Modelling Dynamic Conditional Correlation
with Multivariate GARCH Models
Empirical application to 30 assets in 6 industry sectors:
Oil industry, Automotive industry, Electronics industry, Pharmaceutics industry, Banking industry and Beverage industry
Dirk Verburg
August 20, 2014
Advisor: Simon Broda Second advisor: Peter Boswijk
Contents
1 Introduction 3
2 Conditional Correlation 5
2.1 Constant conditional correlation . . . 5
2.2 Introduction of dynamic conditional correlations . . . 6
2.3 Estimation details of DCC-GARCH models . . . 8
3 DCC-GARCH model extensions and testing 10 3.1 The Asymmetric DCC-GARCH model . . . 10
3.2 The Flexible DCC-GARCH model . . . 11
3.3 Empirical residual test statistic . . . 12
3.4 Value at Risk . . . 13 4 Data 15 4.1 Descriptive Statistics . . . 15 4.2 Normality assumption . . . 17 4.3 No Autocorrelation assumption . . . 19 4.4 The portfolio . . . 20 5 Results 22 5.1 Conditional Variance, GARCH(1,1) . . . 22
5.2 Dynamic Conditional Correlation, DCC(1,1) . . . 22
5.3 Asymmetric Dynamic Conditional Correlation, ADCC(1,1,1) . . . 24
5.4 Flexible Dynamic Conditional Correlation, FDCC(1,1) . . . 25
6 Model comparison: post-estimation statistics 27 6.1 Likelihood Function . . . 27 6.2 Likelihood-ratio test . . . 28 6.3 Value at Risk . . . 30 6.4 Estimated correlation . . . 33 6.5 Residual test . . . 36 7 Conclusion 38 8 Bibliography 40
1 Introduction
Over the last decade, there has been a remarkable development in the analysis of time series data, with the availability of newly developed econometric models. Financial time series models, in particular, have seen numerous adjustments and new model specifica-tions. Data of all financial returns have one characteristic in common—the presence of heteroskedasticity.
The first models, which were addressed to capture conditional variance, were ARCH and GARCH. They were developed many years ago by Engle (1982) and Bollerslev (1986). Univariate GARCH model specifications were widely spread in different extensions, such as T-GARCH, GJR-GARCH, E-GARCH and others. There is a rich literature available which discusses these models in detail, see for example the papers of Nelson (1991) and Zakoian (1993).
Multivariate GARCH modelling is a topic that has attracted a lot of attention from both academicians and practitioners. Research on correlation structures in asset returns used to be at a low level because of limitations in computer power. Nowadays, the power of computers has become virtually unlimited. Hence, it is now possible to estimate more complicated models such as multivariate GARCH models, which are nowadays the standard in risk management.
Gaining knowledge about correlation structures among financial assets is very useful for financial institutions in such a way that they are able to diversify and optimize their respective portfolios. However, multivariate GARCH models require higher numbers of unknown parameters to be estimated than those of univariate GARCH models. More powerful computers have become available because of the recent technological revolution.
Therefore, it is now possible to develop different versions of the dynamic conditional correlation (DCC) GARCH model. Note that the family of DCC-GARCH models ex-perienced some extensions in the past few years. The DCC-GARCH model has been generalized in different ways and all the drawbacks of the model tried to be solved with adjusting the original model specification.
This article uses several types of multivariate GARCH models to investigate the dy-namic conditional correlation structure of a broad selection of financial assets in different industries. The generalized DCC-GARCH model forms the basis for this research from a theoretical point of view. One could argue that this model specification is too limited to fully grasp complex conditional correlation structures. This motivates us to implement
two more complex DCC-GARCH model specifications.
An asymmetric extension of the DCC-GARCH model, the A-DCC GARCH model has been taken into account. The motivation behind this lies in the paper of Cappiello, Engle and Sheppard (2006) where they describe the so-called ‘asymmetric volatility phenomenon’. They found out that volatility is more likely to increase after a negative shock—rather than after a positive shock— of the same magnitude.
A flexible specification of the DCC-GARCH model is also used to grasp the correlation structure. The Flexible (F-DCC) GARCH model allows for different structures in the dynamics of the correlation. This could be beneficial when assets are grouped into different industries, because the dynamics of the correlation could differ among each industry. This also solves the limitation of the DCC-GARCH and the A-DCC GARCH model, where the dynamics of the correlation are constant.
Ideally, a more sophisticated model should be slightly better in grasping correlations, especially during a financial crisis. For pre-estimation analysis, descriptive statistics are performed on the data. The normality assumption, which is fundamental for all multi-variate GARCH models is checked by the Jarque-Bera test. For post-estimation analysis, residual tests on the models will be performed to address their ability to remove the correlation from the financial data. Together with the analysis of the Value at risk (VaR) of the multivariate GARCH models, this will be fundamental in testing the credibility and relevance of the models. Validation of the Value at Risk results is ‘backtested’ by the Engle-Manganelli test, which is described in detail in their working paper of 1999. On top of that, Likelihood-ratio tests are performed together with Akaike Information Criteria.
This article is structured as follows: Chapter 2 gives more background information about the purpose of using Multivariate GARCH models. The theoretical framework of the DCC-GARCH model which is available in the current literature will be discussed in detail. Chapter 3 continues with presenting different model specifications for the A-DCC GARCH in Section 3.1 and the F-DCC GARCH model in Section 3.2. Value at Risk theory is also discussed in detail in Section 3.4. Chapter 4 describes and analyses the data which has been used throughout this research in terms of basic statistics, source and composition. Chapter 5 gives a presentation of all the empirical results which were obtained by the estimation of the models and test statistics. Chapter 6 provides an overview of the results which are used for model comparison purposes. Chapter 7 presents the conclusion.
2 Conditional Correlation
2.1 Constant conditional correlation
Prior to the understanding of DCC-GARCH models, it is useful first to introduce a model which assumes the correlation dynamics of a time series to be constant. Thus, the Constant Conditional Correlation (CCC) GARCH model was developed years ago by Bollerslev [1990]. This model provides a good representation of the fundamental aspects of this research, namely volatility and correlation analysis of financial time series. Before any correlation model could be estimated, it is necessary to start with a volatility analysis of each individual asset time series in the portfolio. GARCH models are generally employed to address heteroskedasticity. We can use the different GARCH specifications that have been discussed in the literature to estimate the conditional variance part in the CCC-GARCH model. Most common and well performing is the GARCH (1,1) model, as shown in equations (2.1) and (2.2):
rt∼ N (0, σ2t) (2.1)
σt2 = α0+ α1ε2t−1+ β1σ2t−1, (2.2)
where rt stands for the demeaned logarithmic returns of the asset at period t and σt
represents the conditional volatility. A volatility analysis is important to analyse risks. Knowledge about the behaviour of correlation among the assets could be a big issue. This leads us to a multivariate approach to GARCH modelling. The CCC-GARCH model of Bollerslev [1990] holds the basic structure for more advanced multivariate GARCH models, which will be discussed later. Strong assumptions on the distribution of asset returns are made. The demeaned returns from k assets are conditionally multivariate normal with zero expected value and covariance matrix Ht, as represented in Equations
(2.3) and (2.4):
Et|Ft−1∼ N (0, Ht) (2.3)
Ht= DtRDt, (2.4)
where R is a time-constant correlation matrix and Dtis a diagonal matrix of conditional volatilities. The matrix R can be simply estimated using a two-step procedure. In the first step, GARCH models are used to estimate conditional variances. In the second step, conditional correlation can be estimated when filtering out conditional variances by pre-multiplying Et with the inverse of Dt. The sample estimator for constant conditional
correlation is presented in equation (2.5). R = T X t=1 Dt−1EtEtDt−1 T (2.5)
The big limitation of the CCC-GARCH model is that the correlation is assumed to be constant. However, the assumption that correlations are stable over long periods of time is very unlikely according to Billio and Caporin [2009]. Much evidence is found that correlations vary over time by Longin and Solnik [1995] and Fischer [2007]. To overcome this limitation, a time-varying specification of the correlation matrix was suggested by Engle and Sheppard [2001]. This leads us to the DCC-GARCH model that will be discussed in Section 2.2.
2.2 Introduction of dynamic conditional correlations
Both the CCC-GARCH and the DCC-GARCH model are based on the same fundamental parts. One part models the asset-specific variance evolution and the second part models the conditional behaviour of the correlation. Eventually, it all comes down to whether the assumption of constant correlations is reliable. Conditional correlations are proven not to be stable over long periods of time (Cappiello et al. [2006]). This motivated researchers to choose for the DCC-GARCH model and similar dynamic extensions. Engle and Sheppard [2001] proposed new dynamics of correlation that addresses the limitations of the CCC-GARCH model. By plugging in a time-varying correlation matrix Rt, instead
of a time-constant correlation matrix, and introducing a quadratic structure in the correlation matrix, the limitation of the CCC-GARCH model is resolved. In this research the specification of Engle and Sheppard [2001] and Billio and Caporin [2009] is used to model the variance-covariance matrix Ht as shown in equation (2.6).
Ht= DtRtDt, (2.6)
where Rtis the time-varying correlation matrix and Dtis the k×k diagonal matrix of
time-varying standard deviations, which has been estimated from univariate GARCH models with √hiton the ithdiagonal. For each individual asset time-series, the estimation of the
conditional variance can be done with GARCH-type models. Once the univariate volatility models are estimated, the standardized residuals are used to estimate the correlation parameters. Engle also made the assumption that the time-varying correlation has a quadratic structure. The reason for this is to ensure that we end up with a correlation matrix. To construct it, a time-varying Q-matrix is introduced, as shown in equations (2.7) and (2.8): Rt= (Q∗t) −1 Qt(Q∗t) −1 (2.7) Qt= (1 − α − b) ¯R + aEt−1Et−10 + bQt−1, (2.8)
where ¯R = E[EtEt0] is the unconditional sample correlation, Q∗t = [qiit∗ ] =
√
qiit is a
diagonal matrix composed of square roots of the diagonal elements of the Qt matrix,
a and b are scalar DCC parameters that measure the effect of the lagged correlation
matrices and the innovation term Qt−1, respectively.
In order to rule out explosive patterns, the scalars need to satisfy the following restric-tion: a + b < 1. An important condition which will be satisfied by this restriction is the positive definiteness of Qt. This necessary condition ensures that Rt is positive definite
and hence is indeed a correlation matrix. As long as Qt is positive definite matrix, Rt
will be a correlation matrix with ones on the diagonal and the other elements either smaller or equal than one in absolute value.
This representation of the DCC-GARCH model is obviously parsimonious. It requires only two parameters to be estimated in order to describe the dynamics of the correlation. This implies several strong restrictions, such as constancy of the dynamics across all correlations. Another restriction to this model is that there should be no interdependence among variances, co-variances and correlations.
The limitation caused by the constancy of the dynamics over all correlations has been already addressed in the economic literature. Engle and Sheppard [2001] solved the constrained of equal dynamics over all correlations by generalizing the DCC-GARCH model according to Equation (2.9).
Qt= [ ¯R − A0RA − B¯ 0RB] + A¯ 0Et−1Et−10 A + B0Qt−1B, (2.9)
where A and B are k × k parameter matrices and the expression for the unconditional correlation is replaced by ¯R = T1 PT
t=1EtEt0. Furthermore, the ‘correlation targeting’
prop-erty holds because the unconditional correlations are equal to the sample correlations. If the matrices A and B are assumed to be diagonal, the generalized DCC-GARCH representation can be reformulated in reduced form as in Equations (2.10) and (2.11).
Qt= [ii0− aa0− bb0] ◦ ¯R + aa0Et−1Et−10 + bb
0◦ Q
t−1 (2.10)
Qt= [ii0− A − B] ◦ ¯R + AEt−1Et−10 + B ◦ Qt−1 (2.11)
Where ◦ stands for the Hadamard product. A sufficient condition for the positive definiteness of Qt is that the square matrices A and B are positive definite. This is in fact the case given that A = aa0 and B = bb0. This feature is laid out in the paper of Engle and Ding [2001].
One way to introduce interdependences among variances, co-variances and correlations is to work with other multivariate GARCH models. In particular, the BEKK and Vech
models are designed by Engle and Kroner [1995] to allow for interdependences among variances and co-variances. Logically, these characteristics imply a dynamic correlation structure. For further details see Engle and Kroner [1995].
Unfortunately, the generalized DCC-GARCH, BEKK and Vech models are empirically unattractive because the number of parameters dramatically increases. Especially when the number of assets increases, the models become no longer suitable. When using the generalized DCC-GARCH model, the number of correlation parameters in our empirical application of 30 financial assets would increase from 2 to 60, because each pair of assets has now its own dynamic correlation structure which needs to be estimated. BEKK and Vech models are even no longer useful in applications to more then 4 or 5 assets, because they have serious optimization problems which lead to unstable and inconsistent parameter estimates Billio and Caporin [2009]. Therefore we do not take such models into account in this research.
Another drawback of the DCC-GARCH model is that it cannot capture asymmetric effects in the correlation. In particular, the scalar DCC-GARCH model does not allow for asset-specific news, as well as smoothing parameters or asymmetries. This leads us to look into more advanced DCC-GARCH models such as Asymmetric DCC-GARCH and Flexible DCC-GARCH model specifications, which are covered in Chapter 3.
2.3 Estimation details of DCC-GARCH models
In the first step of the DCC-GARCH estimation, we model the conditional variance of each individual asset as a univariate GARCH process. According to this method, estimates of the squared volatility hit are obtained. Consequently, the diagonal elements √
hit of the matrix Dt can be computed. In the next step, asset returns, transformed
by their estimated standard deviations, are used to compute the parameters of the Conditional Correlation model.
The parameters of a Conditional Correlation model can be grouped into the set
θ = (ϑ, ϕ) = (ϑ1, ϑ2, . . . , ϑk, ϕ), where ϑi = (ω, α1i, . . . , αPii, β1i, . . . , βQii) are parameters
of the individually estimated GARCH models and the set ϕ comprises the parameters of the correlation model. The relevant quasi-maximum likelihood function for a k-asset model—in relation to GARCH specifications—is specified by Equation (2.12):
L1(ϑ|εt) = − 1 2 k X n=1 (T log(2π) + T X t=1 (log(hit) + ε2it hit )). (2.12)
This equation essentially sums up the log-likelihoods of the individual GARCH Engle and Sheppard [2001].
In the next stage, the estimated residuals—transformed by their standard devia-tion—are used to estimate the parameters of the group ϕ. This is achieved by using a
likelihood function which conditions on the estimated parameters obtained in the prior stage. The Likelihood function is shown in equation (2.13):
L1(ϕ| ˆϑ, εt) = − 1 2 T X t=1 (log(|Rt|) + ε0tR−1t εt), (2.13)
where the matrix Rtonly depends on the DCC parameters group ϕ. In the DCC-GARCH model, ϕ has only two correlation parameters—a and b. In more advanced models, the number of parameters in the group ϕ becomes larger.
3 DCC-GARCH model extensions and
testing
3.1 The Asymmetric DCC-GARCH model
During the financial crisis, news about the state of the economy had predominant im-portance in asset returns. Risks of a prolonged and deep recession would reduce the profitability of assets in all sectors. If the crisis was shorter, on the other hand, asset returns would reflect idiosyncratic risks instead. This association between common downside risks and increased asset correlation is the target of asymmetric correlation models.
There exist several studies that account for asymmetric effects in conditional covariances Cappiello et al. [2006].The AG-DCC GARCH model extends the original DCC-GARCH model in such a way that it permits conditional asymmetries in correlation dynamics. The AG-DCC GARCH correlation evolution equation is presented in equation (3.1):
Qt= ( ¯R − A0RA − B¯ 0RB − G¯ 0N G) + A¯ 0εt−1εt−1A + G0nt1n
0
t1G + B
0Q
t−1B, (3.1)
where A, B and G are k × k parameter matrices. The asymmetric effects are measured with the G-parameters, and nt= I[Et< 0] ◦ Et, where I[·] is a k × 1 indicator function
which takes on the value 0 if the argument is false and value 1 when the argument is true. The Hadamard product is indicated with the symbol ◦. The sample variance-covariance matrix is ¯R = T1 PT
t=1εtε0t. N is the sample variance-covariance matrix of n¯ t which
is equal to T1 PT
t=1ntn0t. A sufficient condition in Equation (3.1) for Qt to be positive
definite is that the intercept matrix [ ¯R −A0RA−B¯ 0RB −G¯ 0N G] and the initial covariance¯
matrix Q0 are positive definite (Cappiello et al. [2006]). This is not so straight forward as in the case of the DCC-GARCH model. For further details about positive definiteness of the matrix Qt see the paper of Engle and Ding [2001].
As discussed previously, the AG-DCC GARCH model is not suitable for systems with a higher number of assets. Therefore, a special case of the AG-DCC GARCH model is used for estimating the correlation structures in this empirical research. The scalar A-DCC GARCH model of Cappiello et al. [2006] requires many fewer parameters to estimate. Therefore it is more appropriate for our empirical application to thirty financial assets. We present the model in Equation (3.2).
In Equation (3.2), the matrices A, B and G from the AG-DCC model are replaced by scalars. This greatly reduces computation because all the thirty parameters from these matrices can now be estimated simultaneously. This means that the number of correlation parameters can be reduced from ninety to only three. The condition which ensures the positive definiteness of matrix Qt is reformulated in Equation (3.3)
a2+ b2+ δg2< 1, (3.3) where δ is equal to the maximum eigenvalue of the matrix [ ¯R−12N ¯¯R−
1
2]. The A-DCC
GARCH model is estimated in similar way as the scalar DCC-GARCH model: First, we estimate GARCH type models, and in the second step the conditional correlation parameters a, b and g are estimated.
3.2 The Flexible DCC-GARCH model
The main drawback of previous scalar DCC-GARCH models is that the dynamics of the correlation is constant among all assets and groups of assets. Therefore Billio, Gobbo and Caporin (2006) and Billio and Caporin (2008) introduced the F-DCC GARCH model. Basically, the model generalizes the DCC-GARCH model of Engle (2002). The F-DCC GARCH model can also be easily derived from the generalized version of the DCC-GARCH model from Franses and Hafner (2003). They suggested the following expression for the Q-matrix in Equation (3.4).
Qt= [1 − n X
i=1
αi− β] + αα0◦ Et−1Et−10 + βQt−1, (3.4)
where α stands for a vector of dimension n. By adding a constant and further generalizing this model, Billio and Caporin proposed the following dynamics of the F-DCC GARCH model in equation (3.5):
Qt= cc0+ αα0◦ Et−1Et−10 + ββ0Qt−1, (3.5)
The following restrictions are imposed to rule out explosive patterns: positive definite-ness is imposed by Billio and Caporin (2006) with the following constraints: αiαj+ βiβj+
cicj = 1 for i, j = 1. . . n; the coefficients also need to satisfy the following stationarity
constraint: αiαj+ βiβj < 1.
Because the dynamics of the correlation can differ among the different industries in the empirical application, this model is adjusted in such way that each industry has its own correlation dynamics build in. Intuitively, this particular F-DCC GARCH model has an attractive trade-off between the limitations of the scalar DCC-GARCH and the generalized DCC-GARCH model. To be precise, α and β are vectors with 6 parameters. In total, two correlation parameters, αi and βi, are estimated for each industry. The
Maximum Likelihood procedure of the econometric estimation is similar to what we have used in the previous models: First, we estimate univariate GARCH models and then conditionally on the parameters obtained in the first step we estimate the 12 parameters of the dynamic correlation structure for each industry, (Billio & Caporin, 2008). These maximum likelihood estimators are consistent and asymptotically normally distributed (Billio & Caporin, 2008).
3.3 Empirical residual test statistic
In order to examine the performance of DCC-GARCH models, we use a test statistic that detects whether correlation has been correctly estimated given the past information. The essential idea behind this test is that if a DCC-GARCH model describes the data sufficiently well, then the cross-products of residuals should have no serially correlated dependence. This empirical test is set up in two steps.
First, the returns are normalized. The normalization transforms the returns into an orthogonal vector of residuals, making use of the eigendecomposition of the variance matrix. More precisely, let Σtbe the variance-covariance matrix of m assets. Since Σt is
a symmetric matrix there exist an orthogonal matrix Utand a diagonal matrix Λt, whose entries are the eigenvalues of Σt, such that Equation (3.6) is satisfied:
Σt= UtΛtUtT (3.6) Here, U ΛtUtT can also can be written as U Λ0.5t (U Λ0.5t )T, given that Λt is a diagonal matrix. Therefore, we have that
rt∼ N (0, Σt) ⇔ rt∼ UtΛ0.5t N (0, I), (3.7)
where I is the identity matrix. This representation of the variance covariance matrix allows us to transform the data from multivariate normal to standard multivariate normal. This transformation is described in Equation (3.8):
inv(UtpΛt) × rt= et∼ N (0, I). (3.8)
Second, a cross-product of the normalized residuals is computed and then examined for autocorrelation structure. This autocorrelation coefficient should be insignificant if the model describes the data well and accounts for all possible correlations between different assets.
In other words, the residuals, defined as e = inv(Ut
√
Λt) × rt, should be distributed
as multivariate standard normal with no serial correlation in the cross product of its elements. The matrix of cross products is defined in Equation (3.9):
The test for autocorrelation is implemented for each entry vt,ij of Vt, estimating the following regression Equation (3.10):
vt,ij = α + βvt−1,ij+ εt,ij, ∀i, j = 1, .., m. (3.10)
The regression coefficient β should be insignificant if the model of conditional correlation explains the correlation across data correctly. We implement this analysis for the residuals of the conditional correlation models.
3.4 Value at Risk
In the last decade, Value at Risk (VaR) has become the standard measure of market risk used by financial institutions and their regulators (Engle and Manganelli 2004). VaR is an important tool for risk managers in order to understand the level of risk of their investments. When investing in a risky asset, it is useful to know what could be the maximum loss on this investment. Basically, Value at Risk measures the potential loss in value of that asset over a defined period for a given confidence interval. To illustrate this numerically, if the VaR with 99% confidence level on an asset is 3 million euros at a 30 days period, there is only a 1% chance that the asset decreases in value by more than 3 million euros in any given 30 days. The Basel Committee of the Bank for International Settlements (2009) prescribe the use of 10-day 99% VaR levels to preserve stability of the banking system. The V aRpt indicates the value in the distribution of losses such that larger losses only occur with a probability p. In other words, VaR represents the maximum loss of an investment, with confidence 1 − p, over horizon l described in Equation (3.11):
P r[Vt+1− Vt≤ −V aR(l)] = p, (3.11)
where Vtwith t ≥ 1 is the daily observed value of the investment. Assuming a cumulative
distribution function F for the losses, the V aRpt can be computed using Equation (3.12):
V aRtp= F (p)−1 (3.12) which in the case of the normal distribution, translates to Equation (3.13):
V aRpt = Φ(p)−1×
q
σ2
t + µt, (3.13)
where Φ(p) is the standard normal cumulative distribution and µt is the mean loss. In
the rest of the paper, the mean loss is ignored and we work with demeaned returns.
To implement this approach in this paper, an equally weighted portfolio of m assets with return Rpt = m1i0rt is constructed, where Rpt stands for the scalar return of the
portfolio, rt = rt1, r2t, ..., rmt 0
is a m × 1 vector of the assets in the portfolio, and
i = (1, 1, ..., 1)0 is the unit vector of size m × 1. Given the m × m variance-covariance matrix of the asset returns Σt at time t the variance of the portfolio is given by the
following equality: σ2
t = m12i0Σti.
An important measure of the success of a VaR model in the setting of conditional correlations, is the dynamics of the violation of the VaR threshold by the actual portfolio return. In the literature, this procedure is called "backtesting" the VaR, and such violations of the VaR threshold are called "hits". If the VaR model is correct the sequence of hits is serially independent. This implies that the autoregression of hits should lack significance. Continuing with the construction of the backtest, a dummy variable It is
introduced, which takes on value 1 if the portfolio return happen to be less than VaR at time t (i.e. a hit occurs), and 0 otherwise. This is shown in Equation (3.14):
It= ( 1, Rpt < −V aR1%t 0, Rpt ≥ −V aR1% t (3.14)
To implement the Engle–Manganelli test, the autoregression shown in Equation (3.15) is performed:
It= β0+ β1It+ t (3.15)
The insignificance of the parameter β1 might indicate lack of dependence among the It and therefore supports a good quality of the model. However, when β1 is significant,
4 Data
4.1 Descriptive Statistics
This section gives an overview of descriptive statistics of the data and checks whether fundamental assumptions hold for the multivariate GARCH models. This article con-tributes to the multivariate GARCH research, with an empirical application of several DCC-GARCH models as discussed in the previous sections. Therefore, a dataset is constructed with 30 assets which are selected from 6 different major industries. The selected industries are respectively, the oil industry, automotive industry, electronics industry, pharmaceutics industry, banking industry, and the beverage industry. Daily frequency data from the biggest five companies in each of these industries are downloaded from DataStream from the European asset exchanges FTSE 100, DAX, AEX and CAC 40. The selected time-period is from 1 January 2002 till 30 December 2011. The whole sample of closing prices consists of 2,500 times 30 assets, which is a total number of 75,000 data points. Estimation and all test results are based on the whole sample. Average returns and standard deviation of the average returns are grouped according to the industries and they are presented in Table 4.1.
Table 4.1: Descriptive statistics of industry returns.
Average return across years St. dev. of average return Oil −6.5158 × 10−5 1.3421 × 10−4 Auto. −7.2171 × 10−5 1.9542 × 10−4 Electron. 8.1068 × 10−5 6.8575 × 10−4 Pharma −51.3820 × 10−5 4.7903 × 10−4 Banking −16.9166 × 10−5 2.5539 × 10−4 Beverage 2.7294 × 10−5 1.3451 × 10−4 Total −11.8659 × 10−5 3.1405 × 10−4
Pharmaceutics industry and beverage industry have respectively the highest and lowest standard deviation. Means are close to zero as expected. The mean of log-returns are not statistically different from zero, as the t-statistic is not significant. This result resembles the evaluation of means in the literature. Pharmaceutics and beverage have respectively the highest and lowest standard deviation. From now on, we work on returns after subtracting the mean daily return. Sample correlation statistics of the different industries are shown as averages in Table 4.2.
Table 4.2: Average sample correlations Oil Auto. Electron. Pharma Banking Beverage Oil 0.58 0.32 0.28 0.33 0.35 0.28 Auto. 0.32 0.37 0.26 0.32 0.22 0.19 Electron. 0.28 0.26 0.26 0.28 0.22 0.18 Pharma 0.35 0.32 0.28 0.46 0.26 0.20 Banking 0.35 0.22 0.22 0.26 0.39 0.29 Beverage 0.28 0.19 0.18 0.20 0.29 0.25
We compute the correlation between all the assets in the sample. Then we take average of correlation between the assets of every pair of industries. The sample averages of correlation within an industry in Table 4.2 are calculated using a 30 rows by 30 columns Excel sheet. This has the representation of a lower-triangular matrix with sample correlations of all ones on the diagonal elements and sample correlations on the off diagonal elements. The correlation is computed for all the assets in the sample. Consequently, the average correlation within each industry is calculated by summing all the off diagonal pairs of assets in that particular sector dividing them by the number of pairs. For the oil sector this is illustrated with formula (4.1):
ˆ
ρindustry1=
ˆ
ρ12+ ˆρ13+ ˆρ14+ ˆρ15+ ˆρ23+ ˆρ24+ ˆρ25+ ˆρ35+ ˆρ45
10 . (4.1)
We present the results in Table 4.2. As we can see the correlation takes typically values between 0.15 and 0.4, while assets within each industry have naturally a higher correlation than between industries (with the exception of electronics and beverage). Industries that are subject to a volatile risk (oil or banking) will have high intra-correlation, while industries that are subject to risks of the economic cycle will show a lower intra-correlation (beverage, electronics).
Looking at the unconditional correlation, it is remarkable that all the assets are posi-tively correlated. This could be due to the fact that all the assets are exposed posiposi-tively to at least one common factor—the overall market. Correlations among the assets from the same industry are the highest. This matches our expectations.
Highest correlations are found in the oil industry, while the other industries also show a high correlation with the oil industry. This can be explained by the fact that the oil industry might be the biggest market and all the other industries are thus more exposed to changes in the oil industry.
Lowest correlations are found in the electronics industry and in the beverage industry. In these industries, management decisions, the dynamics inside the market and other idiosyncratic factors have much more impact then the overall market effect.
4.2 Normality assumption
An important assumption we make when we work with conditional variance models is that asset returns follow a normal distribution. The assumption is relevant because it affects the choice of optimal parameters in maximum likelihood estimation and also the comparison between volatility models. It may invalidate risk models like VaR, because they are usually based on the normality assumption. The exercise of computing the covariance matrix of the returns to compute the tail properties of the portfolio might be insufficient if the normality assumption is wrong.
Alternatively, the Student-t distribution can be used for the underlying assumption of the multivariate GARCH models. The Student-t distribution gives flexibility in the weight of the tails because it has a free parameter, the degrees of freedom, which determines the fatness of the tails. Besides the degrees of freedom it is also compatible with the covariance matrix and the mean of the assets, which are the other two parameters that characterize the distribution.
However, the Student-t distribution might not account well for skewness in the distri-bution. Also the standard deviation of Student-t distribution is slightly above 1, and decreasing in the degrees of freedom, although for practical reasons we omit what we consider a relatively minor correction.
The kurtosis and skewness of the asset returns are presented in Table 4.3. A value of kurtosis in excess of 3 is above the normal distribution value. In that case, while the distribution has the density concentrated around the center, there is a larger probability for distant values.
There are some assets for which the kurtosis is much higher than the normal. Asset 19 has a kurtosis of 229, which is bound to give a bad compatibility with the normal assumption in the estimation of the correlation models. Asset 19 has the single largest daily loss of the whole sample, at −66.4 in percent terms, roughly 50 times the standard deviation.
The values of the skewness are more consistent with the normal distribution, although negative numbers predominate. Large negative values tend to appear often and small positive values tend to appear very frequently. In general, skewness is not an important factor in the performance of risk models. Therefore, we will not put much emphasis on it. Both the Student-t distribution and the normal distributions have a zero skewness, which indicates that they are symmetric.
In order to check the normality of returns, there are several tests available, which can be used at several stages. The Jarque-Bera test of normality is applied to each asset’s returns after they have been filtered by the GARCH model. The Jarque-Bera test relies on the values of the kurtosis and skewness. The Jarque-Bera test is defined in
Table 4.3: Student-t: optimal degrees of freedom asset Kurtosis Skewness
1 10.924 -0.107 2 20.691 -0.617 3 9.120 -0.183 4 9.772 0.222 5 12.684 -0.351 6 11.914 -0.543 7 8.640 0.352 8 6.873 0.018 9 6.091 0.058 10 8.013 0.109 11 7.063 0.099 12 8.627 -0.230 13 5.251 0.074 14 8.641 -0.028 15 7.661 -0.050 16 14.882 0.250 17 31.096 1.033 18 91.101 0.909 19 223.629 -8.300 20 13.305 0.270 21 19.796 -0.920 22 9.988 -0.561 23 7.728 -0.088 24 8.549 0.138 25 8.219 -0.043 26 7.171 -0.346 27 8.116 0.068 28 7.849 -0.033 29 9.446 -0.101 30 5.879 0.230 Equation (4.2): J B = n 6(S 2+1 4(K − 3) 2), (4.2)
where n is the sample size, S is the skewness and K is the kurtosis. If the data are normally distributed, the J B statistic is distributed as a Chi-squared variable with 2 degrees of freedom. The test rejects the normality assumption for all the assets at the 1% significance level.
Given the differences in the returns data from the normal distribution, we do some of the analysis with the Student-t distribution. We have noted above that the Student-t has to some extent the same problems as the normal distribution, although it corrects partially for some of the most serious, namely leptokurtosis.
In Table 4.4 the optimal degrees of freedom for the Student-t distribution are estimated for each asset, using Maximum Likelihood Estimation.
Table 4.4: Student-t: optimal degrees of freedom
asset 1 2 3 4 5 D. of F. 17.16 17.76 19.22 25.57 17.20 asset 6 7 8 9 10 D. of F. 19.69 16.81 16.71 23.35 17.82 asset 11 12 13 14 15 D. of F. 18.15 11.25 18.30 11.56 12.07 asset 16 17 18 19 20 D. of F. 13.37 15.64 13.45 13.21 17.41 asset 21 22 23 24 25 D. of F. 15.20 12.99 13.81 12.74 13.03 asset 26 27 28 29 30 D. of F. 16.22 14.81 16.91 13.24 16.57
The Student-t distribution is only equal to the normal distribution when the degrees of freedom are infinite. The values are finite and smaller than 30 in all cases, while they don’t allow for frequent and large outliers. The conclusion is that the Student-t distribution reflects better the behaviour of returns, and that it should be used in risk models and MLE. For practical reasons we stick to the normal distribution in the model estimation phase and the Student-t distribution is used in the VaR analysis.
We estimate the number of degrees of freedom in the distribution of portfolio residuals, once the conditional variance is applied to returns. The results are shown in the VaR section of each model.
4.3 No Autocorrelation assumption
The no autocorrelation assumption does not have an impact on the reliability of the MLE estimations and conditional variance predictions. However, autocorrelation is a relevant factor in the analysis, for instance to compute VaR or to compute standard errors
of parameters. The autocorrelation in the returns is computed for each of the assets. Consequently, the Breusch-Godfrey test is used to address the extent of the autocor-relation in the returns. The results of the Breusch-Godfrey test are presented in Table 4.5.
When analyzing the values, we can see that the evidence of autocorrelation in returns is mixed. The hypothesis of no autocorrelation is rejected in favour of the hypothesis of correlation at lags smaller than 5 for certain assets, although there is no clear trend. Furthermore, the proportion of the variance of returns that these lags are able to explain is very small. So if we were consistent only with the statistical method we would filter returns using an AR model. But given that it’s reasonable to assume that there is no available information that allows to build portfolios with returns above market we will not apply the AR filter.
4.4 The portfolio
The series of returns of the portfolio is constructed by applying a weighted average on the returns of the 30 assets. The same weight is given to each asset at every period 301. This is not a very realistic procedure, although it is practical and it does not have important implications on results.
A more accurate way to construct the portfolio would be to work with prices of assets, and then spend the same nominal amount on each asset, and possibly rebalance the portfolio with a certain frequency to have the same weights in nominal terms. Then the returns of the portfolio can be computed.
Table 4.5: Breusch Godfrey autocorrelation of returns (p-value of lag coefficient) asset 1 asset 2 asset 3 asset 4 asset 5
Lag 1 0.503 0.013 0.040 0.087 0.012 Lag 2 0.528 0.023 0.039 0.035 0.025 Lag 3 0.436 0.029 0.023 0.026 0.028 Lag 4 0.104 0.030 0.028 0.025 0.054 Lag 5 0.116 0.044 0.048 0.043 0.066 asset 6 asset 7 asset 8 asset 9 asset 10 Lag 1 0.086 0.562 0.160 0.000 0.369 Lag 2 0.210 0.075 0.112 0.000 0.053 Lag 3 0.369 0.154 0.207 0.000 0.084 Lag 4 0.305 0.179 0.262 0.000 0.048 Lag 5 0.401 0.257 0.301 0.000 0.030 asset 11 asset 12 asset 13 asset 14 asset 15 Lag 1 0.059 0.151 0.000 0.092 0.004 Lag 2 0.003 0.355 0.000 0.237 0.010 Lag 3 0.009 0.519 0.000 0.416 0.016 Lag 4 0.001 0.067 0.000 0.538 0.032 Lag 5 0.001 0.036 0.000 0.644 0.028 asset 16 asset 17 asset 18 asset 19 asset 20 Lag 1 0.350 0.160 0.621 0.270 0.121 Lag 2 0.569 0.206 0.192 0.404 0.082 Lag 3 0.569 0.267 0.336 0.375 0.171 Lag 4 0.281 0.260 0.489 0.531 0.197 Lag 5 0.026 0.195 0.276 0.672 0.117 asset 21 asset 22 asset 23 asset 24 asset 25 Lag 1 0.001 0.221 0.061 0.320 0.509 Lag 2 0.002 0.055 0.144 0.437 0.499 Lag 3 0.004 0.021 0.118 0.449 0.099 Lag 4 0.010 0.035 0.108 0.322 0.178 Lag 5 0.019 0.064 0.064 0.265 0.192 asset 26 asset 27 asset 28 asset 29 asset 30 Lag 1 0.001 0.125 0.222 0.842 0.000 Lag 2 0.002 0.074 0.468 0.172 0.000 Lag 3 0.000 0.084 0.471 0.260 0.000 Lag 4 0.001 0.096 0.613 0.328 0.000 Lag 5 0.002 0.113 0.584 0.179 0.000
5 Results
5.1 Conditional Variance, GARCH(1,1)
In this section, we will start with a discussion of the estimation results. We apply a GARCH(1,1) model to all the assets to capture their conditional variance.
A normal distribution is assumed for practical reasons, although as discussed in Sec-tion 4.2 it is not an accurate and not a harmless assumpSec-tion. The implicaSec-tions on the parameter values are not expected to be large. It is likely that there is smaller emphasis in minimizing the probability of large negative returns (possibly reflected in a smaller moving average coefficient).
The GARCH(1,1) results are presented in Table 5.1. Given that the GARCH step is independent of the correlation estimation step, the GARCH parameters estimates are common for all three conditional correlation models.
In general we observe that there is high autocorrelation in conditional variance, and a smaller influence from moving average shocks. This result is similar to what other studies have previously found (see for instance Goyal [2001]). The results are similar across all industries, emphasizing that the Automotive industry and the Electronics industry present the highest autocorrelation in conditional variance. The Pharmaceutics industry presents the lowest autocorrelation in conditional variance.
5.2 Dynamic Conditional Correlation, DCC(1,1)
In Section 4.1 (descriptive statistics) it was shown that the correlation between asset returns is significant. There are certainly differences in the correlation between assets. The factor of being within the same industry is the most important one. Now we test whether the DCC-GARCH model is able to capture these correlation dynamics with one common parameter for the autocorrelation matrix and for the moving average matrix. The DCC-GARCH correlation parameter estimates are shown in Table 5.2.
The autocorrelation term explains most of the dynamics of the correlation matrix, which tends to revert slowly to the mean. The residual shocks have a small coefficient of 0.00394.
Table 5.1: Estimated values of GARCH parameters Asset ω α β 1 0.000 003 0.070 102 0.913 632 2 0.000 006 0.095 044 0.879 615 3 0.000 006 0.078 335 0.903 476 4 0.000 004 0.089 627 0.893 743 5 0.000 006 0.079 176 0.896 605 6 0.000 004 0.061 767 0.926 377 7 0.000 008 0.072 196 0.917 568 8 0.000 003 0.056 899 0.937 596 9 0.000 002 0.032 780 0.964 882 10 0.000 002 0.067 842 0.928 802 11 0.000 007 0.043 352 0.945 770 12 0.000 010 0.054 199 0.931 211 13 0.000 003 0.035 675 0.959 018 14 0.000 002 0.023 926 0.968 665 15 0.000 003 0.032 082 0.962 634 16 0.000 004 0.099 277 0.899 430 17 0.000 009 0.071 862 0.911 594 18 0.000 008 0.091 272 0.904 849 19 0.000 013 0.187 203 0.812 795 20 0.000 004 0.080 719 0.912 640 21 0.000 002 0.065 886 0.922 542 22 0.000 006 0.060 034 0.921 256 23 0.000 004 0.057 527 0.924 598 24 0.000 010 0.099 850 0.841 984 25 0.000 006 0.069 426 0.911 042 26 0.000 005 0.059 132 0.918 086 27 0.000 005 0.115 672 0.855 350 28 0.000 001 0.049 007 0.943 489 29 0.000 004 0.067 802 0.915 488 30 0.000 001 0.029 680 0.965 661
Table 5.2: DCC(1,1) model parameter estimates
DCC a b
Value 0.00394 0.97672 St. Dev 0.00023 0.00202
the likelihood function (2.13) at the optimum, whose negative value is the information matrix, and then invert this matrix to obtain the variance covariance matrix of
coef-ficient estimates. Hence, the standard error of the coefcoef-ficients are the square root of the diagonal elements. The rationale behind the procedure is that coefficient estimates follow distribution (5.1). For further details see Rodriguez [2014]. All coefficients are statistically significant.
e
θ ∼ N (θ, I−1(θ)) (5.1)
When coding the likelihood function, the biggest challenge was to ensure that the correlation matrix is at every period positive semi-definite. A condition that suffices for the correlation matrix to be positive is that a + b < 1. Both the matrix of residual products and the average correlation are positive semi definite. The condition is that all coefficients have a positive sign.
5.3 Asymmetric Dynamic Conditional Correlation,
ADCC(1,1,1)
We obtain the parameter estimates in the A-DCC GARCH model again by maximum likelihood estimation, where we fit residuals from a GARCH(1,1) filter assuming a multi-variate normal distribution. Results of the A-DCC GARCH correlation parameters are presented in Table 5.3.
Table 5.3: ADCC(1,1,1) model parameter estimates
ADCC a2 b2 g2
Value 0.00330 0.97611 0.00216 St. Dev 0.00016 0.00141 0.00036
Results for parameters a and b do not experience large differences compared to the values in the DCC-GARCH model. The asymmetric shock coefficient enters in the equation with a value of 0.00216, and the sum of the coefficients of both shocks in the A-DCC GARCH model is larger than the a coefficient in the DCC-GARCH model. This evidence suggests that the augmented model A-DCC GARCH does not overfit the data but improves the performance of the DCC-GARCH.
Standard errors are presented in Table 5.3. All coefficients are significant at the 5% level. To ensure that the autocorrelation matrix is positive semi-definite at ev-ery period we impose this restriction on the parameters: a2 + b2 + δg2 < 1, where δ = max(eig([ ¯R−1/2N ¯¯R−1/2]) and additionally that the parameters are positive.
Coefficient standard errors have been estimated using the inverse of the Information matrix (the negative of the Hessian of the likelihood function). Additionally, there also needs to be ensured that the Hessian is actually positive semi-definite. This is necessary in order to invert the matrix.
The estimation can be conducted by constrained maximum likelihood in Matlab with an active set algorithm. The restrictions are linear if we instead of estimating a and b we focus on a2 and b2, while δ is a parameter that can be estimated separately using linear algebra.
5.4 Flexible Dynamic Conditional Correlation, FDCC(1,1)
The last model that is taken into account is the F-DCC GARCH model. In this model, the coefficients for the correlation matrix are not equal for all elements of the matrix. They vary between the industries, but they are common within each industry. The F-DCC GARCH correlation parameter estimates are presented in Table 5.4. The a parameters are behind the coefficient matrix of the outer product of residuals. The b parameters are behind the matrix of coefficients of the first lag of the conditional correlation matrix.
There is one parameter of each class for each industry. It is important to stress that each matrix of coefficients is the outer product of a set of parameters. These are being repeated as many times as assets in each industry. Then the matrix of coefficients Hadamard multiplies the corresponding matrix in the F-DCC GARCH dynamics equation (i.e. A = aa0 , where a = [a11, .., a1j1, . . . as1, .., asjs], and js is the number of assets in
industry s).
The parameters of the model are estimated by Maximum Likelihood Estimation, and we impose that the elements in the coefficient matrix for the long term correlation are the complement of the sum of Aij+ Bij. Additionally, we impose also that all the parameters
only take on positive values.
The results in Table 5.4 allow us to distinguish the correlation dynamics between the different industries. Now we are able to answer the question of which industries show the most stable correlation and which show the most persistence. For instance, the correlation in the Automotive industry and the Electronics industry is very stable over time (with a coefficient for lag 1 being very close to 1), while it is low for the Beverage industry. This result, together with the fact that correlation in Beverage industry is the lowest among all industries (0.248 as presented in the table of industry correlations), suggests that the assets in the Beverage industry are heterogeneous and are driven by idiosyncratic or economy-wide factors, rather than by a particular variable in the industry.
Table 5.4: FDCC(1,1) model parameter estimates FDCC Value St. Dev a1 0.09998 0.00396 a2 0.04849 0.00209 a3 0.04563 0.00330 a4 0.07787 0.00336 a5 0.06884 0.00458 a6 0.08757 0.00739 b1 0.97997 0.00191 b2 0.99745 0.00028 b3 0.99211 0.00160 b4 0.98714 0.00146 b5 0.97679 0.00347 b6 0.91306 0.01764
6 Model comparison: post-estimation
statistics
In this section the performance of the three dynamic conditional correlation models is analyzed in detail. We look at the model selection statistics, and we evaluate their performance for risk measurement purposes.
6.1 Likelihood Function
To recover the likelihood function and at the same time ensure that results are transparent and reliable we build the covariance matrix as implied by each of the correlation models and GARCH filter. We present the results in logarithms in Table 6.1. We see that the F-DCC GARCH model attains the highest likelihood while the DCC-GARCH model offers the lowest fit. This is a logical result, since the number of parameters is the smallest in the DCC-GARCH model and the largest in the F-DCC GARCH model. The A-DCC GARCH model and the F-DCC GARCH model both nest the DCC-GARCH model, but the F-DCC GARCH model does not nest the A-DCC GARCH model and conversely. Basically, when all a and b parameters are equal the F-DCC GARCH and the DCC-GARCH model are identical.
To account for the different number of parameters the Akaike Information Criterion (AIC) is computed. The model with the lowest AIC provides the best fit. F-DCC GARCH is again the winner as expected, while the A-DCC GARCH model still dominates the DCC-GARCH model.
Table 6.1: Likelihood and Akaike
DCC ADCC FDCC Log-Likelihood 212256.5 212262.9 212309.3 AIC -424329.0 -424339.8 -424434.6
6.2 Likelihood-ratio test
In this section we extend the comparisons of Section 6.1 to the Likelihood-ratio test. The idea of this test is that some changes in the likelihood function can be explained by statistical variability, by the randomness in the population sampling. Additionally, we can evaluate if the changes across model likelihood values are due to this randomness.
On the other hand, a better parametrization of the population properties also will be reflected in the likelihood. Consequently, better models will have higher likelihoods, and those models that represent a statistically significant improvement are interesting for this research.
Therefore, we can try to separate the differences due to randomness from those due to precision of the model. For this purpose we have the Likelihood-ratio test. It will state a population distribution of the ratio of likelihoods. Whenever this ratio takes values that lie in the tails and are deemed too unlikely, then we can reject the hypothesis of randomness and conclude that one model is better than the other.
The Likelihood-ratio test is constructed as follows. It is important to find two models, where one model nests the other model. To be precise, one model includes the parameters from the other model and, in addition to that, other parameters which can take on a value that makes the two models equivalent. The larger model represents the alternative hypothesis, the smaller model represents the null hypothesis.
In this research, we can elaborate two such comparisons. The A-DCC GARCH model nests the DCC-GARCH model. The A-DCC GARCH model has the parameters of the DCC-GARCH model plus an additional parameter for the asymmetric residuals outer product. In the second comparison the F-DCC GARCH model is the larger model and the DCC-GARCH model is the smaller one. The F-DCC GARCH model is equivalent to the DCC-GARCH model when the coefficients of the F-DCC GARCH are all equal across industries. Both models have a coefficient for the sample correlation, a coefficient for the lag 1 conditional correlation and a coefficient for the outer product of residuals.
The expression for this Likelihood-ratio test statistic is shown in equation (6.1):
D = −2 ln
likelihood for null model
likelihood for alternative model
(6.1)
Under the null hypothesis that both models are equivalent the statistic is distributed according to a chi-squared distribution with k1− k2 degrees of freedom, where k1 is the
number of parameters of the larger model, and k2 is the number of parameters of the nested or smaller model.
Hence, we make use of the fact that the DCC-GARCH model is nested within the A-DCC GARCH model to test whether the hypothesis that the A-DCC GARCH model
is no different from the DCC-GARCH model, i.e. that the coefficient for the asymmetric shocks is zero.
The number of parameters of the DCC-GARCH model is equal to 3 × 30 + 2. In this calculation, the 3×30 account for the parameters estimated in the first stage GARCH(1,1) estimation, and the 2 remaining parameters account for the second stage DCC-GARCH correlation parameters.
In the case of the A-DCC GARCH model, the number of parameters is 3 × 30 + 3. Hence the difference in degrees of freedom is equal to 1. Therefore, the degrees of freedom of the chi-squared distribution is equal to 1. The results for this Likelihood-ratio test are presented in table 6.2.
Table 6.2: Likelihood-ratio test (A-DCC vs DCC) Value 12.81530
p-value 0.01010
Table 6.2 leads us to conclude that we can not reject the hypothesis that the models are equal at the 1% significance level, although the p-value is close to 1%.
In Table 6.3, the Likelihood-ratio test is repeated for the F-DCC GARCH against the null hypothesis of the DCC-GARCH model. The number of parameters of the F-DCC GARCH model are equal to 3 × 30 + 6 + 6, where the additional parameters to the GARCH filter are formed by the two parameters for each industry. In each industry there is one parameter for the lagged conditional correlation and one parameter for the matrix of outer products of residuals.
The F-DCC GARCH additional parameters are statistically significant at the 1% confidence level. In summary, both the A-DCC GARCH model and the F-DCC GARCH model have proven to fit the data better than the DCC-GARCH model. Additionally, we have been able to demonstrate that this difference is not due to statistical variation but to a better adjustment to the population.
Table 6.3: Likelihood-ratio test (F-DCC vs DCC) Value 105.6
6.3 Value at Risk
In VaR predictions the Student-t pdf will be used, because of the fact that the data of returns is more consistent with the Student-t distribution. We estimate by MLE the degrees of freedom of the distribution of portfolio residuals, after filtering the returns of the portfolio with the GARCH model and applying the investment weights. The likelihood function is displayed in Equation (6.2). The result is 24.7 degrees of freedom.
L({rt}Tt=1|θ) =
T Y
t=1
t(rt|θ). (6.2)
We also add for comparison the Value at Risk based on a rolling window. Using an In-Sample Window of 240 days, the sample standard deviation is computed, and then the inverse of the cdf of the Student-t distribution is used to find the VaR at each probability. For the degrees of freedom, we use 24.7.
The Value at Risk is computed at the 0.1%, 1%, and 5% levels for daily returns and the results are presented in Figure 6.1. Value at Risk is presented in figure 6.1 in negative terms.
(a) DCC-GARCH (b) A-DCC GARCH
(c) F-DCC GARCH
Results are very similar between correlation models. The correlation models do not generate large differences in the predicted correlation between assets. In Table 6.4 we also include the percentage of hits. Hits are defined as the observations for which the return is lower than the negative of the VaR. If the model accurately predicts the tail of the returns distribution, the percentage values in the table will be similar to the probability level that the VaR is targeting. Indeed we see small variations from the target probability. Moreover, the results are equal for DCC-GARCH and A-DCC GARCH at all levels. The F-DCC GARCH only shows a small difference from DCC-GARCH and A-DCC GARCH at 5% VaR.
Table 6.4: Summary of hits of VaR
VaR 0.1% 1% 5%
dcc 0.00199 0.01396 0.04188 a-dcc 0.00199 0.01396 0.04188 fdcc 0.00199 0.01396 0.04228 Roll. W. 0.0056 0.0156 0.0403
The three multivariate GARCH models overestimate the weight of the tail at the 5% level and underestimate the tail at the 1% and 0.1%. This suggests that the true distribution of returns is even more leptokurtic than the Student-t distribution.
As a measure of accuracy of the VaR, we test the hypothesis that the proportion of hits is different from the probability associated to the VaR. The test is based on the VaR(1%). We assume a normal distribution on the proportion of hits. The true distribution function is a binomial probability distribution, with T number of trials, and under the null the probability p associated to the V aR(p). Since T × p is large this distribution can be approximated by the normal distribution, with mean p and standard deviationp
p(1 − p).
We find that the p-values of the test against the hipothesis of correct specification are 0.1161 in the DCC-GARCH, A-DCC GARCH and F-DCC GARCH models. Therefore we do not reject the hypothesis that the VaR is correctly specified on the basis that the proportion of hits is different to the probability of the VaR. All three correlation models show the same value because the number of hits is the same. There are small differences in the VaR of them and they are not reflected in this test.
Once we have the series of VaR hits for each model we are interested to know whether there is autocorrelation in them. If the VaR was correct there should not be autocor-relation, since the shocks to prices before multiplying the variance are assumed to be independent and identically distributed. Observing autocorrelation means that the VaR is not reacting adequately to changes in variance.
In order to test the autocorrelation among the hits, we perform the Engle-Manganelli test. In Tables 6.5-6.8 we present the results of the regression of the hit variable on its 3 first lags and a constant term, for the case of 1% probability.
Table 6.5: Engle-Manganelli test: dcc Coefficient St.Dev Constant 0.0155 0.0026 Lag 1 0.0728 0.0210 Lag 2 0.0334 0.0207 Lag 3 0.0909 0.0207
Table 6.6: Engle-Manganelli test: a-dcc Coefficient St.Dev Constant 0.0155 0.0026 Lag 1 0.0728 0.0210 Lag 2 0.0334 0.0207 Lag 3 0.0909 0.0207
Table 6.7: Engle-Manganelli test: fdcc Coefficient St.Dev Constant 0.0155 0.0026 Lag 1 0.0728 0.0210 Lag 2 0.0334 0.0207
Table 6.8: Engle-Manganelli test: Rolling VaR Coefficient St.Dev
Constant 0.0156 0.0024 Lag 1 0.1466 0.0199 Lag 2 0.0929 0.0196
Since the variance covariance estimations for all the three models were similar, the hits series are also very similar, and the results of the test as well.
We can see that there is a significant explanatory power in the three lags of the regression. The coefficients are all positively in the regression. If there is a hit in the
VaR today the probability of there being another hit is 0.07, 0.03 and 0.09 in the next three days respectively. This is a very significant increase, which suggests that the VaR is not reacting quickly enough to changes in volatility. This is a defect that the correlation models are receiving from the GARCH model, since the conditional variance has a much more important effect in the behaviour of VaR.
6.4 Estimated correlation
Correlation between assets can be estimated with a rolling window and a sample cor-relation. Alternatively, we can use the correlation models to track the evolution of the correlation. We use the F-DCC GARCH model to illustrate the correlation of the three pairs of assets with the highest average correlation in Figure 6.2 and the three pairs with the lowest average correlation in Figure 6.3.
Figure 6.2: FDCC correlation, highest correlation assets
Correlation remains stable throughout the sample, and the short term (20 days) fluc-tuations have small amplitude compared to long term cycles (1000 days). This is a point in favour of the correlation models, since more sensitivity in the short term would add noise in the estimations and would be also difficult to justify from a theoretical perspective.
Correlation also fluctuates more when it is higher among the assets. When there is little in common between two assets they will tend to move with independence. Assets
Figure 6.3: FDCC correlation, lowest correlation assets
with high relation will tend to move together when the factor that binds them gains importance, which changes over time.
To compare the performance between models we analyze the correlation between two assets of the same industry and the correlation between two assets from different industries. The correlation between assets 1 and 2 from the same industry are shown in Figure 6.4 and the correlation between assets 1 and 6 from different industries are shown in Figure 6.5.
The Engle-Manganelli regression and the VaR exercises show similar results across models. The GARCH parameters are equal for all three models, and variance of the assets is an important determinant of VaR performance. The other factor is correlation, which could be different across assets.
Along with the correlations predicted by the three dynamic conditional correlation models, we include in Figures 6.4 and 6.5 the 50 days sample correlation between the assets.
This 50 days average correlation or rolling correlation shows wide, quick and unpre-dictable fluctuations. As the window narrows the variability in this correlation is larger. We have chosen 50 days to have an estimation with low standard deviation, and if it still shows large variability is not due so much to a small sample but to a rapidly
changing market. Although this sample correlation might better reflect the instantaneous correlation between assets, it is not very useful in making predictions in correlation because the swings are too drastic. The predictions will be already useless for a one-day or two-days forecast.
On the other side, the three dynamic correlation models are much more stable, and barely change over the whole sample period. The predicted values are correctly estimated at a point near the center of the rolling correlation, and while they are less sensitive to market changes, they evolve in the same direction as the rolling correlation.
The behavior of the three models is very similar, and it is difficult to point at systematic differences that can be tracked to the properties of each model. The F-DCC GARCH model seems to be the most volatile, while the A-DCC GARCH and the DCC-GARCH model move very close to each other. This is also expected given that the asymmetric coefficient in the A-DCC GARCH model was estimated very close to zero. To summarize, the F-DCC GARCH model reacts more quickly to changes in observed correlation, while A-DCC GARCH and DCC-GARCH are very similar.
Figure 6.5: Correlation between assets 1 and 6
6.5 Residual test
In this section we implement a test that allows us to see if the dynamics in the correlation matrix have been completely captured. The test detects autocorrelation in the outer product of residuals.
The residuals are constructed as follows; at every period, and using the covariance matrix estimated by the GARCH filter and a dynamic conditional correlation model, we divide returns by the square root of their covariance matrix. Using the eigenvalue decomposition, we obtain a matrix such that if it premultiplies and postmultiplies the covariance matrix, the result is the identity.
Moreover, if the variance covariance matrix is accurate, the outer product of residuals should approximate the identity matrix in expectation, because all the correlation between returns is theoretically removed already, and the residuals are orthonormal. A more detailed definition can be found in Section 3.3.
The residual test does not measure whether the outer product of the residuals is in expectation the identity matrix. Actually, it tests whether there is correlation between the outer product and its value in the period before, such that if the covariance matrix is not accurate, at least it cannot be improved by using information from the period before with a linear function.
If the test is positive, it means the dynamic conditional correlation is too restricted, and can be improved. For instance by multiplying each entry of the outer product of returns (after GARCH) by a different coefficient in the dynamic conditional correlation equation. The models we use incorporate many fewer parameters to simplify the computation and strengthen the results, and this test shows the costs of this parsimony.
The amount of tests conducted is m × (m + 1)/2, where m is the number of assets. We regress the cross product of residuals of every period on the same variable one period back. In Table 6.9 we present the percent of coefficients that are significantly different from zero, using the normality assumption for the regression residuals and the 5% confidence level.
Table 6.9: Residuals test, significant coefficients Model Significant coef. (%)
dcc 18.9247
a-dcc 18.2796
fdcc 17.4194
Table 6.9 shows that the three models have a very similar performance, and in all three of them the cross product of normalized residuals have excessive predictive power on their step one value. We should see that 5% of the coefficients are statistically significant, and instead we see higher values. The DCC-GARCH model has more significant coefficients than A-DCC GARCH and this in turn more than the F-DCC GARCH model. There is some correlation of the returns which can be predicted with the last period’s returns that these models are not being able to capture.
7 Conclusion
This article discusses the properties of DCC-GARCH models and its extensions in the form of A-DCC GARCH and F-DCC GARCH. Empirical results show that the DCC-GARCH models are useful in estimating correlation on the basis of residual tests. Even during the financial crises in 2003 and 2008, the DCC-GARCH models managed to grasp the correlation between assets rather well.
We have analyzed some of the properties of the returns. They share the feature of high excess kurtosis, and mildly negative skewness. The normality assumption is rejected in all the cases. As a partial solution, the Student-t distribution is used in some parts of the research. The optimal degrees of freedom is in the range between 10 and 25. However, for the purposes of estimating the optimal parameters of the dynamic conditional correlation models and GARCH the normal distribution is used for practical reasons, while the Student-t distribution is applied in VaR.
The DCC-GARCH and A-DCC GARCH have an almost identical performance in the tests which were conducted. This is explained by the fact that the optimal estimate for the asymmetric shock coefficient is very close to zero and everything else in the A-DCC GARCH model is similar to the DCC-GARCH model.
Performance of Value at Risk estimations using the family of DCC-GARCH models is very similar across each other, and is quite accurate: the error in the percent of hits is up to 20% respect the target. The risk models overestimate the weight of the tail at the 5% level and underestimate the tail at the 1% and 0.1% levels. This is consistent with the fact that the true distribution of returns is even more leptokurtic than the Student-t distribution.
In terms of likelihood of each model, the F-DCC GARCH model dominates the A-DCC GARCH model and consequently also dominates the DCC-GARCH model. The differ-ences are not substantial, as we see in the other evaluations. However, the differdiffer-ences are sufficient to reject the null hypothesis, in a Likelihood-ratio test, that the F-DCC GARCH model is equivalent to the DCC-GARCH model, and that the A-DCC GARCH model is equivalent to the DCC-GARCH model. We reject the hypothesis that the F-DCC GARCH is not different from the DCC-GARCH at the 1% significance level, while we reject that the A-DCC GARCH and DCC-GARCH are equivalent at the 5% significance level.
