ification of a suitable ARCH-GARCH model
2. Estimation (using Maximum Likelihood)
3. Testing/diagnostic checking of the model to ensure it provides an adequate representation of the actual DGP (see Seminar 5).Warwick Business School 19
Identification of ARCH-GARCH models
First perform a simple ARCH-test(see Appendix) to test for ARCH effects.
If there are ARCH effects present, identify a particular ARCH-GARCH model as follows:
i) The AR and ARMA representations for the squared residuals suggest the ACF and PACFsof the squared residuals can be used to identify a specific ARCH-GARCH model.
An ARCH(q) model is indicated by:
a) An infinite decay in the ACF of the squared residuals.
b) q spikes in the PACF of the squared residuals.
A GARCH(p,q) model is indicated by an infinite decay in both the ACF and the PACF of the squared residuals.
ii) In practice its more common to use information criteria such as the Schwarz Criterionto help select a model (see lecture 5).
iii) Many authors simply assumea GARCH(1,1) specification.Warwick Business School 20
Maximum Likelihood (ML) (see Brooks Chp8, Appendix)
Suppose we have a sample of independent observations drawn from a knowndensityHowever the parameters are unknownand there is a given sample of data. Therefore re-interpretthe joint distribution as the likelihood function:()Tyy,...,1()()()θθθTTyfyfyyf××=...,...,11()()()θθθTTyfyfyyL××≡...,...,11()()Σ==TttyfL1loglogθθThe probability of observing different realizations of y for givenparameters θ.The likelihood of observing the (given) data for different values of θML estimator found by maximizingthe log-likelihood functionwith respect to θML chooses θto maximize the likelihood of observing the sample data. ML estimators are:1) Consistent.2) Asymptotically efficient.3) Asymptotically normally distributed.Warwick Business School 21
ML estimation of ARCH-GARCH model
The ARCH-GARCH likelihood function involves the conditional error density. If v~NID(0,1) then:The log likelihood function is:()()θσσεπσθεttttttvff2222121exp21=⎭⎬⎫⎩⎨⎧−=()()()()()ΣΣΣ===−−−=⎥⎦⎤⎢⎣⎡−=TtttTttTtttTvfL122121221log212log2log21loglogσεσπσθθ()22με−=ttr2tσis a function of the unknown parameters of the conditional variance.is a function of the unknown parameters of the conditional mean.The ML estimates of the conditionalmean/variance parameters are the values which maximize log L(θ).Warwick Business School 22
Alternative conditional distributions (see Seminar 5)
What happens if the standardized residuals, v,are non-normal (but normality is assumed)?
The ML estimator is still consistent and asymptotically normalbut the standard errorsare inconsistent.
Eviewsgives an option to estimate ARCH-GARCH models with a conditional normal distribution but with a var-covmatrix which is robust to non-normality. This is known as Quasi-ML (QML)
Alternatively we can choose a different conditional distribution. Eviewsallows the choice of a:
i) Student’s tdistribution
ii) Generalized error distributionWarwick Business School 23
Extensions to ARCH-GARCH
Asymmetric GARCHStandard GARCH models force a symmetric response of the conditional variance to shocks (since depends on lagged squaredresiduals).However, typically bad news (negative shocks) may be expected to increase volatility more than good news (positive shocks) of the same
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