Changes of Structure in Financial Time Series and the Garch Model

• In this paper we propose a goodness of fit test that checks the resemblance of the spectral density of a GARCH process to that of the log-returns. The asymptotic behavior of the test statistics are given by a functional central limit theorem for the integrated periodogram of the data. A simulation study investigates the small sample behavior, the size and the power of our test. We apply our results to the S&P500 returns and detect changes in the structure of the data related to shifts of the unconditional variance. We show how a long range dependence type behavior in the sample ACF of absolute returns might be induced by these changes. Key-Words: • integrated periodogram; spectral distribution; functional central limit theorem; Kiefer– Müller process; Brownian bridge; sample autocorrelation; change point; GARCH process; long range dependence; IGARCH; non-stationarity. AMS Subject Classification: • Primary: 62P20; Secondary: 60G10 60F17 60B12 60G15 62M15. ∗This research supported in part by a grant of the Dutch Science Foundation (NWO), by DYNSTOCH, a research training network under the programme Improving Human Potential financed by The 5 Framework Programme of the European Commission, the Danish Research Council Grant No 21-01-0546, and MaPhySto, the Danish Network for Mathematical Physics and Stochastics, funded by The Danish National Research Foundation. 42 Thomas Mikosch and Cătălin Stărică Changes of Structure in Financial Time Series 43