The sample autocorrelations of heavy-tailed processes with applications to ARCH

Richard Arnold Davis · Thomas Mikosch

225 citations0/0 files storedThe Annals of Statistics

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Financial Risk and Volatility Modeling Probability and Statistical Research Complex Systems and Time Series Analysis

Abstract

We study the sample ACVF and ACF of a general stationary sequence under a weak mixing condition and in the case that the marginal distributions are regularly varying. This includes linear and bilinear processes with regularly varying noise and ARCH processes, their squares and absolute values. We show that the distributional limits of the sample ACF can be random, provided that the variance of the marginal distribution is infinite and the process is nonlinear. This is in contrast to infinite variance linear processes. If the process has a finite second but infinite fourth moment, then the sample ACF is consistent with scaling rates that grow at a slower rate than the standard $\sqrt{n}$. Consequently, asymptotic confidence bands are wider than those constructed in the classical theory. We demonstrate the theory in full detail for an ARCH (1) process.

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References

1997Modelling Extremal Events 2007Modelling Financial Time Series 1993A geographical model for the daily and weekly seasonal volatility in the foreign exchange market 1997From the bird's eye to the microscope: A survey of new stylized facts of the intra-daily foreign exchange markets 1994Testing the covariance stationarity of heavy-tailed time series: An overview of the theory with applications to several financial datasets 1993Long Memory in Foreign-Exchange Rates 1988Extremal Theory for Stochastic Processes 1989Extremal behaviour of solutions to a stochastic difference equation with applications to arch processes

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