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VOLATILITY MODELS. GARCH. GENERALIZED- more general than ARCH model AUTOREGRESSIVE-depends on its own past CONDITIONAL-variance depends upon past information HETEROSKEDASTICITY- fancy word for non-constant variance. HISTORY. I DEVELOPED THE ARCH MODEL WHEN I WAS VISITING LSE IN 1979

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garch
GARCH
  • GENERALIZED- more general than ARCH model
  • AUTOREGRESSIVE-depends on its own past
  • CONDITIONAL-variance depends upon past information
  • HETEROSKEDASTICITY- fancy word for non-constant variance
history
HISTORY
  • I DEVELOPED THE ARCH MODEL WHEN I WAS VISITING LSE IN 1979
  • IT WAS PUBLISHED IN 1982 WITH MACRO APPLICATION - THE VARIANCE OF UK INFLATION
  • TIM BOLLERSLEV DEVELOPED THE GARCH GENERALIZATION AS MY PHD STUDENT - PUBLISHED IN 1986
the garch model
The GARCH Model
  • The variance of rt is a weighted average of three components
    • a constant or unconditional variance
    • yesterday’s forecast
    • yesterday’s news
parameter estimation
PARAMETER ESTIMATION
  • Historical data reveals when volatilities were large and the process of volatility
  • Pick parameters to match the historical volatility episodes
  • Maximum Likelihood is a systematic approach:
  • Max
diagnostic checking
DIAGNOSTIC CHECKING
  • Time varying volatility is revealed by volatility clusters
  • These are measured by the Ljung Box statistic on squared returns
  • The standardized returns no longer should show significant volatilty clustering
  • Best models will minimize AIC and Schwarz criteria
theorems
THEOREMS
  • GARCH MODELS WITH GAUSSIAN SHOCKS HAVE EXCESS KURTOSIS
  • FORECASTS OF GARCH(1,1) ARE MONOTONICALLY INCREASING OR DECREASING IN HORIZON
forecasting with garch
FORECASTING WITH GARCH
  • GARCH(1,1) can be written as ARMA(1,1)
  • The autoregressive coefficient is
  • The moving average coefficient is
forecasting average volatility
FORECASTING AVERAGE VOLATILITY
  • Annualized Vol=square root of 252 times the average daily standard deviation
  • Assume that returns are uncorrelated.
variance targeting
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
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
the leverage effect asymmetric models
The Leverage Effect -Asymmetric Models
  • 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
exogenous variables in a garch model
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
EXAMPLES
  • Non-linear effects
  • Deterministic Effects
  • News from other markets
    • Heat waves vs. Meteor Showers
    • Other assets
    • Implied Volatilities
    • Index volatility
  • MacroVariables or Events
what is the best model
WHAT IS THE BEST MODEL?
  • The most reliable and robust is GARCH(1,1)
  • For short term forecasts, this is good enough.
  • For long term forecasts, a component model with leverage is often needed.
  • A model with economic causal variables is the ideal
procter and gamble daily returns 2 89 3 99 variance equation garch 1 1
Procter and Gamble Daily Returns 2/89-3/99Variance Equation Garch(1,1)

C 5.11E-06 1.24E-06 4.112268

ARCH(1) 0.047402 0.006681 7.094764

GARCH(1) 0.927946 0.011114 83.49235

AIC -5.7086 , SCHWARZ CRITERION -5.699617

correlogram of squared residuals
CORRELOGRAM OF SQUARED RESIDUALS

AC PAC Q-Stat Prob

1 0.030 0.030 2.3720 0.124

2 -0.018 -0.019 3.1992 0.202

3 -0.010 -0.009 3.4859 0.323

4 0.018 0.018 4.3016 0.367

5 -0.011 -0.013 4.6317 0.462

6 -0.002 -0.001 4.6408 0.591

7 0.013 0.013 5.0865 0.649

8 -0.016 -0.017 5.7164 0.679

9 0.010 0.012 5.9973 0.740

10 0.003 0.002 6.0183 0.814

p g tarch
P&G TARCH

C 0.0000 0.0000 4.6121 0.0000

ARCH(1) 0.0269 0.0093 2.9062 0.0037

(RESID<0)*ARCH(1)0.0520 0.0141 3.6976 0.0002

GARCH(1) 0.9123 0.0139 65.8060 0.0000

AIC -5.7114 SCHWARZ CRITERION -5.7002

p g egarch
P&G EGARCH

C -0.3836 0.0672 -5.7052

|RES|/SQR[GARCH](1) 0.1186 0.0153 7.7645

RES/SQR[GARCH](1) -0.0392 0.0103 -3.8195

EGARCH(1) 0.9656 0.0070 137.9063

AIC -5.7114 SCHWARZ CRITERION -5.7002

p g asymmetric component
P&G Asymmetric Component

Perm: C 0.0002 0.0000 14.1047

Perm: [Q-C] 0.9835 0.0049 201.5877

Perm: [ARCH-GARCH] 0.0335 0.0079 4.2577

Tran: [ARCH-Q] -0.0361 0.018 -2.0045

Tran: (RES<0)*[ARCH-Q] 0.0910 0.0213 4.2838

Tran: [GARCH-Q] 0.8063 0.0819 9.8403

AIC -5.7132, SCHWARZ CRITERION -5.6974

discuss gaussian assumption
DISCUSS GAUSSIAN ASSUMPTION
  • EMPIRICAL EVIDENCE INDICATES THAT INNOVATIONS ARE LEPTOKURTIC
  • GAUSSIAN GARCH IS QMLE
    • CONSISTENT BUT NEEDS BOLLERSLEV WOOLDRIDGE STANDARD ERRORS
    • T-DISTRIBUTION MAY BE MORE EFFICIENT
    • CAN DO SEMI-PARAMETRIC ESTIMATOR OF ENGLE AND GONZALES-RIVERA
bollerslev wooldridge standard errors
Bollerslev Wooldridge Standard Errors

ROBUST TO NON-NORMAL ERRORS

Perm: C 0.0002 0.0000 8.3857

Perm: [Q-C] 0.9835 0.0081 121.9762

Perm: [ARCH-GARCH] 0.0335 0.0107 3.1264

Tran: [ARCH-Q] -0.036 0.0242 -1.4898

Tran: (RES<0)*[ARCH-Q] 0.0910 0.0429 2.1220

Tran: [GARCH-Q] 0.8063 0.1223 6.5919

risk premia
RISK PREMIA
  • WHEN RISK IS GREATER, EXPECTED RETURNS SHOULD BE GREATER
    • HOW MUCH?
    • WHAT COUNTS AS RISK?
  • CAPM GIVES AN ANSWER
  • MULTI-BETA GIVES ANOTHER
  • PRICING KERNEL COVERS ALL