EXTENSIONS OF ARCHAPPLICATIONSAutoregressive Conditional Heteroskedastic Model (ARCH) We have looked at ways of modeling a time series ty by using (past) information on the series up until time t –1: []L,,|21−−tttyyyE For example, the zero-mean AR(1) process: ttyyyεφ+=−1. Where yt has the conditional mean of 1−tyφ and an unconditional mean of zero, and tεis a white noise process with a fixed variance . 2)var(σε= The standard approach to heteroskedasticity is to find an exogenous variable xt which predicts the variance: 1−=tttxyε. This requires knowing and specifying the cause of the time-varying variance, and does not recognize that both the conditional means and variances may change over time. 1Engle (1982), “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation,” Econometrica, 987-1008. Engle (1982) suggests generalizing so that: ),(~,|1 ttttthgNXYy− where {}{}0,1,1≥=≥=−−−sxXsyYsttstt and gt and ht are both functions of Yt-1 and Xt. Let zt be some subset of the variables in (Yt-1 ,Xt ), so that: βttzg′= and ),(~,|,1120ttttttttqiitithgNXYygyh−=−−=+=∑εεαα These together form the ARCH(q) model. 2These can be rewritten by defining: ),,,(),,,1(10221qqtttwααααεεKK=′=′−− so that ).,(~,|1αβtttttwzNXYy′′− EXAMPLE: AR(1) model for yt with ARCH(1) errors. 1,1=+=′−qthatnoteyztttεφβ. [][]{}.0,|var0|2110111≥=+===+=−−−−−sEwherehEEEyystttttttttttεεααεεεφ assume that ,1<φ so yt is stationary. Also, .0and0010≥>>ααrequiresht Engle (1982) shows that the unconditional variance of tε will be finite if :11<α 1021)var(αασε−==t 3and the conditional variance of tε is: )(22112σεασ−=−−tth . The errors tε and τε−t are not correlated, but they are not independent. The squared errors are related by: .)var(2110−+=ttεααε Although yt is conditionally normal, it is not jointly normal, and neither is its marginal distribution. The marginal distribution of yt will be symmetric if the conditional distribution of tε is symmetric. Both of conditional mean and the conditional variance depend on the available information set. Engle (1982) shows that an ARCH(q) model will have a finite, positive variance: ∑=−=qiit101)var(ααε if 0,,,010≥>qαααK , and if all of the roots of the associated characteristic equation lie outside the unit circle so that .1<Σα 4EXTENSIONS OF ARCH Weiss, A.A. (1984) “ARMA Models with ARCH Errors,” Journal of Time Series, 129-143. Assumes tyˆ may be ARIMA. Differences yt for stationarity to produce an ty which is ARMA. This produces the conditional variance: 2021120)()( yyyyhttpiitiqiitit−−+−++=∑∑=−=−εδδεαα with 0≥iδ. This implies an “ARMA-ARCH” or “ARMACH” model. 5Bollerslev, T. (1986) “Generalized Autoregres-sive Conditional Heteroskedasticity,” Journal of Econometrics, 307-27. ______ ,(1988) “On the Correlation Structure for the Generalized Conditional Heteroskedastic Process,” Journal of Time Series Analysis, 121-32. Generalizes ht to .0,1120≥++=∑∑=−=− ipiitiqiitithhββεαα This is called a Generalized ARCH (p,q) process, or simply GARCH(p,q). With βttzg′= it is called a GARCH Regression Model. (Note that q is often quite large.) Engle, R.F., Lilien, D.M., and Robbins, R.P. (1987) “Estimating Time Varying Risk Premia in the Term Structure: The ARCH-M Model.” Econometrica , 391-408. Extend to allow the conditional variance to affect the mean, hence ARCH-M (ARCH in the Mean) Model. γαδβtttttttttzwhhhzNXYy22211),(~,|′+′=+′− 6where the z1t and z2t are two possibly different subsets of the variables in (Yt-1,Xt ). Engle and Bollerslev (1986) “Modeling the Persistence of Conditional Variances” Econometric Reviews, 1-50. Extend to the more general multivariate cases, and also introduce the Integrated GARCH (IGARCH) Model, arising with a unit root in the GARCH(p,q) process. APPLICATIONS Engle (1983) Journal of Money, Credit and Banking, 286-301. Engle, Hendry, and Trumble (1985), Canadian Journal of Economics. Pagan et al. (1983) Review of Economic Studies. Consider inflation varance. Bollerslev (1987) REStat. Considers time varying risk premia in the term structure of interest rates. Engle and Bollerslev (1986) Point to applications in the foreign exchange market to test long bond returns against Shiller’s variance bounds. It has also been used to derive pricing relations for financial assets.