On the Empirical Importance of Periodicity In the Volatility of Financial Time Series

We discuss the empirical importance of long term cyclical effects in the volatility of financial returns. Following Cizek and Spokoiny (2009), Amado and Teräsvirta (2012) and others, we consider a general conditionally heteroscedastic process with stationarity property distorted by a deterministic function that governs the possible variability in time of unconditional variance. The function proposed in this paper can be interpreted as a finite Fourier approximation of an Almost Periodic (AP) function as defined by Corduneanu (1989). The resulting model has a particular form of a GARCH process with time varying parameters, intensively discussed in the recent literature.In the empirical analyses we apply a generalisation of the Bayesian AR(1)-t-GARCH(1,1) model for daily returns of S&P500, covering the period of sixty years of US postwar economy, including the recently observed global financial crisis. The results of a formal Bayesian model comparison clearly indicate the existence of significant long term cyclical patterns in volatility with a strongly supported periodic component corresponding to a 14 year cycle. This may be interpreted as empirical evidence in favour of a linkage between the business cycle in the US economy and long term changes in the volatility of the basic stock market index