Wavelet Theory: for Economic & Financial Cycles

Cycles - their nature in existence, their implications on human-kind and the study thereof have sparked some important philosophical debates since the very pre-historic days. Notable contributions by famous, genius philosophers, mathematicians, historians and economists such as Pareto, Deulofeu, Danielewski, Kuznets, Kondratiev, Elliot and many others in itself shows how cycles and their study have been deemed important, through the history and process of scientific and philosophical inquiry. Particularly, the explication of Business, Economic and Financial cycles have seen some significant research and policy attention. Nevertheless, most of the methodologies employed in this space are either purely empirical in nature, time series based or the so-called Regime-Switching Markov model popularized in Economics by James Hamilton. In this work, we develop a Statistical, non-linear model fit based on circle geometry which is applicable for the dating of cycles. This study proposes a scalable, smooth and differentiable quarter-circular wavelet basis for the smoothing and dating of business, economic and financial cycles. The dating then necessitates the forecasting of the cyclical patterns in the evolution of business, economic and financial time series. The practical significance of dating and forecasting business and financial cycles cannot be over-emphasized. The use of wavelet decomposition in explaining cycles can be seen as an critical contribution of spectral methods of statistical modelling to finance and economic policy at large. Being a relatively new method, wavelet analysis has seen some great contribution in geophysical modelling. This study endeavours to widen the use and application of frequency-time decomposition to the economic and financial space. Wavelets are localized in both time and frequency, such that there is no loss of the time resolution. The importance of time resolution in dating of cycles is another motivation behind using wavelets. Moreover, the preservation of time resolution in wavelet analysis is a fundamental strength employed in the dating of cycles.Thesis (DPhil) -- Faculty of Science, Mathematical Statistics, 201