NEW FRONTIERS FOR ARCH MODELS 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?

1982 ARCH Paper  Weights can be estimated  ARCH(p)

WHAT ABOUT HETEROSKEDASTICITY?

EXPONENTIAL SMOOTHER  Another Simple Model – Weights are declining – No finite cutoff – What is lambda? (Riskmetrics=.06)

The GARCH Model   The variance of r t is a weighted average of three components – a constant or unconditional variance – yesterday’s forecast – yesterday’s news

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

GARCH(1,1) Forecasts

Monotonic Term Structure of Volatility

FORECASTING AVERAGE VOLATILITY  Annualized Vol=square root of 252 times the average daily standard deviation  Assume that returns are uncorrelated.

TWO YEARS TERM STRUCTURE OF PORT

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

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

MORE GARCH MODELS  CONSIDER ONLY SYMMETRIC GARCH MODELS  ESTIMATE ALL MODELS WITH A DECADE OF SP500 ENDING AUG  GARCH(1,1), EGARCH(1,1), COMPONENT GARCH(1,1) ARE FAMILIAR

OLDER GARCH MODELS  Bollerslev-Engle(1986) Power GARCH omega alpha p beta Log likelihood

PARCH  Ding Granger Engle(1993) omega alpha gamma beta Log likelihood

TAYLOR-SCHWERT  Standard deviation model omega alpha beta Log likelihood

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

SQGARCH LogL: SQGARCH Method: Maximum Likelihood (Marquardt) Date: 08/03/01 Time: 19:47 Sample: Included observations: 2927 Evaluation order: By observation Convergence achieved after 12 iterations Coefficient Std. Errorz-StatisticProb. C(1) C(2) C(3) Log likelihood Akaike info criterion Avg. log likelihood Schwarz criterion Number of Coefs.3 Hannan-Quinn criter

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.

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

NLGARCH LogL: NLGARCH Method: Maximum Likelihood (Marquardt) Date: 08/18/01 Time: 11:27 Initial Values: C(2)= , C(4)= , C(1)= Convergence achieved after 32 iterations CoefficientStd. Errorz-StatisticProb. alpha gamma delta Log likelihood Akaike info criterion Avg. log likelihood Schwarz criterion Number of Coefs.3 Hannan-Quinn criter

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

NEWS IMPACT CURVE

Other Asymmetric Models

PARTIALLY NON-PARAMETRIC ENGLE AND NG(1993)

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

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=.055  Mu=0

Long Memory SV  Breidt et al, Hurvich and Deo  d=.47  kappa=.6

Breaking Volatility  Randomly arriving breaks in volatility  mu=-0.5  kappa=1  p=.99