NEW FRONTIERS NEW FRONTIERS FOR ARCH FOR ARCH MODELSMODELS

Prepared for Conference on Volatility Modeling

and Forecasting

Perth, Australia, September 2001

Robert Engle

UCSD and NYU

The First ARCH Model

Rolling Volatility or “Historical” Volatility Estimator

– Weights are equal for j<N– Weights are zero for j>N– What is N?

2

1

1 N

t t jj

h rN

1982 ARCH Paper

Weights can be estimated

ARCH(p)

2

1

p

t j t jj

h r

WHAT ABOUT HETEROSKEDASTICITY?

EXPONENTIAL SMOOTHER

Another Simple Model

– Weights are declining– No finite cutoff– What is lambda? (Riskmetrics= .06)

21 11t t th r h

The GARCH Model

The variance of rt is a weighted average of three components– a constant or unconditional variance– yesterday’s forecast– yesterday’s news

12

1

12

11

tt

tttt

ttt

hr

hhh

hr

T h i s m o d e l c a n b e r e w r i t t e n a s :

1

21

)1( jjt

j

t rh

I t c a n a l s o b e r e w r i t t e n :

121 tttt hhh

T h e f o r e c a s t o f c o n d i t i o n a l v a r i a n c e o n e s t e p a h e a d i s g i v e nb y t h e f i r s t s q u a r e b r a c k e t a n d t h e s u r p r i s e i s g i v e n b y t h es e c o n d . V o l a t i l i t y i s p r e d i c t a b l e b u t n o t p e r f e c t l y .

FORECASTING WITH GARCH

)()() 21

21

21

2ttttt hrhrrr

t

GARCH(1,1) can be written as ARMA(1,1) The autoregressive coefficient is The moving average coefficient is

)(

GARCH(1,1) Forecasts

1

12

1 )()(

kttktt

ttt

hEhE

hrh

Monotonic Term Structure of Volatility

FORECAST PERIOD

FORECASTING AVERAGE VOLATILITY

)(...)(

)(...)(...

1

221

21

ktttt

kttttkttt

hEhE

rErErrE

Annualized Vol=square root of 252 times the average daily standard deviation

Assume that returns are uncorrelated.

TWO YEARS TERM STRUCTURE OF PORT

0.08

0.09

0.10

0.11

0.12

0.13

0.14

2000 2050 2100 2150 2200 2250 2300 2350 2400 2450 2500

TERM2000

0.14

0.16

0.18

0.20

0.22

0.24

2900 2950 3000 3050 3100 3150 3200 3250 3300 3350

TERMEND

0.005

0.010

0.015

0.020

0.025

500 1000 1500 2000

0.162

0.164

0.166

0.168

0.170

0.172

0.174

0.176

0.178

2450 2500 2550 2600 2650 2700 2750 2800 2850 2900

TERMMIL_2411

0.18

0.19

0.20

0.21

0.22

1800 1850 1900 1950 2000 2050 2100 2150 2200 2250 2300

TERMMIL_1800

0.174

0.176

0.178

0.180

0.182

0.184

0.186

0.188

2400 2450 2500 2550 2600 2650 2700 2750 2800 2850

TERMMIL_2357

Variance Targeting

Rewriting the GARCH model

where is easily seen to be the unconditional or long run variance

this parameter can be constrained to be equal to some number such as the sample variance. MLE only estimates the dynamics

)()( 12

11 tttt hhh

)1/(

The Component Model

Engle and Lee(1999) q is long run component and (h-q) is transitory volatility mean reverts to a slowly moving long

run component

)()(

)()(

12

11

1112

1

tttt

tttttt

hrqq

qhqrqh

MORE GARCH MODELS

CONSIDER ONLY SYMMETRIC GARCH MODELS

ESTIMATE ALL MODELS WITH A DECADE OF SP500 ENDING AUG 2 2001

GARCH(1,1), EGARCH(1,1), COMPONENT GARCH(1,1) ARE FAMILIAR

OLDER GARCH MODELS

Bollerslev-Engle(1986) Power GARCH

omega 0.000587 0.002340 0.251006alpha 0.067071 0.009086 7.381861p 1.712818 0.117212 14.61294beta 0.941132 0.005546 169.7078Log likelihood -3739.091

1p

t t th r h

PARCH

Ding Granger Engle(1993)

omega 0.006680 0.001653 4.041563alpha 0.064930 0.005608 11.57887gamma 0.665636 0.082814 8.037719beta 0.941625 0.005211 180.704

Log likelihood -3738.040

21t t th r h

TAYLOR-SCHWERT

Standard deviation model

omega 0.007678 0.001667 4.605529

alpha 0.065232 0.005212 12.51587

beta 0.942517 0.005104 184.6524

Log likelihood -3739.032

1t t th r h

SQ-GARCH MODEL

SQGARCH (Engle and Ishida(2001)) has the property that the variance of the variance is linear in the variance. They establish conditions for positive and stationary variances

21/ 2

1 1 1tt t

t

rh h h

h

SQGARCH

LogL: SQGARCHMethod: Maximum Likelihood (Marquardt)

Date: 08/03/01 Time: 19:47Sample: 2 2928Included observations: 2927Evaluation order: By observationConvergence achieved after 12 iterations

Coefficient Std. Error z-Statistic Prob. C(1) 0.008874 0.001596 5.560236 0.0000C(2) 0.041878 0.003685 11.36383 0.0000C(3) 0.990080 0.001990 497.5850 0.0000Log likelihood -3747.891 Akaike info criterion 2.562960Avg. log likelihood -1.280455 Schwarz criterion 2.569090Number of Coefs. 3 Hannan-Quinn criter. 2.565168

CEV-GARCH MODEL

The elasticity of conditional variance with respect to conditional variance is a parameter to be estimated.

Slight adjustment is needed to ensure positive variance forecasts.

2

1 1 1tt t

t

rh h h

h

NON LINEAR GARCH

THE MODEL IS IGARCH WITHOUT INTERCEPT. HOWEVER, FOR SMALL VARIANCES, IT IS NONLINEAR AND CANNOT IMPLODE

FOR

21t t t t t hh h r h h I

21, (1 )t t th h r h

NLGARCH

LogL: NLGARCH

Method: Maximum Likelihood (Marquardt)

Date: 08/18/01 Time: 11:27

Initial Values: C(2)=0.05464, C(4)=0.00035, C(1)=2.34004

Convergence achieved after 32 iterations

CoefficientStd. Error z-Statistic Prob.

alpha 0.054196 0.004749 11.41158 0.0000

gamma 0.001208 0.001935 0.624299 0.5324

delta 3.194072 4.471226 0.714362 0.4750

Log likelihood -3741.520 Akaike info criterion 2.558606

Avg. log likelihood -1.278278 Schwarz criterion 2.564737

Number of Coefs. 3 Hannan-Quinn criter. 2.560814

Asymmetric Models - The Leverage Effect

Engle and Ng(1993) following Nelson(1989) News Impact Curve relates today’s returns to tomorrows

volatility Define d as a dummy variable which is 1 for down days

112

12

1 ttttt hdrrh

NEWS IMPACT CURVE

NEWS

VOLATILITY

Other Asymmetric Models

E G A R C H : N E L S O N (1 9 8 9 )

1

1

1

11 )log()log(

t

t

t

ttt h

rhr

hh

NGARCH: ENGLE(1990)

1

2

1 )( ttt hrh

PARTIALLY NON-PARAMETRICENGLE AND NG(1993)

NEWS

VOLATILITY

EXOGENOUS VARIABLES IN A GARCH MODEL

Include predetermined variables into the variance equation

Easy to estimate and forecast one step Multi-step forecasting is difficult Timing may not be right

112

1 tttt zhrh

EXAMPLES

Non-linear effects Deterministic Effects News from other markets

– Heat waves vs. Meteor Showers– Other assets– Implied Volatilities– Index volatility

MacroVariables or Events

STOCHASTIC VOLATILITY MODELS

Easy to simulate models Easy to calculate realized volatility Difficult to summarize past information set How to define innovation

SV MODELS

Taylor(1982)

beta=.997 kappa=.055Mu=0

1log logt t t

t t t

r

Long Memory SV

Breidt et al, Hurvich and Deo

d=.47 kappa=.6

1 log

t t t

d

t t

r

B

Breaking Volatility

Randomly arriving breaks in volatility

mu=-0.5 kappa=1 p=.99

1loglog

t t t

tt

r

with probability p

otherwise