FINANCIAL ECONOMETRICS

FINANCIAL ECONOMETRICS

SPRING 2013 WEEK VIIMULTIVARIATE MODELLING OF VOLATILITY

Prof. Dr. Burç ÜLENGİN

MULTIVARIATE VOLATILITYThere may be interactions among the

conditional variance of the return series.Also covariance of the return series may

change over the time.Therefore the full perspective of volatility

modelling requires the treatment of variances and covariances together- simultaneously.

When the variances and covariances are modelled it means that correlations are modelled too.

MOVING CORRELATION OF THE RETURNS OF TWO FINANCIAL ASSETS

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MULTIVARIATE GARCH In multivariate GARCH models, yt is a vector

of the conditional means (Nx1), the conditional variance of yt is an matrix H (NxN).

The diagonal elements of H are the variance terms hii, and the off-diagonal elements are the covariance terms hij.

NNNN

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...

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MULTIVARIATE GARCHThere are numerous different

representations of the multivariate GARCH model.

The main representations are: VECH Diagonal BEKK- Baba, Engle, Kraft, Kroner Constant correlation representation Principle component representation

VECH REPRESANTATION Full treatment of the matrix H In the VECH model, the number of parameters can

be exteremely large. Estimating a large number of parameters is not in

theory a problem as long as there is large enough sample size.

The parameters of VECH are estimated by maximum likelihood and the obtaining convergence of the typical optimization algorithm employed in practice be very difficult when a large number of parameters are involved.

Also estimated variances must be positive and it requires the additional restrictions on parameters

VECH REPRESANTATION 2 Variable Case

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A and B are {Nx(N+1)/2 , Nx(N+1)/2} matrices .

In the case of 2 variables, 3 equations and 21 parameters.

5 variables, 20 equations and 820 parameters.

10 variables, 55 equations and 4025 parameters.

DIAGONAL REPRESENTATION The diagonal representation is based on the

assumption that the individual conditional variances and conditional covariances are functions of only lagged values of themselves and lagged squared residuals.

Bollerslev, Engle and Woodridge (1988) proposed In the case of 2 variables, this representation reduces

the number of parameters to be estimated from 21 to 9.

At the expense of losing information on certain interrelationships, such as the relationship between the individual conditional variances and the conditional covariances.

Also estimated variances must be positive and it requires the additional restrictions on parameters

DIAGONAL REPRESENTATION 2 Variable Case

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RGAZ

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ROIL

DIAGONAL REPRESENTATIONOIL & NATURAL GAS PRICES

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DIAGONAL REPRESENTATION ESTIMATIONOIL & NATURAL GAS PRICES

Estimation Method: ARCH Maximum Likelihood (Marquardt)Covariance specification: Diagonal VECHSample: 1997M02 2007M01Included observations: 120Total system (balanced) observations 240Disturbance assumption: Student's t distributionConvergence achieved after 198 iterations

Coefficient Std. Error z-Statistic Prob. C(1) 1.259 1.082 1.164 0.245C(2) 2.438 0.733 3.328 0.001

Variance Equation CoefficientsC(3) 63.635 41.552 1.531 0.126C(4) 0.837 1.852 0.452 0.651C(5) 9.479 2.224 4.261 0.000C(6) 0.224 0.179 1.256 0.209C(7) -0.056 0.039 -1.421 0.155C(8) -0.200 0.081 -2.467 0.014C(9) 0.352 0.358 0.986 0.324C(10) 0.858 0.065 13.109 0.000C(11) 1.067 0.043 24.956 0.000

t-Distribution (Degree of Freedom) C(12) 18.732 26.610 0.704 0.482

Log likelihood -890.0937 Schwarz criterion 15.31364Avg. log likelihood -3.708724 Hannan-Quinn criter. 15.1481Akaike info criterion 15.03489

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DIAGONAL REPRESENTATION ESTIMATIONOIL & NATURAL GAS PRICES

CoefficientC(1) 1.259C(2) 2.438

Variance Equation CoefficientsC(3) 63.635C(4) 0.837C(5) 9.479C(6) 0.224C(7) -0.056C(8) -0.200C(9) 0.352C(10) 0.858C(11) 1.067

Estimated Equations:=====================RGAZ = C(1)

ROIL = C(2)

Substituted Coefficients:=====================RGAZ = 1.258

ROIL = 2.438

Variance-Covariance Representation:=====================GARCH = M + A1.*RESID(-1)*RESID(-1)' + B1.*GARCH(-1)

Variance and Covariance Equations:=====================GARCH1 = M(1,1) + A1(1,1)*RESID1(-1)^2 + B1(1,1)*GARCH1(-1)

GARCH2 = M(2,2) + A1(2,2)*RESID2(-1)^2 + B1(2,2)*GARCH2(-1)

COV1_2 = M(1,2) + A1(1,2)*RESID1(-1)*RESID2(-1) + B1(1,2)*COV1_2(-1)

Substituted Coefficients:=====================GARCH1 = 63.634 + 0.224*RESID1(-1)^2 + 0.352*GARCH1(-1)

GARCH2 = 9.479 -0.199*RESID2(-1)^2 + 1.066*GARCH2(-1)

COV1_2 = 0.837 -0.055*RESID1(-1)*RESID2(-1) + 0.857*COV1_2(-1)

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Coefficient

M(1,1) 63.63M(1,2) 0.84M(2,2) 9.48A1(1,1) 0.22A1(1,2) -0.06A1(2,2) -0.20B1(1,1) 0.35B1(1,2) 0.86B1(2,2) 1.07

Covariance specification: Diagonal VECHGARCH = M + A1.*RESID(-1)*RESID(-1)' + B1.*GARCH(-1)M is an indefinite matrixA1 is an indefinite matrixB1 is an indefinite matrix

DIAGONAL REPRESENTATION VOLATILITY FORECAST OF OIL & NATURAL GAS PRICES

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97 98 99 00 01 02 03 04 05 06

Var(RGAZ)

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Cov(RGAZ,ROIL)

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Var(ROIL)

Conditional Covariance

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Cor(RGAZ,ROIL)

Conditional Correlation

DIAGONAL REPRESENTATION REVISED MODEL ESTIMATION OIL & NATURAL GAS PRICES

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Estimation Method: ARCH Maximum Likelihood (Marquardt)Covariance specification: Diagonal VECHSample: 1997M03 2007M01Included observations: 119Total system (balanced) observations 238Disturbance assumption: Student's t distributionConvergence achieved after 26 iterations

Coefficient Std. Error z-Statistic Prob. C(1) 0.548 0.873 0.628 0.530C(2) 0.240 0.088 2.730 0.006C(3) 1.513 0.782 1.934 0.053

C(4) 1.974 1.865 1.059 0.290C(5) 0.474 0.107 4.407 0.000C(6) -0.110 0.123 -0.894 0.371C(7) 0.890 0.041 21.543 0.000C(8) 0.983 0.020 48.854 0.000

t-Distribution (Degree of Freedom)

C(9) 6.928 2.791 2.482 0.013

Log likelihood -892.0641 Schwarz criterion 15.35412Avg. log likelihood -3.748168 Hannan-Quinn criter. 15.22928Akaike info criterion 15.14393

Covariance specification: Diagonal VECHGARCH = M + A1.*RESID(-1)*RESID(-1)' + B1.*GARCH(-1)M is a scalarA1 is a rank one matrixB1 is a rank one matrix

Tranformed Variance Coefficients

Coefficient

M 1.974A1(1,1) 0.224A1(1,2) -0.052A1(2,2) 0.012B1(1,1) 0.793B1(1,2) 0.875B1(2,2) 0.966


Covariance specification: Diagonal VECHGARCH = M + A1.*RESID(-1)*RESID(-1)' + B1.*GARCH(-1)M is a scalarA1 is a rank one matrixB1 is a rank one matrix


Coefficient

M 1.974A1(1,1) 0.224A1(1,2) -0.052A1(2,2) 0.012B1(1,1) 0.793B1(1,2) 0.875B1(2,2) 0.966

CoefficientC(1) 0.548C(2) 0.240C(3) 1.513

C(4) 1.974C(5) 0.474C(6) -0.110C(7) 0.890C(8) 0.983

Estimated Equations:=====================RGAZ = C(1)+C(2)*ROIL(-1)

ROIL = C(3)

Substituted Coefficients:=====================RGAZ = 0.548145947197+0.240483266384*ROIL(-1)

ROIL = 1.51272216999

Variance-Covariance Representation:=====================GARCH = M + A1.*RESID(-1)*RESID(-1)' + B1.*GARCH(-1)

Variance and Covariance Equations:=====================GARCH1 = M + A1(1,1)*RESID1(-1)^2 + B1(1,1)*GARCH1(-1)

GARCH2 = M + A1(2,2)*RESID2(-1)^2 + B1(2,2)*GARCH2(-1)

COV1_2 = M + A1(1,2)*RESID1(-1)*RESID2(-1) + B1(1,2)*COV1_2(-1)


GARCH2 = 1.974 + 0.012*RESID2(-1)^2 + 0.965*GARCH2(-1)


DIAGONAL REPRESENTATION VOLATILITY FORECAST OF OIL & NATURAL GAS PRICES

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97 98 99 00 01 02 03 04 05 06

Var(RGAZ)

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Cov(RGAZ,ROIL)

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Var(ROIL)


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Cor(RGAZ,ROIL)


DIAGONAL REPRESENTATION TARCH MODEL ESTIMATION OIL & NATURAL GAS PRICES

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=0 otherwise

DIAGONAL REPRESENTATION TARCH MODEL ESTIMATION OIL & NATURAL GAS PRICESEstimation Method: ARCH Maximum Likelihood (Marquardt)Covariance specification: Diagonal VECH TARCHDate: 08/05/08 Time: 18:36Sample: 1997M03 2007M01Included observations: 119Total system (balanced) observations 238Disturbance assumption: Student's t distributionPresample covariance: backcast (parameter =0.5)Convergence achieved after 169 iterations

Coefficient Std. Error z-Statistic Prob.

C(1) 0.440 1.165 0.378 0.706C(2) 0.213 0.116 1.825 0.068C(3) 1.543 0.775 1.990 0.047

Variance Equation Coefficients

C(4) 1.465 1.116 1.313 0.189C(5) -0.013 0.028 -0.461 0.645C(6) 0.224 0.089 2.506 0.012C(7) -0.234 0.056 -4.155 0.000C(8) 0.971 0.016 61.584 0.000


C(9) 4.810 1.774 2.711 0.007

Log likelihood -898.265 Schwarz criterion 15.45834Avg. log likelihood-3.77422 Hannan-Quinn criter. 15.3335Akaike info criterion15.24815

Covariance specification: Diagonal VECHGARCH = M + A1.*RESID(-1)*RESID(-1)' + D1.*(RESID(-1)*(RESID( -1)<0))*(RESID(-1)*(RESID(-1)<0))'D1.*(RESID(-1)*(RESID(-1)<0)) *(RESID(-1)*(RESID(-1)<0))' + B1.*GARCH(-1)M is a scalarA1 is a scalarD1 is a rank one matrixB1 is a scalar


Coefficient

M 1.465A1 -0.013D1(1,1) 0.050D1(1,2) -0.052D1(2,2) 0.055B1 0.971

DIAGONAL REPRESENTATION TARCH MODEL ESTIMATION OIL & NATURAL GAS PRICES


ROIL = C(3)


ROIL = 1.54264508993

Variance-Covariance Representation:=====================GARCH = M + A1.*RESID(-1)*RESID(-1)' + D1.*(RESID(-1)*(RESID(-1)<0))*(RESID(-1)*(RESID(-1)<0))'D1.*(RESID(-1)*(RESID(-1)<0))*(RESID(-1)*(RESID(-1)<0))' + B1.*GARCH(-1)

Variance and Covariance Equations:=====================GARCH1 = M + A1*RESID1(-1) 2̂ + D1(1,1)*RESID1(-1) 2̂*(RESID1(-1)<0) + B1*GARCH1(-1)

GARCH2 = M + A1*RESID2(-1) 2̂ + D1(2,2)*RESID2(-1) 2̂*(RESID2(-1)<0) + B1*GARCH2(-1)

COV1_2 = M + A1*RESID1(-1)*RESID2(-1) + D1(1,2)*RESID1(-1)*(RESID1(-1)<0)*RESID2(-1)*(RESID2(-1)<0) + B1*COV1_2(-1)

Substituted Coefficients:=====================GARCH1 = 1.464 - 0.013*RESID1(-1)^2 + 0.050*RESID1(-1)^2*(RESID1(-1)<0) + 0.971*GARCH1(-1)

GARCH2 = 1.465 - 0.013*RESID2(-1)^2 + 0.054*RESID2(-1)^2*(RESID2(-1)<0) + 0.971*GARCH2(-1)

COV1_2 = 1.464 - 0.013*RESID1(-1)*RESID2(-1) -0.052*RESID1(-1)*(RESID1(-1)<0)*RESID2(-1)*(RESID2(-1)<0) + 0.971*COV1_2(-1)


Coefficient

M 1.465A1 -0.013D1(1,1) 0.050D1(1,2) -0.052D1(2,2) 0.055B1 0.971

Coefficient

C(1) 0.440C(2) 0.213C(3) 1.543


C(4) 1.465C(5) -0.013C(6) 0.224C(7) -0.234C(8) 0.971

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Cov(RGAZ,ROIL)

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Var(ROIL)


DIAGONAL REPRESENTATION TARCH MODEL FORECAST OIL & NATURAL GAS PRICES VOLATILITY

-.2

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Cor(RGAZ,ROIL)


BEKK REPRESENTATION Engle and Kroner(1995) developed the

Baba(1990) approach. BEKK representation of multivariate GARCH

improves on both the VECH and diagonal representation, since H is almost guaranteed to be positive definite.

BEKK representation require more parameters than Diagonal rep. but less parameters than VECH.

It is more general than diagonal rep. as it allows for interaction effects that diagonal rep. does not.

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BEKK REPRESENTATION OF OIL & NATURAL GAS PRICES

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BEKK ESTIMATION OF OIL & NATURAL GAS PRICES

Estimation Method: ARCH Maximum Likelihood (Marquardt)Covariance specification: BEKKSample: 1997M02 2007M01Included observations: 120Total system (balanced) observations 240Disturbance assumption: Student's t distributionPresample covariance: backcast (parameter =0.5)Convergence achieved after 25 iterations


C(1) 0.890 0.874 1.019 0.308C(2) 1.346 0.786 1.712 0.087


C(3) 1.342 1.367 0.982 0.326C(4) 0.427 0.093 4.605 0.000C(5) -0.079 0.127 -0.623 0.533C(6) 0.908 0.032 28.189 0.000C(7) 0.987 0.014 71.875 0.000


C(8) 6.729 2.788 2.414 0.016


Covariance specification: BEKKGARCH = M + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1M is a scalarA1 is diagonal matrixB1 is diagonal matrix


Coefficient

M 1.342A1(1,1) 0.427A1(2,2) -0.079B1(1,1) 0.908B1(2,2) 0.987

BEKK ESTIMATION OF OIL & NATURAL GAS PRICES

Coefficient

C(1) 0.890C(2) 1.346


C(3) 1.342C(4) 0.427C(5) -0.079C(6) 0.908C(7) 0.987


Coefficient

M 1.342A1(1,1) 0.427A1(2,2) -0.079B1(1,1) 0.908B1(2,2) 0.987

Estimated Equations:=====================RGAZ = C(1)

ROIL = C(2)


ROIL = 1.34556553188

Variance-Covariance Representation:=====================GARCH = M + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1

Variance and Covariance Equations:=====================GARCH1 = M + A1(1,1) 2̂*RESID1(-1) 2̂ + B1(1,1) 2̂*GARCH1(-1)

GARCH2 = M + A1(2,2) 2̂*RESID2(-1) 2̂ + B1(2,2) 2̂*GARCH2(-1)

COV1_2 = M + A1(1,1)*A1(2,2)*RESID1(-1)*RESID2(-1) + B1(1,1)*B1(2,2)*COV1_2(-1)

Substituted Coefficients:=====================GARCH1 = 1.342+0.182*RESID1(-1) 2̂+0.824*GARCH1(-1)

GARCH2 = 1.342+0.0063*RESID2(-1) 2̂+0.974*GARCH2(-1)


REVISED BEKK REPRESENTATION OF OIL & NATURAL GAS PRICES

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REVISED BEKK REPRESENTATION OF OIL & NATURAL GAS PRICES

Estimation Method: ARCH Maximum Likelihood (Marquardt)Covariance specification: BEKKSample: 1997M03 2007M01Included observations: 119Total system (balanced) observations 238Disturbance assumption: Student's t distributionPresample covariance: backcast (parameter =0.5)Convergence achieved after 26 iterations


C(1) 0.548 0.873 0.628 0.530C(2) 0.240 0.088 2.730 0.006C(3) 1.513 0.782 1.934 0.053


C(4) 1.974 1.865 1.059 0.290C(5) 0.474 0.107 4.407 0.000C(6) -0.110 0.123 -0.894 0.371C(7) 0.890 0.041 21.543 0.000C(8) 0.983 0.020 48.854 0.000


C(9) 6.928 2.791 2.482 0.013


Covariance specification: BEKKGARCH = M + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1M is a scalarA1 is diagonal matrixB1 is diagonal matrix



M 1.974 1.865 1.059 0.290A1(1,1) 0.474 0.107 4.407 0.000A1(2,2) -0.110 0.123 -0.894 0.371B1(1,1) 0.890 0.041 21.543 0.000B1(2,2) 0.983 0.020 48.854 0.000

REVISED BEKK REPRESENTATION OF OIL & NATURAL GAS PRICESEstimated Equations:=====================RGAZ = C(1)+C(2)*ROIL(-1)

ROIL = C(3)


ROIL = 1.51272216999

Variance-Covariance Representation:=====================GARCH = M + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1

Variance and Covariance Equations:=====================GARCH1 = M + A1(1,1) 2̂*RESID1(-1) 2̂ + B1(1,1) 2̂*GARCH1(-1)

GARCH2 = M + A1(2,2) 2̂*RESID2(-1) 2̂ + B1(2,2) 2̂*GARCH2(-1)

COV1_2 = M + A1(1,1)*A1(2,2)*RESID1(-1)*RESID2(-1) + B1(1,1)*B1(2,2)*COV1_2(-1)

Substituted Coefficients:

=====================

GARCH1 = 1.974+0.224*RESID1(-1)^2+0.792*GARCH1(-1)

GARCH2 = 1.974+0.012*RESID2(-1)^2+0.965*GARCH2(-1)


REVISED BEKK FORECASTING OF OIL & NATURAL GAS PRICES VOLATILITY

0

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97 98 99 00 01 02 03 04 05 06

Var(RGAZ)

-40

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97 98 99 00 01 02 03 04 05 06

Cov(RGAZ,ROIL)

75

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85

90

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97 98 99 00 01 02 03 04 05 06

Var(ROIL)


-.2

-.1

.0

.1

.2

.3

.4

.5

.6

97 98 99 00 01 02 03 04 05 06

Cor(RGAZ,ROIL)


CONSTANT CORRELATION REPRESENTATIONBollerslev(1990) employes the conditional

corelation matrix R to derive a representation of the multivariate GARCH model.

In his R matrix, Bollerslev restricts the conditional correlations to be equal to the correlation coefficients between variables, which are simply constants. Thus R is constant over time.

This representation has the advantage that H will be positive definite.

CONSTANT CORRELATION REPRESENTATION

1................

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t

t

t

t

t

NNNN

N

N

NN h

h

h

h

h

h

H

The individual variance terms hiit are taken to be individual GARCH processes

CONSTANT CORRELATION REPRESENTATION OF OIL & NATURAL GAS PRICES

Estimation Method: ARCH Maximum Likelihood (Marquardt)Covariance specification: Constant Conditional CorrelationSample: 1997M02 2007M01Included observations: 120Total system (balanced) observations 240Disturbance assumption: Student's t distributionPresample covariance: backcast (parameter =0.5)Convergence achieved after 21 iterations


C(1) 0.938 1.037 0.904 0.366C(2) 2.078 0.605 3.435 0.001


C(3) 53.143 50.532 1.052 0.293C(4) 0.152 0.148 1.027 0.305C(5) 0.449 0.454 0.989 0.323C(6) 8.735 1.906 4.583 0.000C(7) -0.145 0.041 -3.507 0.001C(8) 1.033 0.020 51.343 0.000C(9) 0.154 0.102 1.516 0.130


C(10) 10.97096 8.493965 1.291618 0.1965


Covariance specification: Constant Conditional CorrelationGARCH(i) = M(i) + A1(i)*RESID(i)(-1) 2̂ + B1(i)*GARCH(i)(-1)COV(i,j) = R(i,j)*@SQRT(GARCH(i)*GARCH(j))



M(1) 53.143 50.532 1.052 0.293A1(1) 0.152 0.148 1.027 0.305B1(1) 0.449 0.454 0.989 0.323M(2) 8.735 1.906 4.583 0.000A1(2) -0.145 0.041 -3.507 0.001B1(2) 1.033 0.020 51.343 0.000R(1,2) 0.154 0.102 1.516 0.130

CONSTANT CORRELATION REPRESENTATION OF OIL & NATURAL GAS PRICES


Coefficient

M(1) 53.143A1(1) 0.152B1(1) 0.449M(2) 8.735A1(2) -0.145B1(2) 1.033R(1,2) 0.154


C(3) 53.143C(4) 0.152C(5) 0.449C(6) 8.735C(7) -0.145C(8) 1.033C(9) 0.154


ROIL = 2.07760000977

Variance and Covariance Representations:=====================GARCH(i) = M(i) + A1(i)*RESID(i)(-1) 2̂ + B1(i)*GARCH(i)(-1)

COV(i,j) = R(i,j)*@SQRT(GARCH(i)*GARCH(j))

Variance and Covariance Equations:=====================GARCH1 = C(3) + C(4)*RESID1(-1) 2̂ + C(5)*GARCH1(-1)

GARCH2 = C(6) + C(7)*RESID2(-1) 2̂ + C(8)*GARCH2(-1)

COV1_2 = C(9)*@SQRT(GARCH1*GARCH2)


GARCH2 = 8.735 - 0.145*RESID2(-1)^2 + 1.033*GARCH2(-1)

COV1_2 = 0.154*@SQRT(GARCH1*GARCH2)

CONSTANT CORRELATION FORECAST OF OIL & NATURAL GAS PRICES VOLATILITY

0

100

200

300

400

500

600

97 98 99 00 01 02 03 04 05 06

Var(RGAZ)

0

10

20

30

40

50

97 98 99 00 01 02 03 04 05 06

Cov(RGAZ,ROIL)

0

50

100

150

200

97 98 99 00 01 02 03 04 05 06

Var(ROIL)


.146

.148

.150

.152

.154

.156

.158

.160

.162

97 98 99 00 01 02 03 04 05 06

Cor(RGAZ,ROIL)


REVISED CONSTANT CORRELATION REPRESENTATION OF OIL & NATURAL GAS PRICES

Estimation Method: ARCH Maximum Likelihood (Marquardt)Covariance specification: Constant Conditional CorrelationSample: 1997M03 2007M01Included observations: 119Total system (balanced) observations 238Disturbance assumption: Student's t distributionPresample covariance: backcast (parameter =0.5)Convergence achieved after 26 iterations


C(1) 0.781 1.079 0.724 0.469C(2) 0.199 0.109 1.834 0.067C(3) 2.233 0.573 3.899 0.000


C(4) 64.398 49.848 1.292 0.196C(5) 0.198 0.170 1.163 0.245C(6) 0.314 0.432 0.727 0.468C(7) 7.293 1.229 5.936 0.000C(8) -0.135 0.033 -4.062 0.000C(9) 1.053 0.020 51.973 0.000C(10) 0.120 0.103 1.166 0.244


C(11) 13.509 11.166 1.210 0.226


Covariance specification: Constant Conditional CorrelationGARCH(i) = M(i) + A1(i)*RESID(i)(-1) 2̂ + B1(i)*GARCH(i)(-1)COV(i,j) = R(i,j)*@SQRT(GARCH(i)*GARCH(j))



M(1) 64.398 49.848 1.292 0.196A1(1) 0.198 0.170 1.163 0.245B1(1) 0.314 0.432 0.727 0.468M(2) 7.293 1.229 5.936 0.000A1(2) -0.135 0.033 -4.062 0.000B1(2) 1.053 0.020 51.973 0.000R(1,2) 0.120 0.103 1.166 0.244

REVISED CONSTANT CORRELATION REPRESENTATION OF OIL & NATURAL GAS PRICES


ROIL = C(3)


ROIL = 2.23319899793

Variance and Covariance Representations:=====================GARCH(i) = M(i) + A1(i)*RESID(i)(-1) 2̂ + B1(i)*GARCH(i)(-1)

COV(i,j) = R(i,j)*@SQRT(GARCH(i)*GARCH(j))

Variance and Covariance Equations:=====================GARCH1 = C(4) + C(5)*RESID1(-1) 2̂ + C(6)*GARCH1(-1)

GARCH2 = C(7) + C(8)*RESID2(-1) 2̂ + C(9)*GARCH2(-1)

COV1_2 = C(10)*@SQRT(GARCH1*GARCH2)


GARCH2 = 7.292 - 0.135*RESID2(-1)^2 + 1.052*GARCH2(-1)

COV1_2 = 0.119*@SQRT(GARCH1*GARCH2)


C(4) 64.398C(5) 0.198C(6) 0.314C(7) 7.293C(8) -0.135C(9) 1.053C(10) 0.120


Coefficient

M(1) 64.398A1(1) 0.198B1(1) 0.314M(2) 7.293A1(2) -0.135B1(2) 1.053R(1,2) 0.120

REVISED CONSTANT CORRELATION FORECAST OF OIL & NATURAL GAS PRICES VOLATILITY

0

100

200

300

400

500

600

97 98 99 00 01 02 03 04 05 06

Var(RGAZ)

0

5

10

15

20

25

97 98 99 00 01 02 03 04 05 06

Cov(RGAZ,ROIL)

0

40

80

120

160

97 98 99 00 01 02 03 04 05 06

Var(ROIL)


.112

.114

.116

.118

.120

.122

.124

.126

97 98 99 00 01 02 03 04 05 06

Cor(RGAZ,ROIL)


FACTOR VOLATILITY MODELS

Principal Component Analysis

Principal Component Analysis We collect 120 financial ratios in order to asses

financial health of the firms. How can we reduce these ratios a few indices?

The production control department collect several measures in order to control process. Can we develop some indices in order to summarize the process outcomes?

In order to carry out efficient regression analysis we have to reduce multicollinearity among the explanatory variables if it exists. Can we generate some new indices in order to get orthogonal explanatory series that also contain most of the information of the original variables?

Principal Component Analysis in Finance

To reduce number of risk factors to a manageable dimension. For example, instead of 60 yields of different maturities as risk factors, we might use just 3 principal component.

To identy the key sources of risk. Typically the most important risk factors are parallel shifts, changes in slope and changes in convexity of the curves.

To facilitate the measurement of portfolio risk, for instance by introducing scenarios on the movements in the major risk factors.

Basics & Background

3

2

1

3

2

1

333231

232221

131211

x

x

x

x

x

x

a a a

a a a

a a a

xAx A is square matrix

X is a column vector

is a scalar quantity-eigenvalue

u normalized eigenvector

1uu 0uu

)A(Tr A

1'12

'1

n

1i

i

n

1i

i

Basic properties

Basics & Background

)xx(uy

....................

)xx(uy

)xx(uy

i'nin

i'22i

i'11i

IF matrix A composes of some observed x values

Pricipal Component Scores

MATHEMATICAL BACKGROUND

n'nn2

'221

'11nxn

i'i

n

1i

inxn

uu......uuuuA

uuA

A nxn square matrix

n'n

n2

'2

21

'1

1

1

i'i

n

1i i

1

uu1

......uu1

uu1

A

uu1

A

A BASIC EXAMPLE OF EIGENVALUES AND EIGENVECTORS

3

15

15

5

3

1

43

12

1

11

1

1

1

1

43

12

0.94868330-

0.31622777-,

0.70710678

0.70710678-

9486.

3162.

3

1*101

1031 22

7071.

7071.

1

1*21

2)1(1 22

2

1

2

1

2221

1211

x

x

x

x

a a

a a

xAx

Normalization

MATHEMATICAL EXAMPLE

01*3)4(*)2(

043

12IA

43

12A

88l ®1<,8l ®5<<U1 U2

Basics & Background

• Eigenvalue and Eigenvector:– Eigen originates in the German language and can

be loosely translated as “of itself”– Thus an Eigenvalue of a matrix could be

conceptualized as a “value of itself”– Eigenvalues and Eigenvectors are utilized in a

wide range of applications (PCA, calculating a power of a matrix, finding solutions for a system of differential equations, and growth models)

GEOMETRICAL APPROACH TO PRINCIPAL COMPONENT ANALYSIS

x1

x2

Mean corrected data

AXIS ROTATION

x1

x2

AXIS ROTATIONDIMEMSION REDUCTION

x1

x2 x1’x2

’

AXIS ROTATION

x1

x2

x1’

x2’

X1’ = x1*cos + x2*sin

X2’ = -x1*sin + x2*cos

A

AXIS ROTATION = 0 Observation x1 x2 x1Cor x2Cor 0 0 x1' x2'

1 16 8 8 5 8.000 5.0002 12 10 4 7 4.000 7.0003 13 6 5 3 5.000 3.0004 11 2 3 -1 3.000 -1.0005 10 8 2 5 2.000 5.0006 9 -1 1 -4 1.000 -4.0007 8 4 0 1 0.000 1.0008 7 6 -1 3 -1.000 3.0009 5 -3 -3 -6 -3.000 -6.000

10 3 -1 -5 -4 -5.000 -4.00011 2 -3 -6 -6 -6.000 -6.00012 0 0 -8 -3 -8.000 -3.000

Mean 8 3 0 0 0.00 0.00Variance 23.091 21.091 23.091 21.091 23.091 21.091

Covariance15.083 15.0833 Share of x1'= 52%Correlation 0.746 0.746 Correlation 0.746

Total Variance= 44.182 Total Variance= 44.182

X1’ = x1*cos 0 + x2*sin0

X2’ = -x1*sin 0 + x2*cos0

AXIS ROTATION = 10°

Observation x1 x2 x1Cor x2Cor 10 0.2 x1' x2'1 16 8 8 5 8.746 3.5362 12 10 4 7 5.154 6.2003 13 6 5 3 5.445 2.0874 11 2 3 -1 2.781 -1.5065 10 8 2 5 2.837 4.5776 9 -1 1 -4 0.291 -4.1137 8 4 0 1 0.174 0.9858 7 6 -1 3 -0.464 3.1289 5 -3 -3 -6 -3.996 -5.388

10 3 -1 -5 -4 -5.618 -3.07111 2 -3 -6 -6 -6.950 -4.86812 0 0 -8 -3 -8.399 -1.566

Mean 8 3 0 0 0.00 0.00Variance 23.091 21.091 23.091 21.091 28.656 15.526



X1’ = x1*cos10 + x2*sin10

X2’ = -x1*sin10 + x2*cos10


Observation x1 x2 x1Cor x2Cor 30 1 x1' x2'1 16 8 8 5 9.428 0.3332 12 10 4 7 6.963 4.0643 13 6 5 3 5.830 0.1004 11 2 3 -1 2.099 -2.3655 10 8 2 5 4.231 3.3316 9 -1 1 -4 -1.133 -3.9647 8 4 0 1 0.500 0.8668 7 6 -1 3 0.633 3.0989 5 -3 -3 -6 -5.597 -3.698

10 3 -1 -5 -4 -6.330 -0.96611 2 -3 -6 -6 -8.196 -2.19812 0 0 -8 -3 -8.429 1.400

Mean 8 3 0 0 0.00 0.00Variance 23.091 21.091 23.091 21.091 36.837 7.345



X1’ = x1*cos30 + x2*sin30

X2’ = -x1*sin30 + x2*cos30


Observation x1 x2 x1Cor x2Cor 40 1 x1' x2'1 16 8 8 5 9.343 -1.3092 12 10 4 7 7.563 2.7943 13 6 5 3 5.759 -0.9144 11 2 3 -1 1.656 -2.6945 10 8 2 5 4.745 2.5466 9 -1 1 -4 -1.804 -3.7087 8 4 0 1 0.643 0.7668 7 6 -1 3 1.161 2.9419 5 -3 -3 -6 -6.154 -2.670

10 3 -1 -5 -4 -6.401 0.14711 2 -3 -6 -6 -8.453 -0.74312 0 0 -8 -3 -8.058 2.841

Mean 8 3 0 0 0.00 0.00Variance 23.091 21.091 23.091 21.091 38.468 5.714

Covariance15.083 15.0833 Share of x1'= 87%Correlation0.746 0.746 Correlation 0.127


X1’ = x1*cos40 + x2*sin40

X2’ = -x1*sin40 + x2*cos40


X1’ = x1*cos44 + x2*sin44

X2’ = -x1*sin44 + x2*cos44

Observation x1 x2 x1Cor x2Cor 0.2404 0.755239 x1' x2'1 16 8 8 5 9.252 -1.8432 12 10 4 7 7.711 2.3553 13 6 5 3 5.697 -1.2434 11 2 3 -1 1.499 -2.7845 10 8 2 5 4.884 2.2706 9 -1 1 -4 -2.014 -3.5987 8 4 0 1 0.685 0.7288 7 6 -1 3 1.328 2.8709 5 -3 -3 -6 -6.297 -2.312

10 3 -1 -5 -4 -6.382 0.51511 2 -3 -6 -6 -8.481 -0.25612 0 0 -8 -3 -7.881 3.299

Mean 8 3 0 0 0.00 0.00Variance 23.091 21.091 23.091 21.091 38.576 5.606

Covariance 15.08333 15.08333 Share of x1'= 87%Correlation 0.746 0.746 Correlation 0.000



Observation x1 x2 x1Cor x2Cor 70 1 x1' x2'1 16 8 8 5 7.438 -5.8032 12 10 4 7 7.947 -1.3603 13 6 5 3 4.531 -3.6704 11 2 3 -1 0.088 -3.1615 10 8 2 5 5.383 -0.1666 9 -1 1 -4 -3.415 -2.3107 8 4 0 1 0.939 0.3438 7 6 -1 3 2.476 1.9679 5 -3 -3 -6 -6.665 0.763

10 3 -1 -5 -4 -5.471 3.32711 2 -3 -6 -6 -7.692 3.58112 0 0 -8 -3 -5.559 6.488

Mean 8 3 0 0 0.00 0.00Variance 23.091 21.091 23.091 21.091 31.918 12.264

Covariance15.083 15.0833 Share of x1'= 72%

Correlation0.746 0.746 Correlation -0.669


X1’ = x1*cos70 + x2*sin70

X2’ = -x1*sin70 + x2*cos70

FINDING OPTIMAL °

°

Portion of x1’

over total variance

*

87%

While x1 dimension explains 52% of the total

variance, When ° = 40, new x1

’ dimension explains 87% of total variance

ANALYTICAL APPROACH TO PRINCIPAL COMPONENT ANALYSISAssume there are p variables

1 = w11*x1+w12*x2+....+w1p*xp

2 = w21*x1+w22*x2+....+w2p*xp

........................................

p = wp1*x1+wp2*x2+....+wpp*xp

s are principal components and wij is the weight of the jth variables on the ith principal component

Var(1) > Var(2)> ... >Var(p)wi1

2 + wi22+....+wip

2 = 1 i=1,2,...pwi1*wj1+wi2*wj2+....+wip*wjp = 0 for all ij

ANALYTICAL APPROACH TO PRINCIPAL COMPONENT ANALYSIS• Assume there are p variables

1 = w11*x1+w12*x2+....+w1p*xp

2 = w21*x1+w22*x2+....+w2p*xp

........................................

p = wp1*x1+wp2*x2+....+wpp*xp s are principal components and wij is the weight of the jth variables

on the ith principal component X1

’ = cos * x1 + sin * x2

X2’ = -sin * x1 + cos * x2

1 = w11*x1 + w12*x2

2 = w21*x1 + w22*x2

MATRIX ALGEBRA APPROACH TO PRINCIPAL COMPONENT ANALYSIS

Assume there are p variables1 = w11*x1+w12*x2+....+w1p*xp

2 = w21*x1+w22*x2+....+w2p*xp

........................................

p = wp1*x1+wp2*x2+....+wpp*xp

MATRIX REPRESANTATION 1 = W1

’X 2 = W2

’X W2’*W2=1 W2’*W1=0

MATRIX ALGEBRA APPROACH TO PRINCIPAL COMPONENT ANALYSISVar(1) = Var(W1

’X) = W1’S W1

S = The Variance-Covariance matrix of original variables

Max. Var(1) = W1’S W1

st. W2’*W2=1

If 1, 2, ... , p are the eigenvalues of S

Sw1= 1w1

Var(1)= W1’S W1= W1

’ 1w1= 1W1’w1=1

Variance explained by the first principal component = 1/Trace(S)

Trace(S) = Sum of all isVariance explained by the first k principal components =

(1+2+...+k)/Trace(S)

VARIANCE EXTRACTION METHODS

VARIANCE-COVARIANCE MATRIX THE SIZE EFFECT INCLUDED THE ANALYSIS

CORRELATION MATRIX THE VARIABLES ARE STANDIZED FIRST, SO

THAT MEAN= 0 & VARIANCE= 1 OF ALL VARIABLES

THE VARIANCE COVARIANCE MATRIX OF THE STANDARDIZED VARIABLES IS THE CORRELATION MATRIX

THE SIZE EFFECT EXCLUDED FROM THE ANALYSIS

VARIANCE EXTRACTION METHODS A principal component represantation based on the

variance-covariance matrix has the advantage of providing a linear factor model for the returns, and not a linear factor model for the standardized returns, as is the case when correlation matrix is used. Standardization makes each variable has common mean and variance, 0 and 1 respectively.

A PCA on the covariance matrix captures all the movements in the variables, which may be dominated by the differing volatilities of individual variables. A PCA on the correlation matrix only captures the comovements in returns and ignores their individual volatilities.

It is only when all variables have similiar volatilities that the PCA will have similar characteristics.

STATISTICAL TESTS in PCA

CONFIDENCE INTERVAL FOR THE EIGENVALUES OF

)1(2ˆ of Variance Sample nii

)1(21

ˆ

)1(21

ˆ

]2/[]2/[

nZnZi

ii


2/)2)(1(dof with ~

ˆln

where

)ˆlnln](6/)22()1[(

themof oneleast at :H

....:H

22

1

1

22

1

3210

mmX

m

mmmmknX

mk

kjj

mk

kjj

mkkkk

If X2 > table Reject H0


2/)2)(1(dof with ~

ˆln

and where

)ˆlnln(*

])(6/)222()1[(

22

1

1

1

22

mmX

mkpm

m

mmmknX

mk

kjj

mk

kjj

k

jj

If X2 > table

Reject H0

If we want to test the last p-k roots

themof oneleast at :H

....:H

1

3210 mkkkk

EXAMPLE1= 5.7735 2= 0.9481 3= 0.3564 4= 0.1869

5= 0.1167 6= 0.0967 7= 0.0803 8= 0.0314

n = 292H0: 2=3 H1: 23 hence k = 1 m = 2

0

21

2

2

22

HReject e therefor991.556.66

05.0for 991.5)2(

22/)2)(1(

56.66

)0317.105330.08547.0(289

)3564.0ln9481.0ln6523.0ln2](2*6

222*2)11292[(

mmdof

X

X

X

HOW MANY PRINCIPAL COMPONENTS? IF ALL THE INFORMATION IS EXTRACTED,

P COMPONENTS SHOULD BE SELECTED RESEARCHERS CAN WANT TO ELIMINATE

MARGINAL INFORMATION SO THAT ONLY MAIN INFO. IS UNDERLINEDEXPLAIN RELATIVELY LARGE PERCENTAGE OF THE TOTAL VARIATION. 70-90% ARE USUALLY SUGGESTED FIGURES.EXCLUDE THOSE PRINCIPLE COMPONENTS WHOSE EIGENVALUES ARE LESS THAN AVERAGE EIGENVALUE. IF CORRELATION MATRIX IS USED, EXCLUDE PC WHOSE EIGENVALUES ARE LESS THAN 1. IF SAMPLE SIZE SMALL, THE CUT OFF POINT CAN BE LOWER, 0.7.USE SCREE PLOT TO CATCH THE “ELBOW” AND THE ELBOW POINTS THE NUMBER OF EIGENVALUES SHOULD BE EXCLUDED.

Interpretation of PCsThe first principle component captures a

common trend in assets or interest rates. If the first PC changes at atime when the other components are fixed, then the returns(or changes in interest rates) all move by roughly the same amount.

The second and higher order PC have no intuitive interpretation. But when we use factor rotation, some PC may represent a subgroups of the variables.

A Numerical Example• Original data values & mean centered:

11.511.9

1816.3

16.314.5

12.411

16.317

13.515

1719

14.516

1514

1412

3.35-2.77-

3.151.63

1.450.17-

2.45-3.67-

1.452.33

1.35-0.33

2.154.33

0.35-1.33

0.150.67-

0.85-2.67-

ORIGINAL & MEAN CENTERED

DATA VALUES

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20 -6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

ORIGIN SHIFTING

A Numerical Example• Covariance Matrix results in:

• Which has Eigenvalues and Eigenvectors of:

4.29611114.1227777

4.12277776.3845555

0.6141953-0.7891540-

0.7891540-0.6141953

9.5932970

01.08737

A Numerical Example

• Transformed values = EigenvectorsT x DataT

-0.96912 -0.52988 1.09308 0.96278 1.26804 0.28680 -0.32067 -1.24869 -1.48470 0.942342.62911 0.43660 -0.83461 -4.73756 0.56874 -2.72931 4.40097 -0.75643 -3.22104 4.24351

u1=u2=

Which of the two is the principle component?

Check magnitude of Eigenvalues

9.5932970

01.08737

ORTHOGONAL GARCH

22112

22111

XbXbF

XaXaF

22112

22111

FFX

FFaX

][][

][][

][][][][

][][][][

)])([(],[

][][][][

][][][][

222111

22221111

2222121221211111

2222112222111111

2211221121

2221

2122112

2221

2122111

FVARFVARa

FFCOVFFCOVa

FFCOVFFCOVFFCOVaFFCOVa

FFCOVFFCOVFFaCOVFFaCOV

FFFFaCOVXXCOV

FVarFVarFFVARXVAR

FVarFVarFFaVARXVAR

COV[F1,F2]=0

FACTOR GARCH

1. Calculate eigenvalues and eigenvectors

2. Determine the number of eigenvectors3. Calculate the factor scores and keep

the equations4. Estimate a GARCH models for the each

factor scores.5. Using factor score equations estimate

Variances and Covariances

FINANCIAL ECONOMETRICS

Documents

return series

financial econometricsspring

treatment of variances

financial assetschart20

conditional variance