Abstract
The weak convergence of the empirical process and partial sum process of the residuals from a stationary ARCH-M model is studied. It is obtained for and\(\sqrt n \) consistent estimate of the ARCH-M parameters. We find that the limiting Gaussian processes are no longer distribution free and hence residuals cannot be treated as i.i.d. In fact the limiting Gaussian process for the empirical process is a standard Brownian bridge plus an additional term, while the one for partial sum process is a standard Brownian motion plus an additional term. In the special case of a standard ARCH process, that is an ARCH process with no drift, the additional term disappears. We also study a sub-sampling technique which yields the limiting Gaussian processes for the empirical process and partial sum process as a standard Brownian bridge and a standard Brownian motion respectively.
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Kawczak, J., Kulperger, R. & Yu, H. The empirical distribution function and partial sum process of residuals from a stationary arch with drift process. Ann Inst Stat Math 57, 747–765 (2005). https://doi.org/10.1007/BF02915436
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DOI: https://doi.org/10.1007/BF02915436