1 forecasting correlation and covariance with a range-based dynamic conditional correlation model...

1

Forecasting Correlation and Covariance with a Range-Based Dynamic

Conditional Correlation Model

Ray Y Chou Institute of Economics Academia Sinica

Nathan Liu National Chiao Tung University

Chun-Chou Wu Chung Yuan University

2

Outline

Introduction

Dynamic Conditional Correlation (DCC) Model

Methodology

Data Analysis

Empirical Analysis

Conclusion

3

Introduction

It is of primary importance in the practice of portfolio management, asset allocation and risk management to have an accurate estimate of the covariance matrices for asset prices.

The univariate ARCH/GARCH family of models provides effective tools to estimate the volatilities of individual asset prices, see Bollerslev, Chou and Kroner (1992), Engle (2004).

It is an active research issue in estimating the covariance or correlation matrices of multiple, including VECH model of Bollerslev, Engle and Wooldridge (1988), BEKK model of Engle and Kroner (1995) and the constant correlation model of Bollerslev (1990) among others.

4

Introduction (Cont.)

The constant correlation model is too restrictive as it precludes the dynamic structure of covariance to be completely determined by the individual volatilities.

The VECH and the BEKK models are more flexible in allowing time-varying correlations. Nevertheless the computational difficulty of the estimation of the model parameters has limited the application of these models to only small number of assets.

Engle and Sheppard (2001), Engle (2002a) and Engle, Cappiello and Sheppard (2003) provide a solution to this problem by using a model entitled the Dynamic Conditional Correlation Multivariate GARCH (henceforth DCC).

5

Introduction (Cont.)

Other estimation methods of time-varying correlation are proposed by Tsay (2002) and by Tse and Tsui (2002).

there is a growing awareness of the fact that the high/low range data of asset prices can provide sharper estimates and forecasts than the return data based on close-to-close prices. Studies of suppoting evidences include Parkinson (1980), Garman and Klass (1980), Wiggins (1991), Rogers and Satchell (1991), Kunitomo (1992), and more recently Gallant, Hsu and Tauchen (1999), Yang and Zhang (2000), Alizadeh, Brandt and Diebold (2001), Brandt and Jones (2002), Chou, Liu and Wu (2003) and Chou (2005a, 2005b).

6

Conditional Covariance/Correlation

)( ,2,11,12 tttt rrECOV

1 1, 2,12, 2 2

1 1, 1 2,

( )

( ) ( )

t t tt

t t t t

E r r

E r E r

Traditional Conditional Covariance and Correlation

Moving Average with a 100-week window, MA(100)

1

100,2,1,12 100

1 t

tssst rrCOV

))((

ˆ1

100

2,2

1

100

2,1

1

100,2,1

t

tss

t

tss

t

tsss

t

rr

rr

7

Conditional Covariance/Correlation

Exponentially-Weighted Moving Average (EWMA) model where the weights decrease exponentially

As to the coefficient is usually called the exponential smoother in this model by RiskMetrics.

1

1,2,1

1,12 )1(

t

sss

stt rrCOV

))((

ˆ1

1

2,2

11

1

2,1

1

1

1,2,1

1

t

ss

stt

ss

st

t

sss

st

t

rr

rr

8

Dynamic Conditional Correlation model

The multivariate GARCH model proposed assumes that returns from k assets are conditionally multivariate normal with zero expected value and covariance matrix Ht.

where Dt is the kxk diagonal matrix of time varying standard deviations from univariate volatility models with the square root of on the diagonal, and is the time varying correlation matrix.

),0(~1 ttt HNIr

tttt DRDH

ith tR tR

9

Dynamic Conditional Correlation model (Cont.)

It is important to recognize that although the dynamic of the Dt matrix has usually been structured as a standard univariate GARCH, it can take many other forms.

The standardized series is , hence

are the residuals standardized by their conditional standard deviation.

A general dynamic correlation structure proposed by Engle (2000a) is:

ttt rDZ 1

);0(~ tt RNZ

111 ')'( tttt QBZZABASQ 2/12/1 }{}{ tttt QdiagQQdiagR

10


where S is the unconditional covariance of the standardized residuals. where i is a vector of ones and o is the Hadamard product of two identically sized matrices which is computed simply by element by element multiplication. It’s shown that if A , B, and (ii’-A-B) are positive semi-definite, then Q will be positive semi-definite.

The typical element of Rt will be of the form .

Engle and Sheppard (2001) show the necessary conditions for Rt to be positive definite and hence a correlation matrix with a real, symmetric positive semi-definite matrix, with ones on the diagonal.

jtit

ijt

qqq

11


The loglikelihood of this estimator can be written:

Let the parameters in Dt be denoted and the additional parameters in Rt be denoted .

T

tttttt

T

ttttttttt

T

ttttt

ZRZRDk

rDRDrDRDk

rHrHkL

1

1

1

111

1

1

'loglog2)2log(2

1

'log)2log(2

1

'log)2log(2

1

12


The log likelihood can written as the sum of a volatility part and a correlation part.

The volatility term is:

and the correlation component is:

),(, CV LLL

t

ttttV rDrDnL 22'log)2log(

2

1

tttttttC ZZZRZRL ''log

2

1, 1

13


The volatility part of the likelihood is the sum of individual GARCH likelihood if Dt is determined by a GARCH specification

which can be jointly maximized by separately maximizing each term. If Dt is determined by a CARR specification,

where is the conditional standard deviation as computed from a scaled expected range from the CARR model.

t

k

i ti

titiV h

rhL

1 ,

2,

, )log()2log(2

1

t

k

i ti

titiV

rL

12*,

2,*

, )log(2)2log(2

1

*,ti

14


The second part of the likelihood will be used to estimate the correlation parameters. As the squared residuals are not dependent on these parameters, they will not enter the first order conditions and can be ignored.

The two step approach to maximizing the likelihood is to find

and then take this value as given in the second stage,

VLmaxargˆ

,ˆmax CL

15

Methodology

For bivariate cases, we use the following GARCH and CARR model to perform the first step procedure in estimating the volatility parameters:

GARCH （ return-based conditional volatility model ） ,k=1,2

CARR （ range-based conditional volatility model ） ,k=1,2 ,where ,

tktkr ,, ),0(~| ,1, tkttk hNI

1,2,, tkkitkkktk hh

tktka

tk hrz ,,, /

tktk u ,, );1exp(~| 1, ttk Iu

1,1,, tkktkkktk *

,,, / tktkc

tk rz tkktk adj ,*

, k

kadj

ˆ

16

Methodology (Cont.)

In estimating the correlation parameters, we also allow two specifications of the dynamic structures in the correlation matrix.

MR_DCC (mean-reverting)

111 ')'( tttt QBZZABASQ

2

1,21,21,1

1,21,12

1,1

23

31

2233

3311

12

12

,2,12

,12,1

11

11

1

1

ttt

ttt

tt

tt

zzz

zzz

aa

aa

baba

baba

q

q

qq

qq

T

ttt zz

Tq

1,2,112

1

1,21,12

1,121,1

23

31

tt

tt

qq

qq

bb

bb

17

Methodology (Cont.)

I_DCC (integrated)

111 )1( tttt QAZZAQ

1,21,12

1,121,1

23

31

21,21,21,1

1,21,12

1,1

23

31

,2,12

,12,1

11

11

tt

tt

ttt

ttt

tt

tt

qq

qq

aa

aa

zzz

zzz

aa

aa

qq

qq

18

Methodology (Cont.)

The estimated conditional correlation coefficients are

Like volatilities, the correlation and covariance matrices are also unobservable. We use daily data to construct weekly realized covariance/correlation observations.

The purpose of this is to use these “measured covariance/ correlation” denoted MCOV/MCORR respectively as benchmarks in determining the relative performance of the return-based DCC model and the range-based DCC models.

tttt qqq ,2,1,12,12 /

19

Methodology (Cont.)

We perform the Mincer-Zarnowitz regressions for the in-sample comparison ：

where

treturnttMCORR ,110 ˆ

trangettMCORR ,220 ˆ

tranget

returnttMCORR ,3210 ˆˆ

treturn

tt COVMCOV ,110

trange

tt COVMCOV ,220

trange

treturn

tt COVCOVMCOV ,3210

ttttCOV ,2,1 ˆˆˆ

20

Methodology (Cont.)

Following Engle and Sheppard (2001), the out-of-sample conditional correlations MR_DCC are computed.

For T+h, with , the correlation is

2,2,2,1

,2,12,1

23

31

2233

3311

12

12

1,21,12

1,121,1

11

11

1

1

TTT

TTT

TT

TT

zzz

zzz

aa

aa

baba

baba

q

q

qq

qq

TT

TT

qq

qq

bb

bb

,2,12

,12,1

23

31

1,21,11,121 / TTTT qqq

2233

3311

12

12

,2,12

,12,1

11

11

1

1

baba

baba

q

q

qq

qq

hThT

hThT

1,21,12

1,121,1

2233

3311

hThT

hThT

qq

qq

baba

baba

21

Data Analysis

The data employed in this study comparises 835 weekly observations on the S&P 500 Composite (henceforth S&P 500), NASDAQ stock market index and bond yield for 10 years time to maturities spanning the period 4 January 1988 to 2 January 2004. Two variant DCC structures to estimate the model’s significance are investigated.

Let MCORRt and MCOVt represent the realized multi-correlation coefficients and multi-covariances, respectively. The MCORRt is defined as:

where symbol the tradable days during the t week

1

1

i

ittMCORR

22

Data Analysis (Cont.)

is called the correlation coefficient at the i trading day of the week based on two indices that we have selected. We can use the daily returns data and fit them into the MR_DCC model, so that the can be computed. The multi-covariances (MCOVt) can be expressed as

Where represents the daily returns of k index at the i trading day during the t week.

it

it

121 )(

i

it

itt rrMCOV

iktr

23

Summary Statistics

Index S&P500 Nasdaq 10-year Tbond

Type range return range Type range return

Mean 3.136 0.182 4.204 Mean 3.136 0.182

Median 2.657 0.339 3.225 Median 2.657 0.339

Maximum 14.534 7.492 31.499 Maximum 14.534 7.492

Minimum 0.707 -12.330 0.514 Minimum 0.707 -12.330

Std. Dev. 1.730 2.153 3.224 Std. Dev. 1.730 2.153

Skewness 1.768 -0.458 2.411 Skewness 1.768 -0.458

Kurtosis 8.025 5.733 12.782 Kurtosis 8.025 5.733

Jarque-Bera 1312.227 288.695 4132.758 Jarque-Bera 1312.227 288.695

24

Return- Versus Range-based DCC Models In-sample MCORR Forecasting

Panel A: S&P500 and Nasdaq

MCORR 0̂ 1̂ 2̂ R-squared Return 0.251 (10.425) 0.707 (24.691) 0.409 Range 0.347 (16.937) 0.597 (24.539) 0.389 MR _DCC

BOTH 0.257 (10.524) 0.445 (7.435) 0.258 (5.277) 0.425 Return 0.250 (7.806) 0.705 (18.221) 0.316 Range 0.280 (10.676) 0.672 (21.554) 0.350 I_DCC

BOTH 0.171 (5.689) 0.363 (6.545) 0.440 (10.122) 0.392 MA100 0.259 (9.421) 0.699 (21.22) 0.330

Exp. Smoothing 0.375 (20.745) 0.555 (25.732) 0.489

Panel B: S&P500 and Tbond Return 0.030 (4.524) 0.848 (57.328) 0.777 Range 0.023 (3.789) 0.860 (70.081) 0.800 MR _DCC

BOTH 0.021 (3.621) -0.399 (-3.513) 1.256 (11.066) 0.803 Return 0.023 (3.539) 0.965 (59.105) 0.794 Range 0.007 (1.164) 0.945 (72.435) 0.811 I_DCC

BOTH 0.008 (1.458) 0.128 (1.161) 0.822 (7.730) 0.811 MA100 -0.017 (-1.704) 0.890 (36.625) 0.587

Exp. Smoothing 0.025 (3.947) 0.846 (67.370) 0.792

Panel C: Nasdaq and Tbond Return 0.026 (5.551) 0.851 (57.019) 0.777 Range 0.029 (6.574) 0.818 (67.500) 0.816 MR _DCC


BOTH 0.022 (4.970) 0.351 (3.944) 0.572 (6.777) 0.807 MA100 -0.017 (-2.707) 0.871 (42.423) 0.658

Exp. Smoothing 0.030 (6.283) 0.758 (62.599) 0.776

25

Correlation fitting (S&P500 and Bond)

-.2

-.1

.0

.1

.2

.3

.4

.5

.6

.7

25 50 75 100 125 150 175 200

CORR_SP_BONDRHO_MR_ARHO_MR_C

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The Time-Varying Correlation

27

The Time-Varying Correlation

28

Return- Versus Range-based DCC Models In-sample MCOV Forecasting

Panel A: S&P500 and Nasdaq MCOV 0 1 2 R-squared

Return 0.753 (1.968) 0.939 (10.178) 0.223 Range 0.339 (0.750) 0.867 (9.438) 0.354 MR _DCC


BOTH 0.693 (1.836) -0.186 (-1.206) 0.961 (5.782) 0.354 MA100 0.788 (1.906) 1.011 (10.562) 0.210

Exp. Smooth 1.303 (3.577) 0.862 (9.855) 0.200 CCC 0.669 (1.690) 0.972 (9.994) 0.225

Panel B: S&P500 and Tbond Return -0.239 (-1.654) 0.960 (10.411) 0.230 Range -0.086 (-0.676) 0.860 (10.212) 0.309 MR _DCC

BOTH 0.007 (0.056) -0.559 (-2.025) 1.262 (4.864) 0.320 Return -0.298 (-2.014) 1.102 (10.850) 0.231 Range -0.199 (-1.489) 0.964 (10.312) 0.309 I_DCC

BOTH -0.148 (-1.139) -0.428 (-1.517) 1.262 (4.775) 0.315 MA100 -0.485 (-2.843) 1.070 (8.285) 0.177

Exp. Smooth -0.203 (-1.431) 0.945 (10.040) 0.227 CCC 2.527 (9.138) -3.982 (0.103) 0.103

Panel C: Nasdaq and Tbond Return -0.280 (-1.633) 0.934 (9.682) 0.163 Range -0.025 (-0.149) 0.702 (9.749) 0.220 MR _DCC

BOTH 0.007 (0.043) -0.301 (-1.157) 0.879 (4.418) 0.222 Return -0.264 (-1.556) 1.064 (9.921) 0.168 Range -0.171 (-1.004) 0.782 (9.847) 0.218 I_DCC

BOTH -0.174 (-1.030) -0.107 (-0.412) 0.844 (4.259) 0.218 MA100 -0.477 (-2.317) 0.955 (8.723) 0.152

Exp. Smooth -0.280 (-1.661) 0.814 (9.338) 0.161 CCC 1.696 (5.711) -14.546 (-5.789) 0.078

29

Return- versus Range-based DCC Models Out-of-Sample MCORR Forecasting


R-squared MR_DCC

Forecast horizon return-based range-based MA100

1 0.310 0.548 0.044 2 0.279 0.474 0.076

3 0.248 0.407 0.109

4 0.199 0.325 0.153

Panel B: S&P500 and Tbond

R-squared MR_DCC


1 0.491 0.631 0.165

2 0.441 0.547 0.134

3 0.306 0.463 0.098

4 0.229 0.388 0.065

Panel C: Nasdaq and Tbond

R-squared MR_DCC


1 0.593 0.786 0.443

2 0.673 0.756 0.397

3 0.705 0.706 0.348

4 0.719 0.654 0.299

30

Return- versus Range-based DCC Models Out-of-Sample MCOV Forecasting


R-squared MR_DCC

Forecast horizon return-based range-based MA(100) CCC

1 0.000 0.115 0.000 0.004

2 0.003 0.081 0.004 0.000

3 0.008 0.043 0.008 0.003

4 0.018 0.015 0.018 0.011

Panel B: S&P500 and Tbond

R-squared MR_DCC


1 0.001 0.136 0.001 0.006 2 0.000 0.070 0.000 0.010

3 0.000 0.079 0.002 0.004

4 0.007 0.046 0.006 0.006

Panel C: Nasdaq and Tbond

R-squared MR_DCC


1 0.009 0.114 0.053 0.003

2 0.002 0.083 0.032 0.007

3 0.004 0.066 0.032 0.000

4 0.004 0.046 0.025 0.000

31

Conclusion

A new estimator of the time-varying correlation/covariance matrices is proposed utilizing the range data by combining the CARR model of Chou (2005a) and the framework of Engle (2002)’s DCC model.

We find consistent results that the range-based DCC model outperforms the return-based models in estimating and forecasting covariance and correlation matrices, both in-sample and out-of-sample.

It can be applied to larger systems in a manner similar to the application of the return-based DCC model structures that is demonstrated in Engle and Sheppard (2001).

Future research will be useful in adopting more diagnostic statistics or tests based on value at risk calculations as is proposed by Engle and Manganelli (1999).

1 forecasting correlation and covariance with a range-based dynamic conditional correlation model...

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