Advances in Risk Management

268 TIME-VARYING RETURN CORRELATIONS AND PORTFOLIOS

is the commonly used rolling estimator, where the unconditional means,
variances and co-variances are estimated using a rolling window of fixedN
observations over a sample periodT. The unconditional mean return and
variance of a securityiis estimated as:

Ri=

1

N

∑N

t= 1

Rit (14.6)

σ^2 i=

1

N− 1

∑N

t= 1

(

Rit−

−

Ri

) 2

(14.7)

The co-variance between the returns of two securitiesiandkare estimated
as follows:

σi,k=

1

N− 1

∑N

t= 1

(

Rit−Ri

)(

Rkt−Rk

)

(14.8)

One of the main problems with such rolling estimators is that it does
not capture the time-varying nature of means, variances and co-variances.
To capture the time varying nature of variances and co-variances, the sec-
ond method of estimation uses the Dynamic Conditional Correlation (DCC)
model of Engle (2002). The conditional correlation between two random
variabler 1 andr 2 that have mean zero can be written as:

ρ12,t=

Et− 1 (r1,tr2,t)

√

Et− 1 (r^2 1,t)Et− 1 (r^2 2,t)

(14.9)

Lethi,t=Et− 1 (r^2 i,t)and ri,t=

√
hi,tεi,tfor i=1, 2, whereεi,tis a standardized
disturbance that has zero mean and variance of one.
Substituting the above into equation (14.1) we get:

ρ12,t=

Et− 1 (ε1,tε2,t)

√

Et− 1 (ε^2 1,t)Et− 1 (ε^2 2,t)

=Et− 1 (ε1,tε2,t) (14.10)

Using a GARCH (1,1) specification, the covariance between the random
variables can be written as:

q12,t=ρ 12 +α

(

ε1,t− 1 ε2,t− 1 −ρ 12

)

+β

(

q12,t− 1 −ρ 12

)

(14.11)

The unconditional expectation of the cross product isρ 12 , while for the
variancesρ 12 = 1