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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 31, Fasc. 1,
pages 47 - 59
 

NONLINEARITY OF ARCH AND STOCHASTIC VOLATILITY MODELS AND BARTLETT’S FORMULA

Piotr S. Kokoszka
Dimitris N. Politis

Abstract: We review some notions of linearity of time series and show that ARCH or stochastic volatility (SV) processes are not only non-linear: they are not even weakly linear, i.e., they do not even have a martingale representation. Consequently, the use of Bartlett’s formula is unwarranted in the context of data typically modeled as ARCH or SV processes such as financial returns. More surprisingly, we show that even the squares of an ARCH or SV process are not weakly linear. Finally, we discuss an alternative estimator for the variance of sample autocorrelations that is applicable (and consistent) in the context of financial returns data.

2000 AMS Mathematics Subject Classification: Primary: 62M10; Secondary: 60G42.

Keywords and phrases: ARCH processes, GARCH processes, linear time series, stochastic volatility.

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