Robust Estimation of Nonstationary, Fractionally Integrated, Autoregressive, Stochastic Volatility

Mark J. Jensen

Working Paper 2015-12
November 2015

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Empirical volatility studies have discovered nonstationary, long-memory dynamics in the volatility of the stock market and foreign exchange rates. This highly persistent, infinite variance—but still mean reverting—behavior is commonly found with nonparametric estimates of the fractional differencing parameter d, for financial volatility. In this paper, a fully parametric Bayesian estimator, robust to nonstationarity, is designed for the fractionally integrated, autoregressive, stochastic volatility (SV-FIAR) model. Joint estimates of the autoregressive and fractional differencing parameters of volatility are found via a Bayesian, Markov chain Monte Carlo (MCMC) sampler. Like Jensen (2004), this MCMC algorithm relies on the wavelet representation of the log-squared return series. Unlike the Fourier transform, where a time series must be a stationary process to have a spectral density function, wavelets can represent both stationary and nonstationary processes. As long as the wavelet has a sufficient number of vanishing moments, this paper's MCMC sampler will be robust to nonstationary volatility and capable of generating the posterior distribution of the autoregressive and long-memory parameters of the SV-FIAR model regardless of the value of d. Using simulated and empirical stock market return data, we find our Bayesian estimator producing reliable point estimates of the autoregressive and fractional differencing parameters with reasonable Bayesian confidence intervals for either stationary or nonstationary SV-FIAR models.

JEL classification: C11, C14, C22

Key words: Bayes, infinite variance, long-memory, Markov chain Monte Carlo, mean-reverting, wavelets


The views expressed here are the author's and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the author's responsibility.
Please address questions regarding content to Mark J. Jensen, Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street NE, Atlanta, GA 30309-4470, 404-498-8019, mark.jensen@atl.frb.org.
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