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58;the residuals (ε) are conditionallynormally distributed.This assumption is important for estimation of the model (see below).Warwick Business School 9
ARCH(q) model (Autoregressive Conditional Heteroscedasticity)
Different assumptions about Ωgenerate different models of time varying risk.Suppose thenThis is an ARCH(q) model. A sufficient condition for a positiveconditional variance (variances cannot be negative) is that: Engle shared the Nobel Prize in Economics (2003): “for methods of analyzing economic time series with time-varying volatility (ARCH)”{}2211,...,qttt−−−=Ωεε221102...qtqtt−−+++=εαεαασqii,...,1 ,0 ,00=≥>ααThe ARCH model was firstproposed by Engle (1982)in the context of modelinginflation uncertainty.Warwick Business School 10
ARCH(q) modelThe ARCH(q) model can be written as an AR(q) model in the squared residualsThe process is stationary if all the roots of the characteristicequation lie outside of the unit circle:The long run/unconditionalvariance can be found from the Woldrepresentation:()22122221102,...tttttttqtqttEuuσεεεεαεααε−=Ω−=++++=−−−0...11=−−−qqzazα()Σ=−=⇒−−−+=qiitqqttELLu1021021...1ααεαααεΣ=<qii11αShocks to volatility (u)The unconditional variance is finite, positive and constant (i.e., homoscedastic) if:If this condition holds then volatility is a constantin the long-run. If the process is stationarythenshocks to volatility do not persist⇒the conditional variance returnseventually to it’s long run level.This condition must hold if the process is stationary.Warwick Business School 11
GARCH(p,q) (GeneralisedAutoregressive Conditional Heteroscedasticity)
In practice q may need to be set high to capture all the non-linear dependence in returns.Also with a lot of lags the non-negativity constraints are likely to be violated.Bollerslev(1986) proposed the GARCH model as a parsimoniousalternative to ARCHFor GARCH(p,q):The non-negativity constraints (sufficient restrictions) are: Typically p=q=1 is adequate in most empirical applications.{}2212211,...,,,...,pttqttt−−−−−=ΩσσεεΣΣ=−=−++=pjjtjqiitit121202σβεαασpjqiji,..,1 ,,..,1 ,0,0,00==≥≥>βααWarwick Business School 12
GARCH(1,1)The GARCH(1,1) has an ARMA(1,1) representation in the squared residuals. The unconditional/long-run variance is:()()ttttttttttttuuuu+−++=⇒−++=−⇒++=−−−−−−−1121110212112110221121102βεβααεεβεααεσβεαασ111<+βα()()11021βααε+−=tE111=+βαVolatility is stationaryif:But ifthen shocks to volatility have a permanent effect:⇒INTEGRATED GARCH(IGARCH) process.Apparent IGARCH behaviouris found quite often in empirical work (see below).The long-run variance converges to a constantiff: 111<+βαAn implication of IGARCH is that investors should be frequently altering their portfolios following shocks to reflect permanentchanges in risk. Since this kind of behaviour isn’t observed, IGARCH is incompatible with volatility in the ‘real world’. It’s possible that shocks to volatility are just highly persistent if not permanent.Warwick Business School 13
Multivariate GARCH (MGARCH)
Generalization of GARCH to systems of n-asset returns.The conditional volatility is an n×nvariance-covariance matrix:A widely used formulation of MGARCH is the BEKK model:⎟⎟⎟⎠⎞⎜⎜[