A MULTIVARIATE SKEW-GARCH MODEL

Transcription

1 A MULTIVARIATE SKEW-GARCH MODEL Giovanni De Luca, Marc G. Genton and Nicola Loerfido ABSTRACT Emirical research on Euroean stock markets has shown that they behave differently according to the erformance of the leading financial market identified as the US market. A ositive sign is viewed as good news in the international financial markets, a negative sign means, conversely, bad news. As a result, we assume that Euroean stock market returns are affected by endogenous and exogenous shocks. The former raise in the market itself, the latter come from the US market, because of its most influential role in the world. Under standard assumtions, the distribution of the Euroean market index returns conditionally on the sign of the oneday lagged US return is skew-normal. The resulting model is denoted Skew-GARCH. We study the roerties of this new model and illustrate its alication to time-series data from three Euroean financial markets. 1. INTRODUCTION The ioneering work of Engle (198) has reresented the starting oint of a tremendous scientific roduction with the aim of modeling and forecasting Econometric Analysis of Financial and Economic Time Series/Part A Advances in Econometrics, Volume 0, Coyright r 005 by Elsevier Ltd. All rights of reroduction in any form reserved ISSN: /doi: /S (05)

2 34 GIOVANNI DE LUCA ET AL. the volatility of financial time series. The AutoRegressive Conditional Heteroskedasticity (ARCH) model has been dealt with in deth. Many variants have been roosed. Among them, we emhasize its most oular generalizations, the Generalized ARCH model (Bollerslev, 1986) and the Exonential GARCH model (Nelson, 1991) allowing for the inclusion of the asymmetric effect of volatility. Moving from a univariate to a multivariate ersective, the multivariate GARCH model is quite interesting because it can shed light on the common movements of the volatilities across markets (Bollerslev, Engle, & Wooldridge, 1988; Bollerslev, 1990; Engle, 00). When the analysis focuses on one or more markets, the ossible relevance of an external leading market is usually ignored. Nonetheless, it is an imortant oint which can hel exlaining some emirically detected features. Actually, a wide literature has dealt with the issue of the international transmission of stock markets movements. Eun and Shim (1989) stressed the most influential role of the US stock market. Innovations in the US market are transmitted to the other markets. Conversely, none of the other markets can affect the US market movements. The time horizon of the transmission is very short: the other stock exchanges resonses raidly decrease after one day. The conclusion of Eun and Shim (1989) suggests to consider the US market as the most imortant roducer of information affecting the world stock market. The contemoraneous and lead/lag relationshis among stock markets are also studied in Koch and Koch (1991). The analysis of univariate and multivariate GARCH models has traditionally neglected this asect, with few excetions. For instance, Lin, Engle, and Ito (1994) carried out an emirical investigation of the relationshi between returns and volatilities of Tokyo and New York markets. The eculiarity of the study is the use of a decomosition of daily return into two comonents: the daytime return and the overnight return. The conclusion is the existence of cross-market interdeendence in returns and volatilities. Karolyi (1995) detected the interdeendence between the US and Canadian markets through a bivariate GARCH model. Our analysis starts from the emirical detection of the different behavior of three Euroean markets, according to the erformance of the leading market identified as the US market. From a statistical ersective, we assume that Euroean stock market returns are affected by endogenous and exogenous shocks. The former raise in the market itself, the latter come from the US market, defined as the leading market because of its most influential role in the world. Moreover, the flow of information from the US market to the Euroean markets is asymmetric in its direction as well as in

3 A Multivariate Skew-Garch Model 35 its effects. We recognize a negative (ositive) erformance of the US market as a roxy for bad (good) news for the world stock market. Euroean financial markets returns behave in a different way according to bad or good news. Moreover, they are more reactive to bad news than to good ones. The Euroean and the US markets are not synchronous. When Euroean markets oen on day t, the US market is still closed; when Euroean markets close on the same day, the US market is about to oen or has just oened (deending on the Euroean country). This imlies a ossible causal relationshi from the US market return at time t 1 to the Euroean returns at time t. The distribution of Euroean returns changes according to the sign of the one-day lagged erformance of the US market. Average returns are negative (ositive) in resence of bad (good) news and they are very similar in absolute value. Volatility is higher in resence of bad news. Skewness is negative (ositive) and more remarkable in resence of bad (good) news. In both cases, a high degree of letokurtosis is observed. Finally, bad news involves a stronger correlation between resent Euroean returns and the one-day lagged US return. Allowing for a GARCH structure for taking into account the heteroskedastic nature of financial time series, under standard assumtions, the distribution of the Euroean returns conditionally on news (that is, on the sign of the one-day lagged US return) and ast information turns out to be skew-normal (Azzalini, 1985). This is a generalization of the normal distribution with an additional arameter to control skewness. The two conditional distributions are characterized by different features according to the tye of news (bad or good). In articular, the skewness can be either negative (bad news) or ositive (good news). The resulting model is denoted Skew-GARCH (henceforth SGARCH). The theoretical features of the model erfectly match the emirical evidence. The basic idea can be extended to a multivariate setting. The international integration of financial markets is more remarkable in resence of a geograhical roximity. The Euroean markets tend to show common movements. Under standard assumtions, the joint distribution of Euroean stock market returns conditionally on the sign of the one-day lagged US market return and ast information is a multivariate skewnormal distribution (Azzalini & Dalla Valle, 1996), whose density is indexed by a location vector, a scale matrix and a shae vector. Finally, unconditional (with resect to the erformance of the US market) returns have some features in concordance with emirical evidence.

4 36 GIOVANNI DE LUCA ET AL. The aer is organized as follows. Section describes the theory of the skew-normal distribution. In Section 3, the multivariate SGARCH model is resented. In Section 4, the conditional distribution and related moments are obtained. Some secial cases are described, including the univariate model when the dimensionality reduces to one. Section 5 refers to the unconditional distribution. Section 6 exhibits the estimates of the univariate and multivariate models alied to three small financial markets in Euroe: Dutch, Swiss and Italian. The results show the relevance of the erformance of the leading market suorting the roosal of the SGARCH model. Section 7 concludes. Some roofs are resented in the aendix.. THE SKEW-NORMAL DISTRIBUTION The distribution of a random vector z is multivariate skew-normal (SN, henceforth) with location arameter z, scale arameter O and shae arameter a, that is z SN ðz; O; aþ; if its robability density function (df) is fðz; z; O; a Þ ¼ f ðz z; OÞF a T ðz zþ ; z; z; a R ; O R where F( ) is the cdf of a standardized normal variable and f ðz z; OÞ is the density function of a -dimensional normal distribution with mean z and variance O. For examle, Z SN 1 ðz; o; aþ denotes a random variable whose distribution is univariate SN with df fðz; z; o; aþ ¼ ffiffiffiffi f z ffiffiffiffi z F½aðz zþš o o where f( ) is the df of a standard normal variable. Desite the resence of an additional arameter, the SN distribution resembles the normal one in several ways, formalized through the following roerties: Inclusion: The normal distribution is an SN distribution with shae arameter equal to zero: z SN ðz; O; 0Þ3z N ðz; OÞ Greater norms of a imly greater differences between the density of the multivariate SN ðz; O; aþ and the density of the multivariate N ðz; OÞ: Linearity: The class of SN distributions is closed with resect to linear transformations. If A is a k matrix and b R k ; then

5 A Multivariate Skew-Garch Model 37 z SN ðz; O; aþ ) Az þ b SN k Az þ b; AOA T ; ā AOA T 1AOa ā ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ a T O O 1 A T AOA T 1A Oa It follows that the SN class is closed under marginalization: subvectors of SN vectors are SN, too. In articular, each comonent of an SN random vector is univariate SN. Invariance: The matrix of squares and roducts ðz zþðz zþ T has a Wishart distribution: ðz zþðz zþ T WðO; 1Þ Notice that the distribution of ðz zþðz zþ T does not deend on the shae arameter. In the univariate case, it means that Z SN 1 ðz; o; aþ imlies ðz zþ =o w 1 : All moments of the SN distribution exist and are finite. They have a simle analytical form. However, moments of the SN distribution differ from the normal ones in several ways: Location and scale arameters equal mean and variance only if the shae arameter vector a equals zero. Moments are more conveniently reresented through the arameter d ¼ Oa= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ a T Oa: Tails of the SN distribution are always heavier than the normal ones, when the shae arameter vector a differs from zero. Table 1 reorts the exectation, variance, skewness and kurtosis of the SN distribution, in the multivariate and univariate cases. Multivariate skewness and kurtosis are evaluated through Mardia s indices (Mardia, 1970). Notice that in the univariate case Mardia s index of kurtosis equals the fourth moment of the standardized random variable. On the other hand, in the univariate case, Mardia s index of skewness equals the square of the third moment of the standardized random variable. In the following sections, when dealing with skewness, we shall refer to Mardia s index in the multivariate case and to the third moment of the standardized random variable in the univariate case.

6 38 GIOVANNI DE LUCA ET AL. Table 1. Exectation Variance Skewness Kurtosis Exectation, Variance, Skewness and Kurtosis of the SN Distribution. z SN ðz; O; aþ z þ d Z SN 1 ðz; o; aþ zþ d O ddt o d ð4 Þ d T O 1 3 d ffiffiffi d d T O 1 ð4 Þ d d T O 1 d 8ð 3Þ d T O 1 8ð 3Þ d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o d d o d! 3 3. THE SGARCH MODEL Let Y t be the leading (US) market return at time t. A simle GARCH(,q) model is assumed. Then we can write Y t ¼ Z t t Z t ¼ d 0 þ Pq d i ðz ti ti Þ þ Pqþ d j Z tþqj i¼1 j¼qþ1 (1) where {e t }i.i.d. N(0,1). f t g is the innovation (or shock) of the US market and is hyothesized to be Gaussian. In order to ensure the ositivity of Z t ;d 0 has to be ositive and the remaining arameters in (1) non-negative. After denoting Z t1 ¼½z t1 ; z t ;...Š; it turns out that Y t jy t1 Nð0; Z t Þ Let x t be the 1 return vector of the Euroean markets at time t. We assume that returns at time t deend both on an endogenous (local) shock and an exogenous (global) shock. The endogenous shocks do have relationshis with each other (common movements are usually observed in neighboring markets). The 1 local shock vector is denoted z t. The exogenous or global shock is an event that has an influence across more markets. For the Euroean markets we identify the global shock as the innovation of the US market one-day before, that is t1 : The lag is due to the mentioned nonsynchronicity of the markets. The function fð t1 Þ; secified below, describes the relationshi between the return vector x t and t1 :

7 A Multivariate Skew-Garch Model 39 The local and the global shocks are assumed to be indeendent and to have a joint ð þ 1Þ dimensional normal distribution,!!! t T N z þ1 ; () t 0 0 C where c is a correlation matrix with generic off-diagonal entry r ij. If the hyothesis of Gaussianity is far from true for returns, it aears to be consistent for shocks, even if some authors roose more general distributions (e.g. Bollerslev, 1987). Moreover, we assume that the variances have a multivariate GARCH structure. We can then write x t ¼ D t ½fð t1 Þþz t Š (3) where fð t1 Þ ¼ g þ b t1 gj t1 j (4) 0 s 1t s. t.. D t ¼ C A 0 0 s t (5) s kt ¼ o 0k þ Xq X qþ o ik s k;ti z k;ti þ o jk s k;tþqj þ o qþþ1;kz t1 (6) i¼1 j¼qþ1 The last term takes into account the ossible volatility sillover from the US to the Euroean markets. If at time t 1 the US stock exchange is closed, then Z t1 ¼ Z t The ositivity of s kt is ensured by the usual constraints on the arameters. Assumtions ( 6) comound the SGARCH model. The function fð t1 Þ models the effect of the exogenous shock t1 on the vector x t and {z t }isa sequence of serially indeendent random vectors. The arameter vectors b and g are constrained to be non-negative. Moreover, b g 0: The former describes the direct effect of the ast US innovations on x t, the latter the feedback effect. Volatility feedback theory (Cambell & Hentschel, 199) imlies that news increases volatility, which in turn lowers returns. Hence the direct effect of good (bad) news is mitigated (strengthened) by the

8 40 GIOVANNI DE LUCA ET AL. feedback effect. A oint in favor of the SGARCH model is the formalization of the two effects. The conditional distribution of the return vector is x t ji t1 N ðd t fð t1 Þ; D t CD t Þ where I t1 denotes the information at time t 1: The SGARCH model does not involve a conditional null mean vector. Instead, the mean does deend on the volatility. Moreover, returns are more reactive to bad news than good news. In ¼ D t ðb gþ t1 t1 t ¼ D t ðb þ gþ t1 t1 4. CONDITIONAL DISTRIBUTIONS AND RELATED MOMENTS We are interested in the -variate distribution of x t conditional on D t and on news from the US market, that is on the sign of Y t1 : Theorem 1. Under the SGARCH model s assumtions, the following distributions hold: Good news! D 1 t x t jy t1 40 SN g ; O þ ; a þ where O þ ¼ C þ d þ d T þ ; a C 1 d þ þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and d þ ¼ b g 1 þ d T þ C1 d þ Bad news! D 1 t x t jy t1 o0 SN g ; O ; a where

9 A Multivariate Skew-Garch Model 41 O ¼ C þ d d T ; a C 1 d ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and d ¼ b þ g 1 þ d T C1 d The roof is given in the aendix. Alying the results of linear transformations of a multivariate SN distribution, we can obtain the conditional distributions of returns ðx t jd t ; Y t1 40Þ and ðx t jd t ; Y t1 o0þ: Their formal roerties are found to be consistent with emirical findings. In more detail: Exectation: On average, good (bad) news determines ositive (negative) returns. The effect of both kinds of news is equal in absolute value. Moreover, it is roortional to the direct effect (modeled by the arameter b) mitigated or amlified by the volatility structure (modeled by the diagonal matrix D t ), Eðx t jd t ; Y t1 40Þ ¼ D t b Eðx t jd t ; Y t1 o0þ ¼ D t b This result could be interreted as the resence of an arbitrage oortunity in the market with imlications regarding market efficiency. Actually, our analysis is carried out using returns comuted from two close rices (closeto-close). During the time from the closing to the oening of a Euroean exchange, there is the closing of the US exchange and its effect is reflected mainly in the oen rices of the Euroean exchanges. If we used oen-toclose returns, this aarent arbitrage oortunity would disaear. Variance: Variance is higher (lower) in the resence of bad (good) news. More recisely, the elements on the main diagonal of the covariance matrix are greater (smaller) when revious day s US market returns were negative (ositive) Vx ð t jd t ; Y t1 40Þ ¼ D t C þ d þd T þ D t Vðx t jd t ; Y t1 o0þ ¼ D t C þ d d T D t Skewness: Symmetry of conditional returns would imly that news from the US are irrelevant. In this framework, univariate skewness is negative (ositive) in resence of bad (good) news and higher (smaller) in absolute value. On the other hand, multivariate skewness is always ositive but its

10 4 GIOVANNI DE LUCA ET AL. level cannot be related to the kind of news. Skewness of x t when Y t1 40 can be either lower or higher than skewness of x t when Y t1 o0; deending on the arameters. The two indices are Sðx t jd t ; Y t1 40Þ ¼ 4 ð Þ d T þ C1 d þ þ ð Þd T þ C1 d þ Sðx t jd t ; Y t1 o0þ ¼ 4 ð Þ d T C1 d þ ð Þd T C1 d Kurtosis: In the SGARCH model, relevant news (good or bad) always lead to letokurtotic returns. Again, there is no relationshi between the kind of news (good or bad) and multivariate kurtosis (high or low). However, SGARCH models imly that a higher kurtosis is related to a higher skewness. The two indices are Kðx t jd t ; Y t1 40Þ ¼ 8ð 3Þ Kðx t jd t ; Y t1 o0þ ¼ 8ð 3Þ d T þ C1 d þ þ ð Þd T þ C1 d þ d T C1 d þ ð Þd T C1 d Correlation: Let r + (r ) be the 1 vector whose ith comonent r i+ (r i ) is the correlation coefficient between the ith Euroean return X it and the revious day s US return Y t1 conditionally on good (bad) news, that is Y t1 40; ðy t1 o0þ; and volatility D t. The correlation of X it ; Y t1 jd t ; Y t1 40 (X it ; Y t jd t ; Y t1 o0) is the same under the multivariate and univariate SGARCH model (the former being a multivariate generalization of the latter). Hence, we can recall a roerty of the univariate SGARCH model (De Luca & Loerfido, 004) and write ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi b i g i b r iþ ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i þ g i ; r i ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð Þ b i g i þ ð Þ b i þ g i for i ¼ 1;...; : Imlications of the above results are twofold. In the first lace, r + and r are functions of d þ ¼ b g and d ¼ b þ g; resectively. In the second lace, a little algebra leads to the inequalities 0 r þ r 1 ; where 1 is a -dimensional vector of ones. It follows that bad news strengthen the association between Euroean returns and revious day s US returns.! 3! 3!!

11 A Multivariate Skew-Garch Model 43 The two -variate distributions in Theorem 1 have different characteristics according to the news coming from the US market. As a result, the conditioning aears to be relevant. It is interesting to consider some secial cases. Firstly, if the arameter vector g is zero, then d þ ¼ d ¼ b and O þ ¼ O ¼ O: The two conditional distributions in Theorem 1 differ for the shae arameter which has the same absolute value but a different sign, that is where t x t jy t1 40 SN ð0; O; aþ D 1 O ¼ C þ bb T ; C 1 b a ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ b T C 1 b and D 1 t x t jy t1 o0 SN ð0; O; aþ. In this case, no feedback effect exists. However, it still makes sense to condition on the tye of news and to introduce SN distributions. Secondly, if the arameter vector b is zero (imlying that also g is zero), there is no evidence of any (direct or feedback) effect of the US news on Euroean returns. The two markets are indeendent. The SN distributions in Theorem 1 turn out to have a zero shae arameter which shrinks them to the same normal distribution. As a result, the multivariate skewness and kurtosis indices also shrink to zero. Finally, if the dimensionality arameter equals one, that is if we move from a multivariate framework to a univariate ersective, the multivariate SGARCH model equals the univariate SGARCH model (De Luca & Loerfido, 004). In this case, denoting by X t a Euroean return at time t, we have X t s t ¼ g þ b t1 gj t1 jþ z t where b and g are now scalars and z t is a unidimensional random variable. The main features of the model in a univariate context (arameters of the distribution of X t =s t given the sign of Y t1 and moments of X t given s t and the sign of Y t1 ) are summarized in Table.

12 44 GIOVANNI DE LUCA ET AL. Table. Features of the Univariate SGARCH Model. Location Scale Good News ðy t140þ Bad News ðy t1o0þ Overall r ffiffiffi g q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þðb gþ 1 þðb þ gþ r ffiffiffi g Shae b g q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Exectation Variance Skewness 4 Kurtosis ð 3 Þ s t 1 þ Correlation with Y t1 ðb g Þ q ðb þ gþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þðb þ gþ q 1 þðb gþ r ffiffiffi r ffiffiffi bst bst ðb g Þ 1:5 ðb gþ þð Þðb gþ ðb g Þ þ ð Þðb g Þ ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð Þðb g Þ 4 ð 3 Þ s t 1 þ q ðb þ g Þ 0 s t 1 þ g þ b g g 3b 4g ffiffiffi ðb þ gþ 1:5 þð Þðb þ gþ þð Þg þ b 1:5 ðb þ g Þ þ ð Þðb þ g Þ ffiffiffiffiffiffiffiffiffiffiffi ðb þ g Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð Þðb þ g Þ 3 þ q 8g ð 3 Þ þ ð Þg þ b b ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð Þg þ b

13 A Multivariate Skew-Garch Model UNCONDITIONAL DISTRIBUTIONS We refer to the exression unconditional distribution to indicate the distribution of the vector x t unconditionally on Y t1 : However, it is still conditional on its own ast history. Exectation: The exected value of the returns is the null vector, consistently with emirical findings and economic theory, Eðx t jd t Þ ¼ 0 Variance: The variance of x t can be seen as decomosed into the sum of two comonents, an endogenous comonent, determined by the market internal structure, and an exogenous comonent, determined by news from the US market. The latter can be further reresented as the sum of a comonent deending on the direct effect and another comonent deending on the feedback effect: Vðx t jd t Þ ¼ D t C þ bb T þ ggt D t Skewness: In the multivariate SGARCH model, the feedback effect determines the asymmetric behavior of returns. More formally, multivariate skewness, as measured by Mardia s index, is zero if and only if the feedback arameter is the zero vector: g ¼ 0 ) Sðx t jd t Þ ¼ 0; ga0 ) Sðx t jd t Þ40 The roof of this result is in the aendix. Kurtosis: In the univariate SGARCH model ( ¼ 1), we can show the resence of kurtosis. The same result holds for a linear combination of returns following the model. In the multivariate case we conjecture the existence of multivariate kurtosis as measured by Mardia s index. Correlation: Let r be the 1 vector whose ith comonent r i is the correlation coefficient between the ith Euroean return X it and the revious day s US return Y t1 ; conditionally on volatility D t. It easily follows from the definition of x t as a function of the US shocks and ordinary roerties of covariance that ffiffiffi b r i ¼ i q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; i ¼ 1;... ; þ ð Þg i þ b i as reorted in the last row of Table. This result imlies that 0 r 1 ; where 1 is a -dimensional vector of ones. It also imlies that r i is an

14 46 GIOVANNI DE LUCA ET AL. increasing function of b i and a decreasing function of g i. It follows that association between Euroean returns and revious day s US returns is directly related to the vector of direct effects b and inversely related to the vector of feedback effects g. The same statistics are reorted for the univariate case in the last column of Table. The comlete unconditional distribution could be obtained by simulation. 6. ANALYSIS OF SOME FINANCIAL MARKETS We focus on a univariate and multivariate analysis of three Euroean financial markets: the Dutch, the Swiss and the Italian market. They are small caitalized markets comared to the US market. The differences in sizes between the US market and the three Euroean exchanges are evident. The weight of the caitalization of the US market on the world caitalization is over 40%, while the weights of each of the latter does not exceed 3%. The close-to-close log-returns of reresentative market indexes have been observed in the eriod 18/01/1995 0/05/003. The three markets (Dutch, Swiss and Italian) are reresented by the AEX, SMI and MIB indexes, resectively. The returns of the US market are reresented by the Standard & Poor 500 (S&P), the most oular market index for the New York Stock Exchange Univariate Analysis The analysis roceeds by looking at the most imortant features of the returns in absence and in resence of the conditioning on the erformance of the one-day lagged US market. Table 3 reorts the most salient statistics for the indexes when no conditioning is taken into account. They describe the tyical features of financial returns. The average returns are very close to zero and a certain degree of negative skewness is aarent. Fat tails in the distribution are revealed by the kurtosis indices. Finally, the correlations with the one-day lagged S&P returns are reorted. A dynamic linkage from the US market to the Euroean markets does exist. Then, we divide the entire samles into two subsamles, according to the sign of the one-day lagged S&P return. In order to take into account the

15 A Multivariate Skew-Garch Model 47 Table 3. Descritive Statistics for the Three Stock Indexes. Average Standard deviation Skewness Kurtosis Correlation AEX SMI MIB Table 4. Descritive Statistics for the Three Stock Indexes According to the Sign of One-Day Lagged S&P Returns. Average Standard deviation Skewness Kurtosis Correlation AEX S&P S&Po SMI S&P S&Po MIB S&P S&Po differences in closing days of the stock exchanges, some hyotheses have to be made. We assume that if the Euroean exchange is oen at time t and the US exchange is closed at time t 1; then t1 is set to zero (there is no information from the US market). If the Euroean exchange is closed at time t, the US exchange information at time t 1 is useless; the next Euroean exchange return (at time t þ 1) is related to e t. We comute the same statistics as above in this setting, summarized in Table 4. The resulting statistics are very interesting. They show a different behavior of the Euroean market indexes according to the sign of the last trading day in the American stock exchange. In the three markets the average return is ositive (negative) when the one-day lagged return of the S&P is ositive (negative). The standard deviation of the returns of the Euroean market indexes is always greater in resence of a negative sign coming from the US market. The skewness coefficient is negative and stronger when the American stock exchange return is negative; it is ositive in the oosite case. Finally, the relationshi between resent Euroean returns and ast US returns is clearly stronger

16 48 GIOVANNI DE LUCA ET AL. Table 5. Maximum Likelihood Estimates (Standard Error) of the Univariate SGARCH Models for the Three Indexes Returns. ^o 0 ^o 1 ^o ^o 3 ^b ^g AEX (0.006) (0.016) (0.019) (0.011) (0.0) (0.033) SMI (0.011) (0.019) (0.01) (0.009) (0.04) (0.034) MIB (0.0) (0.018) (0.05) (0.016) (0.01) (0.033) when the S&P return is negative. All these emirical findings match the theoretical features of the univariate SGARCH model (De Luca & Loerfido, 004). Only the observed kurtosis is always smaller in resence of a negative US return, which contradicts the model. The autocorrelation functions of the squared Euroean returns steadily decrease, hinting that ¼ q ¼ 1 in the variance equation. Table 5 contains the maximum likelihood estimates of the arameters of the three univariate models. According to the ratios between estimate and standard error, all the arameters are significant but the volatility sillover coefficients. This involves the advantage of a model including the arameters b and g. The highest value of the quantity bg refers to the AEX returns involving the major distance between the behaviors of the index conditionally on the signals coming from the US market. On the whole, the economic interretation is straightforward: it is relevant to distinguish between bad and good news from the US market. The inclusion of the effect of exogenous news can significantly imrove the redictive erformance. In order to check the stability of the coefficients (articularly b and g) we carried out recursive estimates. The first samle is comosed of the observations from 1 to For each subsequent samle we added an observation. Fig. 1 shows the dynamics of the two arameters. The differences between b and g are aroximately constant for AEX and MIB. For SMI, the distance tends to be slightly more variable. In order to evaluate the erformance of the model in out-of-samle forecasting, we comuted one-ste-ahead forecasts of the volatility, s tjt1 ; for t ¼ 1601;... ; using the SGARCH model. We comared them with benchmark forecasts obtained from a standard GARCH(1,1) model. Following Pagan and Schwert (1990), we ran a regression of log squared returns versus log forecasted volatility, and then comuted the F-test for the

17 A Multivariate Skew-Garch Model Fig. 1. Recursive Estimates of b (To Curve) and g (Bottom Curve) for AEX (Left), SMI (Middle) and MIB (Right). hyothesis of a null intercet and a unit sloe. The F-statistics for SGARCH and GARCH, resectively, are and 69.4 (AEX), 59.5 and 69.4 (SMI), and 7.65 (MIB) and thus favor the SGARCH model. Moreover, we comuted how many times the sign of the return had been correctly redicted with the SGARCH model. The ercentages of correct negative signs are (AEX), (SMI) and (MIB). For ositive sign, they are (AEX), (SMI) and (MIB). 6.. Multivariate Analysis When a multivariate model is considered, the focus on daily data oses some roblems. In fact there are unavoidably some missing values, due to the different holidays of each country. As an examle, on the 1st of May, there is the Labor Holiday in Italy and Switzerland and the stock exchanges do not oerate. But on the same day the stock exchange in the Netherlands is oen. Deleting the day imlies missing a datum. In order to overcome this drawback, we assumed that the Italian and Swiss variance in that day, s 1stMay ; has the same value as on the last oening day of the stock exchange (s 30thAril if not Saturday or Sunday). The next day variance, s ndmay if not Saturday or Sunday, was comuted according to a GARCH (1,1) model without considering the ast holiday. In general, for the k-th market index,

18 50 GIOVANNI DE LUCA ET AL. 8 < o 0k þ o 1k s k;t1 z k;t1 þ ok s s kt ¼ k;t1 þ o 3kZ t1 ; t 1 ¼ oen : o 0k þ o 1k s k;t z k;t þ ok s k;t1 þ o 3kZ t1 ; t 1 ¼ close Maximum likelihood estimation has been erformed using a Gauss code written by the authors. The algorithm converged after a few iterations. The arameters are again all significant, but the sillover arameters. Their values, reorted in Table 6, are not very far from the corresonding estimates in the univariate context. In addition we obtain the estimates of the correlation coefficients, indicated by r ij. The diagnostic of the SGARCH model can be based on the squared norms {S t } of the residuals {r t }, defined as follows: Table 6. Maximum Likelihood Estimates and Standard Errors of the Multivariate SGARCH Model. The Subscrit Letters have the following Meanings: A ¼ AEX; S ¼ SMI; M ¼ MIB. Parameter Estimate Standard Error o 0A o 1A o A o 3A o 0S o 1S o S o 3S o 0M o 1M o M o 3M r AM r AS r MS b A g A b S g S b M g M

19 A Multivariate Skew-Garch Model 51 8 >< S t ¼ r T t r t; r t ¼ >: ^O 1= þ ^O 1= ^D 1 t ^D 1 t qffiffi x t ^g ; Y t1 40 qffiffi x t ^g ; Y t1 o0 If the model is correctly secified, the following results hold: 1. The squared norms {S t } are i.i.d. and S t w :. If n is the number of observed returns, then ffiffi! n 1 X n S t a Nð0; 1Þ (7) n t¼1 The roof of this result is given in the aendix. We comuted the statistic in (7). Its value is 0:031; such that the hyothesis of skew-normality of the conditional distributions cannot be rejected. 7. CONCLUSIONS The multivariate Skew-GARCH model is a generalization of the GARCH model aimed at describing the behavior of a vector of deendent financial returns when an exogenous shock coming from a leading financial market is taken into account. We analyzed returns from three Euroean markets, while the leading market was identified as the US market. It turned out to be significant to consider the effects of the exogenous shock. The distributions of the Euroean returns show different features according to the tye of news arriving from the leading market. When the above assumtions are not consistent, the estimation ste reveals the drawback. In this case, some arameters of the model are zero and the multivariate Skew-GARCH model shrinks to the simle multivariate GARCH model with constant correlation coefficients. A future extension of our roosed model would be to relace the multivariate skew-normal distribution by a multivariate skew-ellitical distribution, see the book edited by Genton (004). For examle, a multivariate skew-t distribution would add further flexibility by introducing an exlicit arameter controlling tail behavior. REFERENCES Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, 1,

20 5 GIOVANNI DE LUCA ET AL. Azzalini, A., & Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika, 83, Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, Bollerslev, T. (1987). A conditional heteroskedastic time series model for seculative rices and rates of return. The Review of Economics and Statistics, 69, Bollerslev, T. (1990). Modelling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH aroach. The Review of Economics and Statistics, 7, Bollerslev, T., Engle, R., & Wooldridge, J. (1988). A caital asset ricing model with timevarying covariances. Journal of Political Economy, 96, Cambell, J. Y., & Hentschel, L. (199). No news is a good news: An asymmetric model of changing volatility in stock returns. Journal of Financial Economics, 31, Dalla Valle, A. (004). The skew-normal distribution. In: M. G. Genton (Ed.), Skew-ellitical distributions and their alications: A journey beyond normality (. 3 4). Boca Raton, FL: Chaman & Hall/CRC. De Luca, G., & Loerfido, N. (004). A Skew-in-Mean GARCH model. In: M. G. Genton (Ed.), Skew-ellitical distributions and their alications: A journey beyond normality (. 05 ). Boca Raton, FL: Chaman & Hall/CRC. Engle, R. F. (198). Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation. Econometrica, 50, Engle, R. F. (00). Dynamic conditional correlation. A simle class of multivariate GARCH models. Journal of Business and Economic Statistics, 0, Eun, C. S., & Shim, S. (1989). International transmission of stock markets movements. Journal of Financial and Quantitative Analysis, 4, Genton, M. G. (Ed.) (004). Skew-ellitical distributions and their alications: A journey beyond normality. Boca Raton: Chaman & Hall/CRC. Genton, M. G., & Loerfido, N. (005). Generalized skew-ellitical distributions and their quadratic forms. Annals of the Institute of Statistical Mathematics, 57, Karolyi, G. A. (1995). A multivariate GARCH model of international transmissions of stock returns and volatility: The case of United States and Canada. Journal of Business and Economic Statistics, 13, Koch, P. D., & Koch, T. W. (1991). Evolution in dynamic linkages across daily national stock indexes. Journal of International Money and Finance, 10, Lin, W., Engle, R., & Ito, T. (1994). Do bulls and bears move across borders? International transmission of stock return and volatility. The Review of Financial Studies, 7, Loerfido, N. (004). Generalized skew-normal distributions. In: M. G. Genton (Ed.), Skewellitical distributions and their alications: A journey beyond normality ( ). Boca Raton: Chaman & Hall/CRC, FL. Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with alications. Biometrika, 57, Nelson, D. (1991). Conditional heteroskedasticity in asset returns. Econometrica, 59, Pagan, A. R., & Schwert, G. W. (1990). Alternative models for conditional stock volatility. Journal of Econometrics, 45,

21 A Multivariate Skew-Garch Model 53 APPENDIX Proof of Theorem 1. We shall rove the theorem for Y t1 40 only. The roof for Y t1 o0 is similar. By assumtion, the joint distribution of random shocks is!!! t T N z þ1 ; t 0 0 C By definition, O þ ¼ C þ d þ d T þ ; so that!!!! ffiffiffiffiffiffiffiffi t1 0 1 d N g = þ dþ t1 þ z þ1 ffiffiffiffiffiffiffiffi T þ ; t g = d þ O þ ffiffiffiffiffiffiffiffi The conditional distribution of g = þ dþ t1 þ z t given t1 40 is multivariate SN (Dalla Valle, 004): 0 1! B O 1 þ g þ d þ t1 þ z t j t1 40 SN g ; O þ ; d þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia 1 d T þ O1 t d þ By definition, d þ ¼ b g; so that standard roerties of absolute values lead to 0 1! g þ b t1 gj t1 jþz t1 B O 1 þ t 40 SN g ; O þ ; d þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia 1 d T þ O1 þ d þ By definition, ffiffiffiffiffiffiffiffi US returns and Euroean returns are Y t1 ¼ Z t1 t1 and x t ¼ D t ðg = þ bt1 gj t1 jþz t Þ: Hence, 0 1 D 1 t x t jy t1 40 B SN g O 1 þ ; O þ ; d þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia 1 d T þ O1 þ d þ Aly now the Sherman-Morrison formula to the matrix O þ ¼ C þ d þ d T þ : O 1 þ ¼ C1 C1 d þ d T þ C1 1 þ d T þ C1 d þ A little algebra leads to the following equations:

22 54 GIOVANNI DE LUCA ET AL. O 1 þ d þ ¼! C 1 C1 d þ d T þ C1 1 þ d T d þ þ C1 d þ! ¼ C 1 d þ 1 dt þ C1 d þ 1 þ d T þ C1 d þ C 1 d þ ¼ 1 þ d T þ C1 d þ 1 d T þ O1 þ d þ ¼ 1 dt þ C1 d þ 1 1 þ d T ¼ þ C1 d þ 1 þ d T þ C1 d þ Recall now the definition of the vector a + : O 1 þ d þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ C1 d þ = 1 þ d T þ C1 d þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 d T þ O1 þ d þ 1= 1 þ d T þ C1 d þ ¼ C 1 d þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a þ 1 þ d T þ C1 d þ We can then write and the roof is comlete.&! D 1 t x t jy t1 40 SN g ; O þ ; a þ Proof of the ositivity of the unconditional skewness. We shall rove the theorem for z t ¼ D 1 t x t only: the roof for x t is similar. When g ¼ 0 the distribution of z t is multivariate normal, so that it suffices to show that ga0 imlies that Sðz t Þ40: Let the vectors w t,u t,g 1,g and the matrix G be defined as follows: Vðz t Þ ¼ S; w t ðz t jy t1 40Þ; u t ðz t jy t1 o0þ; G ¼ S 1 S 1 S 1 g 1 ¼ Eðw t w t w t Þ g ¼ Eðu t u t u t Þ The distribution of z t is the mixture, with equal weights, of the distributions of w t and u t. Ordinary roerties of mixtures lead to

23 A Multivariate Skew-Garch Model 55 Eðz t z t z t Þ ¼ 1 E ð w t w t w t Þþ 1 E ð u t u t u t Þ ¼ 1 g 1 þ g Hence the multivariate skewness of the vector z t can be reresented as follows: Sz ð t Þ ¼ 1 4 g TG 1 þ g g1 þ g The distribution of w t equals that of u t only when g ¼ 0: By assumtion ga0; so that g 1 þ g a0 ) Sz ð t Þ ¼ 1 4 g TG 1 þ g g1 þ g 40 The last inequality follows from S being ositive definite and from roerties of the Kronecker roduct. The roof is then comlete. & Proof of the asymtotic distribution (7). Let the random variables S t and the random vectors f r t g be defined as follows: 8 r ffiffiffi! O 1= þ D 1 t x t g Y t1 40 >< S t ¼ r T t r t; r t ¼ r ffiffiffi! O 1= D 1 t x t g Y t1 o0 >: From Section 3 we know that the distributions of D 1 t x t jy t1 40 are skew-normal: D 1 t D 1 t x t jy t1 40 SN g D 1 t x t jy t1 o0 SN g! ; O þ ; a þ! ; O ; a Aly now linear roerties of the SN distribution: ð r t jy t1 40Þ SN 0; I ; l þ ; lþ ¼ O 1= þ ð r t jy t1 o0þ SN 0; I ; l ; l ¼ O 1= x t jy t1 o0 and First notice that the vectors f r t g are i.i.d. Moreover, the distribution of r t is the mixture, with equal weights, of two -dimensional SN distributions a þ a

24 56 GIOVANNI DE LUCA ET AL. with zero location arameter and the identity matrix for the scale arameter, that is: fð r t Þ ¼ 1 ðþ ex rt t r t F l T = r t þ F l T þ r t The above density can be reresented as follows: fð r t Þ ¼ f r t ; I o ð rt Þ where f ð; SÞ denotes the density of N ð0; SÞ and wðþ is a function such that 0 wð r t Þ¼1wð r t Þ1: Hence the distribution of r t is generalized SN with the zero location arameter and the identity matrix for the scale arameter (Loerfido, 004; Genton & Loerfido, 005). It follows that the df of even functions gð r t Þ¼gð r t Þ does not deend on wðþ: As an immediate consequence, we have: S t ¼ r T t r t w ) E S t ¼ ; V S t ¼ A standard alication of the Central Limit Theorem leads to ffiffi! n 1 X n S t a Nð0; 1Þ n t¼1 When the samle size is large, the distribution of {r t } aroximates the distribution of f r t g: Hence the squared norms {S t } are aroximately indeendent and identically distributed according to a w distribution. Moreover, ffiffi! n 1 X n S t a Nð0; 1Þ n t¼1 and this comletes the roof.&