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NEW MODELS FOR HIGH AND LOW FREQUENCY VOLATILITY Robert Engle NYU Salomon Center Derivatives Research Project Derivatives Research Project
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FORECASTING WITH GARCH
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DJ RETURNS
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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
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DEFINITIONS r t is a mean zero random variable measuring the return on a financial asset CONDITIONAL VARIANCE UNCONDITIONAL VARIANCE
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GARCH(1,1) The unconditional variance is then
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GARCH(1,1) If omega is slowly varying, then This is a complicated expression to interpret
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SPLINE GARCH Instead, use a multiplicative form Tau is a function of time and exogenous variables
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UNCONDITIONAL VOLATILTIY Taking unconditional expectations Thus we can interpret tau as the unconditional variance.
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SPLINE ASSUME UNCONDITIONAL VARIANCE IS AN EXPONENTIAL QUADRATIC SPLINE OF TIME For K knots equally spaced
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ESTIMATION FOR A GIVEN K, USE GAUSSIAN MLE CHOOSE K TO MINIMIZE BIC FOR K LESS THAN OR EQUAL TO 15
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EXAMPLES FOR US SP500 DAILY DATA FROM 1963 THROUGH 2004 ESTIMATE WITH 1 TO 15 KNOTS OPTIMAL NUMBER IS 7
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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
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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
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ONE VARIABLE REGRESSIONS
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MULTIPLE REGRESSIONS
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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
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HIGH FREQUENCY VOLATILITY
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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?
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ONE MONTH OF DAILY RETURNS
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INTRA-DAILY RETURNS
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ARE THESE DAYS THE SAME?
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CAN WE USE THIS INFORMATION TO MEASURE VOLATILITY BETTER? DAILY HIGH AND LOW DAILY REALIZED VOLATILITY
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PARKINSON(1980) HIGH LOW ESTIMATOR IF RETURNS ARE CONTINUOUS AND NORMAL WITH CONSTANT VARIANCE,
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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
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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
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Absolute returns Insert asymmetric effects (sign of returns) Insert asymmetric effects (sign of returns) Insert other lagged indicators Insert other lagged indicators
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Repeat for daily range, hl t : And for realized daily volatility, dv t :
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Smallest BIC-based selection
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Forecasting one step-ahead one step-ahead multi-step-ahead multi-step-ahead
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Term Structure of Volatility 1
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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.
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