Transcript DYNAMIC CONDITIONAL CORRELATION

DYNAMIC CONDITIONAL
CORRELATION :
ECONOMETRIC RESULTS
AND FINANCIAL
APPLICATIONS
Robert Engle
New York University
Prepared for
CARLOS III, MAY 24, 2004
ABSTRACT
A new model for measuring and predicting correlations as well as volatilities is
examined. This Dynamic Conditional Correlation model or DCC, models the volatilities and
correlations in two steps. The specification of the correlation dynamics is extended to allow
asymmetries important for financial practice. The presentation develops the econometric
methods for estimating the DCC model.
The DCC provides a joint density function with tail dependence greater than the
normal. This is explored both by simulation and empirically. The time aggregated DCC is
presented as a useful copula for financial decision making.
As an example, a covariance matrix is estimated between 34 country equity and bond
returns. The role of asymmetry is examined in both volatilities and correlations. The
dispersion of equity and bond volatilities and their pairwise correlations are examined over
time and after the formation of the EURO.
OUTLINE


WHAT IS DCC?
ESTIMATION THEORY




BEYOND LINEAR DEPENDENCE



TWO STEP
QMLE
STANDARD ERRORS
DEPENDENCE MEASURES
MULTI-STEP DISTRIBUTION
ASYMMETRIC GLOBAL CORRELATIONS
Conditional Correlations

Definition of conditional correlation
 x , y ,t 

Et 1  xt yt 
Et 1  xt2  Et 1  yt2 
yt  hy ,t  y ,t , xt  hx ,t  x ,t
And letting
 x , y ,t 
Et 1  x ,t y ,t 
Et 1 
2
x ,t
 E  
t 1
2
y ,t
 Et 1  x ,t y ,t 
Multivariate Definitions

Let r be a vector of returns and D a diagonal
matrix with standard deviations on the
diagonal
rt Ft 1 ~ F (0, Ht ), Ht  Dt Rt Dt

R is a time varying correlation matrix
 t  D r , Et 1  t t '  Rt
1
t t
DYNAMIC CONDITIONAL
CORRELATION
A NEW SOLUTION in Engle(2002)

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
MODELS FOR CONDITIONAL
CORRELATIONS

Constant

Integrated Processes

Mean Reverting Processes

More complex multivariate processes
The Constant Correlation
Estimator: Bollerslev(1990)

Let
t  Dt rt
1
Be the standardized residuals

Then
R
1
T
t t '

T t
1
Specifications for Rho

Exponential Smoother
T
t 

i.e.
t 
s

 1,t s 2,t s
s 1
 T s 2  T s 2 
   1,t  s     2,t  s 
 s 1
 s 1

q1,2,t
q1,1,t q2,2,t
, where
qi , j ,t  1    i ,t 1 j ,t 1    qi , j ,t 1 .
Mean Reverting Rho

Just as in GARCH
t 
q1,2,t
q1,1,t q2,2,t
, where
qi , j ,t  i , j 1         i ,t 1 j ,t 1    qi , j ,t 1 .
Simple Correlation Models
Qt  1   t 1 't 1   Qt 1
Qt  R 1        t 1 't 1    Qt 1
and
Rt  (Qt )
* 1/ 2
Qt (Qt )
* 1/ 2
Qt *  diag Qt 
Tse and Tsui(2002)
A closely related model for modeling
correlations directly as a weighted average of
three correlation matrices.

Rt  R 1  1   2   1rˆt k1   2 Rt 1 ,
1

ˆrt  diag    s s '
 k s t  k

t
k
1/ 2
1

1

 k   s s ' diag  k   s s '
 s t  k

 s t  k

t
t
1/ 2
Higher Order Models
Engle and Sheppard(2002), Theoretical and
Empirical Properties of Dynamic Conditional
Correlation Multivariate GARCH
 Higher order DCC are estimated
 Applied to 100 S&P industry sectors
 Applied to 30 Dow Stocks
Higher Order DCC

Define DCC(p,q) as
p
q
i 1
i 1
Qt  R  i t i  't i  R    i Qt i  R 

For most of the data sets, DCC(1,1) was adequate.
Generalized DCC

Add parameters for each asset
Qt  R  A  t 1 't 1  R   B  Qt 1  R 


Where A and B are square, symmetric, and  is
Hadamard product
If A,B and (ii’-A-B) are p.s.d and R is p.d., then Q is
p.d. See Ding and Engle(2001)
Diagonal Generalized DCC

Choose a parameterization for A, B.
A   ', B   '

So that for any W
A W  diag  W diag 

Hence for any i and j
Qi , j ,t 1  i , j  i j  i ,t j ,t  i , j    i  j Qi , j ,t  i , j 
Asymmetric DCC



Response to two negative returns is different from
overall response.
Define
1 T
t  min  t ,0  , N  tt '
T t 1
Asymmetry can be introduced with terms that are zero
except when both returns are negative such as:
i ,t j ,t

Or more generally (and averaging to zero):
G
  ' N 
t t
Asymmetric Generalized DCC

The Asymmetric Generalized DCC can
be expressed
Qt  R  A  t1 't1  R   B  Qt1  R   G  
t t ' N 

Qi , j ,t 1  i , j
And assuming a diagonal structure for
A,B and G, the typical equation
becomes
             Q          N 
i
j
i ,t
j ,t
i, j
i
j
i , j ,t
i, j
i
j
i ,t
j ,t
i, j
Log Likelihood
1
L     log(2 )  log H t  rt ' H t1rt 
2 t
1
   log(2 )  log Dt Rt Dt  rt ' Dt1Rt1Dt1rt
2 t
1
    log(2 )  2log Dt  log Rt  t ' Rt1t 
2 t
1
    log(2 )  2log Dt  rt ' Dt2rt  rt ' Dt2rt  log Rt  t ' Rt1t 
2 t


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
ECONOMETRIC QUESTIONS


With non-normal data, are these QMLE
estimators?
How can we construct asymptotically
consistent standard errors?


Let parameters in GARCH be  and call the
likelihood function QL1
Let parameters in DCC be  and call the second
part of the likelihood function QL2
QL( , )  QL1    QL2  , 
Quasi Likelihood
QL  QL1  QL2

  log(2 )  2log Dt  rt ' Dt rt
t
2
  log Rt  t ' Rt t  rt ' Dt rt
1
t
2

GENERAL RESULT


Bollerslev and Wooldridge(1992) show that
any multivariate GARCH model that is
correctly specified in the first two moments,
and satisfies a bunch of regularity conditions,
will be a QMLE estimator.
However this does not imply that two step
estimation is consistent.
Two Step Estimators
See Newey and McFadden(1994) pp.2176-2184



Suppose there are two sets of parameters (, ) that
are have no relation, i.e. are “variation free”.
Let there be k1 moment conditions g1() and k2
moment conditions g2(, ), where these conform
with the number of parameters in (, ) .
Consider the GMM estimation min g’Wg with
g=(g1,g2)’.


It is just identified so W=I is no restriction.
It will be a two step estimator since  will solve only the
first set of moments and  will solve the second using the
first estimate of .
GENERAL GMM RESULTS
1
ˆ  arg min  gT   ' gT    , gT 
T
 
If
g
t
uniformly in p
gT   
 g 0  
g 0   =0 is uniquely solved by  0 ,
gT
GT 

  0
p

 G 0 non-singular,
D
T gT  0  
 N  0,   ,
and some regularity conditions,
then
D
T ˆ   0  
 N  0, G0 1G0 1  ,
KEY ASSUMPTION


The first stage is consistent even if not MLE!
That is GARCH models estimated individually
are consistent but inefficient when all
parameters are variation free.
1
QL1,T     log(2 )  2 log Dt  rt ' Dt2 rt 
2 t
1
g1,T  
   log(2 )  2 log Dt  rt ' Dt2 rt 
2T
t
2
k ,t
r
1

  log  hk ,t  
2T
hk ,t
k
t
For Two Step

Given consistency for all parameters
  g1
G
  g 2
 g
1


T
 g1,t g 2,t
2
1,t
0 

 g 2 
g1,t g 2,t  p

 
2
g 2,t 
 ˆ  0  D
1
1
T
 N  0, G G 
 
 ˆ   
0

APPLICATION TO VARIANCE
TARGETING

Variance Targeting was proposed by Engle and
Mezrich(1995) to constrain an ARCH model to have
a prespecified long run variance. Typically this was
the sample variance.
ht   1        y   ht 1
2


2
t 1
It is easily seen that the long run variance forecast
from this model is 2
There are only two parameters to estimate rather than
three.
ECONOMETRIC ISSUES

Using the sample variance in this model gives
ht  ˆ 1        y
2



2
t 1
1
  ht 1 , ˆ   yt2
T
2
This is not a Maximum Likelihood estimator of
GARCH(1,1) and therefore is asymptotically
inefficient
It will be consistent because the sample variance is
consistent in a wide range of models.
If GARCH is misspecified, the long run variance will
still be consistent.
Univariate Variance Targeting

The first set of moments come from the
likelihood and the second from the sample
variance.
ht   1  1   2    y
2
1 t 1
  2 ht 1
2
 
yt  
1    log  ht   h  
g   
t 
T t 

2

y



 
t

Two Step DCC

One set of 3n moments for the variance
models, , and one set for the correlations, .
hi ,t  i ,0   y
2
i ,1 i ,t 1
 i ,2 hi ,t 1
2
 

yi ,t 
1    log  hi ,t   h  ,  i 
g   
i ,t 

T t 


1
   log Rt   t ' Rt  t  


Three Step DCC
or DCC with Correlation Targeting

One set of 3n moments for variances,
n(n-1)/2 for unconditional correlations, and
two for the correlation process.
2
 

yi ,t 
   log  hi ,t  
 ,  i 
hi ,t 

1  
g  

T t   i ,t  j ,t   i , j ,  i  j 
  log R   ' R 1 
t
t
t
t 
 
BEYOND LINEAR
DEPENDENCE
JOINT DISTRIBUTIONS


Dependence properties are all summarized by
a joint distribution
For a vector of kx1 random variables Y with
cumulative distribution function F
F  y1,..., yk   P Y1  y1,...,Yk  yk 

Assuming for simplicity that it is continuously
differentiable, then the density function is:
k F  y 
f  y1,..., yk  
y1...yk
UNIVARIATE PROPERTIES

For any joint distribution function F, there are
univariate distributions Fi and densities fi
defined by:
Fi  yi   P Yi  yi   F  ,.., , yi , ,...,  
Fi
fi  yi  
yi


Ui  Fi Yi  is a uniform random variable on the
interval (0,1)
What is the joint distribution of
U  U1,...,Uk 
COPULA

The joint distribution of these uniform random
variables is called a copula;



it only depends on ranks and
is invariant to monotonic transformations.
U  U1,...,Uk  ~ C u1,.., uk 
Equivalently
F  y   C  F1  y1  ,..., Fk  yk  
C  u   F  F11  u1  ,..., Fk1  uk  
COPULA DENSITY


Again assuming continuous differentiability,
the copula density is
kC u 
c u  
u1 ,...uk
From the chain rule or change of variable rule,
the joint density is the product of the copula
density and the marginal densities
f  y   c u  f1  y1  f2  y2  ... fk  yk 
BIVARIATE DEPENDENCE
MEASURES

Pearson or simple correlation


E Y1Y2   E Y1  E Y2 
 E Y12   E Y1 2   E Y22   E Y2 2 



Will be sensitive to monotonic transformations
of the data, I.e. to the marginal densities as
well as the copula
Invariant Measures

Kendall’s Tau: For a bivariate vector Y,
let Y1, Y2  and Y1 ', Y2 ' be independent observations
  P Y1  Y1 ' Y2  Y2 '   0  P Y1  Y1 ' Y2  Y2 '   0


τ depends only on the ranks, ie on the copula
Spearman or rank correlation
E U1U 2   E U1  E U 2 
S 
 E U12   E U1 2   E U 22   E U 2 2 



 12 E U1U 2   3
A NEW ESTIMATOR
RANK-DCC




A dynamic correlation estimator can be
constructed based only on the order statistics
of the data
First create standardized residuals (?) then rank
them
Build a DCC model based on rank data
Estimator is less sensitive to outliers but pretty
similar to cardinal DCC.
TAIL DEPENDENCE


When one variable is extreme, will another be
also extreme?
Upper tail dependence is
U  lim P U1  u U2  u   lim P U 2  u U1  u 
u1
 lim C  u, u  / 1  u 
u1
u1

Lower tail dependence is
L  lim C  u, u  / u
u 0
Values of Tail Dependence




Tail dependence is a probability and must be
between zero and one
For joint normal distributions: U  L  0
For other copulas one or both may be nonzero.
It is interesting if lower tail is more dependent
than upper
What is the Distribution of DCC?


To focus on the comovements, let volatilities be
constant and normalized to 1.
Consider bivariate distribution of (y,x)



Conditional correlations are changing but there is only one
unconditional correlation.
This is therefore not a multivariate normal
This is a mixture of normals


with standard normal marginals
With same covariance, on average.
Simulation



100,000 observations
N
DCC
  1 .6  
 y
 x  ~ N  0,  .6 1  
 

 
qx , y ,t  .6  .1 yt 1 xt 1  .6   .85  qx , y ,t 1  .6  , rx , y ,t  qx , y ,t / qx , x ,t q y , y ,t

ADCC
qx , y ,t  .6  .2  yt 1 xt 1d t 1  .6   .85  qx , y ,t 1  .6  , rx , y ,t  qx , y ,t / qx , x ,t q y , y ,t
d  1 if y  0 and x  0
Tail Index
(.90 to .999 quantiles, 100000 reps.)
.40
.35
.30
.25
.20
.15
.10
.05
25
LTINDEXDCC1
LTINDEXN1
50
75
UTINDEXDCC1
UTINDEXN1
100
Discussion




Small increase in tail correlations
Very little evidence of non-zero tail index
Still need to develop standard errors.
Similar results for ADCC
Time Aggregation

Multiperiod correlations include new interesting
effects





They are not individually normal
A large comovement leads to large correlations and a
subsequently large comovement.
Expect two period aggregates to show this
ADCC should show this especially in lower tail
Average upper and lower tail for DCC as it is
symmetric
Symmetric Tail Dependence
P2,T
P1,T
Lower Tail Dependence
P2,T
P1,T
P2,T
K1
Put Option
on asset 1
Pays
P1,T
Both options
Payoff
Option on
asset 2
Pays
K2
P2,T
K1
Put Option
on asset 1
Pays
P1,T
Both options
Payoff
Option on
asset 2
Pays
K2
TWO PERIOD RETURNS



Two period return is the
sum of two one period
continuously
compounded returns
Look at binomial tree
version
Asymmetry gives
negative skewness
Low
variance
High
variance
Two period Joint Returns


If returns are both
negative in the first
period, then correlations
are higher.
This leads to lower tail
dependence
Up Market
Down Market
Time Aggregated DCC Tail Index
.45
.40
.35
.30
.25
.20
.15
.10
.05
25
50
75
.5*(LTINDEXDCC1+UTINDEXDCC1)
.5*(LTINDEXDCC2+UTINDEXDCC2)
.5*(LTINDEXDCC5+UTINDEXDCC5)
100
Time Aggregated Tail Index for
ADCC
.6
.5
.4
.3
.2
.1
25
50
LTINDEXA DCC1
LTINDEXA DCC2
LTINDEXA DCC5
75
UTINDEXA DCC1
UTINDEXA DCC2
UTINDEXA DCC5
100
SOME RESULTS FOR EQUITY
RETURNS – DOW STOCKS


For 1992-2002 take 10 years of equity returns
from the 30 current Dow Jones Stocks.
Calculate Tail correlations and Tail indexes for
several pairs and their time aggregates.
1.2
1.0
0.8
0.6
0.4
0.2
0.0
25
50
LTINDEXAAUTX1
LTINDEXAAUTX5
75
UTINDEXAAUTX1
UTINDEXAAUTX5
100
1.2
1.0
0.8
0.6
0.4
0.2
0.0
25
LTINDEXCITIJPM1
LTINDEXCITIJPM5
50
75
UTINDEXCITIJPM1
UTINDEXCITIJPM5
100
1.2
1.0
0.8
0.6
0.4
0.2
0.0
25
UTINDEXIBMMSFT1
UTINDEXIBMMSFT5
50
75
100
LTINDEXIBMMSFT1
LTINDEXIBMMSFT5
1.2
1.0
0.8
0.6
0.4
0.2
0.0
25
LTINDEXDISKO1
LTINDEXDISKO5
50
75
UTINDEXDISKO1
UTINDEXDISKO5
100
FINDINGS



DCC PROVIDES A FLEXIBLE APPROACH
TO CORRELATION ESTIMATION
ASYMPTOTIC STANDARD ERRORS CAN
BE CONSTRUCTED FOR THE TWO AND
THREE STEP ESTIMATORS
TIME AGGREGATED SIMULATED ADCC
AND REAL DATA SHOW HIGHER TAIL
DEPENDENCE, PARTICULARLY IN
LOWER TAIL
Data



Weekly $ returns Jan 1987 to Feb 2002 (785
observations)
21 Country Equity Series from FTSE AllWorld Index
13 Datastream Benchmark Bond Indices with
5 years average maturity
Europe
AUSTRIA*
BELGIUM*
DENMARK*
FRANCE*
GERMANY*
IRELAND*
ITALY
THE NETHERLANDS*
SPAIN
SWEDEN*
SWITZERLAND*
NORWAY
UNITED KINGDOM*
Australasia
AUSTRALIA
HONG KONG
JAPAN*
NEW ZEALAND
SINGAPORE
Americas
CANADA*
MEXICO
UNITED STATES*
GARCH Models
(asymmetric in orange)









GARCH
AVGARCH
NGARCH
EGARCH
ZGARCH
GJR-GARCH
APARCH
AGARCH
NAGARCH









3EQ,8BOND
0
1BOND
6EQ,1BOND
8EQ,1BOND
3EQ,1BOND
0
1EQ,1BOND
0
Parameters of DCC
Asymmetry in red (gamma) and
Symmetry in blue (alpha)
0.02
0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
Au
s
Be
De
Ge
Ire
Ja
Ne
N
S
S
U
B
D
G
J
S
the orw pain wit z nite e lgi e nm erm a pa n wed
tra lgiu nma rma land pa n
u
e
a
d
en
an
rla
S
m
y
r
S
B
n
lia
r
Kin m B ark
Bo
Sto Sto k Sto y St Stoc tock nds St oc tock land
B o y Bo ond
o
g
nd
n
d
o
s
k
s
c
S
n
n
s
S
k
ck
c
d
o
c
s
k
t
d
d
s
s
s
toc
ks
ks
oc
m
s
s
s
s
ks
Sto
ks
ck
s
CORRELATIONS OF
VOLATILITIES

EQUITIES = .32


EUROPEAN = .55
BONDS = .35

WITHIN EMU = .79
BEHAVIOR DURING US BEAR
MARKETS
RESULTS




Asymmetric Correlations – correlations rise in
down markets
Shift in level of correlations with formation of
Euro
Equity Correlations are rising not just within
EMU-Globalization?
EMU Bond correlations are especially highothers are also rising