Fnance etters, 5, 3 (1, 77-86 Emprcal Evdence on Spatal Contagon Between Fnancal arkets Brendan O. Bradley a and urad S. Taqqu b, a Acadan Asset anagement Inc., USA b Boston Unversty, USA Abstract We say that there s contagon from market X to market Y f there s more dependence between X and Y when X s dong badly than when X ehbts typcal performance, that s, f there s more dependence at the loss tal dstrbuton of X than at ts center. Ths alternatve defnton of contagon between fnancal markets was ntroduced n Bradley and Taqqu (4, where a test for contagon based on local correlaton was presented. Usng ths test, we fnd evdence of contagon from the US equty markets to equty markets of several developed countres. We also fnd evdence of flght to qualty from the US equty market to the US government bond market. We make the software wrtten n support of ths work freely avalable and descrbe ts use n the append. Keywords: Contagon, ocal correlaton, Correlaton breakdown, Crss perod JE classfcaton: C1, C14 1. INTRODUCTION A test for contagon, based on local correlaton, was ntroduced n Bradley and Taqqu (4. The test checks for the presence of more dependence between market X and market Y when market X performs poorly than when market X has typcal performance. We assume that the relatonshp between Y and X s of the form Y = m( X + σ ( X ε, (1 where ε s mean zero, unt varance and s ndependent of X. The local correlaton between the random varables Y and X at X = equals ρ( = σ X β ( ( σ β ( + σ ( where X X σ denotes the standard devaton of X, β ( = m ( s the slope of the regresson functon m ( = ( Y X = and σ ( = Var( Y X = s the condtonal varance. Contagon, usng local correlaton, s then defned as follows: Defnton 1.1. We say that there s contagon from market X to market Y f ρ > ρ( (3 ( where = F X (.5 s the medan of the dstrbuton F X ( = { X } of X and = F X (.5 s a low quantle of the dstrbuton. * Correspondng author: Emal: murad@bu.edu We would lke to thank Ashs Gangopadhyay, Vladas Ppras and Stlan Stoev for many valuable comments whch led to an mproved eposton of ths materal. Ths research was partally supported by the NSF Grants ANI-98563 and DS-141 at Boston Unversty. ISSN 174-64 5 Global EcoFnance All rghts reserved. 77
Bradley and Taqqu 78 Contagon s therefore present when there s more dependence n the loss tal of the dstrbuton, as measured by local correlaton, than there s n the center of the dstrbuton. We use the followng test for contagon: H H 1 : : ρ( ρ( ρ( > ρ( (no contagon (contagon where and are as above. As ndcated n Bradley and Taqqu (4, the choce of the.5% quantle can be modfed n accordance wth one's notons of crss. In some cases the.5% quantle can be reached when the data s hghly concentrated around the medan. In ths case, whle contagon may have been found t may be rrelevant. Ths s why one must eamne the data and see whether the losses ncurred at ths quantle are sgnfcant enough for one to care. Due to the heavy-taled nature of nternatonal equty returns, the losses at ths quantle are lkely to be sgnfcant. Fgure 1 plots the US equty market returns for the tme perod consdered here. The.5% quantle corresponds to a loss of about % of one's poston n a sngle day. Fgure 1. Tme seres plot of the US equty returns from from January 198 to ay All returns below the.5% quantle of the emprcal dstrbuton are ndcated by trangles. We are now gong to test for contagon between mature equty markets n the US and abroad and between the US government bond and equty markets usng our methodology. When eamnng dependence between the US government bond and equty markets, we test for flght to qualty. The estmaton procedure for ρ ( s descrbed n Bradley and Taqqu (5. We make the software wrtten n support of ths work freely avalable and descrbe ts use n the append.. DEVEOPED WORD EQUITY ARKETS We frst consder developed world equty markets. We take the vewpont of an nvestor wth a consumpton stream denomnated n U.S. dollars and calculate returns based on U.S. dollars. We test for contagon between the U.S. and a gven nternatonal market. The markets consdered are Hong Kong, Japan, Australa, Belgum, Canada, France, Germany, Italy, Netherlands, Swtzerland and the Unted Kngdom. The daly prce nde hstores are calculated by Datastream, begnnng n January 198 and endng n ay. From the prce ndces, P, we construct the daly, weekly and monthly return ndces. We eperment wth t both smple net returns, 1 ( P t Pt / Pt %, and contnuously compounded (log returns, 1 log( P t / Pt 1%. We fnd the results to be ndfferent between the two and report results based on contnuously compounded (log returns. In order to account for the non-smultaneous market closngs and for
Bradley and Taqqu 79 any other seral dependences wthn and across markets, we also eamne the resduals vector autoregressve VAR ( p model for p = 1,, 5. The VAR ( p model s gven by US Other T p Φ rt = ( I Φ1 B... Φ p B rt = t a t of a two-dmensonal ( B φ + a (4 where r t = [ r t, rt ] and B s the back-shft operator, Br t = r t 1. The VAR(p model removes seral lnear dependences wthn and across markets. Any concurrent relatonshp between markets s revealed by eamnaton of the resduals a t, t = 1,, n. Ths s done for the daly returns seres only where the ssue of nonsmultaneous market closngs may be relevant. In ths case, p = 1 s the meanngful choce but we also consder values of p up to fve to allow for lnear dependence between and wthn markets for up to a week. Unless stated otherwse, the US market acts as the covarate market X. Fgure. QQ and PP plots for the dstrbuton of ˆ ρ( and ˆ ρ ( versus the normal dstrbuton obtaned from 1 Bootstrap samples Clockwse from upper left: QQ plot for ˆ ρ (, PP plot for ˆ ρ (, QQ plot for ˆ ρ (, PP plot for ˆ ρ (. F In order to perform our test for contagon we need to make two assumptons about our estmators. Frst, we assume that the estmates ˆ ρ ( and ˆ ρ ( of ρ and ρ are ndependent. That s, we assume they are estmated from ndependent samples. We argue as follows. Assumng the sample ( X, Y, = 1,, n are..d. then the estmates ˆ ρ ( and ˆ ρ ( would be ndependent f the sets of data ponts used to compute ( ( them had no ponts n common. In other words, f no data pont ( X, Y receved postve weght n the calculaton of both ˆ ρ ( and ˆ ρ (. Due to the nature of the local polynomal regresson, ths would obvously be the case f and were at least two bandwdths apart. Although ths s not always the case, the ponts common to the estmaton of local correlaton of both and would be assgned very small weght and the problem should therefore be neglgble. Second, we assume that the estmators ˆ ρ ( and ˆ ρ ( are normally dstrbuted. The asymptotc normalty of our estmator was establshed n Bradley and Taqqu (5 under certan regularty condtons and asymptotc decay condtons on the bandwdths h 1 and h. In realty, the regularty condtons are dffcult, f not mpossble, to verfy. Also, due to the fnte samples of data, we have statc bandwdths. The asymptotc theory gves the user confdence, but leaves one uncertan about how to proceed when faced wth the realty of fnte sample data. Therefore we rely on an eamnaton of
Bradley and Taqqu 8 the dstrbuton of the estmators obtaned from 1 Bootstrap samples of ˆ ρ ( and ˆ ρ (. Fgure shows Quantle-Quantle (QQ and Probablty-Probablty (PP plots of the Bootstrapped dstrbuton of local correlaton between US and French equty markets versus the normal dstrbuton. Snce the quantles tend typcally to bunch up n the center of the dstrbuton and spread out n the tals, the QQ plots are used to check goodness of ft n the tals and the PP plots n the center of the dstrbuton. The dstrbutons of ˆ ρ ( and ˆ ρ ( appear to be well appromated by the normal dstrbuton. The plots, n conjuncton wth Theorem 5.1 n Bradley and Taqqu (5 gve us enough confdence to proceed wth our assumpton of normalty and construct a correspondng test statstc, namely, under the null hypothess, the statstc Z = ˆ( ρ ˆ( ρ (5 ˆ σ ˆ + ρ ˆ σ ρ ( ˆ ( s standard normally dstrbuted. We reject H and conclude contagon between markets whenever Z z1 α = 1.65 where α =. 5 and z 1 α s the 1 α quantle of a standard normal dstrbuton. ˆ ρ( andσˆ ˆ ρ are computed respectvely usng equatons (3 and (35 n Bradley and Taqqu (5. ( Results for daly returns are gven n Table 1. The results are smlar when we eamne the resduals of the VAR(p processes. For eample, Table 1 shows seven nstances of contagon between the markets tested and the US usng daly returns. When eamnng the resduals of the VAR(p processes we have 7,7,6,5, 5 nstances of contagon when the lag p n (4 s p = 1,, 3, 4, 5 respectvely. The seven nstances n the VAR(1 and VAR( processes nvolve the same countres as those lsted n Table 1 for the daly returns. As the lag p ncreases, some countres are dropped. When p = 5, the Netherlands and the UK are dropped from the lst. Table 1. ocal correlatons and contagon test statstc for daly returns n developed equty markets arket ˆ ρ ( σ ˆ ρ ( ˆ ˆ ρ ( ˆ σ ˆ ρ ( Ẑ Contagon Hong Kong.87.6.116.831.74 NC Japan.948.174.115.538.3569 NC Australa.59.156.139.45 1.4949 NC Belgum.139.175.445.56.664 C Canada.6436.86.6197.37 -.7493 NC France.371.148.3875.359 3.8749 C Germany.181.151.365.378 3.5474 C Italy.1395.157.494.41.4463 C Netherlands.677.155.44.48 3.197 C Swtzerland.1599.185.384.5 4.1873 C UK.33.15.43.48.69 C Contagon s defned as a stronger dependence, as measured by local correlaton, n the (loss tal of the dstrbuton than n the center. Seven of the eleven tested markets ehbt contagon. Our test shows contagon to be present, for daly returns, n seven of the eleven tested markets. Results for weekly and monthly returns are dfferent. For weekly returns we fnd but a sngle nstance of contagon between the Italan and US markets. For monthly returns we fnd contagon to be present n the German and Italan markets. It s not surprsng that the results are dependent upon the tme frequency. Ths s because nvestors react to new nformaton dfferently dependng on ther nvestment tme horzon. An nvestor plannng for hs or her retrement by allocatng ther portfolo across markets should care lttle about the daly dsturbances of the markets. However banks and others requred to meet daly mnmum cash postons and margn calls need be very concerned about daly market condtons and may only care secondarly about longer horzons.
Bradley and Taqqu 81 We also test for contagon when the US equty market acts as the dependent market. Here, we epect to see fewer nstances of contagon snce the US equty market s clearly the domnant world equty market 1. In fact, Table shows that we have have no nstances of contagon when the US market acts as the dependent market. astly, we test for contagon between markets when usng the daly returns seres after flterng them for heteroscedastcty. That s, nstead of usng the return seres {( X, Y, = 1,, n} n the test for contagon, we ~ ~ use the seres {( X, Y, = 1,, n} where the tlde notaton denotes nnovaton seres after flterng for ~ heteroscedastcty. We assume X t = σ X and model the volatlty usng a GARCH(1,1 model, X, t t σ X, t = α + α1x t 1 + β1σ X, t. (7 After flterng for heteroscedastcty, the resultng nnovaton seres, X ~ t, are closer to beng..d. We fnd that the condton for statonarty ( α 1 + β1 < 1 holds for all markets. For nstance, for the US market, we get, =.113 +.671 1 + ~ σ X t X t. 931σ X, t. We model the covarate market Y by Y t = σ Y, tyt as well. Results for contagon tests usng fltered returns are reported n Table. We see that when the US acts as the covarate market, we have 6 nstances of contagon usng the fltered seres. When the US s the dependent market, we have none. Note that although flterng for heteroscedastcty effects the resultng test for contagon, we see that not all of the ncreased dependence between markets when the US market performs poorly can be attrbuted to dependent condtonal volatlty. Even after flterng for volatlty, Table reveals strong evdence of ncreased dependence between the markets when the US markets s performng poorly. The case of Canada s nterestng. The value of Ẑ s very small and t remans relatvely low even after flterng. The test, therefore, clearly ndcates that there s no contagon. One would have thought perhaps that contagon would be present. But recall that contagon s vewed as an ncrease n dependence when thngs go wrong n the US market. Under regular crcumstances, the dependence between the US and Canadan equty markets s already very strong. When the US market s performng very badly, the dependence between US and Canada may even decrease a lttle. Table. Results of testmg for contagon between world equty markets US as covarate market X US as dependent market Y Unfltered Fltered Unfltered Fltered Unfltered Fltered Unfltered Fltered arket Ẑ C/NC Ẑ C/NC Ẑ C/NC Ẑ C/NC Hong Kong.74 NC 1.514 NC -1.6498 NC -.7459 NC Japan.3569 NC 1.1176 NC.973 NC -.915 NC Australa 1.4949 NC.147 C 1.5773 NC.466 NC Belgum.664 C 1.79 NC -.56 NC -1.71 NC Canada -.7493 NC 1.19 NC -.3656 NC -.897 NC France 3.8749 C 1.44 NC.4681 NC.836 NC Germany 3.5474 C 1.8889 C.59 NC -1.756 NC Italy.4463 C.681 C -.556 NC.8551 NC Netherlands 3.197 C.1177 C -.98 NC -.8838 NC Swtzerland 4.1873 C.698 C -.456 NC -.53 NC UK.69 C 1.9387 C -.638 NC -.867 NC We test usng the Us equty market as both the covarate market X and the dependent market Y. We report results for daly returns and volatlty fltered daly returns. 1 As of ay-, the US equty market had between 5 and 76 tmes the captalzaton of the other markets we nvestgated.
Bradley and Taqqu 8 3. BOND AND EQUITY ARKETS IN THE US In addton to better understandng the dependence between nternatonal equty markets, for the purpose of asset allocaton, t s crucal to understand the relatonshp between bond and equty markets. It s often beleved that government bond markets do well when equty markets do not. Ths phenomenon s often referred to as a flght to qualty, the dea beng that as equty markets crash, nvestors hurry to the relatve safety of the government bond market. Tests for flght to qualty typcally proceed n the same fashon as test for contagon, that s, crss and non-crss tme perods are defned, and sample correlatons are computed for each perod respectvely. In ths case, however, nstead of lookng for an ncrease n the correlaton between US bonds and stocks, flght to qualty would manfest tself as a decrease n correlaton. One mght well epect ths correlaton to become negatve, that s, as equtes perform well below ther average, government bond markets should perform well above ther average. Hstorcally, the correlaton of returns n the US Bond and Equty markets s small but postve. For eample, the sample correlaton between the US equty market and government bond market s about r =. 9 for daly returns durng the perod from November 1986 to ay. Typcal tests for flght to qualty often show a negatve correlaton durng a crss perod, presumably gvng evdence of flght to qualty. For eample, Gulko ( tests for flght to qualty n the US government bond and equty markets by dentfyng and aggregatng over s crss perods snce 197. The crss perods are dentfed wth crashes n the equty market, where a crash s defned by a loss of at least 5 %. After dentfyng a crash, the crss perod s defned as a short perod about the crash. Gulko defnes a prologue perod for each crash as the ten tradng days precedng the crss perod. Then Gulko aggregates all the prologue perods and aggregates all the crss perods. Fttng a separate smple lnear regresson model Y = α + βx + ε to the aggregated prologue and crss perods, whch contan 6 and 79 data ponts respectvely, he fnds the stock-bond correlaton to be +. 57 n the prologue (pre-crss perod and. 445 n the crss perod. Tests of ths nature suffer from the same sort of problems as ther contagon counterparts because one s hand pckng the crss perods after the fact. Instead, we prefer to eamne the ssue of flght to qualty n the contet of the local correlaton measure. We consder returns from the US equty seres used n the contagon tests and returns from a errll ynch US Government Bond nde of representatve one to ten year maturty bonds. The return seres cover from November 1986 to ay. The covarate market X s the equty nde and the dependent market Y s the bond nde. Typcal wsdom tells us that when equty markets do partcularly well, bond markets do poorly and when equty markets do partcularly poorly, bond markets do well. In terms of our local correlaton measure ths behavor should manfest tself by negatve local correlaton at both ends of the covarate's spectrum. Our test for flght to qualty s smlar to that for contagon ecept ths tme our test s: H H 1 : : ρ( ρ( where and are defned as n Defnton 1.1. ρ( < ρ( (no flght to qualty (flght to qualty Table 3. ocal correlatons and flght to qualty test statstcs for daly, weekly and monthly returns n the US government bond and equty markets from November 1986 to ay Frequency ˆ ρ ( σ ˆ ρ ( ˆ ˆ ρ ( ˆ σ ˆ ρ ( Ẑ Flght to Qualty (FTQ Daly.475.193 -.49.811-8.147 FTQ Weekly.3399.48 -.496.1466-4.8591 FTQ onthly.444.141 -.5688.348-3.937 FTQ Flght to qualty s defned as a weaker dependence, as measured by local correlaton, n the (loss tal of the equtes dstrbuton than n the center. We say there s a Flght to Qualty (FTQ f Ẑ < -1.65. the results ndcate a very sgnfcant change from postve assocaton n the center of the dstrbuton to negatve assocaton n the tals of the dstrbuton. Results do not change when lookng at an all maturty government bond nde.
Bradley and Taqqu 83 Fgure 3. The correlaton curve, local mean, slope and resdual standard devaton for the US government bond market as a functon of the (log returns, epressed as a percent, of the US equty market 95% confdence ntervals are attached usng normalty of the estmator and equaton (35 n Bradley and Taqqu (5. Table 3 and Fgure 3 confrm the epected behavor and gves credble evdence of flght to qualty. The local correlaton changes from sgnfcantly postve n the center of the dstrbuton to sgnfcantly negatve n the (equty loss tal of the dstrbuton. An nvestor relyng only on the sample correlaton r to quantfy dependence between the equty and bond markets would come to the erroneous concluson that the markets were nearly uncorrelated snce r =. 9. The local correlaton measure shows that the markets have varyng degrees of condtonal dependence. On a typcal day n the equty market, the assocaton s strong and postve wth a local correlaton of ˆ ρ ( =. 43. On a bad day n the equty market, the assocaton s negatve and farly strong wth a local correlaton of ˆ ρ ( =. 4. The use of an overall correlaton between stocks and bonds, whch gves r =. 9, s much less nformatve because t can be vewed, roughly, as averagng over the whole range of values of X and thus s not adequate for judgng flght to qualty. Smlar behavor holds for weekly and monthly horzons and usng heteroscedastcty fltered daly returns. Fgure 3 s especally nformatve. It shows that the change n the dependence, as measured by local correlaton, s a smooth functon of the equty market returns. Although there s obvously a certan amount of smoothng nvolved n the local correlaton modellng, a sharp change n the dependence between the equty and bond markets would stll reveal tself on the correlaton curve. We performed the followng smulaton to confrm that a sharp quantle based change of dependence would be evdent on the correlaton curve. et X (, 1. Generate n s random varates { X, = 1,, ns} wth dstrbuton F X. Then generate Y condtonally on X = so that Y X Y X = F = < F X X σ 1 (.75 =.44 ρ1 σ X σ (.75 =.44 ρ σ X, σ 1 (1 ρ1 = (.3, (, σ (1 ρ = (1.4, ( where σ 1 = 1, ρ1 =.3, σ = and ρ =. 7. Fgure 4 shows the correlaton curve and ts consttuent parts for the smulaton. Although partally smoothed, there s stll a sharp shft n the local correlaton clearly vsble. The correlaton curve for the equty and bond markets n Fgure 3 reveals no such shft n dependence. Our results show that the dependence structure between equty and government bond markets s consderably more.91.4, (7
Bradley and Taqqu 84 complcated than that whch any smple lnear model(s can capture. Fnally, we note that the results are not an artfact of the crash of 1987. They do not change when we eclude t: smlar results were obtaned lookng at the seres from January 199 to ay. Fgure 4. The correlaton curve, local mean, slope and resdual standard devaton of the smulated data {(X, Y 1, = 1,, n s } where n s = 5, X ~N(, 1 and Y s generated condtonally on X = accordng to equaton (7 4. CONCUSION Understandng dependence n the tals of the dstrbuton of returns of fnancal markets s of great concern for those lookng to dversfy fnancal rsk across nternatonal boundares or across dfferent asset classes. Issues such as contagon between nternatonal markets and flght to qualty from equty to government bond markets can have a dramatc effect on how to best allocate one's assets. In ths paper we defne contagon and flght to qualty based on a measure of local correlaton. The measure s based on a general non-lnear model whch does not presuppose, ether mplctly or eplctly, any specfc probablty law on the dstrbuton of returns. Snce t s not eplctly temporal, t avods the hand pckng of crashes and the defnng of crss perods, whch we beleve to be a serous mpedment to most tradtonal tests for contagon and flght to qualty. Instead, local correlaton measures the dependence between Y and covarate X locally throughout the support of the dstrbuton of X. Ths allows one to gan a better understandng of the relatonshp between markets n the tals of the dstrbuton. Ths understandng s crucal for those concerned wth guardng aganst catastrophc losses. Our emprcal study based on local correlaton suggests both contagon and flght to qualty to be prevalent. For daly returns, our tests for contagon wth the US equty market reveal contagon n about half of the markets tested. ower return frequences revealed dfferent amounts of contagon between markets. We found a sngle nstance of contagon for weekly returns and two nstances of contagon for monthly returns. A smlar defnton for flght to qualty between US equty and government bond markets revealed sgnfcant evdence of flght to qualty for daly and weekly returns. Addtonally, the correlaton curve revealed a surprsng relatonshp for the dependence between (US stocks and bonds. The assocaton, as measured by local correlaton, s qute hgh on a typcal day n the equty market and smoothly decreases as equtes perform partcularly well or partcularly poorly. The assocaton s strongly negatve at both ends of the equty return spectrum. The dramatc dfference n the strength of dependence between equty and government bond markets from a typcal day n the equty markets and a very bad day, as measured by local correlaton, lend support to the popular theory of a quck, nearly nstantaneous transton from postve to negatve assocaton durng a crash n the equty market.
Bradley and Taqqu 85 6. USING THE SOFTWARE The software used to perform the contagon analyss n ths paper was wrtten n ATAB and may be obtaned from the authors. We llustrate ts use n ths secton. The ATAB functons necessary to use the software are gathered n a drectory called ContagonDr. The user should put hs data n a ATAB data fle named, for eample, ReturnData.mat. We assume here that the data fle contans a num_obsnum_asset ATAB data array called AssetReturns. The array represents return data where num_obs=the number of observatons and num_asset=the number of assets. Each row of ths array corresponds to a jont observaton of the num_assets assets. We wll analyze a par of assets at a tme. A sample sesson usng ths software mght proceed as follows. Invoke ATAB. Add the ContagonDr drectory to ATAB's workng path. Here, we assume the drectory s located n the yhomepath subdrectory: addpath('c:\yhomepath\contagondr'; oad the data set nto ATAB workspace: load('c:\yhomepath\returndata'; The ATAB workspace wll now contan the array AssetReturns. Net, pck a par of assets from the array. We pck, 1 num _ assets to be the covarate market X and j, 1 j num _ assets, j to be the dependent market Y. Choose = 1 and j = for eample: = 1; j = ; X = AssetReturns(:,; Y = AssetReturns(:,j; Defne a set of target ponts for whch we would lke local correlaton estmates. For eample, suppose we want 11 equally spaces estmates of the local correlaton from mn = F X (.5 to ma = F X (.975. Then enter num_targets = 11; _mn = prctle(x,.5; _ma = prctle(x, 97.5; _ = lnspace(_mn, _ma, num_targets'; s now a column vector of target ponts. The followng command estmates the local correlaton at the target ponts and plots the correlaton curve and ts assocated parts (see Fgure 3: plot_flag = 1; [Rho, Beta, Sgma, StdRho] = CorrCurve(Y, X, _, plot_flag; The functon CorrCurve returns the followng data. ˆ Rho (num_targe ts 1 array of local correlaton estmates ρ (. Beta (num_targe ts 3 array of local regresson coeffcents. The frst column corresponds to local mean estmates m (, the second corresponds to the local slope estmates ˆ( β and the ˆ thrd corresponds to 1 /! tmes the estmate of the second dervatve of the regresson functon, ( ˆ m ( /!, at the target ponts (see equaton (8 n Bradley and Taqqu, 5. Sgma (num_targe ts 1 array of local resdual standard devaton estmates σ (. StdRho (num_targe ts 1 array of local standard devatons of the estmator Rho, ˆ σ ˆ ρ( (see equaton (35, n Bradley and Taqqu, 5 to be used n establshng confdence ntervals. To eamne the data, type Rho, Beta, Sgma or StdRho at the ATAB command prompt. To save the results to a ATAB data fle called Results.mat type: save 'C:\yHomePath\Results' Rho Beta Sgma StdRho To eamne QQ and PP plots of the dstrbuton of ˆ ρ ( and ˆ ρ ( obtaned from num_boot Bootstrap ˆ
Bradley and Taqqu 86 resamples, enter the followng. num_boot = 1; BootstrapocalCorr(Y, X, num_boot; The plots wll be dsplayed automatcally. Tests for contagon then proceed by constructng the statstc Ẑ as n (5. If has been constructed as above, then (lower quantle s the frst element of the array and (medan pont s the st 51. If we are testng for contagon at the 1 α =. 95 confdence level then type: confdence_level =.95; lower_d = 1; medan_d = 51; Contagon = TestContagon(Rho, StdRho, lower_d, medan_d, confdence_level; The scalar Contagon s ether 1, f the null hypothess s rejected (contagon, or f the null hypothess s not rejected (no contagon. To perform the analyss on the resduals of the VAR(p (see equaton (4 we nstead call the functon CorrCurveVARp nstead of CorrCurve. The VAR(p modelng uses functons from the econometrcs toolbo wrtten by James P. esage, Dept of Economcs at the Unversty of Toledo. It s freely avalable at http://www.spatal-econometrcs.com/. Once downloaded, add the drectory and subdrectores to ATAB path as above. To estmate the local correlaton on the resduals of a VAR(p, p = model type: p_lag = ; [Rho, Beta, Sgma, StdRho] = CorrCurveVARp(Y, X, _, p_lag, plot_flag; REFERENCES Bradley, B. and. Taqqu (4 Framework for analyzng spatal contagon between fnancal markets, Fnance etters, (6, 8-15. Bradley, B. and. Taqqu (5 Emprcal evdence on spatal contagon between fnancal markets, Fnance etters, 3 (1, 64-76. Gulko,. ( Decouplng, Techncal Report, Paloma Partners, USA (lgulko@paloma.com.