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

2. FORECASTING WITH GARCH

3. DJ RETURNS

4. DOW JONES SINCE 1990 Dependent Variable: DJRET Method: ML - ARCH (Marquardt) - Normal distribution Date: 01/13/05 Time: 14:30 Sample: 15362 19150 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. C0.0005520.0001354.0934780.0000 Variance Equation C9.89E-071.84E-075.3809130.0000 RESID(-1)^20.0664090.00447814.828440.0000 GARCH(-1)0.9249120.005719161.73650.0000 R-squared-0.000370 Mean dependent var0.000356 Adjusted R-squared-0.001163 S.D. dependent var0.010194 S.E. of regression0.010200 Akaike info criterion-6.557778 Sum squared resid0.393815 Schwarz criterion-6.551191 Log likelihood12427.71 Durbin-Watson stat1.985498

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

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

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

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

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

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

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

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

17. RESULTS LogL: SPGARCH Method: Maximum Likelihood (Marquardt) Date: 08/04/04 Time: 16:32 Sample: 1 12455 Included observations: 12455 Evaluation order: By observation Convergence achieved after 19 iterations CoefficientStd. Errorz-StatisticProb. C(4)-0.0003197.52E-05-4.2466430.0000 W(1)-1.89E-082.59E-08-0.7294230.4657 W(2)2.71E-072.88E-089.4285620.0000 W(3)-4.35E-073.87E-08-11.247180.0000 W(4)3.28E-075.42E-086.0582210.0000 W(5)-3.98E-075.40E-08-7.3774870.0000 W(6)6.00E-075.85E-0810.263390.0000 W(7)-8.04E-079.93E-08-8.0922080.0000 C(5)1.1372770.04356326.106660.0000 C(1)0.0894870.00241837.008160.0000 C(2)0.8810050.004612191.02450.0000 Log likelihood-15733.51 Akaike info criterion2.528223 Avg. log likelihood-1.263228 Schwarz criterion2.534785 Number of Coefs.11 Hannan-Quinn criter.2.530420

25. 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

26. ONE VARIABLE REGRESSIONS

27. MULTIPLE REGRESSIONS

28. 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

29. HIGH FREQUENCY VOLATILITY

30. 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?

31. ONE MONTH OF DAILY RETURNS

32. INTRA-DAILY RETURNS

33. ARE THESE DAYS THE SAME?

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

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

36. TARCH MODEL WITH RANGE  C1.07E-062.03E-075.2680490.0000  RESID(-1)^2-0.1009170.011398-8.8535490.0000  RESID(-1)^2*(RESID(-1)<0)0.0967440.0109518.8342090.0000  GARCH(-1)0.8799760.01051883.659950.0000  RANGE(-1)^20.0759630.0082819.1726900.0000   Adjusted R-squared-0.001360 S.D. dependent var0.010323  S.E. of regression0.010330 Akaike info criterion-6.616277  Sum squared resid0.404010 Schwarz criterion-6.606403  Log likelihood12550.46 Durbin-Watson stat2.001541

37. 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

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

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

40. Smallest BIC-based selection

41. Forecasting one step-ahead one step-ahead multi-step-ahead multi-step-ahead

42. Term Structure of Volatility 1

43. 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.