Unconditional Mean, Volatility, and the FOURIER-GARCH Representation

Recently there has been an upsurge interest in modeling the nonstationarities present in the volatility of financial data. The clustering and the persistence of volatility of asset returns have been well documented. The IGARCH model of Engle and Bollerslev (1986), for instance, describes in a parsimonious way the high persistence in the conditional volatility of stock returns while the underlying process remains strictly stationary. Alternatively, Granger (1980) and Granger and Joyeux (1980) model the long memory or the long-range dependence of a series of log returns as a fractionally integrated process to allow the autocorrelation functions to decay very slowly, in a fashion characteristic of stock returns. However, seminal papers from Granger and Joyeux (1986), Lamoureux and Lastrapes (1990), and, more recently, from Diebold and Inoue (2001), Mikosch and Starica (2004), Starica and Granger (2005), and Perron and Qu (2007) argue that the high persistence close to unit root and long memory both in the first and the second moments may actually be caused by structural changes in the level or slope of an otherwise locally stationary process of the long-run volatility. Diebold and Inoue (2001) argue that this is due to switching regimes in the data. Mikosch and Starica (2004) provide theoretical evidence that changes in the unconditional mean or variance induce the statistical tools (e.g., sample ACF, periodogram) to behave the same way they would if used on stationary long-range dependent sequences.