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.