Statistics of Financial Markets by Jürgen Franke Wolfgang Karl Härdle & Christian Matthias Hafner

Author:Jürgen Franke, Wolfgang Karl Härdle & Christian Matthias Hafner
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

(13.39)

(13.40)

(13.41)

where (13.39) substitutes for since , (13.40) uses the fact that while (13.41) neglects the constant , since is very small.

The approximation (13.41) is in fact an exponential smoother, the exponentially weighted moving average (EWMA) applied to ɛ t 2. Forecasting with EWMA is optimal in the mean-square error sense for the state space model

(13.42)

(13.43)

where η t and ν t are i.i.d. sequences

and .

Assuming

(13.44)

the GARCH(1,1) model can be written in the form of (13.42):

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