Suppose that we have an ARCH(2) model for financial returns

"Bill which does not besure that the variant Take Conditionally Heteroscedastic (ARC photoably depend tions? of west ang Models Conditionally Heteroscedastic (ARC - Our example of a full amodel would be where of - outme , lose Condi eteroscedastic (ARC . This is an ARCH() model. . Instead of calling the variance of. in the other Way of Writing ARCH Models . For illustration, coredder an ARCH(1). Inared of the . The two are different ways of expriming routly esting for 'ARCH Effects' 1 the residuals of an estimated model can following step run sity postulated Boost rearendon string the residuals, de. The for EACH of anbe s in run the regresam sting for 'ARCH Effects' 11 Obtain f from this respondon The test statitle Is defined on The the writerof correlation) from the last regresshome and b alerted em The mall and allerzathe hypotheses now value from the y' distributor. then reject the bull Note that the ARCH bed & ple practi go wepp died diner le ARCH effect in S&:P500 daily returns roblemns with ARCH (o) Models around some of these probasis is a GARCit model alised ARCH (GARCH) Models I Bollersdey (1956) developed the GARCH model which of - est end.. + al- period (on? ) want the fitted variance from the model THE CARCHILD model which Is eralised ARCH (GARCH) Models 1I CC therefore a for amore widely cheralised ARCH (GARCH) Models III of - outand , tou + mad at float wings caeralised ARCH (GARCH) Models IV . An infinite number of successive substitutions would ytek Heralised ARCH (GARCH) Models V "GARCHO thei GARCE(1 1) model to a ility clustering in the data . Why is GARCH better then ARCH? Unconditional Variance under the GARCH . The useandalocal vetinure of to is given by when on + 3 0. We assume that {m} is weakly stationary. a) Derive the uneonditional mean and variance of {H1}. [10 marks] 13) Derive a condition on the parameters which ensures weak stationarity of the {m} pro- cess. [5 marks] c) Suppose that we have estimated the ARCH(2) model and obtained the parameter estimates: an = 1, e; = 0.5, and a2 = 0.1. Assume that 113}. = 1, and uh = o are observed. Denote by 0;.\"13. the h-step-ahead forecast of the conditional variance. Compute the one and twostepahead forecasts of the conditional variance of. [10 marks]