Abstract
 Top of page
 Abstract
 1. Introduction
 2. Methodology
 3. Simulation
 4. Real data examples
 5. Conclusional remark
 Acknowledgements
 References
 Appendix
Summary. We propose to model multivariate volatility processes on the basis of the newly defined conditionally uncorrelated components (CUCs). This model represents a parsimonious representation for matrixvalued processes. It is flexible in the sense that each CUC may be fitted separately with any appropriate univariate volatility model. Computationally it splits one high dimensional optimization problem into several lower dimensional subproblems. Consistency for the estimated CUCs has been established. A bootstrap method is proposed for testing the existence of CUCs. The methodology proposed is illustrated with both simulated and real data sets.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Methodology
 3. Simulation
 4. Real data examples
 5. Conclusional remark
 Acknowledgements
 References
 Appendix
One of the most prolific areas of research in financial econometrics literature in the last two decades has been to model time varying volatility of financial returns. Many statistical models, most designed for univariate data, have been proposed for this. From a practical point of view, there are at least two incentives to model several financial returns jointly. First, time varying correlations between different securities are important and useful information for portfolio optimization, asset pricing and risk management. Secondly, models for a single security may be improved by incorporating the relevant information in other related models. The quest for modelling multivariate processes, which are often represented by conditional covariance matrices, has motivated the attempts to extend univariate volatility models to multivariate cases, aiming for practical and/or statistical effectiveness. We list some of the endeavours below.
Let {X_{t}} be a vectorvalued (return) time series with
where ℱ_{t} is the σalgebra that is generated by {X_{t},X_{t−1},…}, and Σ_{t} is an ℱ_{t−1}measurable d×d semipositive definite matrix. One of the most general multivariate generalized autoregressive conditional heteroscedasticity GARCH(p,q) models is the BEKK representation (Engle and Kroner, 1995)
 ( (1.1))
where C, A_{ij} and B_{ij} are d×d matrices, and C is positive definite. Although the form of this model is quite general especially when m is reasonably large (proposition 2.2 of Engle and Kroner (1995)), it suffers from overparameterization. Similar to multivariate autoregressive moving average models, not all parameters in model (1.1) are necessarily identifiable even when m=1. Overparameterization will also lead to a flat likelihood function, making statistical inference intrinsically difficult and computationally troublesome (Engle and Kroner, 1995; Jerez et al., 2001).
To overcome the difficulties due to overparameterization, a dynamic conditional correlation (DCC) model (Engle, 2002; Engle and Sheppard, 2001) has been proposed. It is based on the decomposition
 ( (1.2))
where , σ_{t,ii} is the conditional variance of the ith component of X_{t} and R_{t}≡(ρ_{t,ij}) is the conditional correlation matrix. A simple way to facilitate such a model is to model each σ_{t,ii} with a univariate volatility model and to model conditional correlation by using a rolling exponential smoothing as follows:
where and λ_{i},λ_{j} ∈ (0,1) are constants. Even with such a simple specification, the estimation typically involves solving a high dimensional optimization problem as, for example, the Gaussian likelihood function cannot be factorized into several lower dimensional functions. To overcome the computational difficulty, Engle (2002) proposed a twostep estimation procedure as follows: first fit each σ_{t,ii} in equation (1.2) with a univariate GARCH(1,1) model, and then model the conditional correlation matrix R_{t} by the simple GARCH(1,1) form
 ( (1.3))
and ɛ_{t} is a (d×1)vector of the standardized residuals that are obtained in the separate GARCH(1,1) fittings for the d components of X_{t} and R is the sample correlation matrix of ɛ_{t}. There are only two unknown parameters θ_{1} and θ_{2} in model (1.3), so it can be easily implemented even for large or very large d. However, it may not provide adequate fitting when the components of X_{t} exhibit different dynamic correlation structures; see the real data examples in Section 4 later. Indeed, the conditional correlation matrix in model (1.3) is a linear combination of the static sample correlation matrix R and the exponential smoothing of , which is a nonparametric estimator. When θ_{1}+θ_{2}=1, it is a pure nonparametric (exponential smoothing) estimator. The biases are inevitable in such an estimation for the conditional correlation.
Alexander (2001) proposed an orthogonal GARCH model which fits each principal component with a univariate GARCH model separately and treats all principal components as conditionally uncorrelated random variables. Since principal components are only unconditionally uncorrelated, such a misspecification may lead to nonnegligible errors in the fitting; see the first example in Section 4.
In this paper, we propose a new alternative for modelling multivariate volatilities. The basic idea is to assume that X_{t} is a linear combination of a set of conditionally uncorrelated components (CUCs); see Section 2.1. One fundamental difference from the orthogonal GARCH model is that we use CUCs, instead of PCs, which are genuinely conditionally uncorrelated. The advantages of the new approach include
 (a)
the CUC decomposition leads to a parsimonious and identifiable representation, and the number of parameters in the model is significantly reduced comparing with for example, the BEKK representation or the vectorized multivariate GARCH models,
 (b)
it has the flexibility to model each CUC separately with any appropriate univariate volatility models,
 (c)
computationally it splits a high dimensional optimization problem into several lower dimensional subproblems and
 (d)
it allows the volatility model for one CUC to depend on the lagged value of the other CUCs.
However, the estimation of CUCs involves solving a nonlinear optimization problem with d(d−1)/2 variables, where d is the dimension of X_{t}. This poses some limitation on the dimensionality d with the available computing capacity. We view the CUC as a model that is capable of catching sophisticated dynamical correlation structures, but its potential may only be fully capitalized with further development in computing power and/or high dimensional optimization algorithms.
The idea of using CUCs is similar to the socalled independent component analysis (Hyvärinen et al., 2001). However, instead of requiring that all the component series are independent of each other, we impose only a weaker condition that the component series are conditionally uncorrelated; see condition (2.1) below. This relaxation is critical for the problem that is of concern in this paper. Of course, like independent components, CUCs may not always exist. We propose a bootstrap test to assess the existence of CUCs. Our empirical experience indicates that, for a large number of practical examples with small or moderately large d, there is no significant evidence to reject the hypothesis on the existence of CUCs.
The rest of the paper is organized as follows. Section 2 contains a detailed description of the new methodology proposed and the associated theoretical results. Simulation results are reported in Section 3. Illustration with two real data examples of dimension d=4 and d=10 is presented in Section 4. Applicability of the CUC method beyond its standard setting is discussed in Section 5. Technical proofs are relegated to Appendix A.
The data that are analysed in the paper and the program that was used can be obtained from
3. Simulation
 Top of page
 Abstract
 1. Introduction
 2. Methodology
 3. Simulation
 4. Real data examples
 5. Conclusional remark
 Acknowledgements
 References
 Appendix
We conduct a Monte Carlo experiment to illustrate the CUC approach proposed. In particular we check the accuracy of the estimation for the transformation matrix A in equation (2.2).
We consider a CUC extended GARCH(1,1) model with d=3:
 ( (3.1))
where , i=1,2,3, and parameter values given in Table 1. It is easy to see that A^{T}A=I_{3} and γ_{i}=1−α_{i1}−α_{i2}−α_{i3}−β_{i}. Thus the variances of the CUCs are 1. Since α_{11}+α_{12}+α_{13}+β_{1}=0.98, the volatility for the first CUC is highly persistent. In contrast, the volatility persistence in the third component is less pronounced, as α_{31}+α_{32}+α_{33}+β_{3}=0.72 only.
Table 1. Parameter values A  i  γ_{i}  β_{i}  α_{i1}  α_{i2}  α_{i3} 

0  0.500  0.866  1  0.02  0.90  0.04  0  0.04 
0  0.866  −0.500  2  0.10  0.80  0  0.10  0 
−1  0  0  3  0.28  0.60  0  0  0.12 
For each of 800 samples with size n=500 or n=1000 generated from the above model, we estimated A by minimizing Ψ_{n}(A) defined in expression (2.5). As far as the estimation of A is concerned, two orthogonal matrices are treated as identical if the Ddistance between them is 0; see equation (2.6). The coefficients α_{ij}, β_{i} and γ_{i} were estimated by using QMLE based on Gaussian likelihood. The estimates are summarized in Table 2 and Fig. 1. Estimation errors for α_{12}, α_{21}, α_{23}, α_{31} and α_{32} are all very close to 0 and are not reported here for brevity.
Table 2. Summary statistics of the estimation errors in simulation n  Parameter         

500  Mean  0.130  0.842  0.035  0.041  0.761  0.076  0.616  0.084 
Median  0.128  0.884  0.030  0.036  0.803  0.072  0.668  0.076 
Standard deviation  0.080  0.147  0.030  0.029  0.175  0.045  0.257  0.058 
Bias  —  −0.058  −0.005  0.001  −0.039  −0.024  0.016  −0.036 
Rootmeansquared error  —  0.158  0.031  0.029  0.180  0.052  0.258  0.068 
1000  Mean  0.114  0.869  0.037  0.037  0.782  0.077  0.616  0.089 
Median  0.102  0.885  0.036  0.035  0.804  0.076  0.641  0.087 
Standard deviation  0.077  0.078  0.019  0.019  0.119  0.033  0.214  0.043 
Bias  —  −0.031  −0.003  −0.003  −0.018  −0.023  0.016  −0.032 
Rootmeansquared error  —  0.084  0.020  0.019  0.120  0.041  0.215  0.054 
Both the means and the standard deviations of are small. This indicates that the estimation for A seems to be reasonably accurate. The coefficients in each CUC model were also estimated accurately. The estimators are almost unbiased, as the biases are negligible in comparisons with the corresponding variances. The errors in estimation decrease as the sample size increases from 500 to 1000, roughly by a factor of √2.
Since most of the biases that are reported in Table 2 are negative (see also Fig. 1), the coefficients in the GARCH models for CUCs were slightly underestimated. Also note that the estimation errors decrease when the volatility persistence (which is measured by α_{i1}+α_{i2}+α_{i3}+β_{i}) increases; see Fig. 1(a) with the sample size 1000. Fig. 1(b) presents the estimation errors of the GARCH coefficients with A given. The difference between the estimation errors of the two cases is small.
4. Real data examples
 Top of page
 Abstract
 1. Introduction
 2. Methodology
 3. Simulation
 4. Real data examples
 5. Conclusional remark
 Acknowledgements
 References
 Appendix
In this section we illustrate the method proposed with two real data examples with d=4 and d=10. First we analyse the 2527 daily logreturns (in percentages) of the Standard and Poors 500 index S&P500, the stock prices of Cisco Systems, Intel Corporation and Sprint in the period from January 2nd, 1991, to December 31st, 2000. This data set was downloaded from Yahoo!Finance. The close prices adjusted for dividends and splits were used to produce the return series that are plotted in Fig. 2. We use the first 2275 observations (i.e. the data up to the end of 1999) for estimating the parameters in the models, and we leave the last 252 data points (i.e. the data in 2000) for checking the postsample forecasting performance.
To account for the conditional mean of the return series, a vector AR(2) model, which was selected by both M(i) (Tiao and Box, 1981) and the Akaike information criterion, was first fitted to the data. We denote by Y_{t}, t=1,2,…,2273, the residuals that resulted from this fitting. In what follows, we focus on modelling the conditional covariance matrix process of Y_{t}.
Let S be the sample covariance matrix of Y_{t}, and X_{t}=S^{−1/2}Y_{t}. The estimator was obtained by minimizing Ψ_{n}(A). For comparison, the estimator that is obtained by maximizing the likelihood function of the GOGARCH(1,1) model (van der Weide, 2002) was also computed. We applied the bootstrap test that was described in Section 2.4, with bootstrap sampling repeated 400 times, to test for the existence of the CUCs and we obtained the Pvalue 0.34. This indicates that there is no significant evidence against the hypothesis that the CUCs exist for this data set. The 95% bootstrap confidence set for the transformation matrix A is . Since , I_{4} is not contained in the confidence sets. Thus the principal components cannot be taken as the CUCs. Also ; therefore is not contained in the confidence set either. This suggests that the MLE that is based on the GOGARCH(1,1) model does not lead to CUCs and, therefore, it would be inappropriate to assume that the conditional covariance matrix of is diagonal, as implied by the GOGARCH(1,1) approach.
Table 3 lists the estimated extended GARCH(1,1) models for the estimated CUCs. The models were selected by the algorithm that was specified in Section 2.3.3. There is a causalityinvariance relationship from the fourth CUC to the second CUC. Also the last two CUCs are highly persistent as the sum of all the GARCH and ARCH coefficients is close to 1 for both of them. On the basis of the fitted volatility (i=1,2,3,4) for the CUCs, the conditional covariance matrix for the original residuals Y_{t} is of the form
 ( (4.1))
Table 3. Extended GARCH model for CUCs of the S&P500–Cisco Systems–Intel Corporation–Sprint data j  j_{i}  

1   
2  4  
3   
4   
where .
For the comparison, we also computed the estimated volatility processes for Y_{t} based on the OGARCH(1,1) model of Alexander (2001), the DCCGARCH(1,1) model of Engle (2002) and the GOGARCH(1,1) model of van der Weide (2002). We also included the CUCGARCH(1,1) model in our comparison, i.e. we fitted for each CUC a standard GARCH(1,1) model without incorporating the lagged values from the other CUCs. As we have pointed out earlier, the GOGARCH(1,1) model does not fit this data set well. In fact the estimated conditional correlation process between the S&P500return and the Intel return based on the GOGARCH(1,1) model is negatively correlated with its counterpart based on any other models that were mentioned above. Therefore we exclude the GOGARCH(1,1) model results in the comparison below.
Fig. 3 displays the time plots of the estimated conditional variance processes of the S&P500return by the OGARCH(1,1) model, the DCCGARCH(1,1) model and the CUCGARCH(1,1) model. Whereas the estimated processes by the DCCGARCH(1,1) and the CUCGARCH(1,1) models look similar, the OGARCH(1,1) model certainly leads to a very different volatility profile. Comparing with the original return series in Fig. 2(a), the two peaks around t=850 should not be there. They were caused by the extreme negative returns of the Cisco Systems price in the same period; see Fig. 2(b). Such a misleading phenomenon resulted from treating the principal components as CUCs in the OGARCH(1,1) model. The estimated conditional correlation processes between the S&P500return and the Intel price return are plotted in Fig. 4. The conditional correlation estimated by the CUCGARCH(1,1) model is more volatile than those estimated by the OGARCH(1,1) and the DCCGARCH(1,1) models. In particular the CUCestimated conditional correlation is small in the middle period before it peaks up twice towards the end. Those two peaks correspond to the two peaks in the volatility process of the S&P500return. Note that the estimated correlations by the DCC and the CUC models are quite different numerically from each other.
We now apply two diagnostic checking statistics to assess the various fitted models. Following the lead of Tse and Tsui (1999), we use the Box–Pierce statistic to check the crossproduct of the standardized residuals. For this, let be the standardized residual for the ith component, where is the (i,i)th element of the fitted conditional variance of Y_{t}. Put
 ( (4.2))
where is the estimated conditional correlation between Y_{ti} and Y_{tj}. If the model is correctly specified, there is no autocorrelation in {C_{t,ij},t1} for any fixed i and j. Define
 ( (4.3))
where r_{ij,k} is the sample autocorrelation of C_{t,ij} at lag k. It is intuitively clear that large values of Q(i,j;M) are indicative for the lack of fit for the conditional correlation between the ith and the jth components Y_{t} when i≠j, and for the lack of fit for the conditional variance of the ith component when i=j. We also employ a multivariate portmanteau statistic (section 5.5 of Reinsel (1997)) to test for the autocorrelation in the vectorized crossproduct of residuals , where . Let be the autocovariance matrix of ξ_{t} at lag l. The multivariate portmanteau statistic is defined as
 ( (4.4))
This may be seen as a multivariate extension of McLeod and Li (1983) which applied a univariate portmanteau test to squared residuals.
Table 4. Q(i, j; M) with M=5 for the S&P500–Cisco Systems–Intel Corporation–Sprint data i,j  Results for the following models: 

OGARCH  DCC  GOGARCH  CUCGARCH  CUCEx GARCH 


1,1  69.37†  5.00  5.23  5.19  5.19 
2,2  10.38  9.05  8.91  8.22  8.12 
3,3  2.11  4.67  5.88  1.55  1.59 
4,4  1.29  1.08  0.97  0.46  0.41 
1,2  48.11†  10.91‡  10.31‡  8.36  8.31 
1,3  54.44†  15.79†  10.67‡  4.73  4.55 
1,4  18.69§  1.86  1.51  1.40  1.38 
2,3  1.05  5.15  7.72  4.34  4.25 
2,4  6.99  3.04  3.35  3.11  2.93 
3,4  2.15  4.11  2.31  2.83  2.82 
Table 5. P(k) for the S&P500–Cisco Systems–Intel Corporation–Sprint data k  Results for the following models: 

OGARCH  DCC  GOGARCH  CUCGARCH  CUCEx GARCH 

1  182.76  117.32  99.83  96.69  96.75 
2  307.64  210.99  190.85  186.95  184.49 
3  439.22  325.91  302.53  302.87  295.92 
4  523.74  412.77  392.74  395.79  387.39 
5  634.51  507.46  486.91  494.16  489.16 
Tables 4 and 5 indicate that the OGARCH(1,1) model provided overall the poorest fit among the five models concerned according to both Q(i,j;M) and P(k); in particular four of its Qstatistics are significant at the 0.05level. However, the tests with Q(1,2;5) and Q(1,3;5) for both DCCGARCH(1,1) and GOGARCH(1,1) models are significant at least at level 10%, whereas both the CUCGARCH(1,1) and CUCextended GARCH(1,1) models passed all the tests with the statistic Q(i,j;M). Note that the values of P(k) for the two CUCbased models are smaller than those for the OGARCH(1,1) and the DCCGARCH(1,1) models. Overall both the diagnostic statistics indicate that the CUCextended GARCH(1,1) model is the best model for this particular data set.
To make a postsample comparison between these models, we need to construct proxies for unobserved conditional covariance matrices by using the daily returns. Let be the pdaysahead forecast of the covariance matrix at t. Following the lead of Pelletier (2006) and Fan et al. (2007), we gauge the quality of forecasting on the basis of the adaptive mean absolute deviations:
 ( (4.5))
where v is a nonnegative integer, and the sum over t is over the n^{*} postsample points. When v=0, AMAD reduces to the mean absolute deviation that was used in Pelletier (2006) and the proxy for the covariance matrix at time t+p is just the crossproduct of the return vector at that day; when v>0, the adjacent 2v+1 days returns are used to average out the stochastic error in the proxy.
We use the last 252 observations in the data to compute the AMADs. On the basis of the pstepahead forecast of the univariate GARCH model (see, for example, page 94 in Tsay (2002)) for each component and the transformation matrix, can be constructed in a straightforward way for the OGARCH, GOGARCH and CUCGARCH models. The forecast for the DCC model follows the procedure of Pelletier (2006). The lengths of the samples that were used for parameter estimation are 500 and 1000, and the estimates are updated every 5 days, and no causal component is considered for the CUCs. Table 6 lists the results for p=1 and p=5.
Table 6. AMAD for the S&P500–Cisco Systems–Intel Corporation–Sprint data T  p  Results for the following models: 

OGARCH  DCC  GOGARCH  CUCGARCH 

v=0 
500  1  7.3095  7.2187  7.9227  7.2104 
5  7.2869  7.2383  7.7048  7.1909 
1000  1  7.2094  7.1636  7.8361  7.0859 
5  7.1781  7.1646  7.6988  7.1436 
v=1 
500  1  4.5650  4.5897  4.8360  4.6894 
5  4.9543  4.9634  5.4663  4.8978 
1000  1  4.5856  4.5921  4.7972  4.6747 
5  4.9158  4.9343  5.7790  4.9142 
AMAD for the CUCGARCH model is always the smallest when v=0. When v=1, this is still true for the 5dayahead forecast, but for the 1dayahead forecast AMADs for the OGARCH and DCC models are both smaller than those for the CUCGARCH model. In contrast, the GOGARCH model provides the worst forecasts for this data set. Overall the CUCGARCH model outperforms the other three models in this forecasting comparison.
Our second example concerns the daily logreturn of the exchange rates of the 10 European currencies against the US dollar during January 2nd, 1990–December 31st, 1998, immediately before the introduction of the euro. The currencies concerned are from Austria, Belgium, Finland, France, Germany, Ireland, Italy, the Netherlands, Portugal and Spain. For this data set, n=2263 and d=10. Diagnostic checks similar to those in the first example were carried out. For brevity, in Table 7 we list only the multivariate portmanteau statistics for the various models. It is not surprising that P(k) of the extended CUCGARCH model is the smallest in each row and the values of P(k) for the CUCGARCH model are smaller than those for the OGARCH, DCC and GOGARCH models as k>3. The DCC model may be too simple to catch the dynamical structure of a 10dimensional volatility process. The extension to incorporate more flexibility into the DCC structure would present an interesting line for further development.
Table 7. P(k) for the exchange rates data k  Results for the following models: 

OGARCH  DCC  GOGARCH  CUCGARCH  CUCEx GARCH 

1  6783  10271  6352  7297  6316 
2  11224  16792  10918  11562  9871 
3  15736  23530  15043  15288  12706 
4  20538  30701  19871  19448  16205 
5  23655  37077  23313  22839  19022 
10  41631  62943  41770  41197  35928 
Again the comparison that is based on postsample forecasting was also in favour of the CUC approach. In fact we reserve the wholeyear data in 1998 (with 252 observations) for checking the postsample forecasting performance. Both 1dayahead and 5dayahead forecasts are made on the basis of the fitted models using 500 observations in the immediate past. Table 8 lists the AMADvalues (see equation (4.5)) for the forecasts that are based on the four different models. Except for one case with v=1 and p=1, the CUCGARCH model provides the best forecasts among the models concerned. On the basis of Tables 7 and 8, we would conclude that the CUC provides an alternative parsimonious representation for the dynamics of conditional covariance processes which is more accommodating than, for example, the simple DCC model when the dimension of the underlying process is large.
Table 8. AMAD for the exchange rates data T  p  Results for the following models: 

OGARCH  DCC  GOGARCH  CUCGARCH 

v= 0 
500  1  0.3097  0.3098  0.4074  0.2978 
2  0.3112  0.3109  0.4034  0.2987 
3  0.3089  0.3083  0.3885  0.2958 
4  0.3097  0.3089  0.3502  0.2974 
5  0.3097  0.3087  0.3752  0.2991 
v= 1 
500  1  0.2061  0.2038  0.2475  0.2088 
2  0.2127  0.2115  0.3439  0.2108 
3  0.2117  0.2102  0.3193  0.2087 
4  0.2138  0.2121  0.3145  0.2086 
5  0.2150  0.2133  0.3043  0.2094 
5. Conclusional remark
 Top of page
 Abstract
 1. Introduction
 2. Methodology
 3. Simulation
 4. Real data examples
 5. Conclusional remark
 Acknowledgements
 References
 Appendix
It is extremely effective for analysing multivariate time series to find an appropriate linear transformation such that the components of the transformed series exhibit certain ‘unrelatedness’. There are at least three types of unrelatedness. For modelling conditional covariance processes, the conditional uncorrelatedness is the correct measure which serves the purpose adequately, whereas the unconditional uncorrelatedness that is required in the orthogonal GARCH model (Alexander, 2001) is too weak and the independence in the independent component analysis is too strong.
Modelling multivariate volatility processes is a practically important and methodologically challenging problem. The CUCbased method that is proposed in this paper attempts to catch sophisticated conditional heteroscedasticity structures while maintaining a parsimonious representation for matrix processes. One natural question arises: do the CUCs that are so defined exist? Empirical experiments with various real data sets indicate that the Pvalue of the bootstrap test which is described in Section 2.4 tends to decrease as d increases. However, with small or moderately large d the hypothesis of the existence of CUCs has rarely been rejected in our empirical experiments.
In the event that the CUCs do not exist, we argue that it is very natural to find the linear transformation such that the resulting components are the least conditionally correlated, especially if we take the viewpoint that any statistical model is merely an approximation to the reality. In this sense, our CUC estimation leads to the least conditionally correlated directions and we build up an (approximate) volatility model by assuming that the conditional correlations between those least conditionally correlated directions are 0. The least conditionally correlated directions are the directions which minimize Ψ(·) that is defined in equation (2.8); also see equation (2.9). Theorem 1 indicates that the columns of are the consistent estimators for the least conditionally correlated directions. Note that both theorem 1 and theorem 2 still apply when the CUCs do not exist (i.e. Ψ(A_{0})≠0); see condition (c) in Appendix A. Even if the CUCs do not exist, a CUCGARCH(1,1) model, for example, still provides a more relevant fit than OGARCH(1,1) and GOGARCH(1,1) models.
Finally we point out that, for any multivariate time series X_{t}, there is always an ℱ_{t−1}measurable orthogonal matrix A_{t−1} for which the components in are conditionally uncorrelated. The CUC requires a further constraint A_{t−1}≡A. It is reasonable to assume that A_{t−1} varies smoothly in t (see also equation (1.3)). Therefore we may assume that the CUCs exist for a short time period in which A_{t−1}≈A. This further extends the scope of the applicability of our method.