The empirical distribution function and partial sum process of residuals from a stationary arch with drift process |
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Authors: | Janusz Kawczak Reg Kulperger Hao Yu |
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Institution: | (1) Department of Mathematics and Statistics, University of North Carolina, 28223 Charlotte, NC, USA;(2) Department of Statistical and Actuarial Science, University of Western Ontario, N6A 5B7 London, Canada |
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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
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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Keywords: | and phrases" target="_blank"> and phrases Weak convergence residuals ARCH drift empirical distribution |
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