Macquarie University
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Log quantile differences and the temporal aggregation of alpha-stable moving average processes

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posted on 2022-03-28, 22:07 authored by Adrian Walter Barker
The modelling of the time-varying volatility of financial market asset log returns has attracted considerable interest from researchers and market participants. Prominent amongst these models are the generalized autoregressive conditional heteroskedastic (garch) models and the stochastic volatility (sv) models. Generally, such models use the conditional variance as the measure of dispersion. This thesis advocates for the use of the log quantile di¤erence (lqd) as an alternative measure of dispersion where the variance of the intraday log returns does not exist. Use of the LQD rather than the variance can present analytical and computational challenges. In this thesis we show that the impact can be mitigated by assuming that the log returns are from an alpha-stable moving average (sma) process. The formulae derived for the LQD of the temporal aggregation of an sma process allow the lqd shape to be examined as a function of aggregation level. Asymptotically normal estimators are proposed for the lqd of the temporal aggregation of an SMA process, which require asymptotically normal estimators of the sma process. The quantile-based stable distribution parameter estimators of McCulloch (1986) are adapted for use from an sma process rather than an independent process. Traditionally such estimators have been calculated at the standard quantile levels originally proposed by McCulloch (1986). In this thesis, the quantile levels are identified which optimise estimators from a selection of SMA processes. We find that in many cases these optimised quantile-based estimators significantly outperform the quantile-based estimators using the standard quantile levels. Improved evaluations of the maximum likelihood estimator asymptotics are made to calculate the relative asymptotic e¢ ciency of the optimal quantile-based estimators. Methods for order identi…cation of an sma process are developed and studied. An extension to the SV model is proposed, which we call the stable stochastic volatility (SSV) model, where the conditional distribution of the daily log returns is alpha-stable. Estimation of the SSV model parameters is done utilising the LQD estimators of the temporal aggregation of the intraday log return process together with an allowance for measurement error. The methods proposed in this thesis are illustrated in an empirical study carried out on ASX00 index data from 2009 and 2010. A similar study was carried out by the author on the same data in Barker (2014).


Table of Contents

1. Introduction -- 2. Log quantile difference of the temporal aggregation of stable moving average processes -- 3. Estimation of the parameters of a stable moving average process -- 4. Estimation of the log quantile difference of the temporal aggregation an SMA(q) process -- 5. Measurement of realised volatility using log quantile differences -- 6. Application -- 7. Conclusion -- A. Proofs -- B. Stochastic convergence -- C. Time series -- D. Quantile estimation -- E. Stable distributions.


Includes bibliographical references Empirical thesis.

Awarding Institution

Macquarie University

Degree Type

Thesis PhD


PhD, Macquarie University, Faculty of Science and Engineering, Department of Statistics

Department, Centre or School

Department of Statistics

Year of Award


Principal Supervisor

Nino Kordzakhia


Copyright Adrian Walter Barker 2015. Copyright disclaimer:




1 online resource (xxii, 206 pages)

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