresses problems o nonsynchroncty whch can be thought o as the random, or determnstc, arrval o data wthn the nterval o measurement and ts consequences on the measurement o beta (See Atchson, Butler and Smonds, [] and Dmson []). More recent works nclude Barndor-Nelsen et al. [3] and Fung, Lu and Tse [4]. Recent works tend to ocus on the econometrcs method o nonsynchronous tradng problem. A careul nvestgaton o exstng lterature n nonsynchronous tradng eects reveals that there are several potental aspects that could be urther nvestgated. () The cross autocorrelaton s much less explored compare to the autocorrelaton wthn the ndex due to ts complex nature. The latter nduces autocorrelaton manly rom thnly traded securtes that have less lqudty whle the rst nduces autocorrelaton due to equty market operaton tme derence. Ths paper s nterested n the cross market nonsynchronous tradng eect and reers t to INTE. There could be two consequences o nonsynchronous tradng to nternatonal equty market data. () Most o the exstng lterature nvestgate return seres nstead o prce seres. Although postve prce autocorrelaton means postve return autocorrelaton n AR process, the drect eect o nonsynchronous tradng s prce autocorrelaton. Return autocorrelaton s a product o the prce autocorrelaton. Ths paper tackles the prce autocorrelaton. (3) Wth more ntegrated world s nancal markets, such nonsynchroncty problem could be nvestgated n a derent angle. Schotman and Zalewska [5] address the ssue o nonsynchronous tradng assocated wth nancal market ntegraton. They argue that controllng or tme derences n tradng hours o stock markets s mportant and show that tme-adjustment mproves estmates o market ntegraton. Ths study shows that nonsynchronous tradng eects may be aected by the degree o nancal market ntegraton. The current paper ntends to expand the scope o Schotman and Zalewska [5] and nvestgate the relatonshp between the equty market ntegraton and INTE. The rest o the paper s organzed as ollows. Secton ntroduces the theoretcal model. Secton 3 emprcally tests the two hypotheses. Secton 4 renvestgates the theoretcal model to explan the dscrepancy between developed and emergng markets. Secton 5 concludes the paper.. The Model In modelng shock emssons among nternatonal equty markets, there are some domestc shocks that are rrelevant to nternatonal markets. These shocks should be excluded rom the analyss; hence the term shock n ths paper reers to the shocks that have nternatonal nluence only. For modelng purpose, I take a common practce n lterature nvestgatng nternatonal equty market volatlty spll over (See Baele [6] and Bekaert and Harvey [7]) and assume that there exsts a common actor that s aected by all shocks mmedately a shock occurs n country j, the common actor s aected smultaneously, whle the shock s only updated to the equty market o country when ts market opens. Ths paper employs a moded verson o the model developed by Lo and Wang [8]. The bass or the model analyzed n ths paper orms the bvarate Ornsten Uhlenbeck wth some modcatons permttng resduals o asset and market return to be correlated. Ths correlaton may correspond to common actors, due to the ndces dependence on actors. Assume that the logarthm o the ndex levels log ( P ( t )) and log ( P ( t )) have lnear trends µ and µ t respectvely. We consder the de-trended log prce process, t : log and : log q t = P t µ t q t = P t µ t () Assume that Q( t ) satses the ollowng par o stochastc derental equatons, dq ( t) q ( t) q ( t) dt σ dw t ( ) = + + () 58

3 σ dq t = q t dt+ dw t (3) where P stands or prce, subscrpt ndcates that the parameter s or country, subscrpt ndcates that the parameter s or the common actor and and σ are the parameters o the OU process wth the obvous use o subscrpts. Furthermore, we assume σ > 0, and W( t ) denotes a Wener process. W ( t ) and W ( t ) are correlated standardzed Wener processes wth correlaton coecent κ,.e. EW ( tw ) ( t) = κdt. Thereore κ can be seen as the degree o equty market s ntegraton wth the common actor. It could be also seen as the underlyng systemc rsks o the market wth respect to the common actor, over and above the market model term gven by. For most nternatonal equty markets, κ s expected to be postve. Accordng to Hong and Satchell [9], the soluton can be wrtten as ( ) ( tu) ( tu) ( tu) e e e t e e e Q( t) Q( t ) = + HdE( u) (4) t ( tu) 0 e 0 e σ A κ κσ de A A t where H = and d E 0 σ ( t) = ~ ( 0, Idt) M dem ( t) All results are condtonal on tme zero normaton. Remark : The condtonal covarance o ndex prce and actor prce can be wrtten as ( + ) t t ( + ) t e e e Cov q( t), q ( t) σσ κ = + σ + + See Appendx A o Lo and Wang [8] or the proo or Remark. Remark : The condtonal t -perod autocovarance o ndex return and actor return can be wrtten as ( + ) t t ( + ) t e e e s s s Cov q( t s), q ( t) e σσ κ e e + = + σ + + A proo o Remark s n the Appendx. Equaton (6) s the amount o autocovarance nduced rom nonsynchronous tradng. The system o Equatons () and (3) represents the one actor model that the demeaned log prce o ndex s nluence by the demeaned log prce o the actor. Beta o ths one actor model measures the responsveness o the ndex prce to the actor prce. Snce we are nterested n lagged relatonshp, the beta s computed wth lagged tme seres, q ( t+ s) and q ( t ), and wll be denoted as β ( s). I reer β ( s) as auto-beta because t captures the beta-type relatonshp o a one actor model between lagged terms, lke autocorrelaton. The β and autocorrelaton s that the ormer captures the lagged relatonshp o two derent prces whle the latter captures the lagged relatonshp wthn one prce. Although the notaton beta s used to ndcate ths s a measure o responsveness, current model s very derent rom the CAPM n many aspects. Current model should be consdered as prce one actor model. derence between ( s) Proposton : The nonsynchronous tradng eect, β ( s) β ( s), can be wrtten as ( + ) t ( + ) t e σ e s s s = e κ + e e + e σ + e t t Proposton s an mmedate result o Remark, dvdng the both sdes by the varance. Proposton : The mpact o equty market ntegraton to nonsynchronous tradng eects can be expressed as ( s) β ( + ) t s e s dβ = = e > 0 t κ + e σ σ (5) (6) (7) (8) 59

4 Proposton s a partal dervatve o Proposton wth respect to kappa. Ths measure becomes postve as long and have the same sgn. Ths condton ndcates that both ndex and actor prces revert back to mean or both do not revert back to mean. It would be reasonable to assume that ths s true snce de-trended equty ndces are known to be mean-revertng. When both the ndex and the actor prces exhbt mean-revertng eature, Equaton (8) s postve. Proposton mples that more ntegrated nternatonal nancal equty markets ncreases the level o INTE. Intutvely, ths s because more ntegrated nancal markets allow shocks rom one market to be transerred more wdely to other markets. Proposton 3: The senstvty o β ( s) wth respect to nancal ntegraton and lag tme perod can be wrtten as ( + ) t s e β σ = s > κ s + σ t e s e 0 Proposton 3 s a partal dervatve o Proposton wth respect to s. Gven and have the same sgn, Equaton (9) becomes negatve s postve and becomes postve s negatve. Snce t s more lkely that the de-trended ndex equty prce seres to be mean-revertng, hence > 0, Equaton (9) s lkely to be negatve. Ths ndcates that the sze o the INTE monotoncally decreases over tme. Ths should ntutvely make sense. From the theoretcal ndng o Proposton and 3, two hypotheses could be drawn. H: The more ntegrated nancal market equty market ncreases the INTE. H: INTE monotoncally decreases wth respect to tme. In next secton, the two hypotheses are emprcally tested. 3. Emprcal Test 3.. Data and Measures In the emprcal nvestgaton o INTE, daly data s more sutable than weekly or monthly data. Schotman and Zalewska [5] nd that usng weekly requency does not sdestep the consequences o the tme-match problem but leads to sgncant loss o normaton. Hence ths paper uses the daly data o nternatonal nancal equty markets rom the rst quarter o 998 to the rst quarter o 03. Nonsynchronous tradng s not the only source o prce autocorrelaton. Table presents the lst o nternatonal nancal equty markets n the sample. Intervallng eect (Dmson []) and overlappng data (Hansen and Hodrck [0]) could also nduce the autocorrelaton. These are closely related but derent ssues to nonsynchronous tradng eects. Ths paper uses non-overlappng data wth daly measurement nterval to overcome these problems. Thereore cross autocorrelaton between an ndex and the actor could be nterpreted as the INTE. Ths paper takes ths approach o Baele [6] and have S&P 500 ndex as the common actor and nvestgate the relatonshp o other ndces to the actor. Hence there would be 0relatonshps wth S&P 500 to be captured n the current sample. β as a measure o INTE and Pearson correlaton as a measure o equty market ntegraton. Exstng papers n nternatonal equty market ntegraton oten use smple Pearson correlaton n measurng the degree o equty market ntegraton. Moreover, Hong and Satchell [9] showed that prce correlaton s closely related to the κ o the bvarate OU process model. Thereore, ths paper ollows the exstng lterature and use ndex prce correlaton wth S&P 500 n measurng Two measures are computed accordng to the theoretcal secton: ( s) how equty markets are ntegrated and denote the correlaton o ndex as ρ. Table shows the descrptve statstcs o ndex prce correlaton wth respect to S&P 500 over the sample perod. Snce quarterly non-overlappng prce correlatons o most lqud stock ndces usng daly requency, the data s less lkely to suer rom autocorrelaton nduced by overlappng data and ntervallng eect. The autocorrelaton and cross autocorrelaton observed n the sample data would be mostly due to INTE between equty markets. We see that Canadan the European stock markets move more closely wth the US stock market compare to those o East Asan countres. Ths conorms to the general expectaton. All correlaton dstrbutons are negatvely skewed and ths ndcates that relatvely hgh level o correlatons are more lkely compare to relatvely low level o correlatons. Ths s also consstent wth the ndngs o the prevous studes. ( s) β s computed up to the order o 5 and aggregated addtvely. The lag o 5 busness days s chosen based (9) 50

5 Table. Lst o sample nternatonal equty markets. The table below presents the lst o sample nternatonal equty markets used n ths paper. Country s the country that the ndex belongs to, Index s the name o the ndex and Notaton s the notaton or the ndex that ths paper reer to. Country Index Notaton Unted States S&P 500 Index S&P 500 Canada S&P/TSX Composte Index S&P/TSX Germany Deutsche Borse Ag German Stock Index DAX Unted Kngdom FTSE 00 Index FTSE00 France CAC40 Index CAC40 Netherland Amsterdam Exchange ndex AEX Japan NIKKEI Index NIKKEI Chna HANGSENG Index HANGSENG Korea KOSPI Index KOSPI Australa ASX Index ASX Brazl IBOVESPA Index IBOVESPA Table. Descrptve statstcs o hstorcal correlatons. The table below presents the mean, standard devaton, skewness and kurtoss o the hstorcal quarterly non-overlappng correlatons between 0 nternatonal ndces and S&P 500. The sample used n ths analyss conssts o 0 nternatonal equty ndces rom Q o 998 to Q o 03. Index S&P/TSX DAX FTSE00 CAC40 AEX Mean Standard Dev Skewness Kurtoss Index NIKKEI HANGSENG KOSPI ASX IBOVESPA Mean Standard Dev Skewness Kurtoss on the results that the nonsynchronous tradng problem could be resolved by usng weekly data nstead o daly 5 data. The sum o squared auto-betas, N = β s s, s used as the degree o INTE. Ths measure s equva- = lent to Ljung-Box Q statstc usng lagged betas nstead o autocorrelatons. 3.. Emprcal Result: H In order to test the rst hypothess that The more ntegrated nancal market equty market ncreases the INTE, Followng regresson or each ndex s ran. N = α + λ ρ + ε (0) t, t, t, where = {S&P/TSX, DAX, FTSE00, CAC40, AEX, NIKKE, HANSENG, KOSPI, ASX, IBOVESPA}. Equaton (0) explans the ve-day aggregated auto-betas usng equty market cross correlaton. Table 3 presents the result. 5

6 Table 3. Regresson result o Equaton (0). The table below presents the regresson result o Equaton (0) or the 0 sample nternatonal equty ndces. R Square s the R o the regresson equaton, α and λ are the regresson coecents. p-values are shown n parentheses. ***, **, * ndcate statstcal sgncance at the %, 5%, and 0% levels, respectvely. The sample used n ths analyss conssts o 0 nternatonal equty ndces rom Q o 998 to Q o 03. Index S&P/TSX DAX FTSE00 CAC40 AEX R Square λ ** 57.5 *** 3.5 *** 7.5 *** 0.65 *** (0.03) (0.004) (0.0000) (0.00) (0.006) α ** (0.970) (0.680) (0.00) (0.530) (0.6600) Index NIKKEI HANGSENG KOSPI ASX IBOVESPA R Square λ 9.7 *** * 43.9 *** ** (0.00) (0.390) (0.050) (0.006) (0.086) α ** (0.7430) (0.0760) (0.390) (0.87) (0.000) Frst we note that all but one constant term, α, are not statstcally sgncant at 5% condence level. Ths ndcates that equty markets are not ntegrated at all, there would be no statstcally sgncant INTE. Ths ntutvely makes sense n that all equty markets move ndependently, nonsynchronous tradng hour should not matter. λ o HANGSENG, KOSPI and IBOVESPA are not statstcally sgncant at % condence level. The λ o the rest o the sample ndces are statstcally sgncant at % condence level. All λ s are postve. Ths ndcates that the degree o equty market s lkely to have meanngul mpact on INTE. More ntegrated nancal equty markets emprcally show larger INTE. It s nterestng to see that the emergng markets, Chna, Korea and Brazl, do not have sgncant λ. Ths mples that even the equty markets o these countres are more ntegrated wth the US equty market, the INTE do not ncrease. It s nterestng to see that only equty markets o emergng economes suer less rom the ncrease n INTE due to nancal ntegraton. The contrbuton o equty market ntegrty to INTE should be Rs o the developed markets are hgher than those o captured by the R o the regresson equaton. emergng markets. The numercal sze o the parameters s less meanngul snce the nterpretaton o the squared beta s not clear. The sgn and the statstcal sgncance are the mportant normaton. The overall emprcal nvestgaton supports the theoretcal ndng o Proposton or developed countres whle not or emergng economes. Ths wll be urther nvestgated n Secton Emprcal Result: H The second hypothess states that the INTE monotoncally decreases wth respect to tme, hence predcts monotoncally decreasng β ( s) wth respect to s. For the ve auto-betas, probablty dctates that ths should occur 6.5% o the tmes. Table 4 presents the number o monotoncally decreasng quarterly auto-betas or all ve lag perods. Note that there are total 6 quarters durng the sample perod. The result o the ormal test s reported n Table 5. Observed Mean and SD are the parameters or the number o the observed monotonc relatonshps. Random Mean and SD are the parameters when the sze o β ( s) takes random value. The emprcal result tells us that the percentage o the monotoncally decreasng relatonshps should be statstcally derent rom 6.5%. Hence we do not have enough emprcal evdence to reject the second hypothess. Ths mples that or the gven sample, we may conclude that INTE monotoncally decreases wth respect to the length o the tme lag. And the monotoncty s presented consstently n both emergng and developed economes. 5

7 Table 4. Monotoncally decreasng sample relatonshps. The table below presents the number o monotoncally decreasng sample betas or all 5 laggng perods and ts percentage wth respect to the total sample perod o 6 quarters. β > β > β 3 > β 4 > β 5 ndcates the case when all the ve betas are monotoncally decreasng wth respect the number o lag perod. % o Monotonc Relatonshps s the porton o quarters that showed the monotonc relatonshp out o total 6 quarters. The sample used n ths analyss conssts o 0 nternatonal equty ndces rom Q o 998 to Q o 03. β β β( 3) β( 4) β( 5) S&P/TSX DAX FTSE00 CAC40 AEX > > > > Otherwse % o Monotonc Relatonshps 83.6% 88.5% 88.5% 88.5% 86.89% β β β( 3) β( 4) β( 5) NIKKEI HANGSENG KOSPI ASX IBOVESPA > > > > Otherwse 4 6 % o Monotonc Relatonshps 65.57% 77.05% 65.57% 80.33% 73.77% Table 5. Test o the monotoncty. The table below presents the result o testng the statstcal sgncance o the number o quarters that all ve betas showed monotonc relatonshp. Observed Mean s the actual percentage o monotonc relatonshps, Observed SD s the standard devaton o the dstrbuton o the actual monotonc relatonshps, Random Mean s the percentage o monotonc relatonshps betas ollow random order and Random SD s the standard devaton o the dstrbuton o the monotonc relatonshps betas ollow random order. The sample used n ths analyss conssts o 0 nternatonal equty ndces rom Q o 998 to Q o 03. S&P/TSX DAX FTSE00 CAC40 AEX Observed Mean 83.6% 88.5% 88.5% 88.5% 86.89% Observed SD 37.33% 3.4% 3.4% 3.4% 34.04% Random Mean 6.5% 6.5% 6.5% 6.5% 6.5% Random SD 4.% 4.% 4.% 4.% 4.% t-stat NIKKEI HANGSENG KOSPI ASX IBOVESPA Observed Mean 65.57% 77.05% 65.57% 80.33% 73.77% Observed SD 47.9% 4.40% 47.9% 40.08% 44.35% Random Mean 6.5% 6.5% 6.5% 6.5% 6.5% Random SD 4.% 4.% 4.% 4.% 4.% t-stat Developed vs. Emergng Economes: Theoretcal Model Revsted Sectons and 3 o ths paper nvestgate the senstvty o the most recent ve day actor prce wth respect to the current ndex and actor prce correlaton. The theoretcal ndngs usng a bvarate OU process model predct a postve sgncant relatonshp between the degree o equty market ntegraton and INTE n nternatonal equty markets. However the emprcal evdences show that the more ntegrated equty market ncreases INTE n developed economes but t does not have sgncant eect to emergng economes. In order to resolve the dscrepancy, we need to take advantage o usng a theoretcal model and analyze exactly what nduces the senstvty o INTE wth respect to the equty market ntegrty. Although they are not statstcally sgncant, we stll observe postve estmated parameters. Ths ndcates that nose n the data could hde the relatonshp to be emprcally observed. Ths could be the reason why the emprcal result or developng countres s not consstent wth the theoretcal outcome. One potental explana- 53

8 ton could be that developed markets have parameters that make emprcal ( s) dβ postve and sgncant whle emergng markets do not. Hence we are nterested n whch values o country specc parameters, σ and, make Equaton (0) large and postve. Proposton 4: The senstvty o β ( s) wth respect to country specc parameters, σ and, can be wrtten as t s e + e s β = > 0 σ + σ t e ( s) s t β σ se e ( s t) + = + ( s+ t ) t t < + σ e e e 0 We can see that the larger σ and smaller dβ s. An equty market wth lager σ ndcates that the equty market has hgher volatlty. The smaller ndcates the larger prce autocorrelaton, hence less normaton ecent market. Ths s consstent wth the emprcal ndngs. Emergng markets tend to have hgher volatlty and larger prce autocorrelaton n ther domestc equty markets. Whle nancal market ntegraton worsens the problem o nonsynchronous tradng among the nternatonal equty markets n general, emergng markets suer less rom t compared to developed markets. The paper nds that ths s due to emergng markets hgher level o volatlty and hgher prce autocorrelaton compared to those o developed markets. These could be ndcators o less normaton ecent market. Ironcally speakng, less ecent equty markets o emergng countres mtgate the ncrease n INTE rom more ntegrated nancal equty markets. 5. Concluson ncreases Ths paper bulds a theoretcal model, usng a bvarate Ornsten Uhlenbeck process, to nvestgate the mpact o the nternatonal equty market ntegraton to INTE. The paper nds that the nancal market ntegraton would ncrease the eect, n general, and the mpact monotoncally decreases over the lag length. However the paper emprcally nds that the ncrease s asymmetrc between developed and emergng markets. Further theoretcal nvestgaton reveals that the level o volatlty and autocorrelaton n equty market are postvely related to the ncrease n INTE due to nancal market ntegraton. The paper nds that relatvely hgher level o equty ndex volatlty and autocorrelaton n emergng markets could mtgate the ncrease n INTE rom nancal market ntegraton. Reerences [] Atchson, M.D., Butler, K.C. and Smonds, R.R. (987) Nonsynchronos Securty Tradng and Market Index Autocorrelaton. The Journal o Fnance, 4, [] Dmson, E. (979) Rsk Measurement When Shares Are Subject to Inrequent Tradng. Journal o Fnancal Economcs, 7, [3] Barndor-Nelsen, O.E., Hansen, P.R., Lunde, A. and Shephard, N. (0) Multvarate Realsed Kernels: Consstent Postve Sem-Dente Estmators o the Covaraton o Equty Prces wth Nose and Nonsynchronous Tradng. Journal o Econometrcs, 6, [4] Fung, H.G., Lu, Q. and Tse, Y. (00) The Inormaton low and Market Ecency between the US and Chnese Alumnum and Copper Futures Markets. Journal o Futures Markets, 30, [5] Schotman, P.C. and Zalewska, A. (006) Non-Synchronous Tradng and Testng or Market Integraton n Central European Emergng Markets. Journal o Emprcal Fnance, 3, [6] Baele, L. (005) Volatlty Spllover Eects n European Equty Markets. Journal o Fnancal and Quanttatve Analyss, 40, [7] Bekaert, G. and Harvey, C.R. (997) Emergng Equty Market Volatlty. Journal o Fnancal Economcs, 43, [8] Lo, A.W. and Wang, J. (995) Implementng Opton Prcng Models When Asset Returns Are Predctable. Journal o Fnance, Amercan Fnance Assocaton, 50, () () 54

9 [9] Hong, K. and Satchell, S. (04) The Senstvty o Beta to the Tme Horzon When Log Prces Follow an Ornsten- Uhlenbeck Process. European Journal o Fnance, 0, [0] Hansen, L.P. and Hodrck, R.J. (980) Forward Exchange Rates as Optmal Predctors o Future Spot Rates: An Econometrc Analyss. Journal o Poltcal Economy, 88, Appendx Let A : ( ) ( t+ s) ( t+ s) ( t+ s) q e e e t+ s Q q ( t) = t 0 e ( 0) ( ) ( t+ su) ( t+ su) ( t+ su) t+ se e e + 0 ( tu) ( ) 0 e ( t+ s) ( t+ s) ( t+ s) e e e a b 0 c t 0 e = = HdE u and K: Covarance Matrx o ( + ) ( t) q t s q K t+ s t+ s 0 0 = AHH A du a b σ κσσ a 0 a σ + abκσσ + b σσ acκσσ + bcσ K K : = = 0 c κσ σ σ b c acκσ σ + bcσ c σ 0 K Usng that t s only dened or t > 0, ( + ), Cov q t s q t t+ s t+ s 0 0 ( κσ σ σ ) = K du = ac + bc du t+ s ( + )( tu) ( ) s e e e s tu s + tu = κσ σ + e e e σ du 0 t t t ( ) s + tu s ( tu) s = e κσ σ e du+ e σ e due 0 σ e 0 0 ( + ) t t ( + ) t e e e s s s = e κσ σ + e e σ. + + ( + )( tu) du 55

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