D YNAMIC E CONOMETRIC M ODELS DOI: hp://dx.doi.org/10.12775/dem.2013.004 Vol. 13 (2013) 69 85 Submied Ocober 10, 2013 ISSN Acceped December 30, 2013 1234-3862 Milda Maria Burzała * Deerminaion of he Time of Conagion in Capial Markes Based on he Swiching Model A b s r a c. This aricle aemps o compare conclusions made abou marke conagion based on he periods indicaed by using he Markov-swiching model and based on a range for uncondiional correlaions as well as on arbirary arrangemens. DCC-model was used o conrol for correlaion change over ime. Deerminaion of exremely high correlaions by using a range for uncondiional correlaions and he MS(3) swiching model yields similar resuls regarding conclusions abou he occurrence of he process of conagion in a marke. Conclusions abou conagion are, however, made a a higher significance level in he case of he swiching model. K e y w o r d s: swiching model, DCC-GARCH model, conagion. J E L Classificaion: G01, G15, C24. Inroducion Curren economical and financial crises in general have inernaional characer. Propagaion mechanisms across counries and markes are called he ransmissions for fundamenal linkages. In lieraure conagion erm is applied only o he financial markes, however i should no be idenified only wih he financial linkages i can also concern he markes which are no significanly financially conneced. Many auhors claim ha increase in financial inegraion inensifies conagion effecs. On he subjec of inerde- * Correspondence o: Milda Burzala, Deparmen of Economerics, Faculy of Informaics and Elecronic Economy, Poznan Universiy of Economics, Towarowa 53, 61-896 Poznań, Poland, e-mail: m.burzala@ue.poznan.pl. 2013 Nicolaus Copernicus Universiy. All righs reserved. hp://www.dem.umk.pl/dem
70 Milda Maria Burzała pendence beween markes, conagion effecs and ransmission channels rea, among ohers, he works of: Eichengreen e al. (1995, 1996), Goldsein (1998), Masson (1998), Kaminsky and Reinhar (2000, 2002), Forbes and Rigobon (2002), Pericoli and Sbracia (2003), Pesaran and Pick (2004), Dungey e al. (2005). The mos resricive definiion provided by he World Bank assumes ha marke conagion occurs when he correlaion beween markes in a ime of crisis is significanly higher han during he period of ranquilliy 1. I is possible o conrol for correlaion change over ime by using, for example, a dynamic condiional correlaion model. Researchers ofen also adop an addiional definiion of conagion ha would sui he purpose of a given research mehod. If volailiy models are used, hen conagion is idenified wih he spread of uncerainy across financial markes. The assessmen of he significance of a conagion process requires dividing a sample ino observaions from he ime of ranquilliy and from he ime of crisis in financial markes. A period of ranquilliy is a benchmark period for deermining connecions beween markes, which is a poin of reference for changes observed during a crisis. The ransiion from a ranquilliy period o he period of crisis is usually esablished based on evens which may change he behaviour of cerain indicaors. The resuls of research sudies depend on he division which has been made and he ime of crisis ofen covers boh high and low correlaions beween researched markes. Esablishing a poenial ime of marke conagion by using he Markovswiching model makes i possible o make an assumpion abou he differences in a sochasic process ha deermines correlaions in paricular regimes. The main hypohesis refers o he possibiliy of using he onedimensional Markov swiching model o deermine he ime of conagion in capial markes. Resuls were compared wih conclusions made abou marke conagion based on a range for uncondiional correlaions as well as on arbirary arrangemens. The consequences of adoping paricular divisions are, in fac, imporan informaion for researchers. Research resuls presened in his paper concern he assessmen of he significance of conagion in cerain capial markes in he years 2007 2009. Seleced sock exchange indices represen he siuaion in securiies markes 2. In empirical sudies ha are described laer, he concep of marke 1 Conagion of Financial Crises, World Bank, hp://www.worldbank.org/economicpolicy/managing%20volailiy/conagion/definiions.hm (14 May 2012). This definiion is cied based on Forbes and Rigobon s paper (2002). 2 Capial marke crisis is idenified wih sharp decline in sock prices, mainaining for an exended period of ime. Role of sock marke indexes is broadly described by Jajuga (2006).
Deerminaion of he Time of Conagion in Capial Markes... 71 conagion means a conagion spreading from an index represening he U.S. marke o an index represening he sudied marke 3. Secion 1 presens he DCC-GARCH model and secion 2 describes he Markov-swiching model which has been used in he research. Secion 3 conains informaion on he esed sock exchange indices as well as he crieria for an arbirary division of he se of observaions ino hose relaing o he ime of crisis and hose relaing o he period of ranquilliy in securiies markes. The obained research resuls are presened in secion 4. 1. A dynamic Condiional Correlaion Model Le us assume ha an n-dimensional vecor of raes of reurn s ( = 1,,T) can be decomposed ino he following form: s = μ + ε, (1) 1/ 2 ξ ε = H, (2) where μ is he vecor of condiional expeced values of vecor s based on model VAR(p). In empirical research i is usually assumed ha p =1 4. The Suden s -disribuion was used because of an increased kurosis for process ξ. The dynamic condiional correlaion (DCC) model can be formulaed as (Engle, 2002): H = D R D, (3) D = diag( h,..., h ), (4) 11, NN, 3 Causaliy ess are someimes used for esablishing he direcion of conagion (Cheung, Ng, 1996; Coporale, Piis and Spagnolo, 2002). Their usefulness, however, is limied. This is because hese ess are based on Granger s concep relaed o analysing correlaions beween sudied processes and he consequences of evens. I is ofen a researcher who decides wheher o es a paricular causal relaionship exiss based on his or her knowledge and experience (Osińska, 2006; Fiszeder, 2009). 4 A vecor auoregression model also conrols for he muual inerdependence beween markes hrough connecions beween he delayed values of endogenous variables. Empirical sudies described in lieraure have found ha linear relaionship beween sock reurns are low significan. Some researchers sugges ha i is beer o resign from expeced value model han include incorrecly specified model, especially in he case of oal model for expeced values and variances (Doman, Doman, 2009).
72 Milda Maria Burzała q p 2 2 i, = i+ αε ij i, j+ γij εi, j < εi, j + βij i, j j= 1 j= 1 (5) h c ( I( 0) ) h, i= 1,... N, R Q Q Q (6) 1/2 1/2 = ( diag( )) ( diag( )), K L K L ' = αk βl + αk k k + βl l k= 1 l= 1 k= 1 l= 1 Q 1 Q ξ ξ Q, (7) Marix R is a posiively defined symmeric marix wih ones along he main 1 diagonal; vecor ξ = D ε in his case denoes he vecor of sandardised residuals from model VAR(1). Marix D was esimaed based on he onedimensional GJR-GARCH(1,1) model (Glosen, Jagannahan, Runkle, 1993). In equaion (5) I ( ) is an indicaor funcion and i assumes he value of 1 for ε i, j < 0 and he value of 0 for ε i, j 0 ( i = 1,..., N ). Posiive values of parameer γ ij which are significanly differen from zero prove ha he leverage effec occurs 5. Covariance saionariy and hus a finie variance in equaion (5) is ensured by saisfying hese condiions: k k= 1 l= 1 q p c > 0, α, β 0 and ( α + γ / 2) + β < 1. (8) i ij ij ij ij ij j= 1 j= 1 In equaion (6) Q denoes a square marix of uncondiional covariances of he vecor ξ variables. In addiion, i is required ha α k, βl 0, K L α + βl < 1. The model s parameers are esimaed in wo sages. The logarihmic likelihood funcion is he sum of likelihood funcions for a volailiy model and likelihood funcions for he parameers of dynamic correlaions (Engle, 2002). 5 The leverage effec resuls from an asymmeric response of raes of reurn o posiive and negaive informaion reaching he financial marke. I is a consequence of a negaive correlaion beween securiies prices and he volailiy of raes of reurn. The higher he value of paramer γ ij, he sronger he leverage effec (he addiional impac of negaive informaion).
Deerminaion of he Time of Conagion in Capial Markes... 73 2. The Markov-Swiching Model In Markov-swiching models i is assumed ha a swich beween he behaviours of raes of reurn in regimes (periods), and hus he process of conagion in a marke, depend on cerain hidden facors which are no direcly observable. One can only observe he exernal sympoms of regime change by observing, for example, he muual correlaions beween raes of reurn. Theoreically, for a ime series of dynamic condiional correlaions ρ, a one-dimensional swiching model can be used, in which swiches occur as a resul of changes in he expeced value μ, variance σ 2 or he expeced value and variance of he sudied correlaions (Hamilon, 1989; Davidson, 2013). If a swiching model is only consruced for he purpose of classifying he already obained heoreical values of correlaions, i can be assumed ha, in each regime, values are generaed by independen processes wih a differen consan expeced value and consan variance: ρ = μ( r = i) + ε ε ~ N(0, σ ( r = i), i = 0,1,2, (9) i 2 i i where μ ( r = i), σ ( r = i) denoe he expeced value and variance of condiional correlaions, respecively, in he i-regime. Such an approach allows one o use a one-dimensional model, in which i is assumed ha hree regimes will be analysed, i.e. r = i (i = 0, 1, 2), which are relaed o an exremely low, average and exremely high correlaion beween raes of reurn. The proposed sequenial procedure enails esimaing he model of dynamic condiional correlaions and he swiching model separaely, which makes i possible o avoid many problems relaed o esimaing mulidimensional models 6. The series of random variables r in he subsequen momens in ime (=1,..., T) has he Markov propery, i.e. is value a he ime momen +1, i.e. r +1, depends only on he regime a he momen, raher han on all he preceding regimes, which is formally formulaed as: 2 6 In pracice, he use of mulidimensional swiching models is associaed wih many problems because of he number of esimaed parameers which grows exponenially, as in he mulidimensional VechGARCH model (Billio, Lo Duca, Pelizzon, 2005). If one assumes ha only wo saes (of high and low volailiy) can occur in each of wo sudied markes, han one already allows for he occurrence of four regimes, and he marix of condiional probabiliies has dimensions [4 x 4]. The final number of parameers depends on he assumpions regarding he differences beween processes deermining he behaviour of raes of reurn in paricular regimes.
74 Milda Maria Burzała Pr ( + 1 = j r = i, r 1 = k,...) = P( r+ 1 = j r = i) = pij, (10) i, j = 0,1,2. Probabiliies p ij denoe he probabiliy of ransiion of he dependence beween raes of reurn from regime i o regime j. If a he 1 momen he process was under he r -1 = i regime, hen he condiional densiy funcion of he explained variable ρ can be represened as f ( s r = i, I 1), where I 1 denoes he hisory of he process unil he 1 momen. Any supposiions on he i regime may be made by means of a condiional probabiliy: f ( s r = i, I ) P( r 1) 1 = i I P ( r = i I ) =, (11) 2 f ( s r = j, I ) P( r = j I ) j = 0 1 1 where 2 ) = = p ( = j 0 ij P r 1 j I 1 P ( r = i I 1 ). (12) The model s parameers are esimaed by using he maximum likelihood mehod (Davidson, 2013) 7. 3. The Saisical Maerial and an Arbirary Division of he Sample In he empirical research, daily coninuously compounded raes of reurn on six indices represening he siuaion on sock exchanges during he period from Augus 17, 2005 o July 31, 2009 were used (1022 observaions for each sock exchange): s i = 100 (ln( Pi, ) ln( Pi, 1)). Two indices from srong EU economies DAX and CAC, represening he siuaion on he German and French sock exchanges, as well as wo indices from weaker economies from he old European Union he Spanish IBEX and he Greek ATEX, and wo indices from he counries of Cenral and Easern Europe he Hungarian BUX and he Polish WIG20 were seleced for he purpose of he analysis (source: he Sooq daabase). The Dow Jones 7 The relevan likelihood funcion is presened as par of he descripion of he TSM (Time Series Modelling) program. I is no easy o esimae he model s parameers. Numerical problems resul from he occurrence of local exrema of he logarihmic likelihood funcion. This is why normally wo, up o hree, regimes under which a process may be are disinguished.
Deerminaion of he Time of Conagion in Capial Markes... 75 Indusrial Average (DJIA) index represened he siuaion on he U.S. sock exchanges. Gaps in he daa were filled by using he linear inerpolaion mehod. Due o he differen quoaion imes, daa were smoohed by using a wo-period moving average (Dungey e al., 2007). 15000 14000 17-08-2005 he beginning of he sample 15-09-2008 bankrupcy of he bank Lehman Brohers 31-07-2009 end of he sample 13000 12000 11000 10000 9000 8000 7000 6000 2005-08-17 2005-11-17 2006-02-17 2006-05-17 2006-08-17 09-08-2007 BNP Paribas canno deermine securiy values march-2009 min on he sock 2006-11-17 2007-02-17 2007-05-17 2007-08-17 2007-11-17 2008-02-17 2008-05-17 2008-08-17 2008-11-17 2009-02-17 2009-05-17 6 4 2 0-2 -4-6 05 06 07 08 09 Figure 1. The Dow Jones Indusrial Average index during he period from Augus 17, 2005 o July 31, 2009 (lower figure daily raes of reurns from DJIA index) The behaviour of he Dow Jones Indusrial index during he sudied period is shown in Figure 1. An arbirary division of he observaion se ino wo subses relaed o he period of crisis (high volailiy of raes of reurn) and he period of ranquilliy (low volailiy) in securiies markes should be made in such a way ha he ime of ranquilliy immediaely precedes he ime of crisis. This is because i consiues a benchmark for comparisons. Quoaions preceding, for example, he collapse of Lehman Brohers, hardly represened a ime of ranquilliy as he Dow Jones Indusrial Average had already been declining for some ime and he rae of reurn on he index had been characerised by increased volailiy. Therefore, i was decided ha he informaion abou he difficulies relaed o evaluaing asses which was announced by he French BNP Paribas would be used when dividing he se of observaions. On 9 Augus 2007 his bank suspended paymens from hree funds invesing in he marke of bonds secured by subprime morgages. The period of crisis was exended beyond he ime when securiies were rading a he lowes level because of he increased volailiy of raes of reurn which persised unil he end of July 2009. There were 511 observaions for he ime of crisis deermined in his way. In order o ensure comparabiliy of resuls, 511 former observaions were analysed for evens conribuing o
76 Milda Maria Burzała financial urmoil. I was a ime of rising asse prices, wih minor adjusmens, which is why i has been assumed o be a period of ranquilliy on he sock exchange. 4. Research Resuls The esimaion of a model of dynamic condiional correlaions should be preceded by a es jusifying heir use. The resuls of such ess depend on he adoped specificaion of volailiy models. Therefore, wo ess were used in he research, i.e. he Tse es (2000) as well as Engle and Sheppard es (2001) in wo versions wih delays p=5 and p=10. For all indices a leas one es indicaed he reasonableness of consrucing a dynamic condiional correlaion model 8. A significan increase of correlaions in he ime of crisis confirms he occurrence of he process of marke conagion. Forbes and Rigobon (2002) propose using Fisher s ransformaion of correlaion coefficiens while esing he significance of change in correlaion beween raes of reurn. Afer Fisher s ransformaion, he sample correlaion coefficien can be reaed as he realizaion of a random variable wih a normal disribuion 1 1+ ˆ ρ wih he expeced value of E( ρ) = ln, where ρˆ is he esimaed correlaion coefficien. The variance of his variable is Var( ρ ) =. The null 2 1 ˆ ρ 1 T 3 hypohesis : K S H0 ρ ρ is esed agains an alernaive hypohesis, i.e. : K S H1 ρ > ρ (index S means he ranquiliy period, K - he ime of poenial marke conagion). The empirical saisic in he es for wo expeced values is as follows: 1 ˆ 1 ˆ 0,5ln K S + ρ 0,5ln + ρ K S 1 ˆ ρ 1 ˆ ρ FR =. (13) 1 1 + T 3 T 3 K S The FR-saisic have a normal disribuion N(0,1) and even if he sample is small i allows one o use he criical values of a sandard normal disribuion. 8 Calculaions were carried ou by using he OxMerics 6.10 program.
Deerminaion of he Time of Conagion in Capial Markes... 77 The parameers of he GJR-GARCH(1,1) model are presened in he upper par of Table 1. Le us remember ha he models were esimaed on he basis of residuals from model VAR(1). Therefore, a slighly differen model for he DJIA index was conneced wih each index. In all he volailiy models for he DJIA index, he alfa parameer which describes he impac of posiive residual impulses was insignifican. In he models for he European indices, he alfa parameer was only significan for he Greek (ATH), he Hungarian (BUX) and he Polish (WIG20) indices. The parameers ha were significan were bea and gamma which describe he impac of pas variance as well as he leverage effec (an addiional impac of negaive informaion reaching he marke). The model s assumpions require ha he alfa parameer be significanly greaer han zero. Thus, in order o sandardise he residuals from model VAR(1), i was finally he GARCH(1,1) models ha were used 9 : where h q p ij, = ci + ij i, j ij ii, j = N j= 1 j= 1 q p ij j= 1 j= 1 2 α ε + β h, i 1,..., (14) α + βij < 1, c i > 0, α ij 0, βij 0. This ime he obained parameer esimaes mee he models assumpions; he arch effec is significan in all of he models (he lower par of Table 1). This means ha he impac of negaive informaion on many markes was considerably sronger han he impac of posiive informaion. Such markes could be idenified by esimaing he GJR-GARCH model a he firs sage of he research. Esimaes of parameer β exceed he value of 0.8, which confirms he volailiy clusering phenomenon. Boh in he GARCH(1,1) model and in he condiional correlaion model DCC-GARCH(1,1) he requiremen of covariance saionariy is saisfied. Also he condiions for variance (nonnegaive, significan model parameers) are me. The sum of parameers ( α + β ) in he GARCH model is close o one, which means he occurrence of persisence (long-erm dependencies) and sugges es of inegraed model (IGARCH, FIGARCH) in furher sudies. The highes uncondiional correla- 9 The same volailiy model was adoped for wo ime series due o program limiaions. In he research GARCH (1,2), GARCH (2,1) and GARCH (2,2) models were also esed only in GARCH (1,1) model all parameers were significan and was he lowes informaion crierion value (AIC). Condiional correlaions from wo models (DCC-GJR-GARCH (1,1) and DCC-GARCH) were slighly differen.
78 Milda Maria Burzała ion wih he DJI index was recorded for he CAC and he DAX indices and he lowes for he BUX and he WIG20 indices. Table 1. The DCC-GARCH model s parameers GJR GARCH(1,1) Index for index i for index DJIA cons(i) alfa(i) bea(i) gamma(i) cons alfa bea gamma ATH 0.017 *** 0.069 ** 0.842 *** 0.146 *** 0.006 *** 0.003 0.894 *** 0.211 *** CAC 0.010 *** 0.022 0.900 *** 0.123 *** 0.006 *** 0.002 0.891 *** 0.201 *** DAX 0.013 *** 0.032 0.868 *** 0.162 *** 0.006 *** 0.005 0.892 *** 0.214 *** IBEX 0.015 ** 0.043 0.853 *** 0.164 *** 0.007 *** 0.006 0.880 *** 0.212 *** BUX 0.037 ** 0.113 *** 0.798 *** 0.135 *** 0.006 *** 0.005 0.881 *** 0.209 *** WIG20 0.027 ** 0.059 ** 0.889 *** 0.064 ** 0.006 *** 0.003 0.889 *** 0.221 *** GARCH(1,1) Index for index i for index DJIA cons(i) alfa(i) bea(i) cons alfa bea ATH 0.016 ** 0.165 *** 0.823 *** 0.005 ** 0.111 *** 0.882 *** CAC 0.009 ** 0.107 *** 0.883 *** 0.005 ** 0.116 *** 0.877 *** DAX 0.011 ** 0.129 *** 0.861 *** 0.005 ** 0.112 *** 0.881 *** IBEX 0.015 ** 0.159 *** 0.824 *** 0.006 ** 0.118 *** 0.875 *** BUX 0.034 ** 0.172 *** 0.806 *** 0.004 ** 0.116 *** 0.880 *** WIG20 0.026 * 0.098 *** 0.883 *** 0.005 ** 0.114 *** 0.879 *** DCC GARCH(1,1) Index Alfa bea df uncondiional correlaions Log likelihood ATH 0.022 *** 0.958 *** 16.960 *** 0.384 *** 2060.58 CAC 0.016 * 0.962 *** 12.636 *** 0.641 *** 1820.90 DAX 0.027 ** 0.951 *** 10.960 *** 0.649 *** 1801.13 IBEX 0.016 ** 0.971 *** 11.360 *** 0.590 *** 1845.05 BUX 0.075 0.829 *** 16.020 *** 0.319 *** 2292.27 WIG20 0.018 ** 0.967 *** 16.971 *** 0.378 *** 2346.02 Noe: a parameer s significance for α = 0.01 is marked wih hree aserisks, for α = 0.05 wih wo aserisks and for α = 0.1 wih one aserisk. The parameers of he MS(3) swiching model are provided in Table 2. Regime 2 is relaed o an exremely high correlaion (he shaded area in Figure 2). As for he wo indices CAC and DAX, regime 2 covered boh significanly high and significanly low correlaion beween markes. This probably resuled from a very high volailiy of exreme correlaions. A correc classificaion was obained by simplifying he process o a model in which only he expeced value would change. For he remaining four indices, he variance of exremely high and exremely low correlaions was significanly higher han he variance of condi-
Deerminaion of he Time of Conagion in Capial Markes... 79 ional correlaions in he ime of ranquilliy, and he model made i possible o make a correc classificaion 10. Table 2. The esimaes of he MS(3) swiching model s parameers Swiches occur as a resul of changes he expeced value and variance he expeced value Regime ATH IBEX BUX WIG20 CAC DAX 0 283 423 178 312 223 222 Number of observaions 1 379 387 405 336 614 461 2 360 212 439 374 185 339 0 0.264 0.511 0.103 0.255 0.567 0,520 Expeced value 1 0.372 0.598 0.271 0.372 0.638 0,632 2 0.482 0.672 0.440 0.468 0.691 0,710 FR-saisic 1,820 ** 1.449 * 2.811 *** 1.545 * 1.132 1.989 ** 0 0.060 0.031 0.085 0.047 x x Variance 1 0.027 0.017 0.052 0.029 x x 2 0.049 0.029 0.078 0.038 x x H-saisic 601.6 *** 589.2 *** 854.1 *** 596.2 *** 492.4 *** 650.4 *** D-saisic 0 15.89 *** 12.90 *** 16.69 *** 13.77 *** 10.83 *** 11.61 *** 1 24.53 *** 23.10 *** 29.23 *** 23.78 *** 19.97 *** 22.72 *** p_{0/0} 0.977 0.983 0.880 0.987 0.978 0,961 p_{0/1} 0.023 0.017 0.120 0.013 0.000 0,039 p_{0/2} 0.000 0.000 0.000 0.000 0.022 0,000 p_{1/0} 0.017 0.018 0.052 0.012 0.008 0,018 p_{1/1} 0.962 0.970 0.880 0.964 0.984 0,965 p_{1/2} 0.021 0.012 0.068 0.024 0.008 0,017 p_{2/0} 0.000 0.000 0.061 0.019 0.000 0,000 p_{2/1} 0.022 0.021 0.000 0.000 0.046 0,021 p_{2/2} 0.978 0.979 0.939 0.981 0.954 0,979 Log-likelihood 1708,9 2257.9 1063.5 1824.5 2307.3 1797.5 AIC 3,325 4.399 2.062 3.551 4.499 3.502 Probabiliy of ransiion Noe: he FR-saisic refers o he difference in correlaion beween regimes 2 and 1; he D-saisic refers o he difference disribuions beween regime 2 and regime 0 or 1. The Jarque-Berra es rejecs a convenional significance level he normaliy of correlaion in hree regimes (no repored). I is he reason of he use of nonparameric variance analysis (Kruskal and Wallis-es) o evaluae he qualiy of classificaion (division of he sample ino observaions from he ime of ranquilliy and from he ime of poenial marke conagion). In he firs research sage, H-saisic indicaes he diversificaion of disribuion a leas in wo regimes. In he second sage, D-saisics indicaes he 10 High variance in 0 regime (exremely low correlaions) indicaes, ha differeniaion of he sign of reurns on wo markes causes he increase of uncerainy among invesors.
80 Milda Maria Burzała diversificaion of disribuion in all regimes for all sudied indices. Relevan saisics are provided in Table 2. Resuls confirms he legiimacy of he use of one-dimensional swiching model. Simple model can give saisfacory resuls. Corr_ATH_DJIA Corr_IBEX_DJIA Corr_BUX_DJIA Corr_WIG20_DJIA Corr_CAC_DJIA Corr_DAX_DJIA Figure 2. Poenial periods of marke conagion as deermined based on condiional correlaions from he DCC-GARCH(1.1) model Noe: he shaded area: he ime of exremely high correlaions (regime 2); he average value of condiional correlaions in he ime of crisis (Augus 9, 2007 o July 31, 2009); - - - he upper limi of he range: uncondiional correlaion + 2 error.
Deerminaion of he Time of Conagion in Capial Markes... 81 The occurrence of exreme correlaions under regime 2 only means he ime of poenial marke conagion. Only he rejecion of he null hypohesis in he es for wo expeced values means ha he process of marke conagion has occurred. The expeced value of correlaion in regime 2 is significanly higher han he expeced value of correlaion in regime 1 (he ime of ranquilliy in he marke) in all securiies markes excep for he French marke (CAC). The significance level ha allows one o rejec he null hypohesis is, however, varied, which is highlighed in he able. The FRsaisics assumes he highes value for he Hungarian marke. I can be assumed ha he value of saisics reflecs he effecs of conagion. The higher he value of saisics, he more severe he effecs of marke conagion. The probabiliy of remaining under each of he regimes, which is provided in Table 2, is high, which means ha he highlighed regimes are persisen. The analysis of chars in Figure 2 allows one o compare he frequency of an index being under regime 2 and he ime of remaining here. The addiional, horizonal and dashed line makes i possible o relae indicaed periods o he ime of poenial marke conagion as idenified based on a range for uncondiional correlaions. In his case, hose correlaions which fall ouside he upper and he lower limis deermined by a double esimaion error are assumed o be exremely high and exremely low correlaions, respecively. The ime of ranquilliy in a marke is represened by correlaions from a range deermined in his way. As for arbirary arrangemens, i should be remembered ha he sample was only divided ino wo subses. Toal duraion imes of he poenial marke conagion period are provided in Table 3. The longes ime is for an arbirary division and he shores for he range for uncondiional correlaions. The swiching model indicaed ha he period of poenial conagion in he Hungarian marke was he longes. Table 3. The number of observaions during he poenial period of marke conagion Meod ATH IBEX BUX WIG20 CAC DAX Arbirary arrangemens 511 511 511 511 511 511 Swiching model regime 2 360 212 439 374 185 339 Range for uncondiional correlaions 161 89 243 101 57 81 A comparison of he resuls of ess of he significance of occurrence of conagion spreading from he U.S. marke o a given marke is presened in Table 4.
82 Milda Maria Burzała Table 4. A comparison of he resuls of esing he significance of conagion in a marke Mehod Index ATH IBEX BUX WIG20 CAC DAX Conagion 0.416 0.594 0.360 0.427 0.650 0.654 Arbirary Tranquiliy 0.349 0.562 0.267 0.315 0.614 0.613 arrangemens FR-saisic 1.309 * 0.770 1.641 * 2.072 ** 0.966 1.096 Conagion 0.482 0.672 0.440 0.468 0.691 0.710 Swiching model Tranquiliy 0.372 0.598 0.271 0.372 0.638 0.632 FR-saisic 1.820 ** 1.449 * 2.811 *** 1.545 * 1.132 1.989 ** Range for uncondiional correlaions Conagion 0.535 0.702 0.492 0.574 0.731 0.771 Tranquiliy 0.384 0.584 0.321 0.405 0.639 0.653 FR-saisic 2.019 ** 1.788 ** 2.647 *** 1.572 * 1.237 2.030 ** Noe: conagion means he ime of poenial marke conagion. For five indices he values of he FR-saisics recorded in he case of he swiching model are lower han he corresponding saisics in he analysis of uncondiional correlaions, which has an effec on conclusions abou conagion. The occurrence of he conagion process is regisered more ofen (for lower significance levels) in he analysis of a range for uncondiional correlaion. The opposie is rue only for he BUX index, which probably resuls from small differences beween he duraion imes of marke conagion ha are deermined by using differen mehods. The lowes values of he FRsaisics are usually recorded for an arbirary division. Deailed resuls and significance levels are provided in Table 4. Conclusions I is relaively difficul o dae a crisis in financial markes. During periods deermined based on evens ha change he behaviour of raes of reurn, boh high and low correlaion beween markes can be observed. This paper proposes indicaing he periods of poenial marke conagion on he basis of a one-dimensional swiching model. Tess made confirm he legiimacy of he use of simple swiching model o deermine poenial marke conagion periods. Furher sudies should be conduced i is imporan o compare obained resuls wih he resuls from mulidimensional model, where swiches are deermined based on he changes of expeced value, variance, and covariance. In he paper such comparisons were no made because of he lack of appropriae sofware. Occurrence of persisence sugges, ha furher sudies should also include inference based on he inegraed model. Resuls confirm he conclusions made by he auhor on he subjec of conagion on he basis of logi model for panel daa (Burzała, 2012). Significan conagion effecs were observed on German marke, less significan
Deerminaion of he Time of Conagion in Capial Markes... 83 on Polish, and he lack of significan conagion effecs were observed on French marke. Deerminaion of exremely high correlaions by using a range for uncondiional correlaions and he MS(3) swiching model yields similar resuls regarding conclusions abou he occurrence of he process of conagion in a marke. Conclusions abou conagion are, however, made a a higher significance level in he case of he swiching model. I is worh emphasising ha i is necessary ha he appropriae ess be conduced which would confirm he significance of he increase of correlaion beween markes. Also, he ime of poenial marke conagion deermined on he basis of a regime wih an exremely high correlaion (he swiching model) is longer. In furher sudies more aenion should be paid o he issue of deermining he direcion of conagion as well as exremely low correlaions which may be a harbinger of a crisis. References Aczel, A. D. (2000), Saysyka w zarządzaniu (Saisic in Business), PWN, Warszawa. Billio, M., Lo Duca, M., Pelizzon, L. (2005), Conagion Deecion wih Swiching Regime Models: a Shor and Long Run Analysis, Working Paper 05.01, Universiy Ca Foscari of Venice, Ialy, DOI: hp://dx.doi.org/10.2139%2fssrn.676956. Burzała, M. M. (2012), Efeky zarażania giełd europejskich w czasie kryzysu finansowego 2008-2009 - model logiowy dla danych panelowych (Conagion Effecs on European Sock Exchanges During 2008 2009 Financial Crisis Logi Model for Panel Daa), in: Appenzeller D. (ed.), Maemayka i informayka na usługach ekonomii: meody, analizy, prognozy (Mahemaics and compuer science a he service of economics: mehods, analysis, forecass), 31 43, Wydawnicwo Naukowe UEP, Poznań. Cieciura, M., Zacharski, J. (2007), Meody probabilisyczne w ujęciu prakycznym (Probabilisic Mehods: a Pracical Approach), Vizja Press&IT, Warszawa. Davidson, J. (2013), Time Series Modelling, Version 4.38, Universiy of Exeer, hp://www.imeseriesmodelling.com/smod4doc.pdf, (13.06.2013), DOI: hp://dx.doi.org/10.2307%2f2231972. Doman, M., Doman, R. (2009), Modelowanie zmienności i ryzyka. Meody ekonomerii finansowej (Volailiy and Risk Modeling. Mehods of Financial Economerics), Oficyna, Kraków. Dungey, M., Fry, R. A., Gonzalez-Hermosillo, B., Marin, V. L. (2005), Empirical Modeling of Conagion: A Review of Mehodologies, Quaniaive Finance, 5(1), 9 24, DOI: hp://dx.doi.org/10.1080%2f14697680500142045. Dungey, M., Fry, R., A., González-Hermosillo, B., Marin, V., L. (2007), Conagion in Global Equiy Markes in 1998: The Effecs of he Russian and LTCM Crises, Norh American Journal of Economics and Finance, 18(2), 155 174, DOI: hp://dx.doi.org/10.1016%2fj.najef.2007.05.003. Eichengreen, B., Rose, A., Wyplosz, C., Dumas, B., Weber, A. (1995), Exchange Marke Mayhem: The Anecendens and Afermah of Speculaive Aacks, Economic Policy, 21, 249 312, DOI: hp://dx.doi.org/10.2307%2f1344591.
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Deerminaion of he Time of Conagion in Capial Markes... 85 Wyznaczanie czasu zarażania rynków kapiałowych na podsawie modelu przełącznikowego Z a r y s r e ś c i. W arykule podjęo próbę porównania wnioskowania o zarażaniu rynków na podsawie okresów wskazanych przez model przełącznikowy Markowa z wnioskowaniem oparym na przedziale dla korelacji bezwarunkowych i usaleniach arbiralnych. W celu konrolowania zmieniających się w czasie korelacji wykorzysano model DCC. Usalenie eksremalnie wysokich korelacji przy wykorzysaniu przedziału dla korelacji bezwarunkowych lub modelu przełącznikowego MS(3) prowadzi do podobnych rezulaów w zakresie wnioskowania o wysąpieniu procesu zarażania rynku. Wnioski o zarażaniu są jednak sawiane przy wyższym poziomie isoności w przypadku modelu przełącznikowego. S ł o w a k l u c z o w e: model przełącznikowy, model DCC, zarażanie.