Elliptical Distributions and Dynamic Conditional Correlation Models

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“Common Features in London”

Cass Business School, 16-17 December 2004

Elliptical Distributions and Dynamic

Conditional Correlation Models

Juan P. Cajigas and Giovanni Urga

Centre for Econometric Analysis (CEA@Cass) Cass Business School, London

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Outline of the presentation

• DCC/ADCC models and the implication of the assumption of normality

• Elliptical distributions and the DCC model

• An empirical application using FTSE-All World • Some further theoretical developments:

– Misspecification of the pdf and Newey-Steigerwald (1997)

– Preliminary Monte Carlo evidence

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Dynamic Conditional Correlation (S,L)

[

Engle (2002), Engle and Sheppard (2001)

]

t t t t t t t D R D H H N I r = − ) , 0 ( ~ / ₁ 1 * 1 * ) ( ) ( − − = _t _t _t t Q Q Q R

∑

= − = − − = = + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ₋ ₋ = S s s t s L l l t l t l S s s L l l t Q Q Q 1 1 1 1 ' 1 α β α ε ε β ] ' [ _t _t E Q = ε ε position. diagonal i its on of element diagonal i the of root square with the matrix, diagonal a is th * th * t t Q Q

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Extensions

[

Cappiello, Engle and Sheppard (2003)

]

Asymmetric DCC (ADCC, also ADDCC)

where

A, B, and G are diagonal parameter matrixes

(

Q A Q A B Q B G NG

)

A A B Q B G n n G Q_t = − ' − ' − ' + 'ε_t₋₁ε'_t₋₁ + ' _t₋₁ + ' _t₋₁ '_t₋₁ t t t I n = [ε < 0]Dε ] [n_tn_t' E N =

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Further Models

Four special cases:

• CCC model (Bollerslev 1990): A=B=G=[0] • DCC (1,1) model (Engle 2002): • A(symmetric) DCC (1,1): • G(eneralized) Diagonal DCC (1,1): G=[0]

[ ]

, [ ]

[ ]

, ] [ ], 0 [ A a a B b b G = = _ij = = _ij =

[ ]

, [ ]

[ ]

, [ ]

[ ]

, ] [g g A a a B b b G = _ij = = _ij = = _ij =

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Extensions

[

Hafner and Franses (2003)

]

(Generalized) Diagonal DCC (1,1) model:

the variance targeting approach is sacrificed by replacing with in order to assure that Q_t will be positive-definite

(

a b

)

A A B Q B

Q

Q_t = 1− 2 − 2 + 'ε_t₋₁ε '_t₋₁ + ' _t₋₁

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Implications of the assumption of

normality

• Normality-MLE/QMLE = feasible + consistent but

inefficient DCC coefficients (Bollerslev and Wooldridge, 1992)

• Normality is not a satisfactory property for financial time series.

• Non normal distribution to achieve efficiency with implication for the first stage

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Implications of the assumption of

normality

Efficiency loss in volatility parameters (univariate mode, % changes)

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Implications of the assumption of

normality

Efficiency loss in correlation parameters

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Alternative I:

Semiparametric (sp) estimation

• Univariate case:

Engle and González-Rivera (1991), Drost and Klaassen (1997), González-Rivera (1997),

González-Rivera and Drost (1999).

• Multivariate case:

Hafner and Rombouts (2004)

Remarks:

– More efficient than QMLE

but

– Estimation and inference quite difficult, and only feasible with small dimensions

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Alternative II:

Assuming tick distributions

• Univariate case:

– Bollerslev (1987): Student-t distribution

– Baillie and Bollerslev (1989): Student-t and exponential-power distributions

– Nelson (1991): exponential-power distribution

– Verhoeven and McAleer (2003): Asymmetric Student-t,

asymmetric generalized error, asymmetric generalized Student-t, Gram-Charlier, and Pearson Type IV distribution.

• Multivariate case:

– Fiorentini, Sentana, Calzolari (2003, JBES): Multivariate Student-t in MGARCH

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The main contribution of this paper:

elliptical-ADDCC (1,1)

We propose an ADDCC (1,1) model and its nested versions, using elliptical distributions for the

vector of standardized residuals.

We consider three distributions:

• Multivariate Laplace • Multivariate Student-t

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Elliptical distributions

• Multivariate Laplace

• Multivariate Student-t

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Two-Step estimation: feasible (1…)

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Two-Step estimation: feasible (…2)

• Normal case

• Engle (2002): uses Newey-McFadden (1994, HoE) results on GMM to justify the use of MLE for consistency

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Two-Step estimation: feasible

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Two-step estimation

No feasible: lack identification of the degree of freedom parameter v

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Two-Step estimation

No feasible: complexity of the functional form

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Empirical application

Data

FTSE All-World weekly indices converted to US

denominated returns for 21 countries from 31/12/1993 to 09/04/2004 (T=538 weekly observations).

The countries are:

Australia Germany Netherlands Switzerland

Austria Hong Kong New Zealand United Kingdom Belgium Ireland Norway United States Canada Italy Singapure

Denmark Japan Spain France Mexico Swedem

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Empirical results

• Six models were estimated: – Normal-DCC (1,1).

– Normal-DDCC (1,1)

– Normal-ADDCC (1,1) still running – Laplace-DCC(1,1)

– Laplace-DDCC (1,1)

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Empirical results

• Step 1: Univariate volatilities were estimated with GARCH (1,1) models (Vol. parameters)

• Step 2: Estimation of the DCC and its variants (Corr. Parameters) using results from Step 1

• Table: note the trade-off between flexibility and feasibility

Model Vol. Parameters Corr. Parameters Total

DCC 63 2 65

DDCC 63 42 105

ADDCC 63 63 126

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Empirical results

Variance parameters for the Normal Models

Australia s.e. Hong Kong s.e. Normway s.e.

omega 2.23E-05 1.02E-05 3.73E-06 2.52E-06 4.42E-06 2.85E-06

alpha 0.14605 0.067485 0.081495 0.034236 0.064193 0.025001

beta 0.69088 0.11078 0.90827 0.032557 0.91538 0.032557

Austria s.e. Ireland s.e. Singapure s.e.

omega 5.77E-06 3.00E-06 1.96E-05 9.60E-06 5.12E-06 4.14E-06

alpha 0.15053 0.042651 0.16321 0.078719 0.09062 0.038479

beta 0.82918 0.039649 0.73073 0.09833 0.89841 0.036273

Belgium s.e. Italy s.e. Spain s.e.

omega 1.89E-05 7.96E-06 0.00028691 0.000111 1.07E-05 5.77E-06

alpha 0.14596 0.042945 0.17492 0.072041 0.11659 0.03723

beta 0.74157 0.071355 0.20017 0.25074 0.83832 0.04905

Canada s.e. Japan s.e. Sweden s.e.

omega 1.62E-06 2.00E-06 5.94E-06 3.99E-06 6.78E-06 4.39E-06

alpha 0.11095 0.032954 0.090254 0.033284 0.11856 0.044855

beta 0.88905 0.033239 0.88638 0.040049 0.87031 0.038701

Denmark s.e. Mexico s.e. Switzerland s.e.

omega 5.31E-07 6.33E-07 9.05E-06 5.57E-06 3.78E-05 1.58E-05

alpha 0.059524 0.020538 0.079256 0.026151 0.18951 0.079165

beta 0.94047 0.019624 0.90382 0.029507 0.52119 0.15448

France s.e. Netherlands s.e. U.K. s.e.

omega 5.12E-06 3.22E-06 5.90E-06 2.88E-06 1.88E-06 1.43E-06

alpha 0.098613 0.034479 0.14257 0.043979 0.070166 0.024429

beta 0.87295 0.041849 0.82979 0.043862 0.91492 0.028555

Germany s.e. New Zeland s.e. USA s.e.

omega 1.90E-06 1.42E-06 1.47E-05 6.98E-06 7.41E-07 7.16E-07

alpha 0.11917 0.036376 0.12868 0.043759 0.093355 0.027173

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Empirical results

Variance parameters for the Laplace Models

Australia s.e. Hong Kong s.e. Norway s.e. omega 2.25E-05 1.19E-05 5.70E-06 3.99E-06 7.83E-06 4.49E-06

alpha 0.13858 0.069143 0.075956 0.03585 0.077512 0.031291

beta 0.74244 0.10637 0.92361 0.03233 0.89433 0.040378

Austria s.e. Ireland s.e. Singapore s.e. omega 6.72E-06 3.45E-06 4.30E-06 2.83E-06 7.04E-06 4.99E-06

alpha 0.16198 0.047779 0.05587 0.029179 0.085002 0.039688

beta 0.833 0.040301 0.926 0.030964 0.90501 0.037034

Belgium s.e. Italy s.e. Spain s.e. omega 1.90E-05 8.31E-06 0.00015011 4.04E-05 1.73E-05 8.90E-06

alpha 0.17791 0.051261 0.16397 0.054078 0.14404 0.049907

beta 0.75896 0.065477 0.15052 0.15538 0.82288 0.057375

Canada s.e. Japan s.e. Sweden s.e. omega 3.94E-06 3.16E-06 4.70E-06 3.77E-06 7.38E-06 4.71E-06

alpha 0.11268 0.039156 0.082615 0.031753 0.10864 0.045311

beta 0.88732 0.040363 0.91738 0.031572 0.89136 0.036487

Denmark s.e. Mexico s.e. Switzerland s.e. omega 1.22E-06 1.01E-06 9.34E-06 6.01E-06 4.99E-05 2.36E-05

alpha 0.050246 0.020454 0.084133 0.02812 0.19755 0.09165

beta 0.94975 0.01951 0.91056 0.028187 0.49496 0.19161

France s.e. Netherlands s.e. U.K. s.e. omega 2.20E-06 1.90E-06 5.16E-06 2.72E-06 2.15E-06 1.69E-06

alpha 0.060811 0.023445 0.12643 0.042101 0.076305 0.028137

beta 0.93919 0.022732 0.86793 0.036157 0.92318 0.027638

Germany s.e. New Zeland s.e. USA s.e. omega 3.23E-06 1.97E-06 2.09E-05 9.41E-06 2.14E-06 1.17E-06

alpha 0.087564 0.033901 0.15096 0.053489 0.087155 0.030793

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Empirical results

• Correlation parameters from the DCC (1,1) model

Parameters Normal DCC s.e. Laplace DCC s.e.

α 0.011518 0.002333 0.078904 0.01791 β 0.90327 0.027541 0.92109 0.023458

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Empirical results

• Correlation parameters from the Normal-DDCC (1,1) model

Australia s.e. Hong Kong s.e. Norway s.e.

alpha 0.00896 0.00983 0.01723 0.01115 0.00675 0.00789

beta 0.90668 0.04939 0.85866 0.06246 0.91990 0.10491

Austria s.e. Ireland s.e. Singapore s.e.

alpha 0.01447 0.01263 0.02815 0.02184 0.00571 0.00755

beta 0.88716 0.13879 0.94368 0.05519 0.89168 0.06822

alpha 0.05449 0.02645 0.06467 0.03358 0.03411 0.02412

beta 0.90249 0.03864 0.93533 0.04094 0.93180 0.03711

alpha 0.02123 0.01600 0.00287 0.00543 0.03512 0.02200

beta 0.90909 0.05732 0.89469 0.07912 0.92594 0.03202

alpha 0.00725 0.00756 0.01026 0.01144 0.05763 0.03640

beta 0.92246 0.04624 0.93240 0.02613 0.91009 0.06829

alpha 0.08221 0.02968 0.06931 0.02700 0.05608 0.03024

beta 0.91778 0.03317 0.92375 0.02804 0.90111 0.07283

alpha 0.06568 0.02650 0.00219 0.00355 0.05173 0.03139

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Empirical results

• Correlation parameters from the Laplace-DDCC (1,1) model

Australia s.e. Hong Kong s.e. Normway s.e.

alpha 0.0503 0.0303 0.3122 0.3727 0.4333 0.5545

beta 0.9497 0.2975 0.6878 0.0978 0.5667 0.2204

Austria s.e. Ireland s.e. Singapure s.e.

alpha 0.2312 0.5029 0.4022 0.3673 0.3202 0.5580

beta 0.7153 0.4389 0.5978 0.2403 0.6798 0.2279

alpha 0.3278 0.4142 0.2212 0.0410 0.4962 0.3361

beta 0.6723 0.1392 0.7788 0.4702 0.5038 0.2047

alpha 0.3171 0.5621 0.3671 0.3601 0.1653 0.1799

beta 0.6829 0.0481 0.6329 0.0834 0.8347 0.3087

alpha 0.2929 0.5182 0.2559 0.0635 0.2001 0.0379

beta 0.7072 0.4522 0.7441 0.3817 0.7999 0.3083

alpha 0.3258 0.3145 0.4515 0.5771 0.3812 0.4881

beta 0.6742 0.0086 0.5485 0.2140 0.6188 0.2206

alpha 0.2879 0.3050 0.2668 0.2972 0.2588 0.0325

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Empirical results:

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Main findings from the empirical

application

• Higher persistence in univariate variances

when the Laplace distribution is used

instead of the normal one

• The size of standard errors varies across

models.

• High levels of persistence in the correlation

under both distributions

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Robust Conditional Moment Test.

Kroner and Ng (1998) in the framework of Wooldridge (1998):

u_t is a “generalized residual” given by the distance

between a point of the scatter plot of and a corresponding point in the news impact surface.

If then u_ijt should be uncorrelated with any

variable known at time t-1 and the model will be correct.

Empirical results: diagnostic test

ijt jt it t h u = ε ε − 0 ) ( 1 = − ijt t u E jt itε ε

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10 indicator variables are established, as in the sign and size bias tests of Engle and Ng (1993):

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Wooldridge (1990):

where is the residual from the regression,

and is the vector of parameters in the MGARCH model. Under general regularity conditions the C test has an asymptotic distribution

Testing

[

]

∑

= − = − = _T t ijt mt T t ijt mt u T u C 1 2 1 2 2 1 1 λ λ

∑

= − − _∂ + ∂ + = K k mt k ijt k mt h x 1 1 0 1 α α θ λ 10 ,..., 1 , 1 = − m mt λ ) ,..., (θ1 θK θ = ) 1 ( 2 χ

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Empirical results: tests

Indicator Normal-DCC Laplace_DCC Normal-DDCC Laplace_DDCC Normal-ADDCC Laplace-ADDCC

x1 22.53% 21.46% 9.72% 18.94% 13.70% 11.02% x2 17.05% 14.30% 7.57% 11.92% 8.63% 5.81% x3 18.86% 16.36% 8.08% 14.46% 9.33% 6.76% x4 25.06% 24.28% 10.87% 21.90% 14.12% 11.77% x5 1.37% 2.19% 1.06% 2.32% 1.73% 1.76% x6 1.79% 2.16% 1.66% 1.88% 2.18% 1.95% x7 3.92% 3.07% 1.31% 1.59% 2.38% 1.42% x8 2.91% 2.20% 1.48% 1.78% 1.49% 1.27% x9 3.14% 2.08% 1.61% 1.54% 2.41% 1.12% x10 3.79% 3.13% 1.84% 3.02% 2.07% 1.91%

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Main findings from testing

• DCC model: Laplace outperforms

Normality

• DDCC model: Normality outperforms

Laplace

• ADDCC: Laplace outperforms Normality

Promising but too preliminary!!! Sample may be too short!!

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First theoretical development:

misspecification of the pdf

• In the QMLE case we can obtain consistent but inefficient estimates (Bollerslev and Wooldridge, 1992), because of Gaussian hypothesis.

• QMLE might not be appropriate when we assume a non-Gaussian distribution (Newey and

Steigerwald, 1997, and Straumman, 2004) which is not the correct one.

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Misspecification of the pdf

[

Intuition for the consistency of QMLE]

• Only for the case of the Gaussian distribution the scale and location parameters coincide with the variance and mean parameters of the stochastic process

.

• For a density g(y) the natural location parameter µ and scale parameter σ are those that maximize,

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + − σ µ σ g y E ln ln _Y

When g_Y ( y)is Gaussian then µ = E(y_t) and σ 2 = E(y_t − µ)2 .

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Newey and Steigerwald (1997)

Two main results:

1. If the true and assumed distributions of residuals are both symmetric then the identification

condition for consistency of the non-Gaussian QMLE is satisfied.

2. If this is not the case then an extra location parameter must be added to the standardized residuals

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Monte Carlo experiments

• We design a Monte Carlo Experiment to analyze some of the theoretical results of Newey and

Steigerwald (1997)

• Because of the symmetric form of elliptical distributions we analyze only Result 1.

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Conclusions

• The DCC model needs to be extended and

alternative distributions to be considered, more adequate than the normal to model financial time series.

• The fitting of the global equity markets improves when the Laplace distribution is used instead of the normal. Though the properties of the

estimators needs more analysis.

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Future developments I

Consistency of the elliptical DCC model in cases of misspecification of the multivariate probability density function

The applicability and benefits of the

elliptical-DCC model should be evaluated via application to finance such VaR and asset allocation models.

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Future developments II

• QMLE for conditional quantiles (Komunjer, 2004; Koenker and Bassett, 1978)

Asymmetric Laplace density and the corresponding QMLE reduces to non-linear quantile estimator

• Consistency as an identification issue (Wu, 1981)

Provides necessary conditions for asymptotic consistency for nonlinear models

• Use of robust procedures in estimating and testing (Hampel, 1974; Huber, 1981)