NEW MODELS FOR HIGH AND LOW FREQUENCY VOLATILITY

# NEW MODELS FOR HIGH AND LOW FREQUENCY VOLATILITY

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## NEW MODELS FOR HIGH AND LOW FREQUENCY VOLATILITY

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1. NEW MODELS FOR HIGH AND LOW FREQUENCY VOLATILITY Robert Engle NYU Salomon Center 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) Coefficient Std. Error z-Statistic Prob. C 0.000552 0.000135 4.093478 0.0000 Variance Equation C 9.89E-07 1.84E-07 5.380913 0.0000 RESID(-1)^2 0.066409 0.004478 14.82844 0.0000 GARCH(-1) 0.924912 0.005719 161.7365 0.0000 R-squared -0.000370 Mean dependent var 0.000356 Adjusted R-squared -0.001163 S.D. dependent var 0.010194 S.E. of regression 0.010200 Akaike info criterion -6.557778 Sum squared resid 0.393815 Schwarz criterion -6.551191 Log likelihood 12427.71 Durbin-Watson stat 1.985498

5. DEFINITIONS • rt is a mean zero random variable measuring the return on a financial asset • CONDITIONAL VARIANCE • UNCONDITIONAL VARIANCE

6. GARCH(1,1) • The unconditional variance is then

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

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

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

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

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

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

13. 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 Coefficient Std. Error z-Statistic Prob. C(4) -0.000319 7.52E-05 -4.246643 0.0000 W(1) -1.89E-08 2.59E-08 -0.729423 0.4657 W(2) 2.71E-07 2.88E-08 9.428562 0.0000 W(3) -4.35E-07 3.87E-08 -11.24718 0.0000 W(4) 3.28E-07 5.42E-08 6.058221 0.0000 W(5) -3.98E-07 5.40E-08 -7.377487 0.0000 W(6) 6.00E-07 5.85E-08 10.26339 0.0000 W(7) -8.04E-07 9.93E-08 -8.092208 0.0000 C(5) 1.137277 0.043563 26.10666 0.0000 C(1) 0.089487 0.002418 37.00816 0.0000 C(2) 0.881005 0.004612 191.0245 0.0000 Log likelihood -15733.51 Akaike info criterion 2.528223 Avg. log likelihood -1.263228 Schwarz criterion 2.534785 Number of Coefs. 11 Hannan-Quinn criter. 2.530420

14. 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

15. ONE VARIABLE REGRESSIONS

16. MULTIPLE REGRESSIONS

17. 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

18. HIGH FREQUENCY VOLATILITY

19. 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?

20. ONE MONTH OF DAILY RETURNS

21. INTRA-DAILY RETURNS

22. ARE THESE DAYS THE SAME?

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

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

25. TARCH MODEL WITH RANGE • C 1.07E-06 2.03E-07 5.268049 0.0000 • RESID(-1)^2 -0.100917 0.011398 -8.853549 0.0000 • RESID(-1)^2*(RESID(-1)<0) 0.096744 0.010951 8.834209 0.0000 • GARCH(-1) 0.879976 0.010518 83.65995 0.0000 • RANGE(-1)^2 0.075963 0.008281 9.172690 0.0000 • Adjusted R-squared -0.001360     S.D. dependent var 0.010323 • S.E. of regression 0.010330     Akaike info criterion -6.616277 • Sum squared resid 0.404010     Schwarz criterion -6.606403 • Log likelihood 12550.46     Durbin-Watson stat 2.001541

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

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

28. Repeat for daily range, hlt: And for realized daily volatility, dvt :

29. Smallest BIC-based selection

31. Term Structure of Volatility 1

32. 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.