Frequency domain aspects of aggregation

Abstract

In this dissertation, I focus on various issues related to aggregation in the frequency domain. The problems tackled include the effects of skip sampling and temporal aggregation on the long memory property, the choice of the sampling frequency, and the asymptotic properties of the discrete Fourier transform (dft). The first chapter considers the effects of temporal aggregation for stochastic volatility model. I provide the link between the spectral density function of the squared low and high-frequency returns. I also analyze the properties of the spectral density function of realized volatility series constructed from squared returns with different frequencies under temporal aggregation. The theoretical results allow explaining many findings reported and uncover new features about volatility in financial market indices. The second chapter deals with the dft of generalized fractional processes, including the case where a singularity in the spectrum can occur at any frequency. This work extends Philips' (1999) results about the dft of a fractional process for which the spectrum is unbounded only at frequency zero. I study the asymptotic properties of the dft and their asymptotic distributions. Applications to semi-parametric estimation methods of the long-memory parameter are also presented. The third chapter studies aggregation pertaining to skip sampling of stock variables as well as temporal aggregation of flow variables. I derive the dft and the periodogram of the aggregated series in terms of the original dft. I further analyze the limit of the expectation of the periodogram of aggregated series in the nonstationary case for a generalized fractional process. I show that for skip sampling a long memory feature at the zero frequency can arise from the aggregation of a generalized fractional series, while temporal aggregation does not induce such an effect. Simulation results pertaining to the estimates of the memory parameter are included to demonstrate the practical relevance of my theoretical results.