NEW MODELS FOR HIGH AND LOW FREQUENCY VOLATILITY Robert Engle NYU Salomon Center Derivatives Research Project Derivatives Research Project

FORECASTING WITH GARCH

DJ RETURNS

DOW JONES SINCE 1990 Dependent Variable: DJRET Method: ML - ARCH (Marquardt) - Normal distribution Date: 01/13/05 Time: 14:30 Sample: Included observations: 3789 Convergence achieved after 14 iterations Variance backcast: ON GARCH = C(2) + C(3)*RESID(-1)^2 + C(4)*GARCH(-1) CoefficientStd. Errorz-StatisticProb. C Variance Equation C9.89E E RESID(-1)^ GARCH(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat

DEFINITIONS  r t is a mean zero random variable measuring the return on a financial asset  CONDITIONAL VARIANCE  UNCONDITIONAL VARIANCE

GARCH(1,1)  The unconditional variance is then 

GARCH(1,1)  If omega is slowly varying, then  This is a complicated expression to interpret

SPLINE GARCH  Instead, use a multiplicative form  Tau is a function of time and exogenous variables

UNCONDITIONAL VOLATILTIY  Taking unconditional expectations  Thus we can interpret tau as the unconditional variance.

SPLINE  ASSUME UNCONDITIONAL VARIANCE IS AN EXPONENTIAL QUADRATIC SPLINE OF TIME  For K knots equally spaced

ESTIMATION  FOR A GIVEN K, USE GAUSSIAN MLE  CHOOSE K TO MINIMIZE BIC FOR K LESS THAN OR EQUAL TO 15

EXAMPLES FOR US SP500  DAILY DATA FROM 1963 THROUGH 2004  ESTIMATE WITH 1 TO 15 KNOTS  OPTIMAL NUMBER IS 7

RESULTS LogL: SPGARCH Method: Maximum Likelihood (Marquardt) Date: 08/04/04 Time: 16:32 Sample: Included observations: Evaluation order: By observation Convergence achieved after 19 iterations CoefficientStd. Errorz-StatisticProb. C(4) E W(1)-1.89E E W(2)2.71E E W(3)-4.35E E W(4)3.28E E W(5)-3.98E E W(6)6.00E E W(7)-8.04E E C(5) C(1) C(2) Log likelihood Akaike info criterion Avg. log likelihood Schwarz criterion Number of Coefs.11 Hannan-Quinn criter

ESTIMATION  Volatility is regressed against explanatory variables with observations for countries and years.  Within a country residuals are auto- correlated due to spline smoothing. Hence use SUR.  Volatility responds to global news so there is a time dummy for each year.  Unbalanced panel

ONE VARIABLE REGRESSIONS

MULTIPLE REGRESSIONS

IMPLICATIONS  Unconditional volatility varies over time and can be modeled  Volatility mean reverts to the level of unconditional volatility  Long run volatility forecasts depend upon macroeconomic and financial fundamentals

HIGH FREQUENCY VOLATILITY

WHERE CAN WE GET IMPROVED ACCURACY?  USING ONLY CLOSING PRICES IGNORES THE PROCESS WITHIN THE DAY.  BUT THERE ARE MANY COMPLICATIONS. HOW CAN WE USE THIS?

ONE MONTH OF DAILY RETURNS

INTRA-DAILY RETURNS

ARE THESE DAYS THE SAME?

CAN WE USE THIS INFORMATION TO MEASURE VOLATILITY BETTER?  DAILY HIGH AND LOW  DAILY REALIZED VOLATILITY

PARKINSON(1980)  HIGH LOW ESTIMATOR  IF RETURNS ARE CONTINUOUS AND NORMAL WITH CONSTANT VARIANCE,

TARCH MODEL WITH RANGE  C1.07E E  RESID(-1)^  RESID(-1)^2*(RESID(-1)<0)  GARCH(-1)  RANGE(-1)^   Adjusted R-squared S.D. dependent var  S.E. of regression Akaike info criterion  Sum squared resid Schwarz criterion  Log likelihood Durbin-Watson stat

Robert F. Engle Giampiero M. Gallo A MULTIPLE INDICATOR MODEL FOR VOLATILITY USING INTRA-DAILY DATA Robert F. Engle Giampiero M. Gallo Forthcoming, Journal of Econometrics

Absolute returns Insert asymmetric effects (sign of returns) Insert asymmetric effects (sign of returns) Insert other lagged indicators Insert other lagged indicators

Repeat for daily range, hl t : And for realized daily volatility, dv t :

Smallest BIC-based selection

Forecasting one step-ahead one step-ahead multi-step-ahead multi-step-ahead

Term Structure of Volatility 1

IMPLICATIONS  Intradaily data can be used to improve volatility forecasts  Both long and short run forecasts can be implemented if all the volatility indicators are modeled  Daily high/low range is a particularly valuable input  These methods could be combined with the spline garch approach.