FINANCIAL ECONOMETRICS
SPRING 2013
WEEK VII
MULTIVARIATE MODELLING OF
VOLATILITY
Prof. Dr. Burç ÜLENGİN
MULTIVARIATE VOLATILITY
 There 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
1.0
0.8
0.7
0.6
0.5
0.4
0.3
cor50
cor100
985
944
903
862
821
780
739
698
657
616
575
534
493
452
411
370
329
288
247
206
165
124
83
42
0.2
1
CORRELATION
0.9
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.
h11h12 .......h1N 
h h ......h 
21 22
2N 

H
....................


hN 1hN 2 ...hNN 
MULTIVARIATE GARCH
There 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
2
 b b b  h11 
h11t  a110  a11 a12 a13   1t 1
11
12
13
t 1


   0 





2
2
h12t   a12   a 21 a22 a23   1t 1  2t 1   b 21 b22 b23  h12t 1 
 
   a 0  a a a   2



b
b
b
h
h
  31 32 33   22t 1 
 22t   22   31 32 33   2t 1
h11t  a110  a 11 12t 1  a12  1t 1  2t 1  a13 22t 1  b11 h11t 1  b12 h12t 1  b13 h22t 1
h12t  a120  a 21 12t 1  a22  1t 1  2t 1  a23 22t 1  b 21 h11t 1  b22 h12t 1  b23 h22t 1
0
h22t  a22
 a 31 12t 1  a32  1t 1  2t 1  a33 22t 1  b 31 h11t 1  b32 h12t 1  b33 h22t 1
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
2
 b 0 0  h11 
h11t  a110  a11 0 0   1t 1
t 1
11


   0 





2
2
h12t   a12   0 a22 0   1t 1  2t 1   0 b22 0  h12t 1 
 0 0 b  
   a 0  0 0 a   2

h
h
33   2
33 
 22t   22  
 22t 1 
 t 1  
h11t  a110  a 11 12t 1  b11 h11t 1
h12t  a120  a 22  1t 1  2t 1  b22 h12t 1
0
h22t  a 22
 a33 22t 1  b33 h22t 1
H    A 
'
t 1 t 1
 BH(1)
DIAGONAL REPRESENTATION
OIL & NATURAL
GAS PRICES
ROIL
40
40
30
30
20
20
10
10
0
0
-10
-10
-20
-20
-30
-30
-40
-40
-50
RGAZ
-50
97
98
99
00
01
02
03
04
05
06
07
97
98
99
00
rgast  1   1t
roilt   2   2t
h11t  a110  a11 12t 1  b11 h11t 1
h12t  a120  a22  1t 1  2t 1  b22 h12t 1
0
h22t  a22
 a33 22t 1  b33 h22t 1
01
02
03
04
05
06
07
DIAGONAL REPRESENTATION ESTIMATION
OIL & NATURAL GAS PRICES
Estimation Method: ARCH Maximum Likelihood (Marquardt)
Covariance specification: Diagonal VECH
Sample: 1997M02 2007M01
Included observations: 120
Total system (balanced) observations 240
Disturbance assumption: Student's t distribution
Convergence achieved after 198 iterations
rgast  1   1t
C(1)
C(2)
Coefficient
1.259
2.438
Std. Error z-Statistic
1.082
1.164
0.733
3.328
Prob.
0.245
0.001
C(3)
C(4)
C(5)
C(6)
C(7)
C(8)
C(9)
C(10)
C(11)
Variance Equation Coefficients
63.635
41.552
1.531
0.837
1.852
0.452
9.479
2.224
4.261
0.224
0.179
1.256
-0.056
0.039
-1.421
-0.200
0.081
-2.467
0.352
0.358
0.986
0.858
0.065
13.109
1.067
0.043
24.956
0.126
0.651
0.000
0.209
0.155
0.014
0.324
0.000
0.000
C(12)
t-Distribution (Degree of Freedom)
18.732
26.610
0.704
0.482
Log likelihood
Avg. log likelihood
Akaike info criterion
-890.0937 Schwarz criterion
-3.708724 Hannan-Quinn criter.
15.03489
15.31364
15.1481
roilt   2   2t
0
h11t  a11
 a 11 12t 1  b11 h11t 1
0
h12t  a12
 a22  1t 1  2t 1  b22 h12t 1
0
h22t  a22
 a33 22t 1  b33 h22t 1
DIAGONAL REPRESENTATION ESTIMATION
OIL & NATURAL GAS PRICES
Covariance specification: Diagonal VECH
GARCH = M + A1.*RESID(-1)*RESID(-1)' + B1.*GARCH(-1)
M is an indefinite matrix
A1 is an indefinite matrix
B1 is an indefinite matrix
C(1)
C(2)
rgast  1   1t
ROIL = C(2)
h11t  a110  a11 12t 1  b11 h11t 1
Substituted Coefficients:
=====================
RGAZ = 1.258
h12t  a120  a22  1t 1  2t 1  b22 h12t 1
roilt   2   2t
0
h22t  a22
 a33 22t 1  b33 h22t 1
ROIL = 2.438
Coefficient
1.259
2.438
Coefficient
C(3)
C(4)
C(5)
C(6)
C(7)
C(8)
C(9)
C(10)
C(11)
Estimated Equations:
=====================
RGAZ = C(1)
Variance Equation Coefficients
63.635 M(1,1)
63.63
0.837 M(1,2)
0.84
9.479 M(2,2)
9.48
0.224 A1(1,1)
0.22
-0.056 A1(1,2)
-0.06
-0.200 A1(2,2)
-0.20
0.352 B1(1,1)
0.35
0.858 B1(1,2)
0.86
1.067 B1(2,2)
1.07
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)
DIAGONAL REPRESENTATION VOLATILITY
FORECAST OF OIL & NATURAL GAS PRICES
Conditional Correlation
Conditional Covariance
Cor(RGAZ,ROIL)
Var(RGAZ)
1.00
800
0.75
600
0.50
0.25
400
0.00
-0.25
200
-0.50
0
97
98
99
00
01
02
03
04
05
-0.75
06
-1.00
Cov(RGAZ,ROIL)
97
Var(ROIL)
300
98
240
200
200
160
100
120
80
0
40
-100
0
97
98
99
00
01
02
03
04
05
06
97
98
99
00
01
02
03
04
05
06
99
00
01
02
03
04
05
06
DIAGONAL REPRESENTATION REVISED MODEL
ESTIMATION OIL & NATURAL GAS PRICES
rgast  1  roilt 1   1t
roilt   2   2t
0
h11t  a11
 a 11 12t 1  b11 h11t 1
h12t  a  a22  1t 1  2t 1  b22 h12t 1
0
12
h22t  a  a33
0
22
2
2t 1
 b33 h22t 1
DIAGONAL REPRESENTATION REVISED MODEL
ESTIMATION OIL & NATURAL GAS PRICES
Covariance specification: Diagonal VECH
GARCH = M + A1.*RESID(-1)*RESID(-1)' + B1.*GARCH(-1)
M is a scalar
A1 is a rank one matrix
B1 is a rank one matrix
Estimation Method: ARCH Maximum Likelihood (Marquardt)
Covariance specification: Diagonal VECH
Sample: 1997M03 2007M01
Included observations: 119
Total system (balanced) observations 238
Disturbance assumption: Student's t distribution
Convergence achieved after 26 iterations
Tranformed Variance Coefficients
Coefficient
C(1)
C(2)
C(3)
Coefficient
0.548
0.240
1.513
Std. Error
0.873
0.088
0.782
z-Statistic
0.628
2.730
1.934
Prob.
0.530
0.006
0.053
C(4)
C(5)
C(6)
C(7)
C(8)
1.974
0.474
-0.110
0.890
0.983
1.865
0.107
0.123
0.041
0.020
1.059
4.407
-0.894
21.543
48.854
0.290
0.000
0.371
0.000
0.000
t-Distribution (Degree of Freedom)
C(9)
Log likelihood
Avg. log likelihood
Akaike info criterion
6.928
2.791
-892.0641 Schwarz criterion
-3.748168 Hannan-Quinn criter.
15.14393
2.482
0.013
15.35412
15.22928
M
A1(1,1)
A1(1,2)
A1(2,2)
B1(1,1)
B1(1,2)
B1(2,2)
1.974
0.224
-0.052
0.012
0.793
0.875
0.966
DIAGONAL REPRESENTATION REVISED MODEL
ESTIMATION OIL & NATURAL GAS PRICES
Covariance specification: Diagonal VECH
GARCH = M + A1.*RESID(-1)*RESID(-1)' + B1.*GARCH(-1)
M is a scalar
A1 is a rank one matrix
B1 is a rank one matrix
C(1)
C(2)
C(3)
C(4)
C(5)
C(6)
C(7)
C(8)
ROIL = C(3)
Substituted Coefficients:
=====================
RGAZ = 0.548145947197+0.240483266384*ROIL(-1)
Coefficient
0.548
0.240
1.513
ROIL = 1.51272216999
Variance-Covariance Representation:
=====================
GARCH = M + A1.*RESID(-1)*RESID(-1)' + B1.*GARCH(-1)
1.974
0.474
-0.110
0.890
0.983
Tranformed Variance Coefficients
Coefficient
M
A1(1,1)
A1(1,2)
A1(2,2)
B1(1,1)
B1(1,2)
B1(2,2)
Estimated Equations:
=====================
RGAZ = C(1)+C(2)*ROIL(-1)
1.974
0.224
-0.052
0.012
0.793
0.875
0.966
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)
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)
COV1_2 = 1.974 -0.052*RESID1(-1)*RESID2(-1) + 0.875*COV1_2(-1)
DIAGONAL REPRESENTATION VOLATILITY FORECAST
OF OIL & NATURAL GAS PRICES
Conditional Correlation
Conditional Covariance
Cor(RGAZ,ROIL)
Var(RGAZ)
.6
800
.5
.4
600
.3
400
.2
.1
200
.0
0
97
98
99
00
01
02
03
04
05
-.1
06
-.2
Cov(RGAZ,ROIL)
97
Var(ROIL)
120
95
80
90
40
85
0
80
-40
98
75
97
98
99
00
01
02
03
04
05
06
97
98
99
00
01
02
03
04
05
06
99
00
01
02
03
04
05
06
DIAGONAL REPRESENTATION TARCH MODEL
ESTIMATION OIL & NATURAL GAS PRICES
rgast  1  roilt 1   1t
I1=1 if 1t-1<0
=0 otherwise
roilt   2   2t
I2=1 if 2t-1<0
=0 otherwise
h11t  a110  a11 12t 1  d11 I1 12t 1  b11 h11t 1
h12t  a  a22  1t 1  2t 1  d 21 ( I  )(I 
0
12
2
1 1t 1
2
2 2t 1
)  b22 h12t1
0
h22t  a22
 a33 22t 1  d 33 I 2 22t 1  b33 h22t 1
H    A t 1 t'1 A  D t 1I ( t 1  0)( t 1I ( t 1  0))D  BH(1)
DIAGONAL REPRESENTATION TARCH MODEL
ESTIMATION OIL & NATURAL GAS PRICES
Estimation Method: ARCH Maximum Likelihood (Marquardt)
Covariance specification: Diagonal VECH
TARCH
Date: 08/05/08 Time: 18:36
Sample: 1997M03 2007M01
Included observations: 119
Total system (balanced) observations 238
Disturbance assumption: Student's t distribution
Presample covariance: backcast (parameter =0.5)
Convergence achieved after 169 iterations
Coefficient Std. Error z-Statistic
C(1)
C(2)
C(3)
0.440
0.213
1.543
1.165
0.116
0.775
0.378
1.825
1.990
Prob.
0.706
0.068
0.047
Variance Equation Coefficients
C(4)
C(5)
C(6)
C(7)
C(8)
1.465
-0.013
0.224
-0.234
0.971
1.116
0.028
0.089
0.056
0.016
1.313
-0.461
2.506
-4.155
61.584
0.189
0.645
0.012
0.000
0.000
t-Distribution (Degree of Freedom)
C(9)
4.810
1.774
2.711
Log likelihood -898.265 Schwarz criterion
Avg. log likelihood
-3.77422 Hannan-Quinn criter.
Akaike info criterion
15.24815
Covariance specification: Diagonal VECH
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)
M is a scalar
A1 is a scalar
D1 is a rank one matrix
B1 is a scalar
0.007
15.45834
15.3335
Tranformed Variance Coefficients
Coefficient
M
A1
D1(1,1)
D1(1,2)
D1(2,2)
B1
1.465
-0.013
0.050
-0.052
0.055
0.971
DIAGONAL REPRESENTATION TARCH MODEL
ESTIMATION OIL & NATURAL GAS PRICES
Coefficient
Estimated Equations:
=====================
RGAZ = C(1)+C(2)*ROIL(-1)
C(1)
C(2)
C(3)
0.440
0.213
1.543
Tranformed Variance Coefficients
Coefficient
Variance Equation Coefficients
ROIL = C(3)
Substituted Coefficients:
=====================
RGAZ = 0.44021432107+0.212553531477*ROIL(-1)
ROIL = 1.54264508993
C(4)
C(5)
C(6)
C(7)
C(8)
1.465
-0.013
0.224
-0.234
0.971
M
A1
D1(1,1)
D1(1,2)
D1(2,2)
B1
1.465
-0.013
0.050
-0.052
0.055
0.971
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
Variance and Covariance Equations:
=====================
GARCH1 = M + A1*RESID1(-1)^2 + D1(1,1)*RESID1(-1)^2*(RESID1(-1)<0) + B1*GARCH1(-1)
GARCH2 = M + A1*RESID2(-1)^2 + D1(2,2)*RESID2(-1)^2*(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)
DIAGONAL REPRESENTATION TARCH MODEL
FORECAST OIL & NATURAL GAS PRICES VOLATILITY
Conditional Correlation
Conditional Covariance
Cor(RGAZ,ROIL)
Var(RGAZ)
600
.8
500
.6
400
.4
300
.2
200
.0
100
97
98
99
00
01
02
03
04
05
06
-.2
97
Cov(RGAZ,ROIL)
98
99
Var(ROIL)
160
140
120
120
80
100
40
80
0
-40
60
97
98
99
00
01
02
03
04
05
06
97
98
99
00
01
02
03
04
05
06
00
01
02
03
04
05
06
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.
2
12t 1  22t 1  a11 a12  b11 b12  h11t 1 h12 t 1  b11 b12 
h11t h12 t  a110 a120  a11 a12  1t 1

 2 2


 0 0 


 h h  b b 
2
h
h
a
a
a
a
b
b
  21 22   21 22   12 t 1 22 t 1   21 22 
 12 t 22 t  a12 a22   21 22  1t 1  2t 1  2t 1
H    A  A  BH(1)B
'
t 1 t 1
BEKK REPRESENTATION OF OIL &
NATURAL GAS PRICES
rgast  1   1t
roilt   2   2t
h11t    a 
2 2
11 1t 1
b h
2
11 11t 1
h12t    a11a22  1t 1  2t 1  b11b22 h12t 1
2 2
h22t    a22
 2t 1  b222 h22t1
BEKK ESTIMATION OF OIL & NATURAL
GAS PRICES
Estimation Method: ARCH Maximum Likelihood (Marquardt)
Covariance specification: BEKK
Sample: 1997M02 2007M01
Included observations: 120
Total system (balanced) observations 240
Disturbance assumption: Student's t distribution
Presample covariance: backcast (parameter =0.5)
Convergence achieved after 25 iterations
Coefficient
C(1)
C(2)
0.890
1.346
Std. Error z-Statistic Prob.
0.874
0.786
1.019
1.712
0.308
0.087
Variance Equation Coefficients
C(3)
C(4)
C(5)
C(6)
C(7)
1.342
0.427
-0.079
0.908
0.987
1.367
0.093
0.127
0.032
0.014
0.982
4.605
-0.623
28.189
71.875
0.326
0.000
0.533
0.000
0.000
t-Distribution (Degree of Freedom)
C(8)
6.729
Covariance specification: BEKK
GARCH = M + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1
M is a scalar
A1 is diagonal matrix
B1 is diagonal matrix
2.788
2.414
Log likelihood -902.4941 Schwarz criterion
Avg. log likelihood
-3.760392 Hannan-Quinn criter.
Akaike info criterion
15.1749
0.016
15.36073
15.25037
Tranformed Variance Coefficients
Coefficient
M
A1(1,1)
A1(2,2)
B1(1,1)
B1(2,2)
1.342
0.427
-0.079
0.908
0.987
BEKK ESTIMATION OF OIL & NATURAL GAS PRICES
Estimated Equations:
=====================
RGAZ = C(1)
ROIL = C(2)
Substituted Coefficients:
=====================
RGAZ = 0.890273745039
ROIL = 1.34556553188
Variance-Covariance Representation:
=====================
Coefficient
C(1)
C(2)
0.890
1.346
Variance Equation Coefficients
C(3)
C(4)
C(5)
C(6)
C(7)
1.342
0.427
-0.079
0.908
0.987
GARCH = M + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1
Variance and Covariance Equations:
=====================
GARCH1 = M + A1(1,1)^2*RESID1(-1)^2 + B1(1,1)^2*GARCH1(-1)
Tranformed Variance Coefficients
GARCH2 = M + A1(2,2)^2*RESID2(-1)^2 + B1(2,2)^2*GARCH2(-1)
M
A1(1,1)
A1(2,2)
B1(1,1)
B1(2,2)
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)^2+0.824*GARCH1(-1)
GARCH2 = 1.342+0.0063*RESID2(-1)^2+0.974*GARCH2(-1)
COV1_2 = 1.342 -0.034*RESID1(-1)*RESID2(-1) + 0.896*COV1_2(-1)
Coefficient
1.342
0.427
-0.079
0.908
0.987
REVISED BEKK REPRESENTATION OF OIL
& NATURAL GAS PRICES
rgast  1  roilt 1   1t
roilt   2   2t
h11t    a 
2 2
11 1t 1
b h
2
11 11t 1
h12t    a11a22  1t 1  2t 1  b11b22 h12t 1
2 2
h22t    a22
 2t 1  b222 h22t1
REVISED BEKK REPRESENTATION OF OIL
& NATURAL GAS PRICES
Estimation Method: ARCH Maximum Likelihood (Marquardt)
Covariance specification: BEKK
Sample: 1997M03 2007M01
Included observations: 119
Total system (balanced) observations 238
Disturbance assumption: Student's t distribution
Presample covariance: backcast (parameter =0.5)
Convergence achieved after 26 iterations
Coefficient Std. Error z-Statistic
C(1)
C(2)
C(3)
0.548
0.240
1.513
0.873
0.088
0.782
0.628
2.730
1.934
1.974
0.474
-0.110
0.890
0.983
1.865
0.107
0.123
0.041
0.020
1.059
4.407
-0.894
21.543
48.854
0.530
0.006
0.053
0.290
0.000
0.371
0.000
0.000
t-Distribution (Degree of Freedom)
C(9)
6.928
2.791
Tranformed Variance Coefficients
Prob.
Variance Equation Coefficients
C(4)
C(5)
C(6)
C(7)
C(8)
Covariance specification: BEKK
GARCH = M + A1*RESID(-1)*RESID(-1)'*A1 + B1*GARCH(-1)*B1
M is a scalar
A1 is diagonal matrix
B1 is diagonal matrix
2.482
0.013
Log likelihood -892.0641 Schwarz criterion
Avg. log likelihood
-3.748168 Hannan-Quinn criter.
Akaike info criterion
15.14393
15.35412
15.22928
Coefficient Std. Error z-Statistic
M
A1(1,1)
A1(2,2)
B1(1,1)
B1(2,2)
1.974
0.474
-0.110
0.890
0.983
1.865
0.107
0.123
0.041
0.020
1.059
4.407
-0.894
21.543
48.854
Prob.
0.290
0.000
0.371
0.000
0.000
REVISED BEKK REPRESENTATION OF OIL & NATURAL
GAS PRICES
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)'*A1 + B1*GARCH(-1)*B1
Variance and Covariance Equations:
=====================
GARCH1 = M + A1(1,1)^2*RESID1(-1)^2 + B1(1,1)^2*GARCH1(-1)
GARCH2 = M + A1(2,2)^2*RESID2(-1)^2 + B1(2,2)^2*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)
COV1_2 = 1.974 -0.052*RESID1(-1)*RESID2(-1) + 0.875*COV1_2(-1)
REVISED BEKK FORECASTING OF OIL & NATURAL
Conditional Correlation
GAS PRICES VOLATILITY
Cor(RGAZ,ROIL)
.6
Conditional Covariance
.5
Var(RGAZ)
800
.4
.3
600
.2
400
.1
.0
200
-.1
0
97
98
99
00
01
02
03
04
05
-.2
06
97
Cov(RGAZ,ROIL)
98
Var(ROIL)
120
95
80
90
40
85
0
80
-40
75
97
98
99
00
01
02
03
04
05
06
97
98
99
00
01
02
03
04
05
06
99
00
01
02
03
04
05
06
CONSTANT CORRELATION
REPRESENTATION
 Bollerslev(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
12 13 .....1N 
1


1

.....

23
2N 
R   12
................................ 




..........
......
1
 N1 N 2

 h11
0
0 .... ..0
t

0
h22 t 0 .......0
H 
. ..................................

0
hNN t
 0

 h11
0
0 .... ..0
t
1


.....



12
13
1
N




 12 1  23 ..... 2 N  0
h22 t 0 .......0






..........
..........
..........
..

. ..................................


   ................1 
 0
0
hNN t
  N 1 N 2

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 Correlation
Sample: 1997M02 2007M01
Included observations: 120
Total system (balanced) observations 240
Disturbance assumption: Student's t distribution
Presample covariance: backcast (parameter =0.5)
Convergence achieved after 21 iterations
Coefficient Std. Error z-Statistic
C(1)
C(2)
0.938
2.078
1.037
0.605
0.904
3.435
Prob.
0.366
0.001
Variance Equation Coefficients
C(3)
C(4)
C(5)
C(6)
C(7)
C(8)
C(9)
53.143
0.152
0.449
8.735
-0.145
1.033
0.154
50.532
0.148
0.454
1.906
0.041
0.020
0.102
1.052
1.027
0.989
4.583
-3.507
51.343
1.516
0.293
0.305
0.323
0.000
0.001
0.000
0.130
t-Distribution (Degree of Freedom)
C(10)
10.97096 8.493965 1.291618
0.1965
Log likelihood -893.8184 Schwarz criterion
Avg. log likelihood
-3.724243 Hannan-Quinn criter.
Akaike info criterion
15.06364
15.29593
15.15797
Covariance specification: Constant Conditional Correlation
GARCH(i) = M(i) + A1(i)*RESID(i)(-1)^2 + B1(i)*GARCH(i)(-1)
COV(i,j) = R(i,j)*@SQRT(GARCH(i)*GARCH(j))
Tranformed Variance Coefficients
Coefficient Std. Error z-Statistic Prob.
M(1)
A1(1)
B1(1)
M(2)
A1(2)
B1(2)
R(1,2)
53.143
0.152
0.449
8.735
-0.145
1.033
0.154
50.532
0.148
0.454
1.906
0.041
0.020
0.102
1.052
1.027
0.989
4.583
-3.507
51.343
1.516
0.293
0.305
0.323
0.000
0.001
0.000
0.130
CONSTANT CORRELATION REPRESENTATION
OF OIL & NATURAL GAS PRICES
Substituted Coefficients:
=====================
RGAZ = 0.937503474235
ROIL = 2.07760000977
Variance and Covariance Representations:
=====================
GARCH(i) = M(i) + A1(i)*RESID(i)(-1)^2 + 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)^2 + C(5)*GARCH1(-1)
Variance Equation Coefficients
C(3)
C(4)
C(5)
C(6)
C(7)
C(8)
C(9)
53.143
0.152
0.449
8.735
-0.145
1.033
0.154
Tranformed Variance Coefficients
Coefficient
GARCH2 = C(6) + C(7)*RESID2(-1)^2 + C(8)*GARCH2(-1)
COV1_2 = C(9)*@SQRT(GARCH1*GARCH2)
Substituted Coefficients:
=====================
GARCH1 = 53.142 + 0.152*RESID1(-1)^2 + 0.448*GARCH1(-1)
GARCH2 = 8.735 - 0.145*RESID2(-1)^2 + 1.033*GARCH2(-1)
COV1_2 = 0.154*@SQRT(GARCH1*GARCH2)
M(1)
A1(1)
B1(1)
M(2)
A1(2)
B1(2)
R(1,2)
53.143
0.152
0.449
8.735
-0.145
1.033
0.154
CONSTANT CORRELATION FORECAST OF
OIL & NATURAL GAS PRICES VOLATILITY
Conditional Correlation
Conditional Covariance
Cor(RGAZ,ROIL)
Var(RGAZ)
600
.162
500
.160
.158
400
.156
300
.154
200
.152
100
.150
0
97
98
99
00
01
02
03
04
05
.148
06
.146
Cov(RGAZ,ROIL)
97
Var(ROIL)
50
200
40
150
30
100
20
50
10
0
0
97
98
99
00
01
02
03
04
05
06
97
98
99
00
01
02
03
04
05
06
98
99
00
01
02
03
04
05
06
REVISED CONSTANT CORRELATION
REPRESENTATION OF OIL & NATURAL GAS PRICES
Estimation Method: ARCH Maximum Likelihood (Marquardt)
Covariance specification: Constant Conditional Correlation
Sample: 1997M03 2007M01
Included observations: 119
Total system (balanced) observations 238
Disturbance assumption: Student's t distribution
Presample covariance: backcast (parameter =0.5)
Convergence achieved after 26 iterations
Coefficient
C(1)
C(2)
C(3)
0.781
0.199
2.233
Std. Error z-Statistic
1.079
0.109
0.573
0.724
1.834
3.899
Prob.
0.469
0.067
0.000
Variance Equation Coefficients
C(4)
C(5)
C(6)
C(7)
C(8)
C(9)
C(10)
64.398
0.198
0.314
7.293
-0.135
1.053
0.120
49.848
0.170
0.432
1.229
0.033
0.020
0.103
1.292
1.163
0.727
5.936
-4.062
51.973
1.166
0.196
0.245
0.468
0.000
0.000
0.000
0.244
t-Distribution (Degree of Freedom)
C(11)
13.509
11.166
1.210
Log likelihood -882.0092 Schwarz criterion
Avg. log likelihood
-3.705921 Hannan-Quinn criter.
Akaike info criterion
15.00856
Covariance specification: Constant Conditional Correlation
GARCH(i) = M(i) + A1(i)*RESID(i)(-1)^2 + B1(i)*GARCH(i)(-1)
COV(i,j) = R(i,j)*@SQRT(GARCH(i)*GARCH(j))
0.226
15.26545
15.11287
Tranformed Variance Coefficients
Coefficient Std. Error
M(1)
A1(1)
B1(1)
M(2)
A1(2)
B1(2)
R(1,2)
64.398
0.198
0.314
7.293
-0.135
1.053
0.120
49.848
0.170
0.432
1.229
0.033
0.020
0.103
z-Statistic Prob.
1.292
1.163
0.727
5.936
-4.062
51.973
1.166
0.196
0.245
0.468
0.000
0.000
0.000
0.244
REVISED CONSTANT CORRELATION
REPRESENTATION OF OIL & NATURAL GAS PRICES
Estimated Equations:
=====================
RGAZ = C(1)+C(2)*ROIL(-1)
Variance Equation Coefficients
ROIL = C(3)
Substituted Coefficients:
=====================
RGAZ = 0.781034818457+0.199474480682*ROIL(-1)
ROIL = 2.23319899793
Variance and Covariance Representations:
=====================
GARCH(i) = M(i) + A1(i)*RESID(i)(-1)^2 + 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)^2 + C(6)*GARCH1(-1)
GARCH2 = C(7) + C(8)*RESID2(-1)^2 + C(9)*GARCH2(-1)
COV1_2 = C(10)*@SQRT(GARCH1*GARCH2)
Substituted Coefficients:
=====================
GARCH1 = 64.398 + 0.197*RESID1(-1)^2 + 0.314*GARCH1(-1)
GARCH2 = 7.292 - 0.135*RESID2(-1)^2 + 1.052*GARCH2(-1)
COV1_2 = 0.119*@SQRT(GARCH1*GARCH2)
C(4)
C(5)
C(6)
C(7)
C(8)
C(9)
C(10)
64.398
0.198
0.314
7.293
-0.135
1.053
0.120
Tranformed Variance Coefficients
Coefficient
M(1)
A1(1)
B1(1)
M(2)
A1(2)
B1(2)
R(1,2)
64.398
0.198
0.314
7.293
-0.135
1.053
0.120
REVISED CONSTANT CORRELATION FORECAST OF
OIL & NATURAL GAS PRICES VOLATILITY
Conditional Correlation
Conditional Covariance
Cor(RGAZ,ROIL)
.126
Var(RGAZ)
600
.124
500
.122
400
.120
300
.118
200
.116
100
.114
0
97
98
99
00
01
02
03
04
05
06
.112
97
Cov(RGAZ,ROIL)
98
Var(ROIL)
25
160
20
120
15
80
10
40
5
0
0
97
98
99
00
01
02
03
04
05
06
97
98
99
00
01
02
03
04
05
06
99
00
01
02
03
04
05
06
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
Ax  x
A is square matrix
a11 a12 a13   x1 
 x1 

 
 
a 21 a 22 a 23   x 2     x 2 
a 31 a 32 a 33   x 3 
 x 3 
n
Basic
properties
A 

 is a scalar quantityeigenvalue
u normalized
eigenvector
n
i
Tr (A) 

i 1
i 1
u1' u 2  0
X is a column vector
u1' u1  1
i
Basics & Background
IF matrix A composes of some observed x values
Pricipal Component Scores
yi1  u1' ( x i  x )
yi 2  u '2 ( x i  x )
....................
yin  u 'n ( x i  x )
MATHEMATICAL BACKGROUND
n
A nxn 

'
i u i u i
i 1
'
'
A nxn  1u1u1   2u 2u 2
A
1
n


i 1
A
1
A nxn square matrix
'
 ......   n u n u n
1 '
uiui
i
1 '
1 '
1 '
 u1u1 
u 2u 2  ...... 
unun
1
2
n
A BASIC EXAMPLE OF EIGENVALUES
AND EIGENVECTORS
Ax  x
a11 a12  x1 
x1 

    
a 21 a 22  x 2 
x 2 
 2 1  1   5   1 

      5 
 3 4  3 15  3
 2 1   1   1   1

      1 
 3 4  1   1   1 
 - 0.70710678
  - 0.31622777

, 

 0.70710678  - 0.94868330
Normalization
12  32  10
1  .3162
1 10 *    

3
.
9486
  

12  (1)2  2
1
 1  .7071
2 *   

1
.
7071
  

MATHEMATICAL EXAMPLE
 2 1

A  
 3 4
2
1
A  I 
0
3
4
8
<
8
<
8 <8<
(2  ) * (4  )  3 *1  0
l ® 1 ,
l ® 5
- 1, 1 ,
1, 3
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
x2






Mean corrected data




x1
AXIS ROTATION
x2










x1
AXIS ROTATION
DIMEMSION REDUCTION
x2’
x1’
x2










x1
AXIS ROTATION
A
x2
x1’
x2’

x1
X1’ =
x1*cos  + x2*sin 
X2’ = -x1*sin  + x2*cos 
AXIS ROTATION  = 0
Observation
1
2
3
4
5
6
7
8
9
10
11
12
Mean
Variance
x2
x1
8
16
10
12
6
13
2
11
8
10
-1
9
4
8
6
7
-3
5
-1
3
-3
2
0
0
3
8
23.091 21.091
Total Variance=
Covariance15.083
Correlation 0.746
X1 ’ =
x1Cor
8
4
5
3
2
1
0
-1
-3
-5
-6
-8
0
23.091
x2Cor
5
7
3
-1
5
-4
1
3
-6
-4
-6
-3
0
21.091
44.182
15.0833
0.746
0
x1'
8.000
4.000
5.000
3.000
2.000
1.000
0.000
-1.000
-3.000
-5.000
-6.000
-8.000
0.00
23.091
0
x2'
5.000
7.000
3.000
-1.000
5.000
-4.000
1.000
3.000
-6.000
-4.000
-6.000
-3.000
0.00
21.091
44.182
Total Variance=
52%
Share of x1'=
Correlation
x1*cos 0 + x2*sin0
X2’ = -x1*sin 0 + x2*cos0
0.746
AXIS ROTATION  = 10°
Observation
1
2
3
4
5
6
7
8
9
10
11
12
Mean
Variance
x1
16
12
13
11
10
9
8
7
5
3
2
0
8
23.091
x2
8
10
6
2
8
-1
4
6
-3
-1
-3
0
3
21.091
Total Variance=
Covariance15.0833
Correlation 0.746
X1’ =
x1Cor
8
4
5
3
2
1
0
-1
-3
-5
-6
-8
0
23.091
x2Cor
5
7
3
-1
5
-4
1
3
-6
-4
-6
-3
0
21.091
44.182
15.0833
0.746
10 0.2
x1'
8.746
5.154
5.445
2.781
2.837
0.291
0.174
-0.464
-3.996
-5.618
-6.950
-8.399
0.00
28.656
Total Variance=
x2'
3.536
6.200
2.087
-1.506
4.577
-4.113
0.985
3.128
-5.388
-3.071
-4.868
-1.566
0.00
15.526
44.182
Share of x1'=
65%
Correlation
x1*cos10 + x2*sin10
X2’ = -x1*sin10 + x2*cos10
0.717
AXIS ROTATION  = 30°
Observation
1
2
3
4
5
6
7
8
9
10
11
12
Mean
Variance
x1
x2
16
8
12
10
13
6
11
2
10
8
9
-1
8
4
7
6
5
-3
3
-1
2
-3
0
0
8
3
23.091 21.091
Total Variance=
Covariance15.083
Correlation 0.746
X1’ =
x1Cor
8
4
5
3
2
1
0
-1
-3
-5
-6
-8
0
23.091
x2Cor
5
7
3
-1
5
-4
1
3
-6
-4
-6
-3
0
21.091
44.182
15.0833
0.746
30
1
x1'
9.428
6.963
5.830
2.099
4.231
-1.133
0.500
0.633
-5.597
-6.330
-8.196
-8.429
0.00
36.837
Total Variance=
x2'
0.333
4.064
0.100
-2.365
3.331
-3.964
0.866
3.098
-3.698
-0.966
-2.198
1.400
0.00
7.345
44.182
Share of x1'=
83%
Correlation
x1*cos30 + x2*sin30
X2’ = -x1*sin30 + x2*cos30
0.448
AXIS ROTATION  = 40°
Observation
1
2
3
4
5
6
7
8
9
10
11
12
Mean
Variance
x1
x2
16
8
12
10
13
6
11
2
10
8
9
-1
8
4
7
6
5
-3
3
-1
2
-3
0
0
8
3
23.091 21.091
Total Variance=
x1Cor x2Cor
40 1
8
5
4
7
5
3
3
-1
2
5
1
-4
0
1
-1
3
-3
-6
-5
-4
-6
-6
-8
-3
0
0
23.091 21.091
44.182
Total Variance=
Covariance
15.083
15.0833
Correlation0.746
0.746
X1’ = x1*cos40 +
Share of x1'=
x1'
9.343
7.563
5.759
1.656
4.745
-1.804
0.643
1.161
-6.154
-6.401
-8.453
-8.058
0.00
38.468
44.182
87%
Correlation
x2*sin40
X2’ = -x1*sin40 + x2*cos40
x2'
-1.309
2.794
-0.914
-2.694
2.546
-3.708
0.766
2.941
-2.670
0.147
-0.743
2.841
0.00
5.714
0.127
AXIS ROTATION  = 44°
Observation
1
2
3
4
5
6
7
8
9
10
11
12
Mean
Variance
x1
16
12
13
11
10
9
8
7
5
3
2
0
8
23.091
x2
8
10
6
2
8
-1
4
6
-3
-1
-3
0
3
21.091
Total Variance=
Covariance 15.08333
Correlation
0.746
X1 ’ =
x1Cor
8
4
5
3
2
1
0
-1
-3
-5
-6
-8
0
23.091
x2Cor
5
7
3
-1
5
-4
1
3
-6
-4
-6
-3
0
21.091
44.182
15.08333
0.746
0.2404 0.755239
Total Variance=
Share of x1'=
x1*cos44 + x2*sin44
X2’ = -x1*sin44 + x2*cos44
x1'
9.252
7.711
5.697
1.499
4.884
-2.014
0.685
1.328
-6.297
-6.382
-8.481
-7.881
0.00
38.576
x2'
-1.843
2.355
-1.243
-2.784
2.270
-3.598
0.728
2.870
-2.312
0.515
-0.256
3.299
0.00
5.606
44.182
87%
Correlation
0.000
AXIS ROTATION  = 70°
Observation
1
2
3
4
5
6
7
8
9
10
11
12
Mean
Variance
x1
x2
16
8
12
10
13
6
11
2
10
8
9
-1
8
4
7
6
5
-3
3
-1
2
-3
0
0
8
3
23.091 21.091
Total Variance=
Covariance
15.083
Correlation0.746
X1 ’ =
x1Cor x2Cor
70 1
8
5
4
7
5
3
3
-1
2
5
1
-4
0
1
-1
3
-3
-6
-5
-4
-6
-6
-8
-3
0
0
23.091 21.091
44.182
Total Variance=
15.0833
0.746
Share of x1'=
x1*cos70 + x2*sin70
X2’ = -x1*sin70 + x2*cos70
x1'
7.438
7.947
4.531
0.088
5.383
-3.415
0.939
2.476
-6.665
-5.471
-7.692
-5.559
0.00
31.918
44.182
72%
Correlation
x2'
-5.803
-1.360
-3.670
-3.161
-0.166
-2.310
0.343
1.967
0.763
3.327
3.581
6.488
0.00
12.264
-0.669
FINDING OPTIMAL
Portion of x1’
over total
variance
°

While x1 dimension explains 52% of the total
variance, When  ° = 40, new x1’ dimension
explains 87% of total variance
87%
*
°
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
Var(1) > Var(2)> ... >Var(p)
wi12 + wi22+....+wip2 = 1
i=1,2,...p
wi1*wj1+wi2*wj2+....+wip*wjp = 0 for all ij
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 variables
1 = 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 ANALYSIS
 Var(1) = Var(W1’X) =
W1’S W1
S=
The Variance-Covariance matrix of original variables

Max. Var(1) = W1’S
st.

W1
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 is
Variance 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 variancecovariance 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
Sample Varianceof ˆi  i 2 (n 1)
CONFIDENCE INTERVAL FOR THE EIGENVALUES OF
ˆi
1  Z[ / 2 ] 2 (n  1)
 i 
S
ˆi
1  Z[ / 2] 2 (n  1)
STATISTICAL TESTS in PCA
H 0 : k 1  k  2  k  3  ....  k  m
H1 : at least one of them
X  [(n  k  1)  (2m  m  2) / 6m](m ln  
2
2
k m
 ln ˆ )
j  k 1
k m
where  
X 2 ~ 2
ˆ
ln

 j
j  k 1
m
with dof  (m  1)(m  2) / 2
If X2 > 2table Reject H0
j
STATISTICAL TESTS in PCA
If we want to test the last p-k roots
H0 : k 1  k  2  k  3  ....  k  m
H1 : at least one of them
k
X 2  [(n  k  1)  (2m 2  2m  2) / 6m   (  j   )]
j 1
* (m ln  
k m
 ln ˆ )
j  k 1
j
k m
where m  p  k and  
X 2 ~ 2
ˆ
ln

 j
j  k 1
If X2 > 2table
m
with dof  (m  1)(m  2) / 2 Reject H0
EXAMPLE
1= 5.7735
2= 0.9481
3= 0.3564
4= 0.1869
5= 0.1167
6= 0.0967
7= 0.0803
8= 0.0314
n = 292
H0: 2=3
H1: 23
hence k = 1 m = 2
2 * 22  2  2
X  [(292  1  1) 
](2 ln 0.6523 ln 0.9481 ln 0.3564)
6*2
X 2  289  (0.8547 0.05330 1.0317)
2
X 2  66.56
dof  (m  1)(m  2) / 2  2
12 (2)  5.991 for   0.05
66.56  5.991 therefore Reject H 0
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 UNDERLINED
 EXPLAIN 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 PCs
 The 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:
 12

 14
 16

 19
 15

 17

 11
14.5

16.3
11.9

14 

15 
14.5

17 
13.5
16.3

12.4
16.3

18 
11.5
 - 2.67

 - 0.67
 1.33

 4.33
 0.33

 2.33

 - 3.67
 - 0.17

 1.63
 - 2.77

- 0.85

0.15 
- 0.35

2.15 
- 1.35
1.45 

- 2.45
1.45 

3.15 
- 3.35
ORIGINAL & MEAN CENTERED
DATA VALUES
20
6
18
4
16
14
2
12
10
0
8
-6
-4
6
-2
0
-2
4
-4
2
0
0
5
10
15
20
ORIGIN SHIFTING
-6
2
4
6
A Numerical Example
• Covariance Matrix results in:
 6.3845555 4.1227777


 4.1227777 4.2961111
• Which has Eigenvalues and Eigenvectors of:
0
1.08737



9.593297
 0
 0.6141953 - 0.7891540


 - 0.7891540 - 0.6141953
A Numerical Example
• Transformed values = EigenvectorsT x DataT
u1=
u2=
-0.96912
2.62911
-0.52988 1.09308 0.96278 1.26804 0.28680 -0.32067 -1.24869 -1.48470
0.43660 -0.83461 -4.73756 0.56874 -2.72931 4.40097 -0.75643 -3.22104
Which of the two is the principle component?
Check magnitude of Eigenvalues
0
1.08737



9.593297
 0
0.94234
4.24351
ORTHOGONAL GARCH
F1  a1 X1  a2 X 2
X1  a1F1   2 F2
F2  b1 X1  b2 X 2
X 2  1F1   2 F2
VAR[ X 1 ]  VAR[a1F1   2 F2 ]  12Var[ F1 ]   22Var[ F2 ]
VAR[ X 2 ]  VAR[ 1F1   2 F2 ]  12Var[ F1 ]   22Var[ F2 ]
COV [ X 1 , X 2 ]  COV [(a1F1   2 F2 )(1F1   2 F2 )]
 COV [a1F11F1 ]  COV [a1F1 2 F2 ]  COV [ 2 F2 1F1 ]  COV [ 2 F2  2 F2 ]
 a11COV [ F1F1 ]  a1 2COV [ F1F2 ]   2 1COV [ F2 F1 ]   2  2COV [ F2 F2 ]
COV[F1,F2]=0
 a11COV [ F1F1 ]   2  2COV [ F2 F2 ]
 a11VAR[ F1 ]   2  2VAR[ F2 ]
FACTOR GARCH
1. Calculate eigenvalues and eigenvectors
2. Determine the number of eigenvectors
3. Calculate the factor scores and keep the
equations
4. Estimate a GARCH models for the each
factor scores.
5. Using factor score equations estimate
Variances and Covariances