Fractionally Integrated Models With ARCH Errors

Abstract: We introduce ARFIMA-ARCH models which simultaneously incorporate fractional differencing and conditional heteroskedasticity. We develop the likelihood function and a numerical estimation procedure for this model class. Two ARCH models - Engle- and Weiss-type - are explicitly treated and stationarity conditions are derived. Finite-sample properties of the estimation procedure are explored by Monte Carlo simulation. An application to the Standard & Poor 500 Index indicates existence of intermediate memory (d<0) for the 1980's and no fractional differencing (d=0) but strong conditional heteroskedastic effects for the 1960's. For the latter time period, contrary to the suggestion of long memory by Mandelbrot, we only found evidence for a positive first-order autoregressive parameter.;