Thursday, December 10, 2015

Long Memory Stochastic Volatility

Check out Mark Jensen's new paper.  Long memory is a key feature of realized high-frequency asset-return volatility, yet it remains poorly understood. Jensen's approach may help change that. Of particular interest are: (1) its ability to handle seamlessly d in [0, 1[, despite the fact that the unconditional variance is infinite for d in ].5, 1[, and (2) closely related, the important role played by wavelets. 


Robust estimation of nonstationary, fractionally integrated, autoregressive, stochastic volatility

Jensen, Mark J. (Federal Reserve Bank of Atlanta)
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 pr! ocesses. 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.