VolatilityThe basic modelsModeling the volatilityARCH modelBasic properties ARCH(1)Basic properties, ARCH(1)Basic properties of ARCHARCH Estimation, Model Fitting and ForecastingGARCH modelK. Ensor, STAT 4211Spring 2004Volatility•Volatility – conditional variance of the process–Don’t observe this quantity directly (only one observation at each time point)•Common features–Serially uncorrelated but a depended process–Stationary–Clusters of low and high volatility–Tends to evolve over time with jumps being rare–Asymmetric as a function of market increases or market decreasesK. Ensor, STAT 4212Spring 2004The basic models•Consider a process r(t) where)()()( tattr ))1(|)(()(  tFtrEt  pjqkkjktajtrt1 10)()()())1(|)(()(2 tFtrVa rtConditional mean evolves as an ARMA processHow does the conditional variance evolve?K. Ensor, STAT 4213Spring 2004Modeling the volatility•Evolution of the conditional variance follows to basic sets of models–The evolution is set by a fixed equation (ARCH, GARCH,…)–The evolution is driven by a stochastic equation (stochastic volatility models).•Notation: –a(t)=shock or mean-corrected return; – is the positive square root of the volatility)(tK. Ensor, STAT 4214Spring 2004ARCH model•We have the general format as before•The equation defining the evolution of the volatility (conditional variance) is an AR(m) process.)()()( ttta)()1()(22102mtatatmWhy would this model yield “volatility clustering”?K. Ensor, STAT 4215Spring 2004Basic properties ARCH(1)0)))(()([))]1(|)()(([))]1(|)(([)]([tEtEtFttEEtFtaEEtaEUnconditional mean is 0.K. Ensor, STAT 4216Spring 2004Basic properties, ARCH(1))1/()]([)]([)]1([)]1([)]1([)]([0)]1(|)(([))]1(|)(([)]([1010102102102taVartaVartaVartaEtaEtEtFtaVarEtFtaEVartaVarUnconditional varianceWhat constraint does this put on 1?K. Ensor, STAT 4217Spring 2004Basic properties of ARCH•01<1•Higher order moments lead to additional constraints on the parameters–Finite positive (always the case) fourth moments requires0 12<1/3•Moment conditions get more difficult as the order increases – see general framework of equation 3.6•Note – in general the kurtosis for a(t) is greater than 3 even if the ARCH model is built from normal random variates.•Thus the tails are heavier and you expect more “outliers” than “normal”.K. Ensor, STAT 4218Spring 2004ARCH Estimation, Model Fitting and Forecasting•MLE for normal and t-dist ’s is given on pages 88 and 89.•The full likelihood is very difficult and thus the conditional likelihood is most generally used.•The conditional likelihood ignores the component of the likelihood that involves unobserved values (in other words, obs 1 through m)•MLE for joint estimation of parameters and degree of the t-distribution is given.•Model selection–Fit ARMA model to mean structure –Review PACF to identify order of ARCH–Check the standardized residuals – should be WN•Forecasting – identical to AR forecasting but we forecast volatility first and then forecast the process.K. Ensor, STAT 4219Spring 2004GARCH model•Generalize the ARCH model by including an MA component in the model for the volatility or the conditional variance.  mjskkjktjtatttta1 12202)()()()()()(  mjskkjktjtatttta1 12202)()()()()()(Proceed as before – using all you learned from ARMA