DYNAMIC CONDITIONAL CORRELATIONS

Robert EngleUCSD and NYU and Robert F. Engle, Econometric Services

DYNAMIC CONDITIONAL

CORRELATIONS

WHAT WE KNOW

VOLATILITIES AND CORRELATIONS VARY OVER TIME, SOMETIMES ABRUPTLY

RISK MANAGEMENT, ASSET ALLOCATION, DERIVATIVE PRICING AND HEDGING STRATEGIES ALL DEPEND UPON UP TO DATE CORRELATIONS AND VOLATILITIES

AVAILABLE METHODS

MOVING AVERAGES– Length of moving average determines

smoothness and responsiveness EXPONENTIAL SMOOTHING

– Just one parameter to calibrate for memory decay for all vols and correlations

MULTIVARIATE GARCH– Number of parameters becomes intractable for

many assets

DYNAMIC CONDITIONAL CORRELATION

A NEW SOLUTION THE STRATEGY:

– ESTIMATE UNIVARIATE VOLATILITY MODELS FOR ALL ASSETS

– CONSTRUCT STANDARDIZED RESIDUALS (returns divided by conditional standard deviations)

– ESTIMATE CORRELATIONS BETWEEN STANDARDIZED RESIDUALS WITH A SMALL NUMBER OF PARAMETERS

MOTIVATION

Assume structure for conditional correlations

Simplest assumption- constancy Alternatives

– Integrated Processes– Mean Reverting Processes

DEFINITION: CONDITIONAL CORRELATIONS1 . D e f i n i t i o n o f c o n d i t i o n a l c o r r e l a t i o n :

2

12

1

1

tttt

tttt

xEyE

xyE

,

w h e r e x , y a r e e x c e s s r e t u r n s .

2 . I f txtxttytyt hxhy ,,,, , , t h e n

txtyt

txttyt

txtyt

t EEE

E,,12

,1

2

,1

,,1

BOLLERSLEV(1990): CONSTANT CONDITIONAL CORRELATION

Assume conditional correlations are constant!i.e. t for all t.

Bollerslev proposed jointly estimating both GARCHmodels and the conditional correlation by maximumlikelihood. Here we propose doing this in two steps:

1. Estimate a GARCH model for each return2. Compute the correlation between the

standardized residuals.

DISCUSSION

Likelihood is simple when estimating jointly Even simpler when done in two steps Can be used for unlimited number of assets Guaranteed positive definite covariances BUT IS THE ASSUMPTION PLAUSIBLE?

CORRELATIONS BETWEEN PORTFOLIOS

txtxtyty hhxyz ,,,,

txtxtyttxty

txtyttxtxz

hhhhh

hhh

,,,,2

,

,,,,,

2

HOWEVER

EVEN IF ASSETS HAVE CONSTANT CONDITIONAL CORRELATIONS, LINEAR COMBINATIONS OF ASSETS WILL NOT

DYNAMIC CONDITIONAL CORRELATIONS

DYNAMIC CONDITIONAL CORRELATIONS

STRATEGY:estimate the time varying correlation between standardized residuals

MODELS– Moving Average : calculate simple correlations

with a rolling window– Exponential Smoothing: select a decay

parameter and smooth the cross products to get covariances, variances and correlations

– Mean Reverting ARMA

Multivariate FormulationMultivariate Formulation

Let r be a vector of returns and D a diagonal matrix with standard deviations on the diagonal

R is a time varying correlation matrix

tttttt DRDHHNr ),,0(~

ttttttt RErD ', 1

1

Log Likelihood

tttttt

ttttttttt

ttttt

RRD

rDRDrDRD

rHrHL

1

111

1

'loglog2)2log(2

1

'log)2log(2

1

'log)2log(2

1

Conditional Likelihood

Conditional on fixed values of D , the likelihood is maximized with the last two terms.

In the bivariate case this is simply

t t

ttttttConstL

2.2,1

2,2

2,12

1

21log

21

Two Step Maximum Likelihood

First, estimate each return as GARCH possibly with other variables or returns as inputs, and construct the standardized residuals

Second, maximize the conditional likelihood with respect to any unknown parameters in rho

Specifications for Rho

Exponential Smoother

i.e.

T

sst

sT

sst

s

T

sstst

s

t

1

2,2

1

2,1

1,2,1

.1

where,

1,,1,1,,,

,2,2,1,1

,2,1

tjitjtitji

tt

tt

qq

qq

q

Mean Reverting Rho

Just as in GARCH

and

.1

where,

1,,1,1,,,,

,2,2,1,1

,2,1

tjitjtijitji

tt

tt

qq

qq

q

1, ii

Alternatives to MLE

Instead of maximizing the likelihood over the correlation parameters:

For exponential smoother, estimate IMA

For ARMA, estimate

t1tt,2t,1 uu1

tttttt uu 12,11,21,12,1,2,1

Monte Carlo Experiment

Six experiments - Rho is:– Constant = .9– Sine from 0 to .9 - 4 year cycle– Step from .9 to .4– Ramp from 0 to 1– Fast sine - one hundred day cycle– Sine with t-4 shocks

One series is highly persistent, one is not

DIMENSIONS

SAMPLE SIZE 1000 REPLICATIONS 200

0.0

0.2

0.4

0.6

0.8

1.0

1000 2000 3000 4000 5000

RHO_SINE

0.0

0.2

0.4

0.6

0.8

1.0

1000 2000 3000 4000 5000

RHO_STEP

0.0

0.2

0.4

0.6

0.8

1.0

1000 2000 3000 4000 5000

RHO_RAMP

0.0

0.2

0.4

0.6

0.8

1.0

1000 2000 3000 4000 5000

RHO_FAST_SINE

CORRELATION EXPERIMENTS

METHODS

SCALAR BEKK (variance targeting) DIAGONAL BEKK (variance targeting) DCC - LOG LIKELIHOOD WITH MEAN

REVERSION DCC - LOG LIKELIHOOD FOR

INTEGRATED CORRELATIONS DCC - INTEGRATED MOVING

AVERAGE ESTIMATION

MORE METHODS

EXPONENTIAL SMOOTHER .06 MOVING AVERAGE 100 ORTHOGONAL GARCH (first series is

first factor, second is orthogonalized by regression and GARCH estimated for each)

CRITERIA

MEAN ABSOLUTE ERROR IN CORRELATION ESTIMATE

AUTOCORRELATION FOR SQUARED JOINT STANDARDIZED RESIDUALS - SERIES 2, SERIES 1

DYNAMIC QUANTILE TEST FOR VALUE AT RISK

JOINT STANDARDIZED RESIDUALS In a multivariate context the joint

standardized residuals are given by

There are many matrix square roots - the Cholesky root is chosen:

ttt rH 2/1

ttttt

tt

ba ,1,2,2

,1,1

TESTING FOR AUTOCORRELATION REGRESS SQUARED JOINT

STANDARDIZED RESIDUAL ON– ITS OWN LAGS - 5– 5 LAGS OF THE OTHER – 5 LAGS OF CROSS PRODUCTS– AN INTERCEPT

TEST THAT ALL COEFFICIENTS ARE EQUAL TO ZERO EXCEPT INTERCEPT

RESULTS-MeanAbsoluteError

0

0.2

0.4

0.6

0.8

1

1.2

1.4

T(4) SINECONSTRAMPSTEPSINEFAST SINE

FRACTION OF DIAGNOSTIC FAILURES(2)

0

1

2

3

4

5

6

SCAL BEKK

DIAG BEKK

DCC LL MR

DCC LL INT

DCC IMA

EX .06

MA 100

O-GARCH

T(4) SINECONSTRAMPSTEPSINEFAST SINE

FRACTION OF DIAGNOSTIC FAILURES (1)

0

0.05

0.1

0.15

0.2

0.25

T(4) SINECONSTRAMPSTEPSINEFAST SINE

DQT for VALUE AT RISK

00.050.1

0.150.2

0.250.3

0.350.4

0.450.5

T(4) SINECONSTRAMPSTEPSINEFAST SINE

CONCLUSIONS

VARIOUS METHODS FOR ESTIMATING DCC HAVE BEEN PROPOSED and TESTED

IN THESE EXPERIMENTS, THE LIKELIHOOD BASED METHODS ARE SUPERIOR

THE MEAN REVERTING METHODS ARE SLIGHTLY BETTER THAN THE INTEGRATED METHODS

EMPIRICAL EXAMPLES

DOW JONES AND NASDAQ

STOCKS AND BONDS

CURRENCIES

0.2

0.4

0.6

0.8

1.0

3/23/90 1/21/94 11/21/97

DCC_INT_DJ_NQ

NASDAQ - DOW JONES CORRELATIONS -TEN YEARS

0.2

0.4

0.6

0.8

1.0

3/21/97 12/26/97 10/02/98 7/09/99

DCC_INT_DJ_NQDCC_MR_DJ_NQDIAG_BEKK_DJ_NQ

NASDAQ - DOW JONES CORRELATIONS - THREE YEARS

0.2

0.4

0.6

0.8

1.0

3/23/98 12/28/98 10/04/99

DCC_INT_DJ_NQOGARCH_DJ_NQ

DIAG_BEKK_DJ_NQDCC_MR_DJ_NQ

NASDAQ - DOW JONES CORRELATIONS - TWO YEARS

0.2

0.3

0.4

0.5

0.6

0.7

12/31 1/14 1/28 2/11 2/25 3/10

DCC_INT_DJ_NQOGARCH_DJ_NQ

DIAG_BEKK_DJ_NQDCC_MR_DJ_NQ

NASDAQ - DOW JONES CORRELATIONS 2000

0

10

20

30

40

50

60

3/23/90 2/21/92 1/21/94 12/22/95 11/21/97 10/22/99

VOL_DJ_GARCH VOL_NQ_GARCH

TEN YEARS OF VOLATILITIES

0

10

20

30

40

50

60

3/21/97 12/26/97 10/02/98 7/09/99

VOL_DJ_GARCH VOL_NQ_GARCH VOL_BOND_GARCH

STOCK - BOND VOLATILITIES - THREE YEARS

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

3/21/97 12/26/97 10/02/98 7/09/99

DCC_INT_DJ_BOND DCC_INT_NQ_BOND

BOND CORRELATIONS - THREE YEARS

0.2

0.4

0.6

0.8

1.0

500 1000 1500 2000

DCC_INT_RDEM_RITL DIAG_BEKK_RDEM_RITL

CORRELATIONS BETWEEN DM AND LIRA 86-95

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

500 1000 1500 2000

DCC_INT_RDEM_RGBPDCC_INT_RFRF_RDEMDCC_INT_RDEM_RITL

DM CORRELATIONS - 86-95

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

10/04/93 9/04/95 8/04/97

DCC_INT_RDEM_RFRFDCC_INT_RDEM_RITL

DCC_INT_RFRF_RITL

CURRENCY CORRELATIONS

CONCLUSIONS

VARIOUS METHODS FOR ESTIMATING DCC HAVE BEEN PROPOSED and TESTED

IN THESE EXPERIMENTS, THE LIKELIHOOD BASED METHODS ARE SUPERIOR

THE MEAN REVERTING METHODS ARE SLIGHTLY BETTER THAN THE INTEGRATED METHODS