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2 Journal of Economic Dynamics & Control 32 (2008) Determining the optimal dimensionality of multivariate volatility models with tools from random matrix theory Bernd Rosenow Institut für Theoretische Physik, Universität zu Köln, Zulpicher Strasse 77, D Koln, Germany Received 1 May 2005; received in revised form 1 December 2006; accepted 26 January 2007 Available online 7 September 2007 Abstract We present a brief review of methods from random matrix theory (RMT), which allow to gain insight into the problem of estimating cross-correlation matrices of a large number of financial assets. These methods allow to determine the optimal number of principal components or factors for the description of correlations in such a way that only statistically relevant information is used. As an application of this method, we suggest two classes of multivariate GARCH-models which are both easy to estimate and perform well in forecasting the multivariate volatility process for more than 100 stocks. r 2007 Elsevier B.V. All rights reserved. JEL classification: C51; C53; G12 Keywords: Correlation estimation; Random matrix theory; Noise filtering; Multivariate volatility forecasts 1. Introduction The multivariate analysis of high dimensional data sets is an important statistical problem and has numerous applications not only in finance and economics, but also in other areas like signal transmission, climate analysis, and denoising of dynamic Tel.: ; fax: addresses: rosenow@physics.harvard.edu,, rosenow@thp.uni-koeln.de (B. Rosenow) /$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi: /j.jedc

3 280 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) imaging data. One typical problem is the estimation of cross-correlation matrices for N time series of length T. In the limit where the ratio T=N tends to infinity, traditional approaches to multivariate statistics are well founded and the traditional maximum likelihood estimators are applicable (Muirhead, 1982; Anderson, 1984). However, in many applications the length of time series and the number of variables are of the same order, hence the number of correlation coefficients is about the same as the number of available data points. Such an (almost) degeneracy of the correlation matrix due to the large number of time series is referred to as curse of dimensionality. Then, the statistical fluctuations of the correlation coefficients are significant and turn the correlation matrix, at least partially, into a random matrix. If a sample correlation matrix is calculated from Gaussian i.i.d. time series with finite ratio T=N, it belongs to the Wishart ensemble of random matrices. In the limit of infinite T, N, but finite T=N, the eigenvalue distribution of Wishart matrices was calculated by (Marc enko and Pastur, 1967; Stein, 1969). There exists a large body of literature, in which this result was rediscovered and generalized, for a review see (Bai, 1999). In the physics literature, the eigenvalue distribution of Wishart matrices was first found in (Dyson, 1971) and later on generalized in (Sengupta and Mitra, 1999). Using insights into this random matrix aspect of high dimensional correlation estimates, a separation of noise and information is possible and has been used to obtain improved predictions of cross-correlations (Laloux et al., 2000; Rosenow et al., 2002). For practical applications, however, multivariate volatility forecasts are needed, i.e. cross-correlations have to be estimated together with volatilities. Several multivariate GARCH (generalized autoregressive conditional heteroscedasticity (MV-GARCH)) models have been developed for this purpose (see Campbell et al., 1997 and the discussion below). In this article, we suggest ways how to incorporate the knowledge about random matrix theory (RMT) based, improved correlation estimates into MV-GARCH models. In recent years, there has been a number of approaches to the study of crosscorrelations which are based on RMT. In a generic model for a financial market without any correlations between time series, the sample cross-correlation matrix is a random matrix R. Agreement between the statistical properties of the empirical cross-correlation matrix C and the random control R is a signature for measurement noise due to the curse of dimensionality, whereas deviations between C and R indicate the presence of true information. In Laloux et al. (1999) and Plerou et al. (1999), the eigenvalue pdf of empirical cross-correlation matrices was compared to the RMT prediction (Sengupta and Mitra, 1999). The largest part of the empirical eigenvalue spectrum was found to agree well with the RMT prediction, only a few large eigenvalues were found to deviate from it. The studies (Laloux et al., 2000; Rosenow et al., 2002) tested correlation forecasts in the framework of Markowitz portfolio theory. Correlation matrices with a reduced dimensionality of the parameter space, which contain only economically meaningful information as defined by an RMT analysis, turned out to provide much better correlation forecasts than traditional estimates. Based on simulations of time series models, it was found that the influence of estimation noise on portfolios with nonlinear constraints is much more pronounced than in the case of linear constraints

4 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) (Pafka and Kondor, 2002). The influence of noise is reduced if the ratio T=N is significantly larger than one (Pafka and Kondor, 2003). In a comparison of different dimensional reduction techniques (Pafka and Kondor, 2004), the filtering method described in Section 3 turned out to be the most robust one. Whereas in the econophysics literature the necessity of dimensional reduction for correlation estimates and its relation to RMT was stressed, less attention was paid so far to an accurate modeling of the volatility process together with the correlation estimates. On the other hand, in the recent economics literature correlation estimates are mostly embedded into multivariate volatility estimates. As the influence of randomness on high dimensional volatility forecasts is not accounted for in these studies, many of the suggested models are not suitable for application to a large number of time series. Often, MV-GARCH processes are used for volatility modeling. In the univariate setting, a GARCH process relates the present volatility to past volatilities and past returns squared (Bollerslev, 1986). In the most general formulation of MV-GARCH, each covariance is linked to all other covariances. Thus, the number of parameters scales like the number N of time series to the forth power, and such a model is clearly not applicable to the description of a large number of time series. While the literature on MV-GARCH models is reviewed in detail in Section 5, we would like to mention some models here which either use only OðNÞ parameters in their original formulation, or which can easily be modified to use noise filtered correlation matrices as described in Section 3. In Engle et al. (1990), a one-factor model which an ARCH description of the factor dynamics was used to describe the pricing of treasury bills. In orthogonal GARCH (O-GARCH) models, the observed components of a multivariate time series are transformed to uncorrelated components via an orthogonal transformation, and the volatility dynamics of these uncorrelated components is described by univariate GARCH processes (Alexander, 2002). In the conditional constant correlation (CCC) model of Bollerslev, only the parameters of the N univariate GARCH processes have to be estimated simultaneously, whereas the time-constant correlation matrix is the unconditional correlation matrix of GARCH residuals (Bollerslev, 1990). In the dynamical conditional correlation model (DCC-GARCH) (Engle and Sheppard, 2001; Engle, 2002), parameter estimation takes place as a two step process. In the first step, univariate GARCH processes for the individual time series are estimated, and in a second step, a GARCH process for the covariance matrix of the standardized residuals is formulated. For practical applications, the Risk Metrics estimator with equally weighted moving averages (Longerstaey and Zangari, 1996) is used frequently. It has no adjustable parameters and hence easily produces large dimensional covariance matrices. In this manuscript, we go beyond previous RMT based approaches to correlation estimates in that we model the volatility dynamics together with the crosscorrelations. Our novel contribution to the MGARCH literature is the formulation of MV-GARCH models with a reduced dimensionality of the correlation structure, which can be adjusted with the help of RMT tools in such a way that only

5 282 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) statistically relevant information is used. In this way, one obtains models in which only OðNÞ parameters have to be estimated sequentially and which at the same time provide reliable and stable covariance estimates. In this study, we use intraday data to judge the economic relevance of our covariance forecasts. We find that sliding correlation models (SCMs) with a one year horizon for correlation estimates are very suitable for tracking the volatility of an equal distributed portfolio. The same models allow the construction of minimum variance portfolios with only half the risk of an equally distributed portfolio. The outline of the manuscript is as follows: In Section 2, we explain how RMT can be used to distinguish information from noise in correlation matrices, while in Section 3 the removal of noise from empirical correlation estimates is described, and this method of noise filtering is empirically illustrated in Section 4. In Section 5, existing MGARCH models are reviewed with emphasis on their effective dimensionality and their applicability to a large number of time series, and two classes of models for multivariate volatility forecasts are defined in Section 6. In Section 7, a method for evaluating their performance is described. We present the test results in Section 8, discuss them in Section 9, and sum up our key conclusions in Section RMT and correlations Traditional multivariate statistics is not able to deal with problems, in which the number N of different time series is comparable to their length T. Traditional estimators for sample correlation matrices are consistent only in the limit of T=N going to infinity (Muirhead, 1982). In typical applications, one is dealing with ratios T=N ranging from one to ten, and hence statistical fluctuations due to the lack of a sufficient number of observations dominate an empirically observed correlation matrix. To identify the effects of randomness on the eigenvalue spectrum of an empirical cross-correlation matrix, one considers the null hypothesis of normally distributed i.i.d. time series. As far as the eigenvalue statistics is concerned, the cross-correlation matrix R of such time series is equivalent to a random Wishart matrix, i.e. to the covariance matrix of standard normally distributed i.i.d. time series. This equivalence holds true because the influence of the N fluctuating diagonal elements in the covariance matrix as compared to the fixed diagonal elements of the crosscorrelation matrix is negligible in comparison to the fluctuating NðN 1Þ offdiagonal elements. Hence, the cross-correlation matrix R has the eigenvalue pdf of random Wishart matrices. In the limit T! 1, N! 1, and Q ¼ T=N fixed, the eigenvalue pdf of Wishart matrices converges to a limiting distribution. This limiting eigenvalue pdf is given by (Marc enko and Pastur, 1967; Stein, 1969; Dyson, 1971; Sengupta and Mitra, 1999) rðlþ ¼ Q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl þ lþðl l Þ (1) 2p l

6 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) for l within the bounds l olol þ, where sffiffiffiffi l ¼ 1 þ 1 Q 2 1 Q (2) are the minimum and maximum eigenvalues of R, respectively. Agreement between R and the empirical cross-correlation matrix C is a sign of randomness, whereas deviations indicate the presence of economically meaningful information. Specifically, eigenvalues larger than l þ and the corresponding eigenvectors contain information about true correlations and should be used for forecasting purposes. Empirically, it was found that more than 95% of the observed eigenvalues fall in the random matrix interval (Laloux et al., 1999; Plerou et al., 1999). In addition to their eigenvalue pdf, which in general does depend on the specific way in which a random matrix is defined, random matrices have universal statistical properties depending on their symmetry only. For example, the distribution of spacings between neighboring eigenvalues is the same for all real symmetric random matrices. Real symmetric random matrices whose elements are independently drawn from a Gaussian distribution constitute the Gaussian orthogonal ensemble (GOE) and are, with respect to universal statistical properties, representative for all real symmetric random matrices. The investigation of universal properties is important as the eigenvalue pdf alone does not prove the randomness and lack of information in the eigenvalue spectrum (Mehta, 1991; Guhr et al., 1998) of a matrix. In Plerou et al. (1999, 2002) the universal properties of empirical cross-correlation matrices were studied. For a cross-correlation matrix calculated from two years of intraday data for 1000 stocks, it was found that both the nearest neighbor distribution as well as long range spectral correlations show good agreement with the universal predictions for the GOE (Plerou et al., 1999, 2002), indicating that the main part of the eigenvalue spectrum indeed does not contain economically meaningful information. In addition, eigenvectors with eigenvalues in the central, random part of the spectrum are found to have components of similar size (Plerou et al., 1999, 2002). The distribution of the components of these eigenvectors is described by a normal distribution (Laloux et al., 1999; Plerou et al., 2002). In contrast, eigenvectors with eigenvalues at the edges of the spectrum were found to be localized, i.e. to be dominated by a few large components. The eigenvectors belonging to the twelve largest eigenvalues were found to have an economic interpretation (Gopikrishnan et al., 2001): the eigenvector with the largest eigenvalue describes correlations permeating the whole stock market and is similar to a market index. The next eigenvector describes correlations between companies with large market capitalization, whereas the remaining eigenvectors with large eigenvalues could be identified with industry sectors or were found to describe geographical correlations (Gopikrishnan et al., 2001; Plerou et al., 2002). These economically meaningful correlations are stable in time and hence suitable for forecasting future correlations. As correlation matrices are not directly influenced by the volatility dynamics on short time scales, they generally show a higher degree of time stability than covariance matrices, which are fluctuating due to time changing volatility.

7 284 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) Noise filtering of correlation matrices As the RMT analysis of correlation matrices discussed in the last section suggests that economically relevant information resides mostly in the large eigenvalues and corresponding eigenvectors, a filtering process keeping only this information and discarding the noise in the rest of the spectrum is called for. In this section, a filtering algorithm based on principal component analysis will be described. A review of traditional methods for forecasting correlation matrices can be found in Elton and Gruber (1995) and Campbell et al. (1997). In order to use only the statistically relevant information in the correlation matrix C, we diagonalize C by an orthogonal transformation U via U T CU ¼ K. Here, the kth column of U is the eigenvector u ðkþ with eigenvalue l k. We construct a filtered correlation matrix C p by keeping only the p largest eigenvalues in the diagonal matrix (Rosenow et al., 2002) ( L p ii ¼ L ii for i4n p; 0 for ipn p; and by transforming back to the original basis C p ¼ UK p U T. To satisfy the requirement that every time series is fully correlated with itself, the diagonal elements of C p are set to one. Here, p should be chosen such that l p is the smallest eigenvalue clearly above the upper bound l þ of the noise spectrum. For instance, in the eigenvalue spectrum Fig. 4 there is only one eigenvalue clearly above the upper edge of the noise part of the spectrum, and two eigenvalues are close to the upper edge. Alternatively, the rational behind this filtering procedure can be understood by applying principal component analysis to the normalized returns ~ i;t ¼ ð i;t h i iþ=s i. Due to normalization, we have C ¼ Covðf ~ i gþ, i.e. the correlation matrix is the covariance matrix of the normalized returns. After rank ordering the eigenvalues, the eigenvectors u ðnþ, u ðn 1Þ ;... ; u ðn pþ1þ correspond to the p largest eigenvalues l N, l N 1 ;... ; l N pþ1. The eigenvectors define principal components w ðkþ ¼ ~e u ðkþ, where ~e is the T by N matrix of time series. Using the p principal components with the largest eigenvalues, the normalized time series can be decomposed as ~ i;t ¼ Xp 1 l¼0 u ðn lþ i w ðn lþ t þ r i;t, (4) where the residuals r i;t are defined by this decomposition. As the principal components w ðkþ, N pokpn, are uncorrelated among each other and with the residuals r i, the correlation matrix C can be expressed as C ¼ UK p U T þ Covðfr i gþ. (5) However, according to the RMT analysis of C, p was chosen such that the p largest principal components capture all the correlation information contained in C. Hence, the off-diagonal elements of Covðfr i gþ only contain noise and the residual time series r i;t should be considered uncorrelated, i.e. their covariance matrix should have variances s 2 r i on the diagonal and be zero elsewhere. Thus, the filtered correlation (3)

8 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) matrix is given by C p ¼ UK p U T þ diagðfs 2 r i gþ. (6) However, adding the variances of the residuals on the diagonal amounts to just setting the diagonal elements equal to one by hand. This filtering approach is similar to principal component based approaches like O-GARCH in that it uses only the most relevant principal components for the description of correlations. However, there are two differences: First, the residual variances fs 2 r i g are kept and only residual covariances are discarded. In this way, the method works well for weakly correlated time series, where O-GARCH may suffer from identification problems. Second, the number of relevant principal components is derived from theory and not from a somewhat arbitrary empirical rule. 4. Empirical illustration of noise filtering The quality of correlation predictions can be judged either with respect to their statistical significance or with respect to their economic significance. While the statistical significance of a correlation prediction is related to the difference between prediction and realization with respect to a given matrix norm, the economic significance is related to the ability of predicting the risk of a portfolio. Most studies so far have investigated the economic significance of correlation forecasts by using the Markowitz (1959) portfolio theory to design minimum variance portfolios and by comparing the predicted risk of these portfolios to their realized risk. To empirically demonstrate the virtues of noise filtering as described in the last section, we perform an analysis in the spirit of Laloux et al. (2000) and Rosenow et al. (2002). Specifically, we study: (i) the error in predicting the variance of an equally distributed portfolio, (ii) the error in predicting the variance of a minimum variance portfolio, and (iii) the realized variance of minimum variance portfolios. In order to disentangle the influence of correlation predictions versus the influence of variance predictions, we always use actually realized variances to calculate covariance matrices from correlation matrices. In this way, only the accuracy of correlation predictions determines the final result and possible errors in the prediction of variances do not interfere. While a variance prediction for equally distributed portfolios is only sensitive to the average correlation strength, variance predictions for minimum variance portfolios depend on the whole structure of the correlation matrix and are a more powerful test for correlation predictions. We define a portfolio as a selection of N stocks with weight m i and the constraint P N i¼1 m i ¼ N. The variance D 2 of such a portfolio is given by D 2 ¼ XN i;jþ1 S ij m i m j with S ij ¼ C ij s i s j. (7) Here, the fs i g are the standard deviations of individual stocks. In an equally distributed portfolio, all stocks have equal weight m i 1=N, whereas in a minimum

9 286 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) variance portfolio, the weights are given by P N j¼1 m i ¼ S 1 i;j P N. (8) i;j¼1 S 1 ij For both equally weighted and minimum variance portfolios, we calculate the estimated variance D 2 est, the realized variance D2 real, and the mean average percentage error (MAPE) jd 2 est D2 real j=d2 real. For our empirical study, we use daily data for the years for the 238 most actively traded German stocks. We compare different predictions for the actually realized correlation matrices C real of five one year testing periods during the years To obtain forecasts for them, we calculate correlation matrices C est for: (i) a one year estimation period prior to the testing period, and (ii) a five year estimation period prior to the testing period. For the testing year 2000, the eigenvalue spectra of the one-year estimate, the five-year estimate, and the realized matrix are shown in Fig. 1 together with the respective random matrix predictions. We make two key observations. First, the largest eigenvalue l max varies significantly from 39.0 for the five year estimate to 22.0 for the one-year estimate 1999 to 16.2 for the one-year realization Second, due to the smaller width of the random matrix part of the eigenvalue spectrum, there are significantly more eigenvalues above the upper bound for the five year estimate (nine eigenvalues) as for the one-year estimate (five eigenvalues). From the first observation it follows that a short estimation period is desirable in order to accurately track changes in the overall correlation strength described by the largest eigenvalue, whereas the second observation implies that estimation noise is more pronounced for short estimation periods. The empirical results are shown in Tables 1 (one year estimation period) and 2 (five year estimation period). For each estimation period, we calculate the number p rmt of eigenvalues larger than the upper bound l þ of the random matrix part of the spectrum. We compare this with the number p opt of eigenvalues kept in the filtering algorithm, which minimizes the MAPE of the variance of minimum variance portfolios. For the one year estimation period, we find that p rmt and p opt agree closely with each other, while we find reasonable agreement for the five year estimation periods. The discrepancy between p rmt and p opt for the five year estimation periods , and the respective testing periods hints at the possibility that including only eigenvalues clearly separated from the bulk (about eight in both cases) might be a better choice for practical applications than the inclusion of all eigenvalues larger than l þ. The full dependence of the MAPE on p for the 2000 testing period is shown in Fig. 2, the MAPE becomes minimal for p p rmt in that case. For both one year and five year forecasts, the relative forecast error MAPE rmt obtained from filtered matrices with p rmt eigenvalues kept is significantly smaller than the forecast error MAPE N for unfiltered estimates with p ¼ N. In addition, we have calculated the predicted risk D 2 rmt of minimum variance portfolios with p ¼ p rmt, and the realized risk D 2 rmt;real for these portfolios. When comparing the one year predictions with the five year ones, we find that the five year

10 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) pdf λ max pdf λ max pdf λ max eigenvalue Fig. 1. Eigenvalue spectrum of the cross-correlation matrix from daily data for 238 German stocks (grey line): (a) for the one year prediction period 1999, (b) for the five year prediction period , and (c) for the testing period The noise part of the spectrum (black line) of the five year period (b) is compressed as compared to the noise part of the spectrum for the one year period (a), and more eigenvalues are outside the noise part in (b) as compared to (a). In addition, the largest eigenvalue l max in (a) is closer to the realization (c) as that of (b).

11 288 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) Table 1 Estimation of the variance of minimum variance (upper part) and of equally distributed (lower part) portfolios, one year estimation period Test period p rmt p opt D 2 est;rmt D 2 real;rmt MAPE rmt D 2 est;n D 2 real;n MAPE N l max;est l max;real MAPE rmt;equal MAPE N;equal For the calculation of minimum variance portfolios, variances realized in the testing period were used. Table 2 Estimation of the variance of minimum variance (upper part) and of equally distributed (lower part) portfolios, five years estimation period Test period p rmt p opt D 2 est;rmt D 2 real;rmt MAPE rmt D 2 est;n D 2 real;n MAPE N l max;est l max;real MAPE rmt;equal MAPE N;equal For the calculation of minimum variance portfolios, variances realized in the testing period were used. predictions have a smaller prediction error four out of five times, but that the one year predictions have an about twenty percent smaller realized risk four out of five times. We interpret the former finding as indication for a higher information content of five year predictions, and the latter finding as evidence for a more accurate prediction of the actual correlation strength of the one year predictions. The dependence of the realized risk of minimum variance portfolios on the number p of eigenvalues kept in the filtering algorithm is shown in Fig. 3, the realized risk

12 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) MAPE p Fig. 2. MAPE of one year prediction period (grey) line and five year prediction period (black line) for forecasting the volatility of minimum variance portfolios for the year 2000, as a function of the number p of eigenvalues kept in the filtering algorithm realized variance p Fig. 3. Realized variance of minimum variance portfolios in the year 2000 for a one year prediction period (grey line) and a five year prediction period (black line), as a function of the number p of eigenvalues kept in the filtering algorithm. becomes minimal for p p rmt. For comparison, we have calculated D 2 est;n and D2 real;n for unfiltered estimates with p ¼ N. For both one and five year predictions, one finds without exception that the filtered estimates have a significantly lower realized risk than the unfiltered prediction. For filtered correlation matrices with p rmt eigenvalues kept, we calculate the prediction error MAPE rmt;equal for the variance of equally distributed portfolios. One sees that this prediction error is small when predicted and realized largest eigenvalue are close to each other, and large when they are far apart. It seems that an accurate prediction of the largest eigenvalue is important for the estimation of the variance of equally distributed portfolios. On average, correlation estimates from one year

13 290 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) periods are more accurate than correlation estimates from five year periods. When comparing with the prediction error MAPE N;equal of unfiltered correlation matrices with that of filtered matrices, we find no significant difference. It is to be expected that noise filtering has little influence on the variance of equally distributed portfolios, as this variance is only sensitive to the average strength of correlations, which is not changed by discarding random contributions in the filtering algorithm. In summary, we draw the following conclusions: (i) filtering of correlation matrices with keeping all eigenvalues larger than the upper bound of the RMT spectrum significantly increases the accuracy of correlation estimates as compared to unfiltered matrices, and (ii) a short estimation period is desirable to capture the dynamical evolution of the correlation strength. 5. MV-GARCH models The estimation of large dimensional covariance matrices is a key element in the process of estimating and minimizing portfolio risk. Although modern risk management is founded on the analysis of huge data bases and the use of sophisticated theoretical models, there has been only partial progress with respect to covariance forecasts. The reason for this lack of progress is the curse of dimensionality, as in modern financial engineering the dimension of the investment universe is comparable to the number of observations. Much research has been devoted to describing the volatility process with the help of MV-GARCH models, which relate present volatilities to both past volatilities and past returns squared. Here, MV-GARCH models are reviewed with emphasis on both the dimensionality of their parameter space and the possibility to estimate them for a large number of time series. In the most general setup of an MV-GARCH model, each covariance is linked to all other covariances, and the number of parameters scales like the number N of assets to the forth power. Linking a given covariance only to past values of itself and the respective product of returns, the number of parameters is reduced to OðN 2 Þ in the diagonal VECH model (Bollerslev et al., 1988). Another model specification with OðN 2 Þ parameters is obtained when the covariance matrices in a GARCH setup are transformed by quadratic forms (Engle and Kroner, 1995). If these transformation matrices are of rank one, one obtains a one-factor model with OðNÞ parameters to be estimated simultaneously. In Engle et al. (1990), such a model was applied to the pricing of Treasury bills by using factors with prespecified weights. In the CCC model of Bollerslev, N univariate GARCH processes for the individual time series are estimated. The covariance matrix is obtained by multiplying the volatilities of the individual time series with the unconditional correlation matrix of the GARCH residuals (Bollerslev, 1990). By transforming the observed components of a time series to uncorrelated components via an orthogonal transformation, the volatility dynamics can be described by specifying univariate GARCH processes for these uncorrelated components and then transforming back to the original basis. If the transformation

14 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) matrix of such an O-GARCH model is estimated from unconditional information like principal components of the correlation matrix, the parameters describing the volatility dynamics of the uncorrelated components can be estimated sequentially. In applications to strongly correlated time series like term structure data or commodity futures with different maturity, it can be sufficient to keep Oð1Þ uncorrelated components (Alexander, 2002). While working well for highly correlated data sets, O-GARCH suffers from an identification problem for weakly correlated time series. By relaxing the orthogonality condition on the transformation relating correlated observed and uncorrelated unobserved components, generalized orthogonal GARCH (GO-GARCH) (van der Weide, 2002) is able to avoid the above-mentioned identification problems. However, the nonorthogonal transformation is estimated from conditional information and thus the estimation complexity increases. The method was shown to work well when applied to modeling a small number of time series. Bollerslev s CCC model has been generalized to allow for a time evolution of the correlation matrix with a GARCH like structure (Tse and Tsui, 2002). Testing this varying correlation MGARCH (VC-MGARCH), the above authors indeed find evidence that the assumption of constant correlations has to be rejected. However, in the present specification of the mode, OðN 2 Þ parameter have to be estimated simultaneously, making VC-MGARCH not suitable for the estimation of high dimensional covariance matrix. In the dynamical conditional correlation model (DCC-GARCH) (Engle and Sheppard, 2001; Engle, 2002), the authors suggest to first estimate GARCH parameters for the individual time series. Next, a GARCH process for the covariance matrix of the standardized residuals is estimated. Here, the unconditional covariance matrix of the residuals is used as the constant term, and the GARCH parameters for all matrix elements are the same. Hence, only O(1) parameters have to be estimated in the second step. For this reason, the model is suitable for application to a large number of financial assets and is the only of the above mentioned models which actually has been empirically studied for up to one hundred time series. In practical applications, the Risk Metrics estimator with equally weighted moving averages (Longerstaey and Zangari, 1996) is used frequently. As it has no adjustable parameters, it can be easily applied to the estimation of high dimensional covariance matrices. 6. Model description The RMT approach to correlation matrices provides a rational for the use of parsimonious models with a reduced dimensionality of the parameter space. A low dimensional parameter space has the double benefit of more reliable forecasts due to the removal of noise and the applicability of models to a large number of time series due to a reduction of estimation complexity. Of the above discussed models, the factor model (Engle et al., 1990) and O-GARCH (Alexander, 2002) employ a dimensionally reduced correlation structure.

15 292 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) Despite the usage of a high dimensional correlation matrix, the CCC model (Bollerslev, 1990) and DCC-GARCH (Engle and Sheppard, 2001) can be easily estimated even for a large number of time series. In the following, we suggest two classes of MV-GARCH models, which combine elements of these models with the possibility to use an optimal number p of principal components or factors as determined by an RMT analysis of the correlation matrix. The basis of the Sliding Correlation Model (SCM) is the decomposition of the covariance matrix into the cross-correlation matrix C and standard deviations fs i g as first suggested in Bollerslev (1990). Each individual time series is described by a univariate GARCH(1,1) (Bollerslev, 1986) process defined by s 2 i;t ¼ a i;0 þ a i;1 s 2 i;t 1 þ b i;1 2 i;t 1. (9) Here, s 2 i;t and i;t are the volatility and the innovation of time series i at time t, respectively. The innovations are modeled as a product i;t ¼ s i;t x i;t with normally distributed x 2 Nð0; 1Þ. As the emphasis of the present article is on comparing different multivariate volatility models, we make the simplifying normality assumption although empirical studies suggest the presence of leptokurtosis in GARCH residuals, see e.g. Nelson (1991). The parameters a i;0, a i;1, and b i;1 are obtained from maximum likelihood estimations. In order to take into account the dynamics of the correlation strength, the cross-correlation matrix C is calculated as a moving average to accommodate the change of correlations in time. We use the filtering method described in Section 3 and keep only the p largest principal components. The covariance matrix in the SCMðpÞ of rank p is described by S ij;t ¼ C p ij;t s i;t s j;t. (10) This model is related to the constant correlation model (Bollerslev, 1990), where a fixed and unfiltered correlation matrix is used in contrast to our C p ij;t. As a second class of models with a reduced dimensionality of the covariance structure, we consider GARCH factor models (Engle et al., 1990). If the market index is one of the factors, these models have the attractive property that the strength of correlations is tied to market volatility. Considering only the market index as a factor for the moment, one finds C ij ¼ s 2 factor b ib j =ðs i s j Þ, where s 2 factor is the volatility of the market factor, s 2 i is the volatility of stock i, and b i is the regression coefficient of stock i on the market factor. When the market volatility increases more strongly than the individual volatilities, the correlation coefficients increase in such a model in agreement with empirical observations (Drozdz et al., 2000; Plerou et al., 2002). We consider GARCH factor models with p factors (abbreviated FactorðpÞ) S ij;t ¼ ~s 2 i;t d ij þ Xp k¼1 l ðkþ i l ðkþ j S ðkþ t. (11) Here, ~s 2 i;t is the variance of the residual of time series i after linear regression on the p factors f ðkþ t with k ¼ 1;... ; p, and l ðkþ i is the regression coefficient (factor loading) of factor k on time series i. The dynamics of the residuals is described by a parsimonious GARCH(1,1)-process, and the dynamics of the factors is

16 similarly given by S ðkþ t ¼ a ðkþ 0 þ b ðkþ 1 SðkÞ Author's personal copy t 1 þ aðkþ 1 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) ðf ðkþ t 1 Þ2. (12) The factors f ðkþ t are defined via an iterative procedure: we use the largest eigenvector u ðnþ of C to define f ð1þ t ¼ P N i¼1 uðnþ i i;t. Next, we perform a linear regression of f ð1þ on the i and recompute the correlation matrix from the residuals. From the new eigenvector corresponding to the largest eigenvalue, we define a factor f ð2þ, perform another linear regression, and so on. Such an iterative procedure of calculating factor loadings was found to provide an intuitive decomposition of correlations in the American stock market, whereas the additional orthogonality condition imposed on eigenvectors of the correlation matrix can lead to less intuitive results like the mixing of two business sectors in one eigenvector (Gopikrishnan et al., 2001; Plerou et al., 2002). In contrast to the correlation matrix of the SCM models, the time interval for the calculation of factors and factor loadings is not moving but fixed. For both the SCM and the factor models, the number p of principal components or factors should be chosen according to the number of eigenvalues of the correlation matrix outside the RMT-interval. We compare the SCMðpÞ and FactorðpÞ models to the commonly used Risk Metrics covariance estimator (Longerstaey and Zangari, 1996) defined by S ij;t ¼ 0:94 S ij;t 1 þ 0:06 i;t j;t. (13) This model is easy to estimate and provides an example of a model which is completely dominated by estimation noise when applied to a large number of assets: fluctuations of the covariance strength have a decay constant of approximately 16 time steps, implying that for significantly more than 16 time series the estimated covariance matrix is almost singular. 7. Test method We judge the quality of a multivariate volatility model by its ability to forecast the daily variance of both equally distributed and minimum variance portfolios. The predicted portfolio volatility at time t is given by D 2 t;predicted ¼ XN i;j¼1 m i;t m j;t S ij;t. (14) Here, m i;t is the fraction of the capital invested in stock i at time t. For an equally distributed portfolio, we have m i;t 1=N, hence D 2 is just the average element of the covariance matrix. This test probes the ability of a model to correctly predict the average covariance between stocks. On the other hand, the choice, m i;t ¼ XN j¼1 S 1 ij;t X N S 1 k;l k;l¼1, (15)

17 294 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) minimizes the variance under the constraint that the total invested money is equal to one. The volatility prediction for minimum variance portfolios is a more powerful test than the prediction for equally distributed portfolios, as it not only probes the prediction of the average covariance, but of the whole covariance structure: the covariance matrix is first used to calculate the portfolio weights, and then used to estimate the variance of that portfolio. For both types of portfolios, we compare the predicted volatility for a given day to the realized volatility on that day, which is calculated from high frequency (hourly) data. This type of comparison was shown to provide for a good assessment of the prediction accuracy of univariate GARCH models (Anderson and Bollerslev, 1997). The realized portfolio variance at time t is calculated as D 2 t;realized ¼ XN i;j¼1 m i;t m j;t ðhg i;t g j;t i hg i;t ihg j;t iþ, (16) where pffiffiffiffiffi the expectation values are taken over 10 rescaled hourly returns g i;t ðhþ ¼ 10 ln½si;t ðh þ 1Þ=s i;t ðhþš per trading day, and s i;t ðhþ is the price of stock i on day t and hour h ¼ 0;... ; 10. We calculated the average predicted portfolio variance, the average realized variance, the mean square error (MSE) of the prediction and the MAPE. For equally distributed portfolios, the quality of a volatility model is judged from its tracking error with respect to the true portfolio volatility calculated from intraday data. With respect to minimum variance portfolios, the different volatility models are judged with respect to their ability to: (i) produce portfolios with a low average variance, and (ii) to correctly predict this variance. 8. Empirical results For our empirical analysis, we use two different data sets. Data set one comprises daily closing prices of the 118 most frequently traded German stocks for the period from 12/01/93 until 08/31/01. Data set two contains hourly returns for the same stocks for the period 09/01/00 until 08/31/01, it is used for the estimation of daily volatilities in an out of sample test. The time period 12/01/93 until 08/31/00 contained in data set one is used for parameter estimation, and the year 09/01/00 until 08/31/01 for out of sample covariance predictions from the SCM and factor models. The 118 stocks have been chosen in such a way that for the less frequently traded ones there is at least one transaction per hour in fifty percent of the one-hour intervals. In a first step, we use the estimation period 12/01/93 until 08/31/00 contained in data set one for the calculation of correlation matrices, whose eigenvalue spectrum is then compared to RMT predictions. We define daily returns i;t ¼ ln s i;t ln s i;t Dt with Dt ¼ 1day, and calculate the sample covariance matrix S ij ¼ 1 X T ð i;t h i iþð j;t h j iþ. (17) ðt 1Þ t¼1

18 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) The cross-correlation matrix p C is defined as C ij ¼ S ij =ðs i s j Þ by normalizing with the standard deviations s i ¼ ffiffiffiffiffi S ii. We calculate cross-correlation matrices Ca, C b from: (a) T ¼ 1711 daily returns i;t starting 12/01/93 until 08/31/00, and (b) a subset of T ¼ 250 daily returns starting 09/01/99 until 08/31/00. We diagonalize C a, C b and rank-order their eigenvalues l i ol k for iok. The eigenvalue pdf of C b is displayed in Fig. 4. The bulk of eigenvalues is due to noise and well described by the prediction of RMT (black line). The largest eigenvalue l max;b ¼ 7:1 is clearly above the RMT prediction, two more eigenvalues are just separated from the bulk. The spectrum of C a looks similar to that of C b, with the main difference that the largest eigenvalue l max;a ¼ 14:5 is much larger than in the case of the one year interval. This confirms the result of (Plerou et al., 1999, 2002; Drozdz et al., 2000) that the strength of market correlations changes in time. In both cases, we find that three eigenvalues lie outside the random matrix interval, but only one of them clearly. In previous empirical studies (Gopikrishnan et al., 2001; Plerou et al., 2002) it was found that the time stability of eigenvectors decreases when the corresponding eigenvalue approaches the upper edge of the RMT spectrum. In agreement with this result, the prediction quality of filtered correlation matrices calculated according to the algorithm described in Section 3 does not change much when eigenvalues barely above the RMT edge are included, whereas it decreases with the inclusion of eigenvalues below the RMT edge, see (Rosenow et al., 2002) and the discussion in Section 4. As a conclusion, it seems best to choose p such that only eigenvalues clearly above the upper RMT edge are included. For the data sets studied here, this implies that p ¼ 1 should yield best results, and p ¼ 2 should be as good. Choosing p43 should lead to a decrease in prediction quality. As the estimation of GARCH processes is too complex for considering many different values of p, we restrict ourselves to p ¼ 1; 2 for both SCM and factor models, and in addition p ¼ N for the SCM models pdf eigenvalue Fig. 4. Probability distribution of the eigenvalues of the cross-correlation matrix C b calculated from 118 daily return time series from 09/01/99 until 08/31/00 (gray line). The bulk of eigenvalues is due to noise and well described by the prediction of random matrix theory (black line). One eigenvalue is clearly separated from the bulk and contains information about market correlations, two more eigenvalues are just separated from the bulk.

19 296 B. Rosenow / Journal of Economic Dynamics & Control 32 (2008) From Eq. (2) one sees that the length of the interval ½l ; l þ Š shrinks to zero when Q! 1. The smaller Q gets, the larger is the influence of noise and fewer eigenvalues of the empirical matrix are expected to lie above the RMT prediction. At first sight, the conclusion seems to be that the time period over which the cross-correlation matrix is estimated should be as long as possible. However, we have seen that the largest eigenvalue and hence the strength of market correlations changes in time as l max;a ¼ 14:5 for the seven year interval is much larger than l max;b ¼ 7:1 for the one year interval. The insight gained from an RMT analysis of cross-correlations helps in striking a compromise between the requirement of a long estimation window for increased statistical accuracy and a short estimation window for capturing the dynamics of cross-correlations. The data set for parameter estimation comprises seven years of daily data from 12/01/93 until 08/31/00. These data are used for the maximum likelihood estimation of all GARCH parameters and for the calculation of factors and factor loadings of the models FactorðpÞ-seven. For the models FactorðpÞ-one, only the last year of daily data is used for estimating factors and factor loadings, but the full data set is used for estimating GARCH parameters. The cross-correlation matrix of model SCMðpÞ-seven is calculated over a sliding time window of length seven years, and a sliding window of length one year is used for model SCMðpÞone. We would like to note that the estimation of a given SCM or factor model takes less than half an hour on a PC. We perform an out of sample test from 09/01/00 until 08/31/01. Results for equally distributed portfolios: all models under study overestimate the realized variance. The SCMs with an estimation window of seven years have the largest forecast error. For these models, the average estimated variance is about 65 percent larger than the average realized variance, and the MAPE ranges from 86% to 92%. It seems that the origin of this error is the overestimation of crosscorrelations, which apparently were much higher in the seven years preceding the test period than in the test period itself, see the discussion of the largest eigenvalue of seven- and one-year correlation estimates. A similarly strong overestimation of the variance of minimum variance portfolios was found for the five year estimation period , when applied to the testing period 2000, see Table 2. The SCMs with a one year estimation window perform much better than those with the seven year window: the average estimated variance is only 22% and 26% higher than the realized one, and the MAPE lies between 46% and 49%. Their prediction accuracy is higher than that of the Risk Metrics estimator, which has a MAPE of 53%. For all SCM models, dimensional reduction is unimportant for forecasting equally distributed portfolios, i.e. the results are similar for p ¼ 1, 2, and N. This is expected, as only the average matrix element is probed, and reducing the covariance matrix to its average is already an extreme form of dimensional reduction in itself. In contrast to SCM models, the performance of factor models with a seven year estimation period is comparable to that of models with a one year estimation period. This result indicates that factor models are indeed able to describe the time dependence of the average correlation strength. The average predicted portfolio