2010 Vol. 4 Num. 2 - GCG: Revista de GlobalizaciÃ³n, Competitividad ...

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Conditional volatility in sustainable and traditional stock exchange indexes: analysis of the Spanish market114 models provide powerful tools to analyse the evolution of the conditional variance-covariancematrix for a group of assets or other financial series. Multivariate conditional volatility modellinginvolves an increase in the number of parameters to be estimated if a large number ofseries is analysed. In consequence, the accuracy of the inference methods decreases and,therefore, the estimates obtained are less robust. In any case, given the small number of seriesto be analysed (two in this case) and the parsimony of the multivariate GARCH modelschosen, it is not thought that the robustness of the estimates will be affected.4.1. Univariate volatility modellingThree univariate GARCH models were selected to estimate the conditional volatility of thehistorical return series of both indexes (IBEX35, FTSE4Good-IBEX). The first is the GARCHmodel proposed by Bollerslev (1986) and specified as GARCH(1,1).⎛ P ⎞tP tis the index closing price in period t, yt= 100log ⎜ ⎟ is the daily continuous return of⎝Pt−1⎠the index and Ω t denotes the information set available in period t. Thus, the GARCH(1,1)process can be represented by the following equations:yt= +µ ε t (2)εt = ησ t t with ηt i.i.d.; E η[ t]= 0 , Var η[ t]=1 (3); (4)It is verified thatis the mean expected return andis the daily conditional volatility of the return in period t, which reflects the short-run persistenceof shocks (ARCH effect - ) and the contribution of shocks to long-run persistence(GARCH effect - ). Parameter measures the persistence in volatility, so thatthe greater the value, the more pronounced the volatility clustering effect appears. Under aGARCH(1,1) model, if , the process is second-order stationary in volatility, in whichcase,shows the unconditional volatility of the return series.This approach supposes that the impact of return shocks ( ) on the volatility is symmetrical.However, there is empirical evidence that, in most financial series, these shocks areasymmetric, and the impact of a negative shock on volatility is greater than a positive one,showing the so-called leverage effect. In order to capture this effect, additional parametershave to be added to the model. The most commonly used models are the GJR and EGARCHproposed by Glosten et al. (1993) and Nelson (1991), respectively.Under the GJR(1,1) model, the volatility of the return series is given by:; (5)GCG GEORGETOWN UNIVERSITY - UNIVERSIA 2010 VOL. 4 NUM. 2 ISSN: 1988-7116
Eduardo Ortas, José M. Moneva & Manuel Salvadorwhere when and 0 otherwise. Coefficient shows the asymmetry of theimpact of the return shocks ( ) on the volatility of the indexes. A leverage effect is displayedwhen coefficient is positive and significant. Additionally, if the conditional distribution of theerror term is symmetrical, persistence on volatility is given by .115Another approach to analysing the asymmetrical effect of return shocks on volatility, whichalso does not impose restrictions of non-negativity on the volatility equation parameters,is the EGARCH model, proposed by Nelson (1991). Under the EGARCH(1,1) specification,conditional variance is determined by the following expression:(6)In this case, refers to the asymmetry in the link between the return shocks ( ) and the volatilityof the series, there being a leverage effect if the parameter is negative and significant.4.2. Multivariate volatility modellingIn recent years, several multivariate conditional volatility models have been developed(Bauwens et al., 2006). Furthermore, there has been recent empirical work applying thesemethods (Smyth and Nandha, 2003; Lee, 2004; Antell and Vaihekoski, 2007; Chuang et al.,2007; León et al., 2007; Hsu, 2008; Saleem, 2008; Tai, 2008; Savva, 2009). In this study, threemultivariate GARCH models have been selected and are described below.Givenas the vector of continuous returns of the indexes in period t (inthis case, N=2), the multivariate GARCH(1,1) is expressed by the following equations:with and (7)with i.i.d. and (8)andis a positive semi-definite matrix (NxN), so that:where the vech (‘vector half’) operator converts the unique upper triangular elements of asymmetric matrix into a column vector; A, B, C and D are matriceswith C symmetric; denotes the Hadamard product; and is the Nx1 vector suchthat the i-th component is equal to 1 if and 0 otherwise. This verifies that.An asymmetric effect will also be present if the D matrix is significantly different from 0.(9)The number of parameters in the model is equal tocase, is 32., which, in thisGCG GEORGETOWN UNIVERSITY - UNIVERSIA 2010 VOL. 4 NUM. 2 ISSN: 1988-7116
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CARTA DEL EDITOR IN CHIEFEditor in
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SUMARIO SUMMARY SUMÁRIO1Ú2Efectos
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StaffDe la Torre, José, Dean, Chap
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Staff2. MULTINACIONALES, INVERSIÓN
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SELECCIÓN PROCEDURE PROCEDIMENTOIN
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Instrucciones para Autores y Proced
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Instrucciones para Autores y Proced
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16 Efectos de la inversión en I+Ds
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Efectos de la inversión en I+D sob
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28El Pacto Mundial de las NacionesU
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El Pacto Mundial de las Naciones Un
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40La política de Industrializació
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La política de Industrialización
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54autorMiguel Blanco-Callejo 1Depar
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Miguel Blanco-CallejoA pesar de que
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Miguel Blanco-CallejoEn 2003, como
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Miguel Blanco-Callejo3.1. Formula 1
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2010 Vol. 4 Num. 2 - GCG: Revista de GlobalizaciÃ³n, Competitividad ...

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