Date of Award:


Document Type:


Degree Name:

Doctor of Philosophy (PhD)


Mathematics and Statistics

Committee Chair(s)

Piotr Kokoszka


Piotr Kokoszka


Richard Cutler


Michael Minnotte


Jürgen Symanzik


We study in this dissertation Generalized Autoregressive Conditionally Heteroskedastic (GARCH) time series. The research focuses on squared GARCH sequences. Our main results are as follows:

1. We compare three methods of constructing confidence intervals for sample autocorrelations of squared returns modeled by models from the GARCH family. We compare the residual bootstrap, block bootstrap and subsampling methods. The residual bootstrap based on the standard GARCH(l,1) model is seen to perform best. Confidence intervals for cross-correlations of a bivariate GARCH model are also studied.

2. We study a test to discriminate between long memory and volatility changes in financial returns data. Finite sample performance of the test is examined and compared using various variance estimators. The Bartlett kernel estimator with truncation lag determined by a calibrated bandwidth selection procedure is seen to perform best. The testing procedure is robust to various GARCH-type models.

3. We propose several methods of on-line detection of a change in unconditional variance in a conditionally heteroskedastic time series. We follow a paradigm in which the first m observations are assumed to follow a stationary process and the monitoring scheme has asymptotically controlled probability of falsely rejecting the null hypothesis of no change. Our theory is applicable to broad classes of GARCH-type time series and relies on a strong invariance principle which holds for the squares of observations generated by such models. Practical implementation of the procedures is proposed and the performance of the methods is investigated by a simulation study.



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