R Khalfaoui
email: r.kalfaoui@gmail.com

M Boutahar
email: boutahar@univmed.fr

Portfolio risk evaluation: An approach based on dynamic conditional correlations models and wavelet multi-resolution analysis

Keywords: Dynamic conditional correlations, Value-at-Risk, wavelet decomposition, Stock prices

 is the most preferable according to statistical loss functions under raw data. The results also suggest that wavelet-based models increase predictive performance of financial forecasting in low scales according to number of violations and failure probabilities for VaR models.

Introduction

Measuring market risk is the most interest of financial managers and traders. The most widely used measure of risk managment is the Value-at-Risk (VaR), which was introduced by Jorion (1996). Forecasting VaR is based on the volatility models. The well-known volatility model is the generalized autoregressive conditional heteroskedasticity (GARCH) model of [START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation[END_REF] and [START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF]. The success of this model has subsequently led to a family of univariate and multivariate GARCH models which can capture different behavior in financial returns. The development of this family of models has led to the development of conditional VaR forecasts. The literature on multivariate GARCH models is large and expanding. [START_REF] Engle | Multivariate simultaneous generalized arch[END_REF] defined a general class of multivariate GARCH (MGARCH) models. The popular ones are diagonal VECH model of [START_REF] Bollerslev | A capital asset pricing model with time-varying covariances[END_REF], the BEKK model of [START_REF] Engle | Multivariate simultaneous generalized arch[END_REF]. While, popular, these models have limitations. 1 In particular, diagonal VECH lacks correlation between the variance terms, BEKK can have poorly behaved likelihood function (making estimation difficult, especially for models with more than two variables), and VECH has a large number of free parameters (which makes it impractical for models with more than two variables). To deal with the curse of the dimensionality [START_REF] Engle | Dynamic conditional correlation: A simple class of multivariate generalizd autoregressive conditional heteroskedasticity models[END_REF] proposed the dynamic conditional correlations (DCC) model which generalizes the specification of [START_REF] Bollerslev | Modelling the coherence in short-run nominal exchange rates: A multivariate generalized arch model[END_REF] by assuming a time variation of correlation matrix. [START_REF] Tse | A multivariate generalized autoregressive conditional heteroscedasticity model with time-varying correlations[END_REF] defined a multivariate GARCH model which includes time-varying correlations and yet satisfies the positive-definite condition. [START_REF] Ling | Asymptotic theory for a vector arma-garch model[END_REF], [START_REF] Mcaleer | Structure and asymptotic theory for multivariate asymmetric conditional volatility[END_REF] and Caporin andMcAleer (2009, 2010) proposed another family of multivariate GARCH models which assume constant conditional correlations, and do not suffer from the problem of dimensionality, by comparison with the VECH and BEKK models. The convenient of these models is that modeling conditional variances allows large shocks to one variable to affect the variances of the other variables. Another important topic in the financial econometrics is the asymmetric behavior of conditional variances. The basic idea is that negative shocks have a different impact on the conditional variance evolution than do positive shocks of similar magnitude. [START_REF] Cappiello | Asymmetric dynamics in the correlations of global equity and bond returns[END_REF], [START_REF] Aielli | Consistent estimation of large scale dynamic conditional correlations[END_REF] and [START_REF] Palandri | Sequential conditional correlations: Inference and evaluation[END_REF] extended the DCC model of [START_REF] Engle | Dynamic conditional correlation: A simple class of multivariate generalizd autoregressive conditional heteroskedasticity models[END_REF] to an asymmetric DCC model which is a generalization of the DCC model (authors develop a model capable of allowing for conditional asymmetries not only in volatilities but also in correlations). Authors used several asymmetric versions of the DCC model. In our article, we based on version which allows asymmetry in conditional variance. This phenomenon was raised by [START_REF] Nelson | Conditional heteroskedasticity in asset returns: A new approach[END_REF] in introducing the Exponential GARCH (EGARCH) model, and was also considered by [START_REF] Glosten | On the relation between the expected value and the volatility of the nominal excess return on stocks[END_REF] (GJR-GARCH models), [START_REF] Rabemananjara | Threshold arch models and asymmetries in volatility[END_REF] and [START_REF] Zakoian | Threshold heteroskedastic models[END_REF] (Threshold GARCH models) for the univariate case. More recently, in the area of finance, several empirical studies based on symmetric and asymmetric multivariate GARCH models have been employed. For instance, [START_REF] Cappiello | Asymmetric dynamics in the correlations of global equity and bond returns[END_REF] used the asymmetric generalized-DCC (AG-DCC) specification to investigate asymmetries in conditional variances and correlation dynamics of three groups of countries (Europe, Australia and North America). They provide evidence that equity returns show strong evidence of asymmetries in conditional volatility, while little is found for bond returns. [START_REF] Chiang | Dynamic correlation analysis of financial contagion: Evidence from asian markets[END_REF] applied a DCC model to nine Asian stocks and confirm a contagian effect during the Asian crisis. [START_REF] Ho | Volatility dynamics of the us business cycle: A multivariate asymmetric garch approach[END_REF] applied various multivariate GARCH models to investigate the evidence of asymmetry and time-varying conditional correlations between five sectors of Industrial Production of the United States. They provide also, strong evidence of asymmetric conditional volatility in all sectors and some support of time-varying correlations in various sectoral pairs. [START_REF] Büttner | Determinants of european stock market integration[END_REF] used a bivariate DCC model to extract dynamic conditional correlations between European stock markets. [START_REF] Chang | Modelling conditional correlations in the volatility of asian rubber spot and futures returns[END_REF] employed several multivariate GARCH models (CCC, DCC, VARMA-GARCH and VARMA-AGARCH) to model conditional volatility in the returns of rubber spot and futures in major rubber futures and rubber spot Asian markets, in order to investigate volatility transmissions across these markets. Their results provide presence of volatility spillovers and asymmetric effects of positive and negative return shocks on conditional volatility. [START_REF] Kenourgios | Financial crises and stock market contagion in a multivariate time-varying asymmetric framework[END_REF] applied the AG-DCC model and a multivariate regime-switching Gaussian copula model to capture non-linear correlation dynamics in four emerging equity markets (Brazil, Russia, India and China) and two developed markets (U.S. and U.K.). [START_REF] Arouri | Return and volatility transmission between world oil prices and stock markets of the gcc countries[END_REF] employed a VAR-GARCH approach to investigate the return linkage and volatility transmission between oil and stock markets in Gulf Cooperation Council countries. [START_REF] Lahrech | U.s. and latin american stock market linkages[END_REF] used DCC multivariate GARCH models to examine the dynamic linkage between U.S. and Latin American stock markets. Their results show an increase in the degree of co-movement between Latin American equity markets and U.S. equity ones. In this study, we employ three multivariate GARCH models, such as the CCC model of [START_REF] Bollerslev | Modelling the coherence in short-run nominal exchange rates: A multivariate generalized arch model[END_REF], the DCC model of [START_REF] Engle | Dynamic conditional correlation: A simple class of multivariate generalizd autoregressive conditional heteroskedasticity models[END_REF] and the cDCC model of [START_REF] Aielli | Consistent estimation of large scale dynamic conditional correlations[END_REF]. These models impose a useful structure on the many possible model parameters. However, parameters of the model can easily be estimated and the model can be evaluated and used in straightforward way. Our empirical methodology follows a two-step approach. The first step applies these dynamic conditional correlation models to model conditional volatility in the returns of three developed stock markets (U.K., U.S. and Japan stock markets), in order to examine the evidence of time-varying conditional variances and correlations between stock markets. Moreover, in order to show the asymmetric effects of positive and negative return shocks on conditional volatility the EGARCH and GJR-GARCH models of [START_REF] Nelson | Conditional heteroskedasticity in asset returns: A new approach[END_REF] and [START_REF] Glosten | On the relation between the expected value and the volatility of the nominal excess return on stocks[END_REF], respectively, were employed for modeling univariate conditional volatility. In the second step, we re-examine the dynamic conditional correlation analysis among the three major developed stock markets through a novel approach, wavelet analysis. This technique is a very promising tool as it is possible to capture the time and frequency varying features of co-movement within an unified framework which represents a refinement to previous approaches. This wavelet-based analysis takes account the distinction between the short and long-term investor. From a portfolio diversification view, there exist a kind of investors whose are more interested in the co-movement of stock returns at higher frequencies (lower scales), that is, short-term fluctuations, and also, there exist a kind of investors whose focuse on the relationship at lower frequencies (higher scales), that is, long-term fluctuations. The study of the co-movement of stock market returns, i.e. dynamics of variances and correlations, across scales is crucial for risk assessment of a portfolio. In terms of risk management, a higher co-movement (higher covariances) among assets of a given portfolio implies lower gains. According to investors or traders, evaluating the co-movement of assets is a great importance to best assess the risk of a portfolio. Several applications of wavelet studying the co-movement of stock indices have been recently applied by [START_REF] Sharkasi | Interrelationships among international stock market indices: Europe, asia and the americas[END_REF], [START_REF] Rua | International comovement of stock market returns: A wavelet analysis[END_REF], [START_REF] Rua | Measuring comovement in the time-frequency space[END_REF] and [START_REF] Masih | Systematic risk and time scales: New evidence from an application of wavelet approach to the emerging gulf stock markets[END_REF]. In this paper, we investigate also the selection of the multivariate GARCH models used in the two approaches (with and without wavelet analysis) to identify which model has the best out-of-sample forecasting performance. The assessment of the forecast performance of these models is based on out-of-sample one-day ahead conditional matrix forecasts. However, to measure model performance we used four statistical loss functions. 2The empirical evidence showed that the conditional variances and correlations of U.K., U.S. and Japan stock market returns were dynamic and the three markets were highly correlated. We showed also, that the cDCC model of [START_REF] Aielli | Consistent estimation of large scale dynamic conditional correlations[END_REF] is preferable than the CCC and DCC models based on one-day ahead out-of-sample forecasts. With regard to wavelet-based multivariate conditional volatility approach, our findings suggested a multi-scale behavior of the three markets under study, which decomposes the total spillover into three sub-spillovers and decomposes the market risk measured by VaR into wavelet VaR (WVaR) estimates. In addition, this new approach help traders and investors to reduce risk management on their investing time horizons. The purpose of the paper is four-fold. First, we estimate multivariate conditional volatility for stock market returns using several recent models of multivariate conditional volatility. Second, we investigate the importance of volatility spillovers on the conditional variance across the three developed markets. Third, we focuse on the forecasting performance of the multivariate conditional volatility models under study for the last 250 days of the data set. Forecast comparison is based on four different loss functions including the mean squared error, the mean absolute error, the mean absolute percentage error and the logarithm loss error. Fourth, we propose a wavelet-based multiresolution analysis in order to combine between traditional multivariate conditional volatility models and wavelet decomposition. The combination of wavelet decomposition and dynamic conditional correlation models was introduced to analyze the comovements and volatility spillovers between three developed markets on multiscale framework, based on behavior of investors. Finally, we compare the performance of wavelet-based multivariate conditional volatility model against the traditional one for one-day ahead forecast. The structure of the paper is organized as follows: Section 2 presents the data used for the empirical analysis and the multivariate conditional volatility methodology. Section 3 reports the empirical results under raw data. Wavelet analysis is discussed in section 4. Finally, concluding remarks are stated in section 5.

Methodology and empirical specifications

Data

Our data on stock market prices consist of the S&P 500, FTSE 100 and Nikkei 225 composite indices for U.S., U.K. and Japan. We collect daily data over the period from January 01, 2003 to February 04, 2011. Indices are obtained from DataStream. We use daily data in order to retain a high number of observations to adequately capture the rapidity and intensity of the dynamic interactions between markets. Returns of market i (index i) at time t are computed as r i,t = log (P i,t /P i,t-1 ) × 100, where P i,t and P i,t-1 are the closing prices for day t and t -1, respectively.

Descriptive statistics

The summary statistics of the data are given in Table 2.1. In panel A (Table 2.1), for each return series, the mean value is close to zero. For each return series the standard deviation is larger than the mean value. Each return series displays a small amount of skewness and large amount of kurtosis and the returns are not normally distributed. In panel B (Table 2.1), unconditional correlation coefficients in stock market index returns indicate strong pairwise correlations. The correlation between S&P 500 and FTSE 100 is positive and larger than the correlation between Nikkei 225 and FTSE 100. This could be due to the high trade share between the two markets. The results of the unit root tests for all sample of level prices (in logs) and returns in each market are summarized in Table 2.2. The Augmented Dickey-Fuller (ADF) and Phillips-Perron (PP) tests are used to explore the existence of unit roots in individual series. The results show that all returns are stationary.

Model specifications

The econometric specification used in our study has two components. To model the stock market return we used a vector autoregression (VAR). To model the conditional variance we used a multivariate GARCH model. A VAR of order p, where the order p represents the number of lags, that includes N variables can be written as the following form:

Y t = Φ 0 + p ∑ i=1 Φ i Y t-i + ε t , t = 1, . . . , T (2.1)
where

Y t = (Y 1t , . . . ,Y Nt )
′ is a column of observations on current values of all variables in the model,Φ i is N × N matrix of unknown coefficients, Φ 0 is a column vector of deterministic constant terms, ε t = (ε 1t , . . . , ε Nt ) ′ is a column vector of errors. 3 Our basic VAR will have the three stationary variables, first log differences of FTSE 100, S&P 500 and NIKKEI 225 stock market prices (will be defined in empirical section). We focused on the modelling of multivariate time-varying volatilities. The most widely used model is DCC one of [START_REF] Engle | Dynamic conditional correlation: A simple class of multivariate generalizd autoregressive conditional heteroskedasticity models[END_REF] which captures the dynamic of time-varying conditional correlations, contrary to the benchmark CCC model [START_REF] Bollerslev | Modelling the coherence in short-run nominal exchange rates: A multivariate generalized arch model[END_REF]) which retains the conditional correlation constant. 4The specification of the DCC model is as follows:

r t = µ + p ∑ s=1 Φ s r t-s + ε t , t = 1, . . . , T, ε t | Ω t-1 ∼ N (0, H t ), (2.2)
ε t = (ε UK,t , ε US,t , ε JP,t ) ′ = H 1/2 t z t , z t ∼ N (0, I 3 ), (2.3)
H t = E(ε t ε ′ t | Ω t-1 ), (2.4)
where, r t is a 3 × 1 vector of the stock market index return, ε t is the error term from the mean equations of stock market indices (Equation 2.2), z t is a 3 × 1 vector of i.i.d errors and H t is the conditional covariance matrix. Equation 2.2 can be re-written as follows: ′ be a 3 × 1 vector of stock market returns, such that, r UK , r US and r JP are the returns of FTSE 100, S&P 500 and Nikkei 225 indices, respectively:

  r UK,t r US,t r JP,t   =   µ UK µ US µ JP   + p ∑ s=1   φ s
r t | Ω t-1 ∼ N (0, H t ).
The DCC-GARCH specification of the covariance matrix, H t , can be written as:

H t = D t R t D t (2.5)
where D t = diag h UK,t , h US,t , h JP,t is 3× 3 diagonal matrix of time-varying standard deviation from univariate GARCH models; i.e. h i,t = ω i + α i ε 2 i,t-1 + β i h i,t-1 , i=UK, US, JP, and R t = ρ i j is the time-varying conditional correlation matrix. The estimation procedure of DCC-GARCH model is based on two stages. In the first stage, a univariate GARCH model is estimated. In the second step, the vector of standardized residuals η i,t = r i,t / h i,t is employed to develop the DCC correlation specification as follows: where Q t = (q i j t ) is a symmetric positive define matrix. Q t is assumed to vary according to a GARCH-type process:

R t = diag q -1/2 11,t , . . . , q -1/2 33,t Q t diag q -1/2 11,t , . . . , q -1/2 33,t
Q t = (1 -θ 1 -θ 2 ) Q + θ 1 η t-1 η ′ t-1 + θ 2 Q t-1 (2.7)
The parameters θ 1 and θ 2 are scalar parameters to capture the effects of previous shocks and previous dynamic conditional correlation on current dynamic conditional correlation. The parameters θ 1 and θ 2 are positive and θ 1 + θ 2 < 1. Q is 3 × 3 unconditional variance matrix of standardized residuals η i,t . The correlation estimators of equation 2.7 are of the form: ρ i j,t = q i j,t √ q ii,t q j j,t .

In the DCC model the choice of Q is not obvious as Q t is neither a conditional variance nor correlation. Although

E(η t-1 η ′ t-1
) is inconsistent for the target since the recursion in Q t does not have a martingale representation. 5 Aielli (2008) proposed the corrected Dynamic Conditional Correlation (cDCC) to evaluate the impact of both the lack of consistency and the existence of bias in the estimated parameters of the DCC model of [START_REF] Engle | Dynamic conditional correlation: A simple class of multivariate generalizd autoregressive conditional heteroskedasticity models[END_REF]. He showed that the bias depends on the persistence of the DCC dynamic parameters. 6In order to resolve this issue, [START_REF] Aielli | Consistent estimation of large scale dynamic conditional correlations[END_REF] introduces the cDCC model, which have the same specification as the DCC model of [START_REF] Engle | Dynamic conditional correlation: A simple class of multivariate generalizd autoregressive conditional heteroskedasticity models[END_REF], except of the correlation process Q t is reformulated as follows:

Q t = (1 -θ 1 -θ 2 ) Q + θ 1 η * t-1 η * ′ t-1 + θ 2 Q t-1 (2.8)
where η * t = diag {Q t } 1/2 η t . To investigate the asymmetric properties of stock market returns we introduce the conditional asymmetries in variance. [START_REF] Cappiello | Asymmetric dynamics in the correlations of global equity and bond returns[END_REF] estimate several asymmetric versions of the dynamic conditional correlation models. The version which we use is based on the following specification:

r t = µ + p ∑ s=1 Φ s r t-s + ε t , t = 1, . . . , T, ε t | Ω t-1 ∼ N (0, H t ) H t = D t R t D t where D t = diag h 1,t , h 2,t , h 3,t h it = ω i + α i ε 2 i,t-1 + γ i 1 {εi,t-1<0} ε 2 i,t-1 + β i h i,t-1 for i = UK,US, JP R t = Q * -1 t Q t Q * -1 t where Q * t = diag √ q 11,t , √ q 22,t , √ q 33,t Q t = (1 -θ 1 -θ 2 ) Q + θ 1 η t-1 η ′ t-1 + θ 2 Q t-1 .

Empirical results

In this section we initially employed a vector autoregressive (VAR) model to examine the relationship among stock market returns of the three developped countries. Our model is estimated on set of stationary variable. These variables are returns in stock market prices for the United Kingdom (U.K), United States (U.S) and Japan. Table 3.1 reports the findings of the VAR(8) model (lags is selected by AIC criterion).

Conditional variance and volatility analysis

This subsection presents the empirical results from symmetric and asymmetric multivariate models. In the first step the univariate GARCH(1,1) model for each stock market is fitted. We model the conditional variance as a GARCH(1,1), EGARCH(1,1) and TGARCH(1,1). In the second step the symmetric multivariate GARCH(1,1) models, such as; Constant Conditional Correlation (CCC), the Dynamic Conditional Correlation (DCC), the corrected Dynamic Conditional Correlation (cDCC) and the asymmetric multivariate GARCH(1,1) models, such as; aCCC, aDCC and a-cDCC are fitted.

Symmetric and asymmetric univariate GARCH analysis. Table 3.2 reports the model estimates (panel A) and related diagnostic tests (panel B) for the three models and for the three stock markets. Firstly, panel A of Table 3.2 shows that the parameters in the conditional variance equations are all statistically significant, except for the α's in the EGARCH model for the three markets. The estimated value of β (GARCH effect) is close to unity (in all models the estimated values are greater than 0.90) and is significant at the 1% level for each model. This indicates a high degree The VAR was estimated using eight lags (the lag was selected using the AIC criterion). p-values show the statistical significance of the results.

of volatility persistence in the U.K, U.S and Japan stock market returns. Secondly, results given by the GARCH(1,1) model assume that positive and negative shocks will have the same influence in conditional volatilities forecasts. In order to identify the asymmetry in conditional volatilities, we fitted the univariate EGARCH(1,1) and TGARCH(1,1) models. This asymmetry is generally reffered to as a "leverage" and a "Threshold" effects. The EGARCH(1,1) model captures this "leverage" asymmetry and the TGARCH(1,1) captures this "threshold" asymmetry. 7 The results showed that the asymmetric parameter γ or δ is positive and significantly different from zero at the 5% level in the EGARCH(1,1) model, indicating that U.K, U.S and Japan stock markets exhibit a leverage effect with positive shocks (good news) and significantly different from zero at 1% level in the TGARCH(1,1) model, identifying that the U.K, U.S and Japan stock markets exhibit a threshold effect with positive effect.

To select the adequate model for our data, we compare between the three models using three criteria; such as the Log-likelihood, Akaike Information Criterion (AIC) and Schwarz Information Criterion (SIC). We showed that the asymmetric GARCH model; TGARCH(1,1) has a superior goodness of fit for the data employed. For instance, AIC in the TGARCH(1,1) is lower than in GARCH(1,1) and EGARCH(1,1) models. Panel B of Table 3.2 depicts the Ljung-Box statistics computed on the standardized residuals and squared standardized residuals. We showed that all the Q(12) and Q s (12) values reject the null hypothesis of serial correlation in the standardized residuals and squared standardized residuals at 1% level.

Symmetric and asymmetric dynamic conditional correlation analysis. In the second step the symmetric and asymmetric multivariate GARCH(1,1) models were estimated in order to investigate the constant and time-varying conditional correlation in the stock markets under study. To fit these models, the standardized residuals of the univariate GARCH(1,1) models specification (discussed in the first step of our study) was employed for the estimation of the symmetric models; CCC, DCC, cDCC and asymmetric models; aCCC, aDCC, a-cDCC. The DCC and cDCC esti- 7 The EGARCH (Exponential GARCH) model of [START_REF] Nelson | Conditional heteroskedasticity in asset returns: A new approach[END_REF] is formulated in terms of the logarithm of the conditional variance, as in the EGARCH(1,1) model,

log(h t ) = ω + α |ε t-1 | √ h t-1 + γ ε t-1 √ h t-1 + β log(h t-1 ).
The parametrization in terms of logarithms has the obvious advantage of avoiding non-negativity constraints on the parameters. The TGARCH (Threshold) GARCH model or GJR-GARCH model defined by [START_REF] Glosten | On the relation between the expected value and the volatility of the nominal excess return on stocks[END_REF] and [START_REF] Zakoian | Threshold heteroskedastic models[END_REF] augments the GARCH model by including an additional ARCH term conditional on the sign of the past innovation, mates of the conditional correlations between the volatilities of the FTSE 100, S&P 500 and NIKKEI 225 returns are given in tables 3.5 and 3.6. Results showed that all coefficients are statistically significant at 1% and 5% levels.

h t = ω + αε 2 t-1 + δ 1 {ε t-1 <0} ε 2 t-1 + β h t-1 .
For the CCC model (Table 3.3 and 3.4), the correlation between FTSE 100 and S&P 500, FTSE 100 and NIKKEI 225 and S&P 500 and NIKKEI 225 are each positive and statistically significant at 1% level, and the highest correlation is between FTSE 100 and S&P 500 followed by the correlation between FTSE 100 and NIKKEI 225. This indicates the positive comovement between the three markets. For instance, we found that the co-movements between U.K and U.S are higher than the co-movements between U.S and Japan. However, we remarked that estimated constant conditional correlation coefficients of the sample stock markets do not seem to be informative on dynamic linkages and co-movements between the abovementioned markets. To evaluate the dynamic (pairwise) correlation structure of U.K., U.S. and Japan stock markets, we employed the DCC and cDCC models in trivariate framework.

For the DCC and cDCC models (Tables 3.5 and 3.6), the estimated parameters θ 1 and θ 2 capture the effect of lagged standardized shocks; η t-1 η ′ t-1 , and η * t-1 η * ′ t-1 and lagged dynamic conditional correlations; Q t-1 , on current dynamic conditional correlations, respectively. We remarked that these parameters are significant, and this statistical significance in each market indicates the presence of time-varying stock market correlations. Following the dynamic conditional variance, the three markets under study produce similar behavior and the estimated conditional variance shows a sharp spikes in the period between 2007 and 2008 (The maximum value of estimated conditional variance is in October 21, 2008 for FTSE 100 returns, in October 16, 2008 for the S&P 500 returns and in November 03, 2008 for NIKKEI 225 returns). This period is related to the U.S subprime financial crisis. This financial crisis led U.K, U.S and Japan capital markets to abrupt downturns, dramatically increasing systematic volatility. We conclude that the estimates of the conditional variances based on DCC and cDCC models suggest the presence of volatility spillovers in the U.K, U.S and Japan stock market returns. As shown in Figure 3.1, we remarked also that the dynamic conditional correlations of the three markets under study show considerable variation, and can vary from the constant conditional correlations (ρ UK-US , ρ UK-JP and ρ US-JP ) indicating that the assumption of constant conditional correlation for all shocks to returns is not supported empirically. We stated that during the period [2003][2004][2005][2006], correlations between U.K. and U.S. decreased (58% to 38%), as indicated in Figure 3.1. Whereas, after 2006 we remarked a substantial increase of correlations between U.K. and U.S., it might be due to the Afghanistan, Iraq and Liban wars and the american subprime financial crisis. The diagnostic tests for standardized residuals of CCC model are shown in Table 3.3 and those of DCC and cDCC models are shown in panel B of Tables 3.5 and 3.6. Ljung-Box Q(12) and Q s (12) statistics for the residuals models indicate no serial correlation in either the standardized residuals Q(12) or the squared standardized residuals Q s (12), inferring that the fitted models are appropriate for the data employed. The Constant Conditional Correlation (CCC) model of [START_REF] Bollerslev | Modelling the coherence in short-run nominal exchange rates: A multivariate generalized arch model[END_REF] assumes that the conditional variance for each return, h it , i=UK, US, JP, follows a univariate GARCH process. The specification is as

r t = Φ 0 + p ∑ s=1 Φ s r t-s + ε t , ε t | Ω t-1 ∼ N (0, H t ) ε t = H 1/2 t z t , z t ∼ N (0, I 3 ) H t = E(ε t ε ′ t | Ω t-1 ), where, r t = r UK,t , r US,t , r JP,t ′ is the vector of stock market index returns, ε t = ε UK,t , ε US,t , ε JP,t ′
is the error term from the mean equation of stock market indices (Equation 2.2), z t is a 3 × 1 vector of i.i.d errors and H t is the conditional covariance matrix, which satisfies the following equation:

H t = D t RD t where D t = diag h 1/2 UK,t , h 1/2 US,t , h 1/2
JP,t , and R = ρ i j , for i, j = UK,US, JP, is the unconditional (time-invariant) correlation matrix. The off-diagonal elements of the conditional covariance matrix are given by To investigate the asymmetry in the conditional volatility we fitted the aCCC, aDCC and a-cDCC models. The results are reported in Table 3.7. In the asymmetric case, we focused on the TGARCH(1,1)-based model to model the asymmetric behavior in the conditional volatilities. 8 The asymmetric dynamic conditional correlation estimates are all significant at 1% level (as shown in Table 3.7). Based on the log-likelihood values and AIC criterion reported in the last two columns of Table 3.7, the asymmetric cDCC (a-cDCC) is supperior to the aCCC and aDCC. Moreover, as it can be seen, the log-likelihood values and AIC criteria of aDCC and a-cDCC are nearly equivalent, hence the choice of model needs to be made on other grounds (see the following subsection). The results from Table 3.2 indicate that TGARCH(1,1) model compared to GARCH(1,1) and EGARCH(1,1), achieves a significant improvement in the log-likelihood function, AIC and SIC. As shown in Table 3.2 the TGARCH(1,1) model has the lowest AIC and BIC, and maximize the log-likelihood value. For instance, the AIC values are -6.5556, -6.4577 and -6.1792 for the FTSE 100, S&P 500 and NIKKEI 225 returns, respectively. The log-likelihood values are 6900.5, 6797.5 and 6504.5 for the FTSE 100, S&P 500 and NIKKEI 225 returns, respectively. The apparent superiority of the TGARCH specification compared with EGARCH models is that the former is more robust to large shocks. This intuition is supported by [START_REF] Nelson | Asymptotic filtering theory for univariate arch models[END_REF]. The authors showed that the TGARCH model is consistent estimator of the conditional variance of near diffusion processes.

H i j = h 1/2 it h 1/2 jt ρ i j , i = j * *
r t = Φ 0 + p ∑ s=1 Φ s r t-s + ε t ε t | Ω t-1 ∼ N (0, H t ) ε t =   ε UK,t ε US,t ε JP,t   = H 1/2 t z t , z t ∼ N (0, I 3 ) H t = D t R t D t where D t = diag h UK,t , h US,t , h JP,t h it = ω i + α i ε 2 i,t-1 + β i h i,t-1 , i = UK,US, JP
The dynamic conditional correlation R t = ρ i j t is a time-varying matrix defined as

R t = Q * -1 t Q t Q * -1 t where Q * -1 t = diag √ q 11,t , √ q 22,t , √ q 33,t Q t = (1 -θ 1 -θ 2 ) Q + θ 1 η t-1 η ′ t-1 + θ 2 Q t-1 where η t = D -1 t ε t .
Q is the unconditional covariance matrix of η. The elements of H t are H i j t = h it h jt ρ i j where ρ ii = 1. * * * , * * and * indicate significance at 10%, 5% and 1%, respectively. Q(12) and Q s (12 respectively represent the Ljung-Box Q statistics of order 12 computed on the standardized residuals and squared standardized residuals.

Q t = (1 -θ 1 -θ 2 ) Q + θ 1 η * t-1 η * ′ t-1 + θ 2 Q t-1 η * t = diag {Q t } 1/2 η t

Forecast performance evaluation

To evaluate the volatility forecasting performance of different multivariate GARCH models and to compare between them, further loss functions can be used. The popular statistical loss functions employed to assess the accuracy of competing models in the forecasting of volatilities over daily trading horizon are Mean squared Error (MSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE) and Logarithm Loss Error (LLE). These loss metrics are expressed as follows:

MSE k = 1 M M ∑ m=1 h 2 m,k -σ 2 m 2 (3.1) MAE k = 1 M M ∑ m=1 h 2 m,k -σ 2 m (3.2) MAPE k = 1 M M ∑ m=1 h 2 m,k -σ 2 m σ 2 m (3.3) LLE k = 1 M M ∑ m=1 log(h 2 m,k ) -log( σ 2 m ) 2 (3.4)
where M is the number of forecast data points; The actual volatility σ 2 m is unobservable. The use of squared returns as a proxy for actual volatility is used in the literature (see [START_REF] Kang | Forecasting volatility of crude oil markets[END_REF], [START_REF] Sadorsky | Modeling and forecasting petroleum futures volatility[END_REF] and [START_REF] Wei | Forecasting crude oil market volatility: Further evidence using garch-class models[END_REF]). In our empirical study, the proxies of actual volatilities are as follows: for forecasts based on the unfiltered return series (original stock market return series), we proxied the actual volatility by the squared returns, r 2 m . Table 3.8 presents the forecast accuracy statistics that consist of the mean of four loss functions: MSE, MAE, MAPE and LLE. In terms of these criteria and based on symmetric models, we find that DCC model produces smaller values than those produced by CCC and cDCC. In terms of these statistics and basing in asymmetric models, the a-cDCC model produces the smallest values. Moreover, we remarked that all loss functions values produced by asymmetric models are smaller than those calculated by symmetric ones. In summary, we can conclude that, in our forecasting study, asymmetric models produce more accurate volatility forecasts as compared to those models with symmetric conditional volatility.

MSE k = M -1 ∑ M m=1 ς 2 m,k -ŝ2 m 2 , MAE k = M -1 ∑ M m=1 ς 2 m,k -ŝ2 m , MAPE k = M -1 ∑ M m=1 (ς 2 m,k -ŝ2 m )/ŝ 2 m and LLE k = M -1 ∑ M m=1 log(ς 2 m,k ) -log(ŝ 2 m ) 2
, where M is the number of forecast data points (M = 250); ς 2 m,k denotes the volatility forecast of the portfolio constitued of three stock market indices and generated by model k for day m, it is defined by equation 3.6; ŝ2 m is the actual portfolio volatility on day m.

Application to Value-at-Risk

In this section, we present a methodology allowing us to compute the VaR of diversified portfolio. A portfolio with weight κ i in stock market index i has return as

r p t = κ ′ r t (3.5)
where r t = (r UK,t , r US,t , r JP,t ) ′ denotes the vector of stock market returns of FTSE 100 (r UK,t ), S&P 500 (r US,t ) and Nikkei 225 (r JP,t ). κ ′ = (κ UK , κ US , κ JP ) is the weight vector of the three given indices.

The portfolio variances are as follows

ς 2 t = κ ′ H t κ (3.6)
where H t is the conditional covariance defined in section 2. The multivaraite GARCH-based VaR estimates for q-day holding periods are computed as follows:

VaR k t+l (n, α) = qκ ′ µ t+l + z α √ qς t+l (3.7)
where the argument (q) of VaR is used to denote the time horizon, z α denotes the corresponding quantile which depends on the chosen distribution. For instance, when computing a 99% VaR using normal distribution, z α = 2.33. ς t+l = (κ ′ H t+l κ) 1/2 is the square root of the daily conditional variance forecast of the portfolio generated from model k (CCC, DCC, cDCC), made at time t + l. µ t+l is the conditional mean forecast at time t + l, generated from VAR(8) model.

Backtesting VaR measures

In order to analyse our results implied by different time-varying volatilities models, we use the above estimation to compute one-day ahead out-of-sample VaR. We performed a backtesting analysis based on likelihood ratio test [START_REF] Kupiec | Techniques for verifying the accuracy of risk measurement models[END_REF] and [START_REF] Christoffersen | Evaluating interval forecasts[END_REF]) and dynamic quantile regression test [START_REF] Engle | Caviar: Conditional autoregressive value at risk by regression quantiles[END_REF]).

Unconditional coverage. [START_REF] Kupiec | Techniques for verifying the accuracy of risk measurement models[END_REF] developped the likelihood ratio test, LR uc as follows:

Let n 1 = ∑ n 0 t=1 1 t be the number of days over a n 0 period that the portfolio loss was larger than the VaR estimate, where

1 t+1 = 1 if r t+1 < VaR t+1 | Ω t 0 if r t+1 ≥ VaR t+1 | Ω t n 1 ∼ B(n 0 , π
) is a Binomial variable representing the number of exceedences or exceptions in the sample of length n 0 .

The null hypothesis of the failure probability, π, is tested against the alternative hypothesis that the failure probability differs from π 0 .10 The LR uc statistic is

LR uc = -2 ln (1 -π) n 0 -n 1 π n 1 + 2 ln (1 - n 1 n 0 )( n 1 n 0 )n 1 (3.8)
Asymptotically, LR uc ∼ χ 2 (1).

Conditional coverage. Christoffersen (1998) developped a likelihood statistic to test joint assumption of unconditional coverage and independence of failures. He tests the null hypothesis of independence against the alternative of first-order Markov structure of {1 t+1 },11 with transition matrix

Π = π 00 1 -π 11 1 -π 00 π 11 where π ii = p(1 t+1 = i | 1 t = i), i = 0, 1.
Let n i j be the number of observations of 1 t+1 assuming value i followed by j, for i, j = 0, 1; and π i j = n i j / ∑ j n i j is the corresponding probability. Hence, we have π01 = n 01 /(n 00 + n 01 ) and π11 = n 11 /(n 10 + n 11 ). The statistic of the test is given by

LR cc = LR uc + LR ind ∼ χ 2 (2) (3.9)
where

LR ind = -2 ln (1 -π) n 0 -n 1 π n 1 + 2 ln (1 -π01 ) n 00 πn 01 01 (1 -π11 ) n 10 πn 11 11
Asymptotically, LR ind ∼ χ 2 (1).

Dynamic Quantile. [START_REF] Engle | Caviar: Conditional autoregressive value at risk by regression quantiles[END_REF] proposed the dynamic quantile (DQ) test to correct for the inefficiency in the conditional coverage test of [START_REF] Christoffersen | Evaluating interval forecasts[END_REF]. They defined an indicator function Hit t (α) = 1

r t <VaR t (α)| Ω t-1
-α to test the VaR of long position as follows:

Hit t+1 (α) = 1 -α if r t+1 < VaR t+1 | Ω t -α, else (3.10)
Engle and Manganelli ( 2004) suggested to test jointly the following hypothesis:

H 0 : E(Hit t+1 ) = 0, (1)
Hit t+1 is uncorrelated with variables included in the information set.

(2) These two tests ( 1) and ( 2) can be done using artificial regression Hit t+1 = X t+1 β + u t+1 , where X t+1 is a N × K matrix whose first column is a column ones and remaining columns are additional explanatory variables.12 Under H 0 , [START_REF] Engle | Caviar: Conditional autoregressive value at risk by regression quantiles[END_REF] show that the dynamic quantile test statistic is given by

DQ = β ′ OLS X ′ X βOLS α(1 -α) ∼ χ 2 (7) (3.11)
The statistical adequacy of VaR forecasts is obtained by the previous tests. It is well-known that these tests have limited power in distinguishing among various models for VaR. Using these metrics, we cannot conclude whether an adequate model is more accurate than another one. Loss functions represent an alternative approach that can be used to compare models. [START_REF] Lopez | Methods for evaluating value-at-risk estimates[END_REF] suggested a loss function based on regulatory needs. He proposed measuring the accuracy of VaR forecasts on the basis of distance between observed return, r t , and forecasted VaR t values. He defined a penalty variable as follows:

LF t+1 = 1 + r t+1 -VaR t+1 | Ω t 2 , if r t+1 < VaR t+1 | Ω t 0, if r t+1 ≥ VaR t+1 | Ω t
In our study we computed the loss function LF as a sum of LF t+1 , with t = 0, . . . , n 0 . Table 3.9 depicts the out-of-sample mean daily VaR of the weighted portfolio consisted of the FTSE 100, S&P 500 and Nikkei 225 stock market returns. We remarked that the a-cDCC model yields the lowest average daily VaR estimates at 1%, 2.5%, 5% and 10% levels. The backtesting analysis considers a comparison over the last 250 days (from February 19, 2010 to February 04, 2011) and focuses mainly on exceptions or violations, i.e. the number of times in which the portfolio returns underperform the VaR measure, and LR uc , LR cc and DQ tests statistics. We also calculated the failure probability and statistic loss functions suggested by [START_REF] Lopez | Methods for evaluating value-at-risk estimates[END_REF]. Tables 3.10 and 3.11 present unconditional, conditional and dynamic quantile test statistics. The results showed that symmetric and asymmetric models have been rejected by the LR uc , LR cc and DQ tests. Hence, these models are slow at updating the VaR values when market volatility changes rapidly.

For symmetric models, at 1% significance level, CCC, DCC and cDCC models provide the same number of violations and failure probabilities. The same remark is shown in case of asymmetric models. The results in Tables 3.10 As shown in Tables 3.10 and 3.11, we also reported the LF criterion, which facilitates the selection of optimal VaR model for risk manager. It is clearly that CCC and aCCC models provide the lowest LF values at 1%, 2.5%, 5% and 10% levels. For symmetric volatilty models, the study identifies the CCC model as the best performing model for a firm, followed by cDCC and DCC models. For the asymmetric volatility models, we identify the aCCC model as the best performing model, followed by DCC and cDCC models. However, we remarked that DCC and cDCC models provide approximately the same performance.

Wavelet-based approach

In this section we employed a new approach, called wavelet transform. 13 We used a discrete wavelet transform (DWT), more specifically, we adopt the maximal overlap discrete wavelet transform (MODWT) 14 in a Multi-Resolution Anal- 13 The wavelet transform has two types of transform, namely, continuous wavelet transform (CWT) and discrete wavelet transform (DWT). Since most of the time series have a finite number of observations, the discrete version of wavelet transform is used in finance and economic applications. The wavelet transform decomposes a time series in terms of some elementary functions, called, wavelets;

ψ u,τ (t) = 1 √ τ ψ [(t -u)/τ].
Where 1 √ τ is a normalization factor, u is the translation parameter and τ is the dilation parameter. ψ(t) must fulfil several conditions (see, [START_REF] Gençay | In: An Introduction to Wavelets and Other Filtering Methods in Finance and Economics[END_REF] and [START_REF] Percival | Wavelet methods for time series analysis[END_REF]): it must have zero mean, +∞ -∞ ψ(t)dt = 0, its square integrates to unity, +∞ -∞ ψ 2 (t)dt = 1 and it should also satisfy the admissibility condition, 0

< C ψ = +∞ 0 | ψ(λ )| 2 λ
dλ < +∞ where ψ(λ ) is the Fourier transform of ψ(t), that is ψ(λ ) = +∞ -∞ ψ(t)e -iλ u dt. Following the latter condition we can reconstrut a time series x(t). 14 We use the MODWT because we can align perfectly the details from the decomposition with the original time series. In comparaison with the DWT, no phase shift will result in the MODWT [START_REF] Gençay | In: An Introduction to Wavelets and Other Filtering Methods in Finance and Economics[END_REF]). Fore more information about the MODWT, please refer to [START_REF] Percival | Wavelet methods for time series analysis[END_REF] and [START_REF] Gençay | In: An Introduction to Wavelets and Other Filtering Methods in Finance and Economics[END_REF]. ysis (MRA) framework. 15 A MRA is a linear reconstruction property of the individual wavelet decomposition. The purpose of this discrete decomposition is to derive the MODWT wavelet coefficients, Wj,t , and MODWT scaling coefficients, Ṽj,t , at levels j, j = 1, . . . , J, which are obtained as follows:

Wj,t = L-b ∑ b=0 h jb r t-b mod T and Ṽj,t = L-b ∑ b=0 g jb r t-b mod T (4.1)
where h jb = h jb /2 j/2 are the MODWT filters, and g jb = g jb /2 j/2 are the MODWT scaling filters, r t is the return series.

In our study, we sample the daily stock market returns at different scale (j) as follows: d1 (2-4 days), d2 (4-8 days), d3 (8-16 days), d4 (16-32 days), d5 (32-64 days), d6 (64-128 days), d7 (128-256 days), d8 (256-512 days). 16 We used Daubechies Least Asymetric wavelet transformation of length L = 8 via LA(L) to obtain multiscale decomposition of the return series. The MRA yields an additive decomposition through MODWT given by

r(t) = s J (t) + J ∑ j=1 d j (t) (4.2)
where s J (t) refers to the smooth series and equals ∑ L-b b=0 g Jb ṼJ,t+b mod T . d j (t) ( j = 1, . . . , J) refers to detail series and equals ∑ L-b b=0 h jb Wj,t+b mod T . The wavelet detail, d j (t), captures local fluctuations over the whole period of time series at each scale, while the wavelet smooth, s J (t), gives an approximation of the original series at scale J.

Notes:

The table reports results of the augmented Dickey-Fuller [START_REF] Dickey | Distribution of the estimators for autoregressive time series with unit root[END_REF]) and Kwiatkowski-Phillips-Schmidt-Shin [START_REF] Kwiatkowski | Testing the null hypothesis of stationarity against the alternative of a unit root : How sure are we that economic time series have a unit root[END_REF]) tests. The KPSS test contains a constant and not a time trend. While, the ADF test without constant and trend. The null hypothesis of ADF test is that a time series contains a unit root, I(1) process, whereas the KPSS test has the null hypothesis of stationarity, I(0) process. * indicate the rejection of the unit root null at 1% significance level.

After computing the MODWT crystals (details and smooths) for every stock market return, and from the decomposed series (d1, . . . , d8, s8) we classify the short, medium and long term series as follows: Short term=d1+d2+d3; Medium term=d4+d5+d6; Long term=d7+d8+s8. This choice of time-horizon decomposition is used to classify three types of investors or traders, such as short, medium and long term ones, i.e. to analyze the behavior of investors among different time-horizons. Here the highest frequency component Short term, (d1+d2+d3) represents the short-term variations due to shocks occuring at a time scale of 2 to 16 days, it provides daily and weekly spillovers, the next component Medium term, (d4+d5+d6) represents the mid-term variations at time scale of 32 to 128 days, it defines the monthly and quarterly spillovers, and the third component Long term, (d7+d8+s8) represents the long-term variations of 256 days and more, it provides the annual spillovers. The main advantage of this classification is to decompose the risk and the volatility spillovers into three investment horizons. Therefore, we focus in three sub-spillovers. All market participants, such as regulators, traders and investors, who trade in stock markets (in our study, U.K., U.S. and Japan stock markets) make decisions over different time scales. In fact, due to the different decision-making time scales among investors, the time-varying volatilities and correlations of stock market indices will vary over the different time scales associated with those horizons (investment strategies). In order to analyze the co-movements in returns and volatilities (the analysis is based on new time series: Short term, Medium term and Long term) between the three markets defined before and to investigate the dynamics and spillover effects, we applied a trivariate dynamic conditional correlation (CCC, DCC and cDCC) model. We proceed the same specifications presented in section 2.3 to model the conditional mean and the conditional variances, we fitted a VAR(1)-MGARCH(1,1) to the three new series as follows:

X t = c 0 + AX t-1 + υ t υ t = D t z t (4.3)
where

X t = (FTSE-Short t , S&P500-Short t , NIKKEI-Short t ) ′ , c 0 is a (3× 1) vector of constants, A is (3× 3) coefficient matrix, υ t is (3 × 1)
vector of error term from the mean equations of decomposed series, and z t refers to a (3 × 1) vector of independenly and identically distributed errors. To ensure the stationarity of our reconstructed series (Short term, Medium term and Long term series), we applied the ADF and KPSS tests to our decomposed data. As shown in Table 4.1 all test statistics are statistically significant at 1% level, therefore indicating stationarity. Table 4.2 presents estimates from two types of conditional volatility regressions: (i) a univariate GARCH model, and (ii) a univariate EGARCH model for each market and at each time scale. A GARCH(1,1) model and EGARCH(1,1) model proved adequate for capturing conditional heteroskedasticity. As shown in Table 4.2 the univariate GARCH models display more statistically significant coefficients than a univariate EGARCH ones. In terms of GARCH model, the α and β coefficients (ARCH and GARCH effects, respectively) are positive and statistically significant in all stock markets, indicating highly persisting volatility dynamics in short-term horizon (lower scales/ higher frequencies). The time scale constant conditional correlations among the three stock market returns from the CCC model are summarized in Table 4.3. The results indicate that the highest constant correlation between markets is shown in mediumterm horizon, which corresponds to monthly and quarterly time horizons. Therefore, the co-movements between the three markets are higher in medium horizon than co-movements in short or long horizons. For instance, the correlations between U.K. and U.S. are as follows: 52.7%, 68.9% and 65% for short, medium and long term, respectively, and the correlations between U.S. and Japan are: 13%, 55.7% and 35.1% for short, medium and long term, respectively. Furthermore, we remark that time scale constant correlations between U.K. and U.S. are stronger than the others in all time scales. Parameter estimates for the conditional variance-covariance equations in the DCC and cDCC models are reported in Table 4.4. Coefficients θ 1 and θ 2 reflect the ARCH and GARCH effects, respectively. The estimates parameters of the DCC and cDCC models are statistically significant at 1% level in all time scales, and the sum θ 1 + θ 2 close to unity for the two models and at each scale, implying high persistent volatility in short, medium and long term horizons.

Based on DCC model, the degree of persistence, θ 1 + θ 2 , are 0.997, 0.894 and 0.995 for short, medium and long term, respectively. Based on cDCC model the degree of persistence is 0.997, 0.995 and 0.999 for the short, medium and long terms, respectively. The short run persistence of shocks on dynamic conditional correlations is the greatest for high scales (0.594 and 0.831), while the largest long run persistence of shocks to conditional correlations is 0.997 (0.004+0.993) for low scales. We remark also that the short run persistence shocks increases by scales, while the long run persistence shocks decreases among scales.

The scale spillover effect in volatility provide strong persistence for all components (Short term, Medium term and Long term), this phenomenon result from the tradings of heterogeneous group of investors. In fact, at finest scales (Short term component), market participants are hedging strategists, speculators and market makers. Therefore, they trade the three markets (U.K., U.S. and Japan) simultaneously. [START_REF] Kim | The relationship between financial variables and real economic activity: Evidence from spectral and wavelet analyses[END_REF] showed that speculators and market makers intensively trade to realize a quick profit (or minimize loss) over short time scales. In the intermediate scales (mid-horizon; Medium term component), the main traders are international portfolio managers who mainly follow index tracking trading strategies. [START_REF] Kim | The relationship between financial variables and real economic activity: Evidence from spectral and wavelet analyses[END_REF] showed that trades typically occurs on a weekly to monthly basis, with little attention paid to daily prices. At high scales (Long term component), the main traders are central banks which operate on long-term horizons and often consider long-term economic fundamentals for their strategy.

The forecasting performance of the CCC, DCC and cDCC models at each scale is evaluated by comparing the four statistical loss functions defined in section 3.2. This forecasting study is based on recursive out-of-sample one-day ahead forecast of variance-covariance matrix of dynamic conditional models at each time horizon, i.

( j) = ∑ M m=1 h 2 m,k ( j) -σ 2 m ( j) 2 /M 1/2 , MAE k ( j) = ∑ M m=1 h 2 m,k ( j) -σ 2 m ( j) /M, MAPE k ( j) = ∑ M m=1 h 2 m,k ( j) -σ 2 m ( j) / σ 2 m ( j) /M and LLE k ( j) = ∑ M m=1 log(h 2 m,k ( j)) -log( σ 2 m ( j)) 2 /M.
Where, h 2 m,k ( j) is the wavelet volatility forecast generated by model k for day m and scale j and σ 2 m ( j) is the actual volatility on day m at scale j. The actual wavelet volatility is not observable, therefore we define three proxies of σ 2 m ( j), such as: (Short term) 2 , (Medium term) 2 and (Long term) 2 , for the short-term horizon, medium-term horizon and long-term horizon, respectively. Table 4.5 summarizes the results of the one-day out-of-sample volatility forecast loss functions. We observe that, at low scales (Short term component) and in terms of MAE, MAPE and LLE criteria, the cDCC model provides better volatility forecasts, it has lower loss function values than CCC and DCC models. While, at intermediate scales (Medium term component) and high scales (Long term component), the three dynamic conditional correlation models show similar accuracy in one-day out-of-sample forecasts (the values of RMSE, MAE and MAPE are equals). Tabel 4.6 presents mean WVaR estimates for each of the dynamic conditional correlation models under 99%, 97.5% and 95% confidence levels over the out-of-sample period from February 22, 2010, to February 04, 2011. This basic statistic can be shown as the preliminary understanding of average performance during the forecasting period before the implementation of backtesting tests and market risk loss criteria. As shown in Table 4.6, the WVaR estimates produced by the cDCC model at short-term horizon (low scales or high frequencies) are smaller than those of CCC

( j) = ∑ M m=1 h 2 m,k ( j) -σ 2 m ( j) 2 /M 1/2 , the MAE is defined as; MAE k ( j) = ∑ M m=1 h 2 m,k ( j) -σ 2 m ( j) /M, the MAPE is defined as; MAPE k ( j) = ∑ M m=1 h 2 m,k ( j) -σ 2 m ( j) / σ 2 m ( j) /M and the LLE is defined as; LLE k ( j) = ∑ M m=1 log(h 2 m,k ( j)) -log( σ 2 m ( j)) 2 /M
, where, h 2 m,k ( j) is the wavelet volatility forecast generated by model k for day m and scale j and σ 2 m ( j) is the actual volatility on day m at scale j. and DCC models in 99%, 97.5% and 95% confidence levels. For instance, at 99% confidence level, the average WVaR is 1.7455 for the cDCC model, while those of CCC and DCC are 1.7684 and 1.7641, respectively. We remark also, that WVaR at medium-term horizon (intermediate scales) and long-term horizon (high scales or low frequencies) produced by the three models are equals, for each confidence level, 99%, 97.5% and 95%. However, the mean WVaR decreases from lower scales to higher ones, such that, Short term component (low scales/ high frequencies) provides high WVaR estimates, followed by Medium term (intermediate scales) component and Long term (high scales/ low frequencies) component, at all confidence levels. To accurately comparing the forecasting ability of the mentioned models in terms of WVaR, 17 backtesting diagnostic 17 The multivariate GARCH-based WVaR estimate for one-day ahead forecasts are defined as follows:

VaR k t+m ( j) = κ ′ d t+m + z α ς t+m ( j).

where tests: unconditional coverage of [START_REF] Kupiec | Techniques for verifying the accuracy of risk measurement models[END_REF], conditional coverage of [START_REF] Christoffersen | Evaluating interval forecasts[END_REF], and dynamic quantile of [START_REF] Engle | Caviar: Conditional autoregressive value at risk by regression quantiles[END_REF] are introduced and used for determining the accuracy of wavelet model-based VaR measurments. We used also, the regularity loss function of [START_REF] Lopez | Methods for evaluating value-at-risk estimates[END_REF] based on wavelet market risk (WVaR) defined before. It is defined in wavelet case as follows:

LF t+1 (j) = 1 + d t+1 -VaR t+1 ( j) | Ω t 2 , if d t+1 < VaR t+1 ( j) | Ω t 0, if d t+1 ≥ VaR t+1 ( j) | Ω t (4.4)
The statistical adequacy of the WVaR forecasts computed by the three dynamic conditional correlation models is obtained by the backtesting procedure described before: we will characterize the model as an adequate one for the volatility forecasting, if the null hypothesis cannot rejected. 18 t=1 LF t (j). At 99%, 97.5% and 95% confidence levels, LF(j) statistic indicates that DCC and cDCC models provide strong WVaR performance (smaller LF(j) values) than CCC model. We can note also, that the three dynamic conditional correlation models have the same WVaR performance at medium-term (intermediate scales) and long-term (high scales) horizons for 99%, 97.5% and 95% confidence levels.

Concluding remarks

The empirical analysis in the paper examined the co-movements and spillover effects in the stock market returns of three developed countries: U.K., U.S. and Japan for the period 01 January 2003 to 04 February 2011. Three multivariate conditional correlation volatility models were used, namely CCC model of [START_REF] Bollerslev | Modelling the coherence in short-run nominal exchange rates: A multivariate generalized arch model[END_REF], DCC model of [START_REF] Engle | Dynamic conditional correlation: A simple class of multivariate generalizd autoregressive conditional heteroskedasticity models[END_REF] and cDCC model of [START_REF] Aielli | Consistent estimation of large scale dynamic conditional correlations[END_REF]. Empirical results show that multivariate estimates were significant for all returns in the CCC, DCC and cDCC models. However, these models showed evidence of volatility spillovers and asymmetric effects of negative and positive shocks of equal magnitude on the conditional variances. The statistical significance of DCC and cDCC estimates indicates that the conditional correlations were dynamic. In fact, the variance-covariance analysis produced useful information on the dynamic correlations between the three developed markets, and for each pairwise series, the dynamic conditional correlations vary considerably from their respective constant correlations, implying the absence of any constant correlation between the stock markets under study. The empirical findings showed that the U.K. and U.S. markets were highly correlated since the end of 2007 (the beginning of subprime crisis), followed by the U.K. and Japan markets and U.S. and Japan markets. These results confirm the presence of spillover effects between pairwise stock market returns. The paper also compared one-day ahead conditional volatility forecasts from the dynamic conditional correlation models used in the study, using 250 one-day out-of-sample forecasts, and showed that asymmetric cDCC model is preferred over the other models, according to the four used statistical loss metrics: mean squared error, mean absolute error, mean absolute percentage error and logarithm loss error. The paper also combined the wavelet analysis and multivariate conditional volatility models to analyse the comovements and volatility spillover effects in a multi-scale framework. Unlike the traditional multivariate dynamic conditional correlation volatility models, the wavelet-based dynamic conditional correlation approach allows one to components of FTSE 100 (d UK,t ), S&P 500 (d US,t ) and Nikkei 225 (d JP,t ), κ ′ = (κ UK , κ US , κ JP ) is the vector of weights in the portfolio. ς t ( j) = κ ′ H t ( j)κ, where H t ( j) is the wavelet conditional variance-covariance matrix defined in section 2.3. Briefly, the wavelet conditional mean and wavelet conditional variance-covariance estimates are calculated from forecasting the model defined in equation 4.3. 18 rejection of the null hypothesis indicates that the computed VaR estimates are not sufficiently accurate.

decompose the spillover effect and co-movement into many sub-spillovers and sub-comovements on various time scales according to heterogeneous groups of traders and investors. However, wavelet analysis help investors to uncover the complex pattern of return and volatility spillovers on their own horizon, and make a good hedging strategy on their risk. The findings of wavelet analysis show that multivariate estimates were significant for all wavelet time series. Moreover, wavelet-based multivariate models highlight volatility spillovers on the conditional variance for all stock markets under study. In fact, the wavelet study successfully decomposes the total spillover into sub-spillovers, respectively for short-term, mid-term and long-term horizons. The out-of-sample forecasts over wavelet scales are evaluated using four statistical loss functions and one-day ahead wavelet VaR (WVaR) forecast accuracy. The out-ofsample forecast results showed that high scales (Long term component) provide smaller statistical loss metrics values than lower scales (Short component).

Figure 2 . 1

 21 Figure 2.1 Time series plots of FTSE 100, S&P 500 and Nikkei 225 stock market indices. We plot the daily level stock market indices (left panel) and corresponding returns (right panel) in the period January 01, to February 04, 2011.

Figure 3 . 1

 31 Figure 3.1 Time-varying conditional variances (left column) and conditional correlations (right column) from DCC model. The pictures on the left column refer to conditional variances of U.K, U.S and Japan stock market indices and those on the right column report conditional correlations among the same group of stock market indices. The blue line is the constant conditional correlation estimate.

  h 2 m,k denotes the volatility forecast generated by model k for day m; σ 2 m is the actual volatility on day m. The assessment of the forecast performance of the various volatility models described in section 2.3 is based on outof-sample one-day ahead prediction conditional variances and correlations. This forecasting study is done by first removing the last 250 observations from our sample starts from 02 January 2003 and ends 04 February 2011. A forecast of volatilities and correlations are generated for the period n + 1, where n is the size of the first sample to be estimated starts 02 January 2003 and ends 19 February 2010 (the estimation is based on three models: CCC, DCC and cDCC). The second sample, starting 03 January 2003 and ending on n + 1, is used to forecast the volatility and correlation of n + 2 based on the estimated models (CCC, DCC and cDCC) for the second sample. The procedure of estimation and forecasting steps is repeated 250 times for the available sample from 02 January 2003 to 04 February 2011. 9 More specifically, we produce the 250 one-day ahead forecasts h m , where h m is the forecast of the conditional variance h m , m = n + 1, . . . , n + 250.

  e. Short term, Medium term and Long term components. The "wavelet" volatility models are estimated based on 1863 observations corresponding to the period 02 January 2003 to 19 February 2010. The variables used in the models are: Short term, Medium term and Long term components. We compare the accuracy of wavelet volatility forecasts based on: RMSE k

Table 2 .1

 2 Descriptive statistics of stock market returns.

	Panel A: Descriptive statistics.
		Mean	Max	Min	Std.dev	Skewness	Kurtosis	Jarque-
								Bera
	FTSE 100 S&P 500 Nikkei 225 0.0097 0.0199 0.0187	9.384 10.957 13.234	-9.265 -9.469 -12.111	1.252 1.314 1.544	-0.0914 -0.256 -0.466	9.019 11.410 8.889	7180.19 * 11508.39 * 7048.35 *
	Panel B: Unconditional (market return) correlation matrix
		FTSE 100	S&P 500	Nikkei 225
	FTSE 100	1.000	0.55	0.34
	S&P 500		1.000	0.11
	Nikkei 225			1.000

Notes: * denotes the rejection of the null hypothesis of normality at the 1% level of the Jarque-Bera test. The data frequency is daily and covers the period from 01 January 2003 to 04 February 2011.

Table 2

 2

	.2
	Unit root tests.

Table 3

 3

	.1
	VAR diagnostic test.
		FTSE 100 [p-value]	S&P 500 [p-value]	NIKKEI 225 [p-value]
	Panel A: Univariate diagnostic test
	Q(12)	3.8854 [0.421]	5.4819 [0.241]	3.2173 [0.522]
	ARCH test	85.192 [0.000]	73.997 [0.000]	53.224 [0.000]
	JB test	811.82 [0.000]	1732.9 [0.000]	964.66 [0.000]
	Std.dev	0.0114	0.0128	0.0123
	Panel B: Multivariate diagnostic test
	Q(12) [p-value]	39.6760 [0.309]	AIC	-18.523
	JB test [p-value]	2841.6 [0.000]	SIC	-18.321
	Log-likelihood	19561.61

Table 3 .

 3 2GARCH parameter estimates stock market indices for raw data.

			United Kingdom			United States			Japan
		GARCH	EGARCH	TGARCH	GARCH	EGARCH	TGARCH	GARCH	EGARCH	TGARCH
	Panel A: model estimates.
	ω α β γ or δ	0.0078 * * 0.0894 * 0.9055 *	-9.2720 * 1.0788 0.9594 * 0.1266 *	0.0101 * 0.0080 0.9267 * 0.1064 *	0.0112 * * 0.0707 * 0.9190 *	-9.0095 * 2.3070 0.9616 * 0.0651	0.0114 * * -0.0078 0.9416 * 0.1055 *	0.0218 * * 0.0753 * 0.9101 *	-8.9373 * 0.1645 0.9306 * 0.2179 *	0.0244 * * 0.0449 * 0.9091 * 0.0562 * *
	Log-likelihood	6875.8	6848.3	6900.5	6764.4	6734.5	6797.5	6497.1	6468.2	6504.5
	AIC	-6.5331	-6.5050	-6.5556	-6.4272	-6.3969	-6.4577	-6.1731	-6.1437	-6.1792
	SIC	-6.5250	-6.4916	-6.5449	-6.4191	-6.3834	-6.4469	-6.1650	-6.1303	-6.1685
	Panel B: diagnostic test for standardized residuals.
	Q(12)	19.8581	13.4106	18.5695	11.0980	8.6872	10.3259	14.3865	10.6390	16.2380
	p-value	0.0698	0.3399	0.0994	0.5205	0.7293	0.5873	0.2767	0.5600	0.1805
	Q s (12)	20.2703	29.1464	18.2849	30.0819	26.7348	26.2148	13.3133	13.8040	10.3462
	p-value	0.0267	0.0011	0.0503	0.0008	0.0028	0.0034	0.2066	0.1821	0.4106

Notes: The estimates are produced by the univariate GARCH(1,1), univariate EGARCH(1,1) and TGARCH(1,1) models. The univariate variance estimates are introduced as inputs in the estimation of the CCC, DCC and cDCC models. The estimated coefficient ω denotes the constant of the variance equation, α represents the ARCH term, β is the GARCH coefficient, γ and δ are the asymmetric effects. * * * , * * and * indicate significance at 10%, 5% and 1%, respectively. Q(12) and Q s (12) respectively represent the Ljung-Box Q statistics of order 12 computed on the standardized residuals and squared standardized residuals. Value of the estimated coefficient ω is multiplied by 10 4 for the EGARCH model. Values in bold indicate the selected model.

Table 3

 3

	.3
	CCC diagnostic under raw data
		Log-likelihood: -7233.3	AIC: 6.8872	SIC: 6.9194
	Diagnostic test for standardized residuals.
		FTSE 100	S&P 500	Nikkei 225
	Q(12)	25.9838	17.2378	13.8918
	p-value	0.0107	0.14086	0.3076
	Q s (12)	26.1324	17.1204	13.3307
	p-value	0.0102	0.1451	0.3454

  * , * * and * indicate significance at 10%, 5% and 1%, respectively. Q(12) and Q s (12) respectively represent the Ljung-Box Q statistics of order 12 computed on the standardized residuals and squared standardized residuals.

Table 3 .4

 3 Constant conditional correlation estimates under raw data.The table summarizes the estimated invariant correlations between the U.K., U.S. and Japan stock markets, as they are produced by the CCC model. Values in (•) are standard errors. * * * , * * and * indicate significance at 10%, 5% and 1%, respectively.

	Stock market returns	FTSE 100	S&P 500	Nikkei 225
	FTSE 100 S&P 500	1	0.5692 * (0.0149) 1	0.2583 * (0.0212) 0.1751 * (0.0222)
	Nikkei 225			1

Table 3

 3

	.5
	DCC estimates under raw data
		θ 1	θ 2	Log-likelihood	AIC
	Panel A: model estimates.
	Coefficient	0.005218 *	0.992655 *	-7211.0	6.8679
	Std.error	0.001416	0.002210
	t-Stat	3.685000	449.1000
	p-value	0.000200	0.000000
	Panel B: diagnostic test for standardized residuals.
		FTSE 100	S&P 500	Nikkei 225
	Q(12)	21.6339	16.2283	13.9273
	p-value	0.04180	0.18100	0.30530
	Q s (12)	25.7240	18.5131	13.3440
	p-value	0.01170	0.10090	0.34450

Estimates of a symmetric version of

[START_REF] Engle | Dynamic conditional correlation: A simple class of multivariate generalizd autoregressive conditional heteroskedasticity models[END_REF]

dynamic conditional correlation model are computed. The specification is as

Table 3

 3

	.6
	corrected DCC (cDCC) estimates under raw data.
		θ 1	θ 2	Log-likelihood	AIC
	Panel A: model estimates.
	Coefficient	0.005282 *	0.992716 *	-7210.5	6.8674
	Std.error	0.001430	0.002047
	t-Stat	3.692000	484.8000
	p-value	0.000200	0.000000
	Panel B: diagnostic test for standardized residuals.
		FTSE 100	S&P 500	Nikkei 225
	Q(12)	21.6228	16.3030	13.9019
	p-value	0.04190	0.17770	0.30700
	Q s (12)	25.6085	18.5792	13.3573
	p-value	0.01210	0.09920	0.34360

[START_REF] Aielli | Consistent estimation of large scale dynamic conditional correlations[END_REF]

proposed a consistent DCC (cDCC) model. He suggested that the DCC correlation Q t should take a slightly different from than given in DCC model. The specification is similar to DCC one, except of Q t which becomes

  * * * , * * and * indicate significance at 10%, 5% and 1%, respectively. Q(12) and Q s (12) respectively represent the Ljung-Box Q statistics of order 12 computed on the standardized residuals and squared standardized residuals.

Table 3 .7

 3 Asymmetric dynamic conditional correlation model under raw. This table presents etimates coefficients for the asymmetric CCC [aCCC] the asymmetric DCC [aDCC] model and the asymmetric cDCC [a-cDCC] model. The specification is similar to those in DCC and cDCC models (given in Tables 3.5 and 3.6), except of the conditional variances follow the asymmetric TGARCH model of[START_REF] Glosten | On the relation between the expected value and the volatility of the nominal excess return on stocks[END_REF]. The value in parentheses are standard errors. * indicate significance at 1% level. Log-likelihood is the log-likelihood value; AIC is the Akaike information criterion. Values in bold indicate the selected model.

		θ 1	θ 2	Log-likelihood	AIC	BIC
	aCCC			-7169.5	6.8293	6.8696
	aDCC a-cDCC	0.0055(0.00162) * 0.9913(0.00295) * -7148.3 0.0056(0.00162) * 0.9913(0.00273) * -7148.0	6.8111 6.8108	6.8568 6.8565

Table 3

 3 The table reports the loss functions over the 250 forecasting observations for the symmetric and asymmetric dynamic conditional correlation models.The MSE, MAE, MAPE and LLE of the weighted portfolio are computed as fol-

	.8
	Volatility forecasts performance.
			Symmetric models			Asymmetric models
		CCC	DCC	cDCC	aCCC	aDCC	a-cDCC
	MSE	0.16923	0.15971	0.15986	0.09237	0.08652	0.08639
	MAE	0.40981	0.39822	0.39844	0.30178	0.29195	0.29170
	MAPE 313.908	304.884	304.990	221.732	214.423	214.199
	LLE	111.794	111.208	111.220	105.430	104.763	104.744

lows:

Table 3

 3

	.9
	Daily VaR forecasts.
	Model	Mean VaR α-level
		1%	2.5%	5%	10%
	Symmetric models
	CCC	1.48782	1.25373	1.05289	0.82008
	DCC	1.46673	1.23595	1.03796	0.80845
	cDCC	1.46716	1.23632	1.03827	0.80869
	Asymmetric models
	aCCC	1.27590	1.07515	0.90292	0.70327
	aDCC	1.25483	1.05739	0.88801	0.69166
	a-cDCC	1.25426	1.05692	0.88761	0.69135

Notes: The table presents the out-of-sample daily VaR of the weighted portfolio consisted of three stock market indices (FTSE 100, S&P 500 and NIKKEI 225). We used quantile for normal distribution. Values in bold show the selected model for better forecasts.

Table 3 .

 3 10 Summary results for daily VaR diagnostic tests in symmetric case.

		Exeedances Fail. prob. (%)	LR uc	LR ind	LR cc	DQ	LF
	1% daily VaR
	CCC	13	5.22	22.403	94.587	116.991	93.013	21.728
	DCC	13	5.22	22.403	94.587	116.991	93.538	22.103
	cDCC 13	5.22	22.403	94.587	116.991	93.441	22.087
	2.5% daily VaR
	CCC	18	7.22	41.188	120.452	161.640	121.155	31.706
	DCC	18	7.22	41.188	120.452	161.640	121.685	32.149
	cDCC 18	7.22	41.188	120.452	161.640	121.586	32.131
	5% daily VaR
	CCC	21	8.43	53.960	129.689	183.649	231.252	40.484
	DCC	23	9.23	63.003	136.722	199.726	340.902	42.967
	cDCC 23	9.23	63.003	136.722	199.726	340.788	42.948
	10% daily VaR
	CCC	33	13.25	113.481	173.093	286.574	391.790	61.424
	DCC	35	14.05	124.484	178.564	303.049	519.173	63.944
	cDCC 35	14.05	124.484	178.564	303.049	519.110	63.924
	and 3.11 showed that all symmetric models produce the lowest number of violations and failure probabilities. For
	instance, we identify 13 and 23 violations and 5.22% and 9.23% p-values for the DCC model at 1% and 5% level,
	respectively. While, for the same model we find 18 and 31 violations and 7.22% and 12.44% p-values at 1% and 5%
	level, respectively.

Notes: The table presents the evaluation of out-of-sample daily VaR for the weighted portfolio (consisted from three stock market indices) generated by symmetric models. it reports test statistics over last 250 days (February 19, 2010 to February 03, 2011). Fail. prob.: The failure probability. LR uc : The LR test of unconditional coverage. LR ind : The LR test of independence. LR cc : The joint test of coverage and independence. DQ: The Dynamic Quantile test. LF: The loss function. Values in bold show the selected model for better forecasts.

Table 3 .

 3 11Summary results for daily VaR diagnostic tests in asymmetric case.

		Exeedances Fail. prob. (%)	LR uc	LR ind	LR ind	DQ	LF
	1% daily VaR
	aCCC	18	7.22	41.188	120.452	161.640	116.833	29.947
	aDCC	18	7.22	41.188	120.452	161.640	116.352	30.462
	a-cDCC 18	7.22	41.188	120.452	161.640	116.394	30.470
	2.5% daily VaR
	aCCC	21	8.43	53.960	130.774	184.734	182.956	38.503
	aDCC	24	9.63	67.671	142.213	209.884	244.406	42.074
	a-cDCC 24	9.63	67.671	142.213	209.884	244.437	42.083
	5% daily VaR
	aCCC	30	12.04	97.505	162.832	260.338	361.907	53.707
	aDCC	31	12.44	102.758	166.311	269.069	375.634	55.327
	a-cDCC 31	12.44	102.758	166.311	269.069	375.646	55.338
	10% daily VaR
	aCCC	37	14.85	135.752	183.844	319.597	576.566	70.127
	aDCC	38	15.26	141.483	186.050	327.534	607.452	71.764
	a-cDCC 18	15.26	141.483	186.050	327.534	607.448	71.776

Notes: The table presents the evaluation of out-of-sample daily VaR for the weighted portfolio (consisted of three stock market indices) generated by asymmetric models. it reports test statistics over last 250 days (February 19, 2010 to February 03, 2011). Fail. prob.: The failure probability. LR uc : The LR test of unconditional coverage. LR ind : The LR test of independence. LR cc : The joint test of coverage and independence. DQ: The Dynamic Quantile test. LF: The loss function. Values in bold show the selected model for better forecasts.

Table 4 .1

 4 Unit root tests of wavelet components.

	FTSE 100	S&P 500	Nikkei 225
	ADF test	KPSS test	ADF test	KPSS test	ADF test	KPSS test
	Short term Medium term -54.386 * -42.986 * Long term -33.796 *	0.006 * 0.016 * 0.520 *	-46.451 * -52.996 * -34.725 *	0.019 * 0.013 * 0.486 *	-43.018 * -49.944 * -47.833 *	0.008 * 0.004 * 0.648 *

Table 4 .2

 4 Univariate GARCH(1,1) estimates for different time horizons. The table summarizes the estimated coefficients produced by the univariate GARCH(1,1) model for short term horizon series and EGARCH(1,1) model medium and long term horizon series. The univariate variance estimates are introduced as inputs in the estimation process of the CCC, DCC and cDCC models. The estimated coefficient ω denotes the constant term, α and β are the ARCH and GARCH terms, respectively, in the conditional variance equations, γ is the asymmetric parameter. The above estimates are for the sample period of 02 January 2003 to 04 February 2011. Significance levels at 1%, 5% and 10% are denoted by * , * * and * * * , respectively. Log-likelihood is the logarithm maximum likelihood function value. AIC is the Akaike information criterion.

		ω	α	β	γ	Log-likelihood AIC
	Short term
	FTSE 100 S&P 500 Nikkei 225	0.0076 * * 0.0089 * * 0.0188 *	0.1056 * 0.0757 * 0.0868 *	0.8896 * 0.9149 * 0.8992 *		-2638.6 -2740.2 -3045.3	2.5027 2.5989 2.8881
	Medium term
	FTSE 100 S&P 500 Nikkei 225 Long term		-0.1052 * -0.1002 * -0.0635 *	0.9351 * 0.9262 * 0.8789 *	2.3467 * * 2.3713 * 2.4242 *	576.00 656.5 -186.02	-0.5416 -0.6179 0.1799
	FTSE 100 S&P 500 Nikkei 225 Notes:		-0.0508 -0.0608 -0.0485	1.0027 * 1.0209 * 1.0232 *	2.5384 * 2.2234 * 2.2854 *	2950.9 2953.6 2008.5	-2.7907 -2.7932 -1.8982

Table 4

 4 The table reports the estimated time scale conditional correlations between the U.K., U.S. and Japan stock markets, as they are produced by the CCC model.

	.3
	Constant conditional correlation estimates under wavelet return series
			Short term			Medium term			Long term
		FTSE 100	S&P 500	Nikkei 225	FTSE 100	S&P 500	Nikkei 225	FTSE 100	S&P 500	Nikkei 225
	FTSE 100 S&P 500	1.000	0.527 * 1.000	0.242 * 0.130 *	1.000	0.689 * 1.000	0.570 * 0.557 *	1.000	0.650 * 1.000	0.283 * 0.351 *
	Nikkei 225			1.000			1.000			1.000

* indicate significance at 1% level.

Table 4 .4

 4 Estimates of the multivariate GARCH(1,1) models (DCC and cDCC) for different time horizons. This table reports the estimates parameters of DCC and cDCC models. Significance levels at 1%, 5% and 10% are denoted by * , * * and * * * , respectively. Log-likelihood is the logarithm maximum likelihood function value. AIC is the Akaike information criterion.

		θ 1	θ 2	Log-likelihood	AIC
	DCC model
	Short term Medium term Long term	0.004457 (0.0017) * 0.719497 (0.0066) * 0.594683 (0.0269) *	0.993156 (0.0032) * 0.175299 (0.0078) * 0.400597 (0.0277) *	-8002.8 14834.1 40909.8	7.59530 -14.0408 -38.7454
	cDCC model
	Short term Medium term Long term	0.004560 (0.0016) * 0.763499 (0.0055) * 0.831890 (0.0242) *	0.993199 (0.0027) * 0.231645 (0.0056) * 0.167528 (0.0244) *	-8002.2 15229.6 43104.5	7.59470 -14.4155 -40.8247
	Notes:

Table 4 .5

 4 One-day out-of-sample volatility forecasts performance for different time horizons.

		Forecast loss functions
		RMSE	MAE	MAPE	LLE
	Panel A: Short term
	CCC	1.488636	0.725253	5.076901	8.356644
	DCC	1.488659	0.723788	5.050991	8.336478
	cDCC	1.489430	0.718159	4.958577	8.273333
	Panel B: Medium term
	CCC	0.139242	0.082996	1.001749	24.867650
	DCC	0.139241	0.082996	1.001749	24.865190
	cDCC	0.139241	0.082996	1.001749	24.864380
	Panel C: Long term
	CCC	0.015626	0.012123	0.999986	73.91793
	DCC	0.015626	0.012123	0.999986	73.91789
	cDCC	0.015626	0.012123	0.999986	73.91788

Notes: The table summarizes the four loss metrics estimates at different time horizons.

The RMSE is defined as; RMSE k

Table 4

 4 Summary results for daily WVaR diagnostic tests. The table presents the evaluation of out-of-sample daily WVaR for the wavelet details. it reports test statistics over last 250 days(February 19, 2010 to February 03, 2011). Exceedance: The term "exceedance" refers to an instance when portfolio losses are greater than corresponding VaR estimates in the backtest, In the literature, the terms "breaches" and "violations" are also used when referring to "exceedances". Fail. prob.: The failure probability. LR uc : The LR test of unconditional coverage. LR ind : The LR test of independence. LR cc : The joint test of coverage and independence. DQ: The Dynamic Quantile test. LF( j): The loss function. The respective critical values of LR uc and LR cc statistics at 5% significance level are 3.84 and 5.99.

	.6
	Daily WVaR forecasts.
		Mean WVaR α-level
		1%	2.5%	5%	10%
	CCC model
	Short term	1.7684	1.4902	1.2514	0.9747
	Medium term	0.0381	0.0321	0.0270	0.0210
	Long term	0.0025	0.0021	0.0018	0.0014
	DCC model
	Short term	1.7641	1.4865	1.2484	0.9723
	Medium term	0.0381	0.0321	0.0270	0.0210
	Long term	0.0025	0.0021	0.0018	0.0014
	cDCC model
	Short term	1.7455	1.4708	1.2352	0.9620
	Medium term	0.0381	0.0321	0.0270	0.0210
	Long term	0.0025	0.0021	0.0018	0.0014

  z α denotes the normal quantile, κ ′ d t+m is the wavelet mean one-day ahead forecast estimate computed by forecasting VAR(1) based on wavelet components series. The wavelet weighted portfolio return is d

	pt f
	t

= κ ′ d t : d t = d UK,t , d US,t , d JP,t ′ denotes the vector of wavelet return

  Table4.7 presents the summary backtesting statistics for the daily WVaR forecasts of wavelet return series. In terms of LR uc , LR ind , LR cc and DQ statistics, the null hypothesis are rejected at 1%, 2.5%, 5% and 10% significance level. According to these metrics, we cannot conclude whether an adequate multivariate model is more accurate than another one.19 Column 9 of Table4.7 shows the mean loss function values. A WVaR model is preferred to another one, if it yields a lower averaging loss criterion value: LF(j), defined as the sum of LF t (j), that is,LF(j) = ∑

	n 0

[START_REF] Hansen | Choosing the best volatility models: the model confidence set approach[END_REF],[START_REF] Hansen | A forecast comparison of volatility models: does anything beat a garch(1,1)[END_REF],[START_REF] Becker | Are combination forecasts of s&p 500 volatility statistically superior?[END_REF] and others, showed that evaluation of univariate volatility forecast is well understood, while for an applied point of view ther are no clear guidelines available on model evaluation and selection in multivariate setting (see[START_REF] Laurent | On the forecasting accuracy of multivariate garch models[END_REF]).

Following Brooks (2002), the main advantage of the VAR is that there is no need to specify which variables are the endogenous variables and which are the explanatory variables because in the VAR, all selected variables are treated as endogenous variables. That is, each variable depends on the lagged values of all selected variables and helps in capturing the complex dynamic properties of the data. Note that selection of appropriate lag length is crucial. If the chosen lag length is too large relative to the sample size, the degrees of freedom will be reduced and the standard errors of estimated coefficients will be large. If the chosen lag length is too small, then the selected lags in the VAR analysis may not be able to capture the dynamic properties of the data. The chosen lag length should be free of the problem of serial correlation in the residuals.

The CCC specification can be presented as: H t = D t RD t , where, D t = diag h i,t is a diagonal matrix with square root of the estimated univariate GARCH variances on the diagonal. R is the time-invariant symmetric matrix of the correlation returns with ρ ii = 1.In CCC model, the conditional correlation coefficients are constant, but conditional variances are allowed to vary in time.

see[START_REF] Aielli | Consistent estimation of large scale dynamic conditional correlations[END_REF] for further details.

[START_REF] Aielli | Consistent estimation of large scale dynamic conditional correlations[END_REF] showed that the lack of consistency of the three-step DCC estimator depends strictly on the persistence of the parameters driving the correlation dynamics and on the relevance of the innovations. The bias is an increasing function of both θ 1 and θ 1 + θ 2 . The parameter estimates obtained from fitting DCC models are small, and close to zero for θ 1 and close to unit for θ 1 + θ 2 .

For instance, forecasting the DCC model is as follows: the initial sample consists of the first 1862 daily observations, i.e. 02 January 2003 to 19 February 2010. The last 250 trading days constitute the sample for which we compute one-day ahead forecasts. We re-estimated the DCC model basing on the initial sample every day using a recursive method (all estimations and forecasts are computed with OxMetrics 6). After that, we computed the conditional variance and conditional correlation forecast values. These forecast values are used to compute the loss functions of the weighted portfolio defined before.

The null hypothesis for[START_REF] Kupiec | Techniques for verifying the accuracy of risk measurement models[END_REF] test is, H 0 : π = n 1 /n 0 = π 0 , where π is the expected proportion of exceedances, which equals the desired coverage level π 0 (usually equal to 1% and 5%).

The null hypothesis for[START_REF] Christoffersen | Evaluating interval forecasts[END_REF] test is, H 0 : π 00 = π 11 .

We include five lags of Hit t and the current VaR as explanatory variables. Hence, Hit t = ι, Hit t-1 , Hit t-2 , Hit t-3 , Hit t-4 , Hit t-5 , VaR t (α)

For more details, see[START_REF] Mallat | A theory for multiresolution signal decomposition: The wavelet representation[END_REF] and[START_REF] Percival | Wavelet methods for time series analysis[END_REF]

We decompose our time series up to scale 8 (scale J ≤ log 2 [(T -1)/(L -1) + 1], where T is the number of observations of stock market returns (T = 2112), and L is the length of the wavelet filter LA(8)). We used the wavelets R package for the MODWT.

Notes: The table presents the out-of-sample daily VaR of the weighted portfolio across wavelet scales. We used quantile for normal distribution.

The abovementioned backtesting tests focused on examining the accuracy of failure frequency and the independence of the failure process for VaR models. However, there are a large number of VaR models that can pass these statistical evaluation tests. How do risk managers choose among alternative VaR models? which model will generate fewer regulatory capital requirements and induce less oppurtunity cost of capital?. To answer these questions we use a regulatory loss function related to market risk.