Financial Data Mining Using Flexible ICA-GARCH Models

Transcription

1 55 Chapter 11 Financial Data Mining Using Flexible ICA-GARCH Models Philip L.H. Yu The University of Hong Kong, Hong Kong Edmond H.C. Wu The Hong Kong Polytechnic University, Hong Kong W.K. Li The University of Hong Kong, Hong Kong AbstrAct As a data mining technique, independent component analysis (ICA) is used to separate mixed data signals into statistically independent sources. In this chapter, we apply ICA for modeling multivariate volatility of financial asset returns which is a useful tool in portfolio selection and risk management. In the finance literature, the generalized autoregressive conditional heteroscedasticity (GARCH) model and its variants such as EGARCH and GJR-GARCH models have become popular standard tools to model the volatility processes of financial time series. Although univariate GARCH models are successful in modeling volatilities of financial time series, the problem of modeling multivariate time series has always been challenging. Recently, Wu, Yu, & Li (006) suggested using independent component analysis (ICA) to decompose multivariate time series into statistically independent time series components and then separately modeled the independent components by univariate GARCH models. In this chapter, we extend this class of ICA-GARCH models to allow more flexible univariate GARCH-type models. We also apply the proposed models to compute the value-at-risk (VaR) for risk management applications. Backtesting and out-of-sample tests suggest that the ICA-GARCH models have a clear cut advantage over some other approaches in value-at-risk estimation. DOI: / ch011 Copyright 010, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

2 IntroductIon In econometrics, volatility modeling of financial time series has received a lot of attention due to its wide applications in finance such as option pricing and risk management. Among the existing volatility models, one of the most important models is the autoregressive conditional heteroscedasticity (ARCH) model proposed by (Engle, 198) which was further extended to generalized ARCH (GARCH) model by (Bollerslev, 1986). After the success of ARCH and GARCH models, researchers further proposed different types of GARCH models such as EGARCH (Nelson, 1991) and GJR-GARCH (Glosten, Jaganathan, & Runkle, 1993), etc. These univariate GARCH models are capable in capturing the dynamics of volatilities from the characteristics of financial time series. Although GARCH models are successful in modeling volatilities of univariate financial time series, the problem of modeling multivariate time series still raises challenges in this research area. It is mainly because in existing multivariate GARCH models, the number of unknown parameters grows very fast with the number of time series in the model. For example, in (Engle, 00), Engle et al. compared the complexity of several multivariate GARCH models and most of them have the complexity of O(N ) or even O(N 3 ), where N is the number of time series. However, in practice, we often need to extend the volatility modeling to high dimensional cases. For instance in portfolio optimization, a portfolio could contain several hundred of stocks. Therefore, new approaches are needed to deal with such situations. Recently, (Wu, Yu & Li, 006) suggested using independent component analysis (ICA) to decompose multivariate time series into statistically independent time series components and then separately modeled the independent components (ICs) by univariate GARCH models. Their experiment results showed that the ICA-GARCH models are more effective in capturing the time-varying features of volatilities and provide better value-at-risk estimate than existing methods including DCC (Engle, 00), PCA-GARCH (Alexander, 001) and RiskMetrics. In this chapter, we extend this class of ICA-GARCH models to allow more flexible univariate GARCH-type models. In addition to the popular ICs extraction methods FastICA algorithms, we also consider other ICA algorithms which can extract ICs from non-stationary data (Hyvärinen, 001) as most financial time series are non-stationary. The rest of this chapter is organized as follows: In Section, we introduce the univariate GARCH process and several of its extended models. Then, we propose flexible ICA-GARCH models for multivariate volatility modeling in Section 3. In Section 4, we consider the estimation of value at risk of a single asset or a portfolio based on the flexible ICA-GARCH models. Experimental results are given in Section 5. Finally, we conclude in Section 6. VolAtIlIty Models In the following, we introduce several prevailing volatility models for financial time series. GArcH (p,q) Model In GARCH models, financial time series {y t } are assumed to be generated by a stochastic process with (conditional) time-varying volatility {σ t }. The general GARCH (p, q) model ( p > 0 and q 0 are integers) is defined as 56

3 y = m+ e = m+ s z, z ~ D(, 01 ) t t t t t (1) s = a + bs + ae t 0 p q å i t-i å j t-j i= 1 j= 1 where α 0 > 0, α i 0 for i = 1,, q and β j 0 for j = 1,, p, and D(0,1) represents a conditional distribution with zero mean and unit variance such as the standard Gaussian distribution N(0,1) and the Student s t distribution (standardized to unit variance). Since GARCH(1,1) model was found to be adequate for many financial time series (Bollerslev, Chou, & Kroner, 199), we focused on this model in our empirical analysis. egarch (p,q) Model Note that the volatility σ t from the GARCH model is symmetric between positive and negative error shocks ε t. However, higher volatilities are often seen for negative shocks than positive shocks. (Nelson 1991) proposed the exponential GARCH (EGARCH) model to allow volatility asymmetry: p q é e ì e ü ù t j t j ln( s ) = a + b ln s + a - -E t 0 i t i j í ï - å - i 1 s s t j î ý ï å = - ï ë ê t-j þï û ú + j å = 1 j= 1 q t-j g e i st-j () where E t - j e { / s } = n- e p for standard Gaussian and E t j - n { s } = - G 1 { } t-j t j p n - { } for Student s t with degrees G of freedom v >. Note that the γ j governs the volatility asymmetry effect. For example, a negative γ j implies that a negative shock increases future volatility while a positive shock reduces future volatility. GJr-GArcH (p,q) Model Another useful GARCH model that can describe the asymmetric conditional volatilities is the GJR- GARCH model (Glosten, Jaganathan, & Runkle 1993). The general GJR-GARCH (p, q) model assumes that the conditional variance at time t follows: s = a + bs + ae + g I e t 0 p q q å i t-i å j t-j å (3) j t-j t-j i= 1 j= 1 j= 1 where I t-j = 1 if ε t-j < 0; otherwise I t-j = 0, and α 0 > 0, β 1 0, α j 0, α j + γ j 0, i = 1,, p; j = 1,, q. So γ j > 0 indicates that future volatility is always higher for a negative shock. We note that GARCH and GJR-GARCH models allow for volatility clustering (i.e., persistence) by a combination of the β i and α j terms, whereas persistence in EGARCH models is entirely captured by the β i terms. 57

4 the FlexIble IcA-GArcH Models In this section, we first introduce the method of ICA and then describe the procedure of applying ICA in multivariate volatility modeling. What is IcA? Independent component analysis (ICA) (Comon, 1994) is a data mining technique which aims to express the observed data in terms of a linear combination of underlying latent variables. These latent variables are assumed to be non-gaussian and mutually independent. A typical ICA model for an N-dimensional multivariate time series {x t = (x it,,x Nt ) : t = 1,,T} is: x t = As t (4) where s t is a vector of statistically independent latent variables called the independent components (ICs), and A is an unknown constant mixing matrix. In this paper, we only consider the case that A is a square matrix. The task of ICA is to identify both the ICs and the matrix A. That is to find W such that the unmixed data y t = Wx t have components of y t as independent as possible and hence y t provides an estimate of s t. In general, W differs from A -1 by a rotation and scale transformation. Various algorithms for parameter estimation have been developed for ICA. Among them, a widely used one is the FastICA algorithm proposed by (Hyvärinen, 1999; Hyvärinen, & Oja, 1997), which is a fast fixed point algorithm (FastICA) for maximizing the non-gaussianity of y it. It was proven that the solutions to this optimization problem give the ICs (see (Hyvärinen, Karhunen, & Oja, 001)). The FastICA algorithm aims to maximize a non-gaussianity measure so-called negentropy which is approximated by the function {E[G(y)] E[G(y gauss )]} where G is a non-quadratic even function, y is an IC and y gauss is Gaussian with the same variance to that of y. Some popular choices of g = G are the derivative of a standard Gaussian density (Guassian), the cubic power function (pow3) and the hyperbolic tangent (tanh). Note that the FastICA algorithm assumes that all ICs are independently and identically distributed over time. However, a time-varying volatility is a common stylized fact in financial time series. It is thus more natural to assume that the ICs are non-stationary with variance changing over time. An alternative approach to separate non-stationary multivariate time series was introduced in (Hyvärinen, 001), where Hyvärinen proposed a cumulant-based approach to find the non-stationary components {y it }. This approach aims to maximize the nonstationarity of y it as measured by the fourthorder cross-cumulant of y it : cum( y, y, y, y ) = E{ y y }-Ey { } Ey { it it it, -t it, -t it it, -t it it, -t } ( Eyy { }) - it it, - t (5) where τ is the time lag. If the series y it is serially uncorrelated, and has zero mean and unit variance, this cumulant is simply the lag-τ autocorrelation ofy it, i.e., corr( y, y ) it it, -t. The ICs are estimated by finding the linear combinations w x t, such that the absolute value of the cross-cumulant is maximized: max cum( w wx ', wx ', wx ', wx ' ) (6) t t t-t t-t 58

5 under the constraint: Var (w x t ) = 1. In (Hyvärinen, 001), Hyvärinen developed a fast fixed-point algorithm similar to the FastICA algorithm for separating ICs by nonstationarity, using cross-cumulants. Hyvärinen showed that the maximally nonstationary linear combinations give the ICs. A second method of separation of nonstationary components is to use a conditional-decorrelation approach proposed by (Matsuoka, Ohya, & Kawamoto, 1995). They showed that if the latent components are conditionally uncorrelated and their local variances fluctuate independently of each other, the components and the mixing matrix can be determined uniquely. In this approach, the ICs are estimated by minimizing the conditional uncorrelatedness of y t as measured by: å Q( W, t) = ln E { y }- ln E{ y y } i t it t t (7) where E t represents the conditional expectation, which can be estimated based on the data around the time point t. building the Flexible IcA-GArcH Models The flexible ICA-GARCH model works as follows: In the first step, we remove the autocorrelation of each return series x it by an autoregressive AR(p i ) model: x = j + j x + + j x + e (8) it 0 1 it, -1 1 it, -p it i where e it is assumed to be a white noise series with mean zero and variance s i. The AR order p i is usually unknown and is determined by choosing the order with the smallest value in the Bayesian information criterion (BIC): BIC = LLF + N para ln(n obs ) (9) where LLF is the value of the maximized log-likelihood function of an AR model under consideration, N para is the number of parameters in the model and N obs is the sample size of the observed return series. After choosing the appropriate AR model for each return series x it, we use ICA to decompose the residual vector e t (obtained from the fitted AR models) into independent components {s i,t }, i = 1,, N, i.e., e t = As t with s t = (s 1,t,,s N,t ). Then, we can model each IC s i,t, by different univariate GARCH-type models mentioned in Section. More specifically, the following six GARCH-type models will be fitted for each IC: GARCH(1,1) with Gaussian error GARCH(1,1) with t error EGARCH(1,1) with Gaussian error EGARCH(1,1) with t error GJR-GARCH(1,1) with Gaussian error GJR-GARCH(1,1) with t error 59

6 By selecting the most suitable GARCH-type model automatically for each IC using the BIC criterion, we end up with a flexible class of ICA-GARCH models. We will describe this in details later. Using the mixing matrix A, the (conditional) covariance matrix of the original return vector x t = (x 1,t,,x N,t ) at time t is given by: H t = AV t A (10) where V t is a diagonal matrix with diagonal elements being the volatilities of independent components s t. Because the N components in s t are independent, such an approach will not significantly increase the computational complexity while retaining a very high accuracy. The ICA-GARCH model allows the multivariate volatilities of N return series to be generated from N univariate GARCH-type models. selecting the best IcA-GArcH Model Notice that the likelihood function of the residual vectors {e t } is given by: T LF = Õ{ p i ( we i t ) det W } t= 1 (11) where p i is the density of the i-th IC fitted by a chosen GARCH-type model, and w i denotes the i-th row of W. As stated in Section 3.1, the ICs and the matrix W can be estimated using any one of the five methods: three FastICA algorithms, cumulant-based and conditional-decorrelation approaches. Also, for each of the five ICA models, the estimated ICs will be estimated by any one of the six GARCH-type models. Our task is to find the optimal ICA model and the most suitable univariate GARCH-type models for each IC. To do so, we propose to use an overall BIC to measure the fitness of an ICA-GARCH model. First of all, for a certain ICA model M m ( 1 m 5), we can use the BIC to select the most suitable GARCH-type model for each IC. As all the ICs in the ICA model are mutually independent, we can determine the likelihood function of e using (11). Because the number of parameters of each GARCH-type model is also estimated, we can use these information to calculate an overall BIC of the ICA-GARCH model with ICs determined by the ICA model M m : å Overall- BIC = BIC m N i= 1 im (1) where BIC im is the BIC value for the most suitable GARCH-type model for the i-th IC estimated using the ICA model M m. Finally, we will choose the best ICA-GARCH model with minimal overall BIC value as the optimal model for the residuals {e t }. Combining with the selected AR models, we obtain a flexible ICA-GARCH model for the original multivariate time series {x t }. 60

7 IcA-GArcH For VolAtIlIty estimation And ForecAstInG In VAr APPlIcAtIons In this section, we introduce the estimation procedure of value at risk (VaR) by using the ICA-GARCH model for multivariate volatility modeling based on univariate GARCH models. There are a lot of researches in using GARCH models for VaR estimation. For example, (So & Yu, 005) gave an empirical studies on VaR estimation using various GARCH models in different market indexes. Value at risk We first briefly introduce the concept of value at risk. Value at risk (VaR) is a widely accepted measure of market risk in many risk management applications. VaR represents the maxial loss of an underlying asset that will occur during a target horizon with a specified probability. Mathematically, it is defined under a probabilistic framework: p = Pr( ΔV t+1 VaR ) (13) where p is the prespecified probability of interest, such as p = 5% or p = 1%. For a single asset, the next day s profit (or loss if negative) is ΔV t+1 = Q 0 (P t+1 P t ), where Q 0 is the quantity of the underlying asset and P t is the asset s market price at time t. Alternatively, we can use log returns to represent ΔV t+1. Our objective is to first estimate the volatilities of multivariate time series by using the proposed models, and then to compute the time-varying VaRs. In essence, the usefulness of VaR estimation relies on the accurate estimation of volatilities. Therefore, we are interested in assessing the performance of ICA-GARCH in VaR estimation. The calculation of VaRs is as follows: we first need to fit one of the above models to the multivariate time series, and then use the estimated model parameters to forecast the next period volatility forecasts. Based on the forecasts of volatilities, we can compute the forecasts of VaRs using Monte Carlo simulation. More specifically, we use Monte Carlo simulation to generate a large number of hypothetical changes ΔV t+1, e.g., 10,000. Then, we obtain the simulated distribution of ΔV t+1. The 5% or 1% VaRs will be the 5% or 1% quantiles of the simulated ΔV t+1. backtesting Vars Backtesting VaRs is a statistical framework that verifies whether the actual losses are consistent with the forecasting losses. It is a crucial model validation step to check whether or not a VaR model is adequate. We can also use backtesting to compare the performance of different volatility models in VaR estimation by using backtesting. A common method to verify the accuracy of a model is to record the failure rate, which represents the proportion of times VaRs exceeded the actual loss in a given sample. Ideally, the failure rate should be an unbiased estimator of p, and the failure rate should converge to p as the sample size increases. That is, when we compare the actual losses with the estimated VaRs, the percentage of losses that exceed the VaRs should be close to the specified levels (e.g., 5% or 1%) if the volatilities forecast is accurate 61

8 Financial Data Mining Using Flexible ICA-GARCH Models Figure 1. Residual series enough. For example, if we forecast the VaRs of the next 1,000 trading days, if the volatility model is good enough, the number of days that exceeds the VaRs should be close to 50 days if p = 0.05 or 10 days if p = experiments There are two parts in this experimental section. In the first part, we test and validate the effectiveness of ICA-GARCH model in multivariate volatilities modeling. Then, we use backtesting to check the performance of the ICA-GARCH models for practical VaR applications. 6

9 Table 1. Selected models and parameter estimates of the flexible ICA-GARCH model Series IC1 IC IC3 IC4 IC5 IC6 IC7 IC8 Best model GARCH-T GARCH-T EGARCH-T GARCH-T GJR-T EGARCH GJR GJR-T α α β γ d.f.* * d.f. stands for the degrees of freedom of a t error data description We used the historical data of MSCI market price Index from eight developed financial markets, including (1) United States, () United Kingdom, (3) Japan, (4) Hong Kong, (5) Singapore, (6) Australia, (7) Germany and (8) Canada. All the indexes are US dollar based. The dataset is from the periods of July 9, 001 to July 6, 006, representing 1,30 daily observations. For model comparison, we divide the dataset into two parts. The first part is the in-sample data consisting of the first 1,10 observations for model training while the remaining 00 observations are out-of-sample data for forecasting evaluation. The daily returns x it are calculated by x it = ln ( P it ) ln ( P i,t-1 ), where P it is the closing price of index i on the trading day t. Then, we employed the AR model to filter the autocorrelation of the return series. The in-sample residuals of the eight series are plotted in Figure 1. Multivariate Volatility Modeling Here, we choose five ICA algorithms: Cumulant-Based (CB), Conditional-Decorrelation (CD), and three versions of FastICA algorithms (FastICA(pow3), FastICA(tanh) and FastICA(Gaussian)). In the GARCH modeling of ICs, we provide six choices: GARCH, GARCH(T), EGARCH, EGARCH(T), GJR and GJR(T). The model selection is based on the BIC criterion we introduce in the previous section. The results are shown in Table 1. According to the overall BIC measure, the flexible ICA-GARCH model selects the CD method as the best ICA algorithm to decompose the eight residual series. The GARCH models selected for each independent component are shown in the second row of Table 1. We note that five different GARCHtype models are chosen to model the eight residual series. The diversity of models selected implies the necessity of using flexible models to better reflect the complexity of multivariate volatility modeling such as heavy tailed distribution and volatility asymmetry. To compare the performance of the flexible ICA-GARCH models with the standard ICA-GARCH models when all ICs are estimated by the same GARCH-type model, we also estimated their conditional volatilities which are shown in Figure, Figure 3, Figure 4, Figure 5 and Figure 6. Note that the ICA- GARCH, ICA-EGARCH, ICA-GARCH(T), ICA-GJR and ICA-GJR(T) models assume a common GARCH-type specification for all ICs. Comparing with the residual series, we can see that the flexible ICA-GARCH is the best model in modeling the dynamic changes of volatilities. For example, the flexible ICA-GARCH indicates that 63

10 Financial Data Mining Using Flexible ICA-GARCH Models Figure. Conditional volatilities by flexible ICA-GARCH series No.1, No. and No.7 demonstrate greater volatilities around the 00-th observation whereas most of the other models are relatively flatten. experiments for backtesting Vars In this section, we implemented the backtesting for the models considered. For risk management purposes, we consider 95 percent (p = 0.05) and 99 percent (p = 0.01) confidence levels of the VaRs. Since one can long (buy) or short (sell) a price index to make profit or loss, we therefore consider both positions for the VaR calculation. Actually, long position focuses on the left-hand tail of the return distribution while short position focuses on the right-hand tail. In this way, we can completely check the effectiveness of models on modeling the tail behavior. 64

11 Financial Data Mining Using Flexible ICA-GARCH Models Figure 3. Conditional volatilities by ICA-EGARCH We also separately assess the performance of model estimation using in-sample data and evaluate the abilities of forecasting by using out-of-sample data described in the previous section. The corresponding results are listed in Table and Table 3. In backtesting, we suggest using the mean rank to measure and compare the overall performance of the seven models: flexible ICA-GARCH (Flex-ICA), ICA-GARCH, ICA-GARCH(T), ICA-EGARCH, ICA-EGARCH(T), ICA-GJR, and ICA-GJR(T). To calculate the mean rank, we first need to sort the values according to a standard from best to worst and assign ranks 1,,,7. The backtesting standard used here is the closeness to the specified level p. If more than one models have the same distance, these models share the ranks assigned to them. The overall mean ranks of the models are listed on the last row of each sub-table. It is clear that the flexible ICA-GARCH model is the winner in most of the cases we considered, especially in prediction tests. From the VaR simulation, we can see that the flexible ICA-GARCH model 65

12 Financial Data Mining Using Flexible ICA-GARCH Models Figure 4. Conditional volatilities by ICA-GJR can improve the modeling quality significantly. The cost is that the computation times of a flexible model will be slightly greater. We also construct an equally weighted portfolio (EWP) consisting with the eight indexes, that is, we give the same amount of investment to each index. Then, we would like to compute the portfolio VaR by using the models. To do so, we can forecast the portfolio s returns based on individual price index s forecasts. The formula is: rn, t +1 ö æ r1, t +1 ççe + +e r p, t +1 = ln ç çç N è ø 66 (14)

13 Figure 5. Conditional volatilities by ICA-EGARCH(T) Where r p,t+1 is the 1-day ahead log return forecast of the equally weighted portfolio with N indexes. The results of backtesting portfolio VaRs are shown in the second last row of each sub-table in Table 3. The results show that the flexible ICA-GARCH model again performs the best in estimating portfolio VaRs. 67

14 Financial Data Mining Using Flexible ICA-GARCH Models Figure 6. Conditional volatilities by ICA-GJR(T) conclusion In this chapter, we have enriched the class of ICA-GARCH models by including more choices of ICA and GARCH models based on the BIC measure. The flexible ICA-GARCH model shows greater adaptability to mine hidden patterns from multivariate time series data. The experimental results validate the usefulness of the flexible ICA-GARCH models in multivariate volatility modeling and VaR estimation. It appears that the flexible ICA-GARCH models have some clear cut advantages over some existing models. Two main advantages of the ICA technique are the suitability of ICA for non-gaussian time series modeling and the independence property of the components. These two features greatly reduce the 68

15 Table. Backtesting VaR methods with in-sample data Index Flex-ICA ICA ICA-E ICA-G ICA-T ICA-ET ICA-GT p = 0.05 Long Position US UK JP HK SG AU GE CA Rank p = 0.05 Short Position US UK JP HK SG AU GE CA Rank p = 0.01 Long Position US UK JP HK SG AU GE CA Rank p = 0.01 Short Position US UK JP HK SG AU GE CA Rank

16 Table 3. Backtesting VaR methods with out-of-sample data Index Flex-ICA ICA ICA-E ICA-G ICA-T ICA-ET ICA-GT p = 0.05 Long Position US UK JP HK SG AU GE CA EWP Rank p = 0.05 Short Position US UK JP HK SG AU GE CA EWP Rank p = 0.01 Long Position US UK JP HK SG AU GE CA EWP Rank p = 0.01 Short Position US UK JP HK SG AU

17 Table 3. continued GE CA EWP Rank problems of model complexity and mis-specification. Moreover, since ICA also serves as a factor model, the independent components may carry financial implications. For example, we can check the proportion of variation explained by each IC and then we can identify some important factors that can interpret the results. Moreover, the relative loadings of each series on each common factor may reveal some interesting financial implications. For instance, we may be able to interpret one common IC as the global market volatility factor shared by all the market indexes series. Other independent factors may be classified as country-specific factors which have different impacts on different markets. Exploring such financial implications may help us to understand better the underlying relationships among the series in terms of their volatilities. references Alexander, C. O. (001). Orthogonal GARCH. In C.O. Alexander (Ed.), Mastering Risk, (Vol. ). London: Financial Times-Prentice Hall. Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics, 31(3), doi: / (86) Bollerslev, T., Chou, R. Y., & Kroner, K. F. (199). ARCH modeling in finance; A review of the theory and empirical evidence. Journal of Econometrics, 5, doi: / (9)90064-x Comon, P. (1994). Independent component analysis: a new concept? Signal Processing, 36, doi: / (94) Engle, R. (198). Autoregressive conditional heteroscedasticity with estimates of the variance of the U.K. inflation. Econometrica, 50(4), doi:10.307/ Engle, R. (00). Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business & Economic Statistics, 0(3), doi: / Glosten, L. R., Jaganathan, R., & Runkle, D. E. (1993). On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks. The Journal of Finance, 48(5), doi:10.307/39067 Hyvärinen, A. (1999). Fast and robust fixed-point algorithms for independent component analysis. IEEE Transactions on Neural Networks, 10(3), doi: /

18 Hyvärinen, A. (001). Blind source separation by nonstationarity of variance: A cumulant based approach. IEEE Transactions on Neural Networks, 1(6), doi: / Hyvärinen, A., Karhunen, J., & Oja, E. (001). Independent Component Analysis. New York: John Wiley & Sons. Hyvärinen, A., & Oja, E. (1997). A fast fixed-point algorithm for independent component analysis. Neural Computation, 9, doi:10.116/neco Matsuoka, K., Ohya, M., & Kawamoto, M. (1995). A neural net for blind separation of nonstationary signals. Neural Networks, 8, doi: / (94)00083-x Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59, doi:10.307/93860 So, M. K. P., & Yu, P. L. H. (006). Empirical analysis of GARCH models in value at risk estimation. Journal of International Financial Markets, Institutions and Money, 16(), doi: /j. intfin Wu, E. H. C., Yu, P. L. H., & Li, W. K. (006). Value at Risk estimation using independent component analysis-generalized autoregressive conditional heteroscedasticity (ICA-GARCH) models. International Journal of Neural Systems, 16(5), doi:10.114/s