Bartlett's formula for a general class of nonlinear processes

Christian Francq · Jean‐Michel Zakoïan

60 citations0/0 files storedJournal of Time Series Analysis

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Financial Risk and Volatility Modeling Complex Systems and Time Series Analysis Time Series Analysis and Forecasting

Abstract

Abstract. A Bartlett‐type formula is proposed for the asymptotic distribution of the sample autocorrelations of nonlinear processes. The asymptotic covariances between sample autocorrelations are expressed as the sum of two terms. The first term corresponds to the standard Bartlett's formula for linear processes, involving only the autocorrelation function of the observed process. The second term, which is specific to nonlinear processes, involves the autocorrelation function of the observed process, the kurtosis of the linear innovation process and the autocorrelation function of its square. This formula is obtained under a symmetry assumption on the linear innovation process. It is illustrated on ARMA–GARCH models and compared to the standard formula. An empirical application on financial time series is proposed.

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References

1986Generalized autoregressive conditional heteroskedasticity 1982Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation 2004Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes 2002NECESSARY AND SUFFICIENT MOMENT CONDITIONS FOR THE GARCH(r,s) AND ASYMMETRIC POWER GARCH(r,s) MODELS 2004Smooth Transition Regression Modeling

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