Long memory in time series : semiparametric estimation and conditional heteroscedasticity
This dissertation considers semiparametric spectral estimates of temporal dependence in time series. Semiparametric frequency domain methods rely on a local parametric specification of the spectral density in a neighbourhood of the frequency of interest. Therefore, such methods can be applied to the analysis of singularities in the spectral density at frequency zero to identify long memory. They can also serve as the basis for the estimation of regular parts of the spectrum. One thereby avoids inconsistency that might arise from misspecification of dynamics at frequencies other than the frequency under focus. In case of long financial time series, the loss of efficiency with respect to fully parametric methods (or full band estimates) may be offset by the greater robustness properties. However, if semiparametric frequency domain methods are to be valid tools for inference on financial time series, they need to allow for conditional heteroscedasticity which is now recognized as a dominant feature of asset returns. This thesis provides a general specification which allows the time series under investigation to exhibit this type of behaviour. Two statistics are considered. The weighted periodogram statistic provides asymptotically normal point estimates of the spectral density at zero frequency for weakly dependent processes. The local Whittle (or local frequency domain maximum likelihood) estimate provides asymptotically normal estimates of long memory in possibly strongly dependent processes. The asymptotic results hold irrespective of the behaviour of the spectral density at non zero frequencies. The asymptotic variances are identical to those that obtain under conditional homogeneity in the distribution of the innovations to the observed process. In semiparametric frequency domain estimation, the choice of bandwidth is crucial. Indeed, it determines the asymptotic efficiency of the procedure. Optimal choices of bandwidth are derived, balancing asymptotic bias and asymptotic variance. Feasible versions of these optimal band-widths are proposed, and their performance is assessed in an extensive Monte Carlo study where the innovations to the observed process are simulated under numerous parametric submodels of the general specification, covering a wide range of persistence properties both in the levels and in the squares of the observed process. The techniques described above are applied to the analysis of temporal dependence and persistence in intra-day foreign exchange rate returns and their volatilities. While no strong indication of returns predictability is found in the former, a clear pattern arises in the latter, indicating that intra-day exchange rate returns are well described as martingale differences with weakly stationary and fractionally cointegrated long memory volatilities.