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77

Table 7.8: GARCH(1,1) models of the principal components

1st PC

2nd PC

3rd PC

4th PC

Coefficient

f-stat

Coefficient

-stat

Coefficient

f-stat

Coefficient

f-stat

Constant

0.00613 0.032609 0.033085 0.934716

0.19446 1.90651 2.69647 35.9676

0.00262 0.066555 0.086002 0.846648

0.09008 3.10594 4.57763 25.9852

-0.0801 0.089961 0.067098 0.841618

-0.26523 2.12915 2.92511

14.4038

0.00267 0.203359 0.070417 0.726134

0.087511 1.80057 2.00423 5.22888

is, the matrix D, is a 4 x 4 diagonal matrix. Table 7.8 reports results from estimating univariate GARCH(1,1) models on each of the four principal components to give the elements of D, at each point in time.

The results of applying the orthogonal GARCH model are the four volatilities and six correlation graphs shown in Figure 7.15. These graphs compare the orthogonal GARCH volatilities and correlations with those estimated from two other multivariate GARCH models, the diagonal vech and the BEKK (§4.5.2).

The reason that this system has been confined to only four variables is that there are no convergence problems with the multivariate GARCH models that are being used for comparison. The four-dimensional diagonal vech model has 10 equations, each with three parameters. The 30 parameter estimates and their f-statistics (in italics) are reported in Table 7.9. The four-dimensional BEKK model has 42 parameters and the estimates of the matrices A, B, and are given in Table 7.10.

It is important to realize that all the 10 graphs in Figure 7.15 come from the models reported in Tables 7.8-7.10. That is, all the BEKK volatilities and correlations come from the same BEKK model estimated in Table 7.10. Similarly, there is only one diagonal vech model, in Table 7.9, generating all the vech series for the graphs and one orthogonal GARCH model in Table 7.8 for the orthogonal GARCH graphs. However, there are cases where the orthogonal GARCH volatilities coincide quite closely with the BEKK volatilities but not the vech volatilities (graphs f, h and j). In some cases the orthogonal GARCH are more similar to the vech volatilities than the BEKK volatilities (graphs d. e and i) and in some cases all three volatilities differ noticeably (graphs a. b. and g). Having said this, there is not a huge difference between the three models in any of the graphs. Given how volatility and correlation estimates can differ when different multivariate GARCH models are used, these graphs are nothing abnormal.

To summarize the discussion, some care must be taken with the initial calibration of orthogonal GARCH. Then it can be used to compute large

Some care must be taken with the initial calibration of orthogonal GARCH. Then it can be used to compute large GARCH covariance matrices that are reliable and computationally efficient on a daily basis



Table 7.9: Diagonal vech parameter estimates

Variance equations Covariance equations

FTSE

AEX-CAC

A EX-DA X

AEX-FTSE

CAC-DAX

CAC-FTSE

DAX-FTSE

«i

«6

«10

5.8 x 1( 6

3.4 x 10"6

5.0 x 10"6

1.8 x 10~6

1.9 x 10~6

9.3 x 10~6

1.8 x 10"6

8.6 x 10"6

3.0 x 10"6

1.6 x 10~6

3.11

2.19

1.71

2.11

2.45

1.63

2.25

3.24

7.54?

2.24?

«i

«3

«5

«6

«7

«8

«9

«10

0.054900

0.028~889

0.028264

0.024601

0.021826

0.031806

0.028069

0.059739

0.022341

0.028377

4.14

3.15

2.24?

2.68

3.75

2.40

3.68

3.56

2.74?

2.4?4?

0.82976

0.89061

0.83654

0.91227

0.95802

0.74503

0.926414

0.829753

0.861387

0.934363

18.23

21.17

9.54?

25.2

79.49

5.20

36.36

17.93

10.67

38.74



Table 7.10: BEKK parameter estimates

FTSE

0.00160 0.00008 0.00094 0.00142

-0.00176 0.00197 -0.00003

-0.0087 -0.00051

0 0 0

2.5 x 10-"

0.22394 -0.07147 -0.06286 -0.016277

-0.04156 0.18757 -0.04764 -0.027589

0.019373 -0.05247

0.29719 -0.017405

0.04785 0.031895 0.07003 0.178563

0.951805 0.033141 0.067985 0.022278

0.027231 0.9615723 0.053024 0.029257

-0.050236 0.023822 0.844291

-0.014482

0.026130 0.013623 0.005211 0.948453

GARCH covariance matrices that are reliable and computationally efficient, on a daily basis. The model calibration will depend on two important factors:

> The assets that are included in the system. PCA works best in a highly correlated system. An asset that has very idiosyncratic properties compared to other assets in the system (such as the 10-year bond in the UK yield curve above) will corrupt the volatilities and correlations of the other assets in the system, because they are all based on the principal components that are common to all assets.

» The time period used for estimation. The GARCH volatilities of the principal components change over time, but it is only their current values that matter for forecasting the covariance matrix. However, the factor weights are also used in this forecast, and these take different values depending on the estimation period. So changing the time period for estimation affects current forecasts of the covariance matrix primarily because it affects the factor weights matrix, not because it affects the parameter estimates for the principal component volatilities in D(.

The main focus of this long subsection on orthogonal GARCH has been to explain and empirically validate a new method of obtaining large GARCH correlation matrices using only univariate GARCH estimation techniques on principal components of the original return series. Empirical examples on commodity futures, interest rates and on international equity indices have been presented and used to explain how best to employ the method in different circumstances. It has been found that when the systems are suitably tailored, the orthogonal method compares very well with the general multivariate GARCH models. In many cases the divergence between the orthogonal GARCH estimates and the BEKK estimates is far less than between the vech and the BEKK estimates.



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