By Christian Gouriéroux (auth.)
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"Gourieroux deals a pleasant stability of thought and alertness during this publication on ARCH modeling in finance…The ebook is definitely written and has huge references. Its specialise in finance will entice monetary engineers and monetary threat managers."
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Extra info for ARCH Models and Financial Applications
2, ... For instance, if Y is strictly stationary, we know that the series may be suitably approximated by its Volterra's expansion. It is then natural to test for the significance of cross terms. Thus, if the ARMA model is reduced to an AR(l) representation the following extended model may be proposed: which includes cross terms in £. This test relies on the usual Fisher F -statistic for the null hypothesis Ho (C II = C 12 = 0), where the additional regressors c;_I' cr-l£r-2 are replaced by the corresponding residuals 8;_1' 8'-18,-2, and where 8, = Yr - pYr - 1 is computed under the null hypothesis.
A h-l] + a hc/_h 2 + U/ + aU/_l + ... + a h-l U/-h+l. c/-h) = - - . I-a Since this variance is time independent, we find that the process £ is a weak white noise. The difference between the conditional variance and the unconditional 32 3. Univariate ARCH Models variance is a simple function of the deviations of squared innovations from the mean. We have indeed V(er/e,-h) - V(e,) = ah[e;_h - E(e;_h)]' Let us suppose that the condition a > 0 is satisfied. If the modulus of the innovation is large, the conditional variance takes smaller values than the unconditional variance.
The QGLS estimator of c, aI, ... ,a p is asymptotically efficient. Finally, we note that the asymptotic variance-covariance matrices of the two types of estimators are such that (Vas v? [ ( Vas v? [ ~~ =; ])-1 = [ t =; ]) -I Jaa 0 0 Jpp J;a =[ J pa J;p Jpp S. ( aT ) is asymptotically efficient, we have mce ~T [ Jaa o [ 0] Jpp Jaa - J;a -1*pa - [J;a Jpa J:p Jpp -J:p ] 0 ]» l l 0, 0 »; by applying the Cauchy-Schwarz inequality, we obtain J;p = O. 20. The QGLS estimators of band c, aI, ... ,a p are asymptotically uncorrelated.
ARCH Models and Financial Applications by Christian Gouriéroux (auth.)