u 3. Multivariate Volatility models - Vaasan yliopisto...

33
3. Multivariate Volatility models Consider a k component multivariate return series r t =(r 1t ,...,r kt ) , where the prime de- notes transpose. As in the univariate case, let r t = μ t + u t , (1) where μ t = E[r t |F t1 ] is the conditional ex- pectation of r t given the past information F t1 . We assume that μ t has a vector AR repre- sentation. 1 The conditional covariance matrix of u t Cov[u t |F t1 ]=Σ t (2) is a k × k matrix, assumed positive definite. There are many ways to generalize the uni- variate models to multivariate. The dimensionality rapidly grows to unman- ageable magnitudes because there are k(k + 1)/2 variance and covariance (or correlation) parameters in Σ t to model. 2

Transcript of u 3. Multivariate Volatility models - Vaasan yliopisto...

Page 1: u 3. Multivariate Volatility models - Vaasan yliopisto |lipas.uwasa.fi/~sjp/Teaching/fts/lectures/ftsc3d.pdf ·  · 2009-04-223. Multivariate Volatility models Consider a k component

3. Multivariate Volatility models

Consider a k component multivariate return

series rt = (r1t, . . . , rkt)′, where the prime de-

notes transpose.

As in the univariate case, let

rt = μt + ut,(1)

where μt = E[rt|Ft−1] is the conditional ex-

pectation of rt given the past information

Ft−1.

We assume that μt has a vector AR repre-

sentation.

1

The conditional covariance matrix of ut

Cov[ut|Ft−1] = Σt(2)

is a k × k matrix, assumed positive definite.

There are many ways to generalize the uni-

variate models to multivariate.

The dimensionality rapidly grows to unman-

ageable magnitudes because there are k(k +

1)/2 variance and covariance (or correlation)

parameters in Σt to model.

2

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3.1 Simple bivariate GARCH(1,1) model

Consider the bivariate covariance matrix

Σt =

(σ11,t σ12,tσ21,t σ22,t

)(3)

The vech operator stacks all columns of a

matrix into a column vector.

In the covariance matrix we account for only

the distinct elements, such that

vech(Σt) =

⎛⎜⎜⎝σ21,t

σ21,t

σ22,t

⎞⎟⎟⎠ ,(4)

where σ2i,t = σii,t, i = 1,2.

3

The simplest GARCH(1,1) parameterization

in a bivariate GARCH is

σ21,t = ω1 + α11u2

1,t−1 + β11σ21,t−1

σ22,t = ω2 + α22u2

2,t−1 + β22σ22,t−1

σ12,t = ω12 + α12u1,t−1u2,t−1 + β12σ12,t−1

(5)

Estimation:

EViews (www.eviews.com),

RATS (www.estima.com),

SAS (www.sas.com),

R (www.r-project.org).

4

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Assuming conditional normality of the ut, the

estimation can be accomplished with the ML

method.

Let θ denote the vector of all the estimated

parameters. Then the likelihood function is

of the form as in the univariate case with

�t(θ) = −k

2log(2π) − 1

2log(|Σt|) − 1

2u′

t Σ−1t ut,

(6)

where ut = yt − μt with μt modeled (usually)

with suitable vector ARMA model.

Example 3.1: Consider the forest and metal industrywith the covariance matrix specification (5) and

y1,t = μ + u1,t

y2,t = μ + u2,t(7)

5

Open Data \Users\Seppo\TeX\Teaching\fts\data\helsinki\omxhex_forest_metaCal(Irregular)Comp Start = 1Comp End = 2207

all endData(org=obs,format=Excel)

** Log return seriesset fr = 100*log(forest/forest{1})set mr = 100*log(metal/metal{1})

garch(p=1, q=1, iter = 200, hmatrices = ht) / fr mr

6

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The results are:

GARCH Model - Estimation by BFGSConvergence in 77 Iterations. Final criterion was 0.0000004 <= 0.00Usable Observations 2206Log Likelihood -7844.72136260

Variable Coeff Std Error T-Stat Signif****************************************************************1. Mean(1) 0.0315998874 0.0325636425 0.97040 0.331845182. Mean(2) 0.1435593579 0.0230536415 6.22719 0.000000003. C(1,1) 0.0209424100 0.0073302967 2.85697 0.004277114. C(2,1) 0.0186451969 0.0053642639 3.47582 0.000509305. C(2,2) 0.0468864922 0.0101689909 4.61073 0.000004016. A(1,1) 0.0496336132 0.0078433444 6.32812 0.000000007. A(2,1) 0.0510392579 0.0078265231 6.52132 0.000000008. A(2,2) 0.0818086550 0.0120582734 6.78444 0.000000009. B(1,1) 0.9450330948 0.0085771093 110.18084 0.0000000010. B(2,1) 0.9286998103 0.0110740983 83.86234 0.0000000011. B(2,2) 0.8956851334 0.0149509667 59.90818 0.00000000****************************************************************

The estimated coefficient seem reasonable.

Nevertheless, the model has inherently severe short-comings.

First, covariance does not include conditional vari-

ances, and thus does not (explicitly) model the ob-

served fact that correlation tends to increase as vari-

ability increases.

7

0

20

40

60

80

100

0.2

0.4

0.6

0.8

1.0

00 01 02 03 04 05 06 07 08

metal forest rho

Vola

tility

(% p

.a)

Correlation

Figure 3.1:. Conditional correlation and variance processes of

the simple bivariate GARCH given in (5)

The (estimated) conditional correlation process de-picted in the Figure is defined as

ρt =σ12,t

σ1,t σ2,t.(8)

As seen from the Figure above, the observed rela-

tionship between volatility and correlation is, however,

captured in some places by the simple model, although

the variances are not explicitly included into the co-

variance equation.

8

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Second, additional constraints should be imposed onthe coefficients to ensure positive definiteness of thecovariance matrix.

Third, and perhaps the most serious drawback is thatthe structure is not invariant with respect to linearcombinations.

This is not a problem with returns, but for examplein exchange rate markets it makes difference in whichcurrency the variables are denominated (e.g. Euro,Dollar, UK pound, Yen, if one is interested in therelationship of these currencies).

Fourth, the model is not invariant under portfolio ag-gregation (this a drawback for most ARCH models).

This means that aggregating individual stocks to port-

folios does not preserve the volatility structure (as an

exercise consider a portfolio of the two stocks with

weights w1 and w2.

9

Before going to the more advanced mod-

els, let us briefly look at an even simpler

model, Exponentially Weighted Moving Av-

erage (EWMA), which sometimes preferred

by practitioners [see e.g., RiskMetricsTM

(www.riskmetrics.com)]

The EWMA is a special case of the simple

GARCH in (5)

σ21,t = (1 − λ)u2

1,t−1 + λσ21,t−1

σ22,t = (1 − λ)u2

2,t−1 + λσ22,t−1

σ12,t = (1 − λ)u1,t−1u2,t−1 + λσ12,t−1,(9)

where the λ-parameter, called persistence is

defined by the user.

10

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A smaller λ implies a higher reaction of the

volatility to the market information in yester-

day’s return.

The range of λ is usually between 0.75 (highly

reactive) and 0.98 (very persistent but not

highly reactive).

11

An n-period moving average of a time seriesxt is

xt(n) =xt−1 + λxt−2 + λ2xt−3 + · · · + λn−1xt−n

1 + λ + λ2 + · · · + λn−1.(10)

Because 0 < λ < 1, n → ∞ the denominator

converges to 1 − λ.

Thus

limn→∞ xt(n) = (1 − λ)

∞∑i=1

λi−1xt−i.(11)

12

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For volatility and correlation one first calcu-

lates the exponentially weighted variance and

covariance estimates

σ2t = (1 − λ)

∞∑i=1

λi−1r2t−i,(12)

and

σ12,t = (1 − λ)∞∑

i=1

λi−1r1,t−1r2,t−i.(13)

Remark 3.1: It is standard to use in above formulas

daily returns, rt, not the residuals, and not deviations

from the mean.

Rewriting (10) and (12) in a recursive form

gives expressions (9)

13

Example 3.2: In the graphs below are volatility esti-

mate series, σt =√

(1 − λ)r2t−1 + λσ2

t−1, and correla-

tion estimate series, ρt = σ12,t/σ1tσ2t with λ = 0.85and λ = 0.95.

2060

100

EWMA 0.85 and 0.95 volatilities and correlations

Volat

2002 2004 2006 2008

-0.4

0.0

0.4

0.8

Cor

2002 2004 2006 2008

Figure 3.2: EWMA series for forest return volatility and correla-

tion between forest and metal returns with λ = 0.85 (blue/green)

and λ = 0.96 (pink/red) (lower panel average in green).

As indicated by the figure, the effect of λ on EWMA

volatility and correlation can be quite substantial.

14

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Remark 3.2: EWMA is a special case of the IGARCH,Integrated GARCH, which is aGARCH(1,1) model with α + β = 1

σ2t = ω + (1 − λ)u2

t−1 + λσ2t−1.(14)

Note that in these models the unconditional variance

is not defined (grows to infinity).

15

3.2 Multivariate GARCH

BEKK model

Generally an n-dimensional vec-model can begiven as

vech(Σt) = W + A vech(ut−1u′t−1) + B vech(Σt−1),(15)

which becomes (5) if the coefficient matrices

are defined as diagonal matrices.

16

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There are altogether

n(n + 1)

2+ 2 ×

(n(n + 1)

2

)2

plus the parameters of the mean equation to

be estimated.

Thus the model can be used in the general

form only in the case of, say, a two or three

variables.

Even in the diagonal case it is hard to ensure

the positive definiteness.

17

Different restrictions may lead substantial dif-

ferences in the estimated model.

This is why the vech model should be em-

ployed with caution.

In order to reduce the number of parame-

ters and guarantee positive definiteness sev-

eral alternatives have been suggested.

18

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A general parameterization that involves the

minimum number of parameters, while im-

posing no gross equation restrictions, and

ensuring positive definiteness is the BEKK

model (Baba, Engle, Kraft and Kroner, see

Engle and Kroner 1995, Econometric Theory

11, 122–150).

The model for n series is of the form

Σt = C′C + A′ut−1u′t−1A + B′Σt−1B,(16)

where C an n × n triangular matrix, B and

A are n × n matrices. It is clear the (50) is

positive definite under fairly general assump-

tions.

19

Example 3.2: A bivariate case

Σt = C′C

+(

a11 a12a21 a22

)′ ( u21,t−1 u1,t−1u2,t−1

u2,t−1u1,t−1 u22,t−1

)(a11 a12a21 a22

)+

(b11 b12b21 b22

)′Σt−1

(b11 b12b21 b22

).

(17)

or, suppressing the time subscripts and the GARCHterms

σ21 = c∗11 + a2

11u21 + 2a11a21u1u2 + a2

21u22

σ12 = c∗12 + a11b12u21 + (a21b12 + a11a22)u1u2 + a21a22u2

2σ2

2 = c∗22 + a212u

21 + 2a12a22u1u2 + a2

22u22,

where c∗ij are the relevant elements of the C∗ = C′C

matrix.

In the general vech model, excluding the constant

terms, there would be 18 parameters to be estimated,

compared to the BEKK which has ”only” 8 parame-

ters to be estimated. Generally the number of pa-

rameters are of order n2 in a system of n variables.

20

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As indicated by the example the formulas be-

come highly non-linear, and there are usually

convergence problems in the estimation pro-

cedures even in this bivariate case.

This was also the case with our forest-metal

data. Even SAS procedure VARMAX which

includes a multivariate GARCH option did

not find the solution.

Below is an example from RATS manual.

21

Several simplifications of the general BEKK

model have been suggested to alleviate the

computational problems.

(a) The Scalar BEKK

In the scalar BEKK the parameters in A are

the same (all series react similarly the market

information).

Similarly if the persistence in the correlations

and variances are assumed to be identical,

the parameters in B are the same.

(b) The diagonal BEKK

Matrices A and B are assumed to be diago-

nal.

22

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Example 3.3: The BEKK model for monthly SP500

and SP Mid Capital stock index (monthly observation

from Jan, 1986 to Dec 1996). A version of RATS

code can be found at www.estima.com >

Proc/Examples > ARCH/GARCH models > garchmv.prg

Modifying the program and running different Σt spec-

ifications yield the following criterion function values

=======================================GARCH Model AIC BIC

---------------------------------------Simple vec 639.97 682.60Full vec 641.49 737.85Constant corr 636.18 671.05Scalar BEKK 766.67 797.67Diagonal BEKK 641.67 688.17BEKK 641.49 695.74

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

The constant correlation model (considered

in more detail below) seems to fit best ac-

cording to bot criterion functions.

23

3.3 Covariance matrices based on univariate

GARCH

Even in the diagonal vech there are 3n(n +

1)/2 parameters to be estimated.

For example in a moderate portfolio problem

with 10 securities there would be 165 param-

eters to be estimated! In the diagonal BEKK

there are n(n + 1)/2 + 2n parameters to be

estimated.

Even in this case with 10 variables there would

be 75 parameters to be estimated.

24

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Thus computational problems become soon

overwhelming even in relatively low dimen-

sional systems.

This has cast sever doubt to the practical

usefulness of full multivariate GARCH mod-

eling.

However, there are some approximations that

allow generate multivariate GARCH matrices

by univariate GARCH.

25

(a) Constant correlation model

Assuming the correlation matrix R time in-

variant, then we can write

Σt = DtRDt,(18)

where Dt is a diagonal matrix of GARCH

volatilities (standard deviations).

Thus the time varying covariances in Σt are

of the form

σij,t = ρij σitσj,t,(19)

where ρij is the correlation of the series i and

j.

26

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Example 3.4: A bivariate constant correlation modelis defined as

σ21,t = α10 + α11u2

1,t−1 + β11 σ21,t−1

σ22,t = α20 + α22u2

2,t−1 + β22 σ22,t−1

σ12,t = ρ σ1,tσ2,t,(20)

where ρ is the contemporaneous correlation of the

return series.

With the above specification and constant mean, i.e.,

ri,t = μi+ui,t, i = 1,2, the following estimation results

are obtained for the data in the previous example.

27

MV_GARCH, CC - Estimation by BFGSConvergence in 34 Iterations.Final criterion was 0.0000052 <= 0.0000100Usable Observations 2206Log Likelihood -7872.25529863

========================================================Variable Coeff Std Error T-Stat Signif--------------------------------------------------------Mean(1) 0.0356503311 0.0404134516 0.88214 0.37770098Mean(2) 0.1405853902 0.0273646717 5.13748 0.00000028C(1) 0.0168377211 0.0064743566 2.60068 0.00930396C(2) 0.0514743777 0.0146868131 3.50480 0.00045695A(1) 0.0435257382 0.0072887789 5.97161 0.00000000A(2) 0.0832994359 0.0132790372 6.27300 0.00000000B(1) 0.9525903493 0.0078680158 121.07123 0.00000000B(2) 0.8913923633 0.0183718999 48.51933 0.00000000R(2,1) 0.4569759043 0.0163659754 27.92231 0.00000000========================================================

Results show strong evidence of GARCH in the volatil-

ity.

28

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(b) Factor ARCH

In order to reduce the dimensionality factor

analysis approach may prove useful.

Consider for simplicity a single factor model,

ri,t = μ + γift + ui,t,(21)

where ri,t is the return of an asset i, and ft

is the common factor for all assets (may be

observable or unobservable).

29

The underlying assumptions are that

Cov[ft, ui,t] = 0,∀i,

E[ui,t|f t−1,ut−1] = 0,∀i,

Cov[ut|f t−1,ut−1] = Ω,

E[ft|f t−1,ut−1] = 0,

σ2t = α0 + α1f2

t−1 + βσ2t−1

(22)

where σ2t = Var[ft|f t−1,ut−1], Ω is the time

invariant contemporaneous covariance matrix

of ut = (u1,t, . . . , un,t)′, the vector of the resid-

ual terms, and f t−1 and ut−1 denote the his-

torical values of the factor and residuals up

to time point t − 1.

30

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The common factor may be observable or

latent.

All the time varying volatility in the returns

are governed by the volatility process of the

single common factor.

The conditional covariance matrix of the as-

set returns becomes then

Σt = Covt−1[rt] = gg′σ2t + Ω,(23)

where g = (γ1, . . . , γn)′.

Engle, et al.∗ consider a more general factor

ARCH and give an application to model bond

yield volatilities.

∗Engle, R.F., V. Ng, and M. Rothschild (1990). As-set pricing with a factor-ARCH covariance structure.Journal of Econometrics,45, 213–237.

31

(c) Orthogonal models

In the orthogonal methods Principal Compo-

nent Analysis (PCA) is utilized to orthogo-

nalize the original returns.

Let

ri = μi + wi1p1 + · · · + winpn,(24)

where ri is the return of the share i = 1, . . . , n,

pj is the jth zero mean principal component,

j = 1, . . . , n. If only, say, m < n first principal

components are utilized, then

ri = μi + wi1p1 + . . . + wimpm + εi,(25)

where the residual εi includes the discarded

components.

32

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Remark 3.3: PCA solution is usually calculated from

the correlation matrix (standardized solution), in which

case the weights must be scaled, such that in place of

wij must be used w∗ijσi, where w∗

ij is the weight of the

standardized solution. Note that in the general case

wij �= w∗ijσi!

33

In terms of (25) the covariance matrix of the

returns becomes

Σ = WDW′ + Vε,(26)

where W = (wij) is the matrix of weights,

and D = diag(Var(p1), . . . , Var(pm)) is the di-

agonal matrix of variances of principal com-

ponents, and Vε is the covariance matrix of

residuals. In PCA Vε is ignored and the ap-

proximation

Σ = WDW′(27)

is used.

34

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Remark 3.4: Formula (25) defines essentially a Fac-

tor Analysis (FA) model with the difference that the

residual terms in (26) are correlated.

35

Orthogonal EWMA

In the basic EWMA model given by equations

(12) and (13) the λ parameter is assumed the

same in all equations. This guarantees that

the covariance matrix is positive semidefinite

(Exercise: prove it).

36

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Utilizing PCA allows use of different weights

in the PCA variances, and the matrix Σ de-

fined via (27) will still be p.s.d. (Note that

it is not p.d. if m < n.)

Remark 3.5: The advantage of PCA to generate risk

covariance matrix is the reduction in equations. In

stead of generating n(n+1)/2 variance and covariance

equations, only m variance equations are needed.

37

Usually the criterion of the successfulness of

the PCA approach is how well it can repro-

duce the direct EWMA series.

Example 3.5: Consider daily returns from Nordic stock

exchanges of Copenhagen (Den), Helsinki (Fin), Oslo

(Nor) and Stockholm (Swe). The sample period is

January 2, 1990 to February 4, 2000.

Below are contemporaneous correlations of the daily

returns

38

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Contemporaneous daily return correlations, means and standarddeviations of Nordic stock exchanges [Jan 2, 1990 to Feb 4, 2000].===============================================

DEN FIN NOR SWE-----------------------------------------------

DEN 1.000FIN 0.420 1.000NOR 0.486 0.452 1.000SWE 0.486 0.550 0.580 1.000

-----------------------------------------------Mean 0.0363 0.0923 0.0379 0.0619

Std. Dev. 0.9625 1.4275 1.1457 1.1866Obs. 2546 2546 2546 2546===============================================

Eigen values and eigen vectors computed from the contemporaneouscovariance matrix===============================================================

Comp 1 Comp 2 Comp 3 Comp 4---------------------------------------------------------------Eigenvalue 3.619747 0.956054 0.562894 0.543779Variance Prop. 0.637002 0.168246 0.099058 0.095694Cumulative Prop. 0.637002 0.805248 0.904306 1.000000===============================================================Eigenvectors:===============================================================Variable Vector 1 Vector 2 Vector 3 Vector 4---------------------------------------------------------------DEN 0.334915 0.270409 0.606230 0.668727FIN 0.640295 -0.749719 0.114759 -0.121550NOR 0.459146 0.536906 0.240941 -0.665480SWE 0.516758 0.276646 -0.749175 0.308488===============================================================

39

Defining the EWMA variances Vart(pi) for the princi-pal components pi as

Vart(pi) = (1 − λ)p2i,t−1 + λVart−1(pi)(28)

we get the conditional variances

σ2i,t =

4∑k=1

w2ikVart(pk),(29)

covariances

σij,t =4∑

k=1

wikwjkVart(pk),(30)

and correlations

ρij,t =σij,t

σi,tσj,t,(31)

i, j = 1, . . . ,4.

Below are graphs of EWMA and PCA-EWMA (annu-alized) volatilities:

Denmark.

0

10

20

30

40

50

60

70

500 1000 1500 2000 2500

DEN_EWMA DEN_EWMA_PCA

Vola

tility

40

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Finland

0

10

20

30

40

50

60

70

500 1000 1500 2000 2500

FIN_EWMA FIN_EWMA_PCA

Vola

tility

Norway

0

10

20

30

40

50

60

70

500 1000 1500 2000 2500

NOR_EWMA NOR_EWMA_PCA

Vola

tility

Sweden

0

10

20

30

40

50

60

70

500 1000 1500 2000 2500

SWE_EWMA SWE_EWMA_PCA

Vola

tility

41

EWMA and PCA-EWMA correlations:

Denmark–Finland

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

500 1000 1500 2000 2500

DE_FI_EWMA DE_FI_EWMA_PCA

Cor

rela

tion

Time

Denmark–Norway

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

500 1000 1500 2000 2500

DE_NO_EWMA DE_NO_EWMA_PCA

Cor

rela

tion

Denmark–Sweden

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

500 1000 1500 2000 2500

DE_SW_EWMA DE_SW_EWMA_PCA

Cor

rela

tion

42

Page 22: u 3. Multivariate Volatility models - Vaasan yliopisto |lipas.uwasa.fi/~sjp/Teaching/fts/lectures/ftsc3d.pdf ·  · 2009-04-223. Multivariate Volatility models Consider a k component

Finland–Norway

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

500 1000 1500 2000 2500

FI_NO_EWMA FI_NO_EWMA_PCA

Cor

rela

tion

Finland–Sweden

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

500 1000 1500 2000 2500

FI_SW_EWMA FI_SW_EWMA_PCA

Cor

rela

tion

Norway–Sweden

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

500 1000 1500 2000 2500

NO_SW_EWMA NO_SW_EWMA_PCA

Cor

rela

tion

43

From the above graphs we observe that the four EWMA-

PCA volatilities pretty well can reproduce the 10 EWMA

volatilities and correlations.

44

Page 23: u 3. Multivariate Volatility models - Vaasan yliopisto |lipas.uwasa.fi/~sjp/Teaching/fts/lectures/ftsc3d.pdf ·  · 2009-04-223. Multivariate Volatility models Consider a k component

The real computational strength of the PCA-

EWMA comes with highly correlated series,

where only few principal components are needed.

For example bond yields of different maturi-

ties are this kind of data.

45

Orthogonal GARCH

As was found earlier it is extremely difficult

to use multivariate GARCH to generate co-

variance matrices even in the case of low di-

mension.

PCA-GARCH is obtained by replacing above

EWMA principal component variances by

GARCH variances.

Note further that PCA is just a special case

of multi-factor model with orthogonal fac-

tors.

46

Page 24: u 3. Multivariate Volatility models - Vaasan yliopisto |lipas.uwasa.fi/~sjp/Teaching/fts/lectures/ftsc3d.pdf ·  · 2009-04-223. Multivariate Volatility models Consider a k component

Let Dt denote the diagonal matrix of m ≤ n

time varying principal component variances,

then the time varying covariance matrix Vt

of the original variables is approximated by

Vt = WDtW′,(32)

where W is, as earlier, the n × m matrix of

eigenvectors (principal component weights).

Model (32) is called orthogonal GARCH.

47

Remark 3.6: The representation (32) is always p.s.d.

Remark 3.7: The principal components are only un-

conditionally uncorrelated (orthogonal), so the assumed

conditional orthogonality is just an approximation.

Example 3.6: Below are GARCH(1,1) models for theprincipal components of the Nordic indices.

Dependent Variable: P1Method: ML - ARCH (Marquardt)Date: 02/25/03Sample(adjusted): 2 2547Included observations: 2546 after adjusting endpointsConvergence achieved after 21 iterationsVariance backcast: ON

Coefficient Std. Error z-Statistic Prob.

Variance Equation

C 0.191986 0.019683 9.753948 0.0000ARCH(1) 0.113826 0.011293 10.07962 0.0000

GARCH(1) 0.828716 0.015554 53.28138 0.0000

R-squared 0.000000 Mean dependent var 1.98E-16Adjusted R-squared -0.000786 S.D. dependent var 1.902937S.E. of regression 1.903685 Akaike info criterion 3.892484Sum squared resid 9215.875 Schwarz criterion 3.899368Log likelihood -4952.133 Durbin-Watson stat 1.699886

48

Page 25: u 3. Multivariate Volatility models - Vaasan yliopisto |lipas.uwasa.fi/~sjp/Teaching/fts/lectures/ftsc3d.pdf ·  · 2009-04-223. Multivariate Volatility models Consider a k component

Dependent Variable: P2Method: ML - ARCH (Marquardt)Date: 02/25/03Sample(adjusted): 2 2547Included observations: 2546 after adjusting endpointsConvergence achieved after 15 iterationsVariance backcast: ON

Coefficient Std. Error z-Statistic Prob.

Variance Equation

C 0.022161 0.003264 6.788933 0.0000ARCH(1) 0.101591 0.009900 10.26179 0.0000

GARCH(1) 0.878827 0.010100 87.01079 0.0000

R-squared 0.000000 Mean dependent var -7.95E-16Adjusted R-squared -0.000786 S.D. dependent var 0.977972S.E. of regression 0.978357 Akaike info criterion 2.625073Sum squared resid 2434.114 Schwarz criterion 2.631957Log likelihood -3338.718 Durbin-Watson stat 1.764620

Dependent Variable: P3Method: ML - ARCH (Marquardt)Date: 02/25/03Sample(adjusted): 2 2547Included observations: 2546 after adjusting endpointsConvergence achieved after 13 iterationsVariance backcast: ON

Coefficient Std. Error z-Statistic Prob.

Variance Equation

C 0.015461 0.003174 4.870846 0.0000ARCH(1) 0.085785 0.008898 9.641193 0.0000

GARCH(1) 0.887398 0.011787 75.28359 0.0000

R-squared 0.000000 Mean dependent var 8.77E-16Adjusted R-squared -0.000786 S.D. dependent var 0.750410S.E. of regression 0.750705 Akaike info criterion 2.106615Sum squared resid 1433.128 Schwarz criterion 2.113499Log likelihood -2678.721 Durbin-Watson stat 1.769628

Dependent Variable: P4Method: ML - ARCH (Marquardt)Date: 02/25/03Sample(adjusted): 2 2547Included observations: 2546 after adjusting endpointsConvergence achieved after 13 iterationsVariance backcast: ON

Coefficient Std. Error z-Statistic Prob.

Variance Equation

C 0.004904 0.001477 3.319334 0.0009ARCH(1) 0.059132 0.006038 9.793347 0.0000

GARCH(1) 0.933042 0.006701 139.2406 0.0000

R-squared 0.000000 Mean dependent var -5.62E-16Adjusted R-squared -0.000786 S.D. dependent var 0.737559S.E. of regression 0.737848 Akaike info criterion 2.098950Sum squared resid 1384.461 Schwarz criterion 2.105834Log likelihood -2668.963 Durbin-Watson stat 1.805472

49

The conditional variances, covariances and correla-

tions again estimated as in (29)–(31).

Below are graphs of the resulted conditional correla-

tions.

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

500 1000 1500 2000 2500

DE_FI_GARCH_PCADE_NO_GARCH_PCADE_SW_GARCH_PCA

Corre

latio

n

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

500 1000 1500 2000 2500

FI_NO_GARCH_PCAFI_SW_GARCH_PCANO_SW_GARCH_PCA

Corre

latio

n

4

5

6

7

8

9

10

500 1000 1500 2000 2500

LOG(DECLOSE)LOG(FICLOSE)

LOG(NOCLOSE)LOG(SWCLOSE)

log(

inde

x)

50

Page 26: u 3. Multivariate Volatility models - Vaasan yliopisto |lipas.uwasa.fi/~sjp/Teaching/fts/lectures/ftsc3d.pdf ·  · 2009-04-223. Multivariate Volatility models Consider a k component

It would be important here to compare the

estimation results with BEKK and vech model

to see how the relatively simple orthogonal

GARCH compares with them.

We however skip the comparison and look at

some other approaches to analyze large scale

systems.

51

Before that let us summarize the orthogonal

approach.

The idea is simply that:

(i) Find a transformation W such that yt =

Wrt, and Cov[yt] = D is diagonal.

(ii )Estimate GARCH for the components

of the yt vector and compile the conditional

variances to the diagonal matrix Dt.

(iii) Under the (strong) assumption Covt−1[yt] =

Dt, we get

Σt = Covt−1[rt] = W−1DtW′−1.(33)

52

Page 27: u 3. Multivariate Volatility models - Vaasan yliopisto |lipas.uwasa.fi/~sjp/Teaching/fts/lectures/ftsc3d.pdf ·  · 2009-04-223. Multivariate Volatility models Consider a k component

3.4. Dynamic Conditional Correlation

In the Dynamic Conditional Correlation (DCC)

proposed by Engle (2002)∗ the conditional

correlation is parameterized directly.

Let ri, i = 1, . . . , n be random variables with

zero mean. The conditional correlations are

defined as

ρij,t =Et−1[ri,t rj,t]√

Et−1[r2i,t]Et−1[r

2j,t]

.(34)

Let σ2i,t = Et−1[r

2i,t], then zi,t = ri,t/σit ∼

WN(0,1), and the correlation can be written

as

ρij,t = Et−1[zi,t zj,t].(35)

Note: Although zit are individually i.i.d it does not follw that they

are jointly independend! That is why the conditional correlation

may depend on the past. However, joint independnece implies

independence of individual zi,ts.

∗Engle, Robert, F. (2002). Dynamic conditionalcorrelation—A simple class of multivariate GARCH.Journal of Business and Economics Statistics. 17,No. 5, 425–446.

53

Engle (2002) suggest to estimate the follow-ing GARCH process

qij,t = ρij + α (zi,t−1 zj,t−1 − ρij) + β (qij,t−1 − ρij),(36)

and

ρij,t =qij,t√

qii,t qjj,t,(37)

where ρij is the unconditional expectation ofthe cross product zi,tzj,t, i.e., unconditionalcorrelations.

Thus the main difference here compared tothe above approaches is that the correlationsare modeled individually as GARCH processeswith common GARCH parameters α and β

and separate unconditional expectations ρij

of the cross products.

Thus there are altogether n(n+1)/2+n+2parameters to be estimated in the model.

Using these and the separately modeled GARCHvariance processes the resulting covariancematrix is obtained as

Σt = DtRtDt,(38)

54

Page 28: u 3. Multivariate Volatility models - Vaasan yliopisto |lipas.uwasa.fi/~sjp/Teaching/fts/lectures/ftsc3d.pdf ·  · 2009-04-223. Multivariate Volatility models Consider a k component

where Dt is the diagonal matrix of conditionalstandard deviations [c.f. formula (18)], andRt is the conditional correlation matrix withelements given by (37).

Remark 3.9: The correlations in (37) will give a p.d.

correlation matrix, because Qt = (qij,t) is a weighted

average of p.s.d and p.d matrix.

Remark 3.9: ωij is the unconditional correlation, and

each term in the denominator of (37) has expected

value one.

Remark 3.10: Model (36) is mean revertingas long as α + β < 1 (α, β ≥ 0)

Estimation

A formulation of the DCC model is

rt|Ft−1 ∼ N(0,DtRtDt)Dt = diag(σ1,t, . . . , σn,t)

zt = D−1t rt

Qt = (qij,t)

Rt = (diag(Qt))−1

2 Qt (diag(Qt))−1

2 ,

(39)

55

where σi,t =√

α0,i + αi,1r2i,t−1 + βiσ2i,t−1, i =

1, . . . , n are the GARCH(1,1) standard devia-

tions, and

(diag(Qt))−1

2 = diag(1/√

q11,t , . . . ,1/√

qnn,t).

Let θ denote the parameters in Dt, and φ

the parameters in Rt, then the log likelihood

function is

�(θ, φ) =T∑

t=1

�t(θ, φ),(40)

where (ignoring the constant term n log 2π)

�t(θ, φ) = −1

2

(log |DtRtDt| + r′tD

−1t R−1

t D−1t rt

).(41)

In order to simplify the maximization proce-dure it is accomplished in two steps. For thepurpose, rearrange terms in (41) such that

�t(θ, φ) = −12

(2 log |Dt| + r′tD

−1t D−1

t rt

−z′tzt + log |Rt| + z′tR−1t zt

)(42)

56

Page 29: u 3. Multivariate Volatility models - Vaasan yliopisto |lipas.uwasa.fi/~sjp/Teaching/fts/lectures/ftsc3d.pdf ·  · 2009-04-223. Multivariate Volatility models Consider a k component

Thus we can decompose the likelihood func-

tion into volatility part

�V,t(θ) = −1

2

(2 log |Dt| + r′tD−1

t D−1t rt

),(43)

and correlation part

�C,t(θ, φ) = −1

2

(log |Rt| + z′tR−1

t zt − z′tzt

),(44)

so that

�t(θ, φ) = �V,t(θ) + �C,t(θ, φ).(45)

Engle (2002) suggest a two step procedure,

where in the first step the variance part

�V (θ) =T∑

t=1

�V,t(θ)(46)

is maximized, and then given the maximizing

value θ, the correlation part

�C(θ, φ) =T∑

t=1

�C,t(θ, φ)(47)

is maximized with respect to φ.

57

Remark 3.11: Because Dt is a diagonal matrix, thevariance part (46) is the sum of individual GARCHlikelihoods

�V (θ) = −1

2

T∑t=1

n∑i=1

(log(2π) + log(σ2

i,t) +r2i,t

σ2i,t

),(48)

which implies that in the first step the GARCH models

can be estimated separately to each series.

Remark 3.12: Because z′tzt terms remain un-

changed in (44) you can ignore it and simplify

�C,t(θ, φ) = −1

2

(log |Rt| + z′tR−1

t zt

),(49)

Example 3.7: Consider the Nordic stock indices. Asnoted above in the first step we simply estimate sepa-rate GARCH(1,1) models for each index return series.Because there is some autocorrelation in the returnseries, we adopt the following specifications

yi,t = φi,0 + φi,1yi,t−1 + ui,t

hi,t = ωi + αiu2i,t−1 + βihi,t−1,

(50)

where in this case hi,t = Vart−1[ui,t] denotes the con-

ditional variance, i = 1, . . . ,4 with 1 = Denmark, 2 =

Finland, 3 = Norway, and 4 = Sweden.

58

Page 30: u 3. Multivariate Volatility models - Vaasan yliopisto |lipas.uwasa.fi/~sjp/Teaching/fts/lectures/ftsc3d.pdf ·  · 2009-04-223. Multivariate Volatility models Consider a k component

RATS estimates for the AR and GARCH parametersare as follows================================================================

Variable Coeff Std Error T-Stat Signif----------------------------------------------------------------

PHI0(1) 0.0258196495 0.0207576877 1.24386 0.21355123PHI0(2) 0.0463076920 0.0366149726 1.26472 0.20597168PHI0(3) 0.0572372304 0.0264378197 2.16498 0.03038957PHI0(4) 0.0844326535 0.0209986432 4.02086 0.00005799PHI1(1) 0.1652603392 0.0218878890 7.55031 0.00000000PHI1(2) 0.1994514707 0.0194654520 10.24643 0.00000000PHI1(3) 0.1843649422 0.0276705032 6.66287 0.00000000PHI1(4) 0.1462929873 0.0169525559 8.62955 0.00000000OMEGA(1) 0.0812695314 0.0360803362 2.25246 0.02429319OMEGA(2) 0.0715641017 0.0280201692 2.55402 0.01064868OMEGA(3) 0.0513847071 0.0247171604 2.07891 0.03762579OMEGA(4) 0.0574232062 0.0176653233 3.25062 0.00115155ALPHA(1) 0.1358098187 0.0352697921 3.85060 0.00011783ALPHA(2) 0.1330450304 0.0231289321 5.75232 0.00000001ALPHA(3) 0.1640905715 0.0434603942 3.77563 0.00015960ALPHA(4) 0.1366265789 0.0234533241 5.82547 0.00000001BETA(1) 0.7759450051 0.0611495501 12.68930 0.00000000BETA(2) 0.8309721748 0.0316890857 26.22266 0.00000000BETA(3) 0.8054756174 0.0505598297 15.93114 0.00000000BETA(4) 0.8244948090 0.0251068472 32.83944 0.00000000

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

Thus e.g., for Finland

y2,t = 0.0463 + 0.1995y2,t−1 + u2,t

(0.037) (0.019)

h2,t = 0.0716 + 0.1330u22,t−1 + 0.8310h2,t−1

(0.028) (0.023) (0.032)with standard errors in parentheses.

59

For the sake of simplicity we use the simplified GARCH(1,1)specification for the correlation part as

qij,t = ρi,j + α(zi,t−1zj,t−1 − ρi,j) + β(qij,t−1 − ρij),(51)

where ρij are estimated by the sample contempora-

neous correlations. That is, only the GARCH(1,1)-

parameters α and β are estimated, and ρij are re-

placed by the sample contemporaneus correlations of

the standardized series zi,t. The correlations are as

follows

Contemporaneous standardized residual correlations========================================

Den Fin Nor Swe----------------------------------------Den 1Fin 0.371 1Nor 0.461 0.416 1Swe 0.464 0.504 0.542 1========================================

These are generally a little bit lower than the return

contemporaneous correlations considered earlier.

The (quasi) ML esitmation of the GARCH(1,1) pa-

rameters yields α = 0.0336(0.001703) and

β = 0.9247(0.004116) with standard errors in paren-

theses. The respective t-values are 19.7 and 224.7,

and thus highly significant.

60

Page 31: u 3. Multivariate Volatility models - Vaasan yliopisto |lipas.uwasa.fi/~sjp/Teaching/fts/lectures/ftsc3d.pdf ·  · 2009-04-223. Multivariate Volatility models Consider a k component

The estimate of β is close to one, which implies that

the correlations should be highly persistent. Neverthe-

less, as indicated by the graphs below there is consid-

erable variation in the correlations. The general ten-

dency is increasing. These can be compared with the

earlier unconditional contemporaneous correlations

Conditional CorrelationsDenmark, Finland, Norway and Sweden [Jan 1990- Jan 2000]

Denmark Finland

Co

rr

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Denmark Norway

Co

rr

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Denmark Sweden

Co

rr

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Finland Norway

Co

rr

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Finland Sweden

Co

rr

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Norway Sweden

Co

rr

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure. Estimated dynamic conditional correlations.

61

Bellow is RATS code. If experimented with another data setsonly those parts indicated by ′ <<<<′ should be changed.¶

** Estimated Dynamic Conditional Correlation DCC Example*Cal(Irregular)*COMPUTE GSTART=4, GEND = 2547 ;* <<<<all gendOpen Data ind012000pg.xls ;* <<<<data(format=xls,org=cols)comp nSeries = 4 ;* number of seies in the analysis <<<<* Define return seriesdec vect[series] y(nSeries) ;* Return seriesset y(1) = 100*log(deClose/deClose{1}) ;* <<<<set y(2) = 100*log(fiClose/fiClose{1}) ;* <<<<set y(3) = 100*log(noClose/noClose{1}) ;* <<<<set y(4) = 100*log(swClose/swClose{1}) ;* <<<<** YOU NEED NOT CHANGE THE PROGRAM BELOW THIS** Parameters for the regression function*dec vect phi0(nSeries) phi1(nSeries) ;* AR-parametersdec vect omega(nSeries) alpha(nSeries) beta(nSeries) ;* GARCH paramsdec vect yt(nSeries)dec vect yt1(nSeries)dec vect[series] z(nSeries) ;* standardized seriesdec vect[series] h(nSeries) ;* GARCH processesdec vect ht(nSeries)dec vect ht1(nSeries)dec frml[vect] resid ;* AR residual vector functionsdec vect residt1(nSeries)dec frml[vect] hf ;* variance vector funcitonsdec symm sigma(nSeries,nSeries)dec vec zVec(nSeries)

¶I want to thank Martin Richter from Danskebank forcareful examination of an earlier version of the codeand helpful suggestions.

62

Page 32: u 3. Multivariate Volatility models - Vaasan yliopisto |lipas.uwasa.fi/~sjp/Teaching/fts/lectures/ftsc3d.pdf ·  · 2009-04-223. Multivariate Volatility models Consider a k component

NONLIN(parmset=arParms) phi0 phi1NONLIN(parmset=garchParms) omega alpha betaFRML RESID = yt=%xt(y,t),yt1=%xt(y,t-1),yt - phi0 - %diag(phi1)*yt1frml hf = residt1=%xt(z,t-1),ht1=%xt(h,t-1),$

residt1=%xdiag(%outerxx(residt1)),$omega + %diag(alpha)*residt1 + %diag(beta)*ht1

frml glogl = %pt(z,t,resid(t)),$%pt(h,t,hf(t)),$sigma = %diag(%xt(h,t)), zVec=%xt(z,t),$%logdensity(sigma,zVec)

** Do initial AR regression.* Copy initial values for regression parameters, and* Initialize GARCH variance seriesdo i=1,nSeriesLINREG(noPrint) y(i) / z(i)# CONSTANT y(i){1}COMPUTE phi0(i) = %BETA(1), phi1(i) = %BETA(2)set h(i) = %seesq**2

end do i** Initialize GARCH estimatescomp omega = alpha = beta = %mscalar(0.05)** Estimate GARCH parametersMAXIMIZE(parmset=arParms+garchparms,$METHOD=SIMPLEX,ITERS=10) GLOGL GSTART GEND ;* Improve starting valMAXIMIZE(parmset=arParms+garchparms,$METHOD=BFGS,ITERS=100,robust) GLOGL GSTART GEND

63

* CORRELATION PART** Standardized variables*do i=1,nSeriesset z(i) = z(i)/sqrt(h(i))

end do idecl symm[series] r(nSeries,nSeries)decl symm[series] q(nSeries,nSeries)decl symm qMat(nSeries,nSeries) qMat1(nSeries,nSeries)decl symm rMat(nSeries,nSeries) zMat1(nSeries,nSeries)decl frml[symm] qfnonlin(parmset=corrParms) alpha_c beta_c** Define the q-functionfrml qf = (zMat1 = %outerxx(%xt(z,t-1))-rMat),$

(qMat1 = %xt(q,t-1)-rMat),$rMat + alpha_c*zMat1 + beta_c*qMat1

** Initialize valuesVCV(MATRIX=rMat,NOPRINT)# z*do i=1,nSeries

do j=1,iset q(i,j) = rMat(i,j)

end do jend do i** Define correlation part of the log likelihoodFRML CLOGL = qMat = qf(t), zVec = %xt(z,t), %pt(q,t,qMat),$

%pt(r,t,%mqform(qMat,inv(%diag(%sqrt(%xdiag(qMat)))))),$qMat = %xt(r,t),$

%logdensity(qMat,zVec)comp alpha_c = 0.05, beta_c = 0.80MAXIMIZE(parmset=corrParms,$

METHOD=SIMPLEX,ITERS=10) CLOGL GSTART+3 GENDMAXIMIZE(parmset=corrParms,$

METHOD=BFGS,ITERS=100) CLOGL GSTART+3 GEND

64

Page 33: u 3. Multivariate Volatility models - Vaasan yliopisto |lipas.uwasa.fi/~sjp/Teaching/fts/lectures/ftsc3d.pdf ·  · 2009-04-223. Multivariate Volatility models Consider a k component

For further details regarding DCC, see En-

gle (2002) and Engle and Sheppard (2001)

(Engle’s home page).

65