Time-spectral density and wavelets approaches. Comparative study. Applications to SP500 returns and US GDP

Various forms of instability can be observed in macroeconomic and financial data including changes in variance, changes in cycle properties, or both. Traditional tests do not allow to distinguish between these different cases. This paper proposes and compares two alternative approaches. The comparison is based on Monte Carlo simulations. The first approach is based on windowed-estimate of the time-spectral density, while the second method is the wavelets theory. We show that the wavelets approach is particularly powerful to detect changes in cyclical properties, while the first approach fails in such a case. In contrast, the wavelets method fails to capture time–interaction effects, while the first approach is more powerful regarding this point. Hence the two methods are complementary. A first application on the SP500 returns shows that there are only changes in variance without altering the cyclical properties of the series. A second application on the US growth rate allows to conclude that there are simultaneous changes in the time and frequency domain.