The Variance of Sample Autocorrelations: Does Barlett's Formula Work With ARCH Data?

We review the notion 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 possess a martingale representation. Consequently, the use of Bartlett’s formula is unwarranted in the context of data typically modelled 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 present an alternative to Bartlett’s formula that is applicable (and consistent) in the context of financial returns data.