Essays on financial time series

This dissertation develops new tools for analyzing financial time series and applies them to real data.; Chapter 1 is motivated by the fact that one important aspect of financial markets has not received due attention in the literature: How the sensitivity of the financial market volatility to shocks varies with the volatility level, which is directly related to such questions as how rapidly and to what extent the market becomes turbulent when volatility-increasing shocks hit the market. This property is determined by the variance of variance as a function of the variance level. Both the popular GARCH(1,1) and the SQGARCH(1,1), newly introduced as an ARCH-analogue of the continuous-time square-root stochastic volatility model, severely restrict the variance-of-variance function. A new model called the CEVGARCH is proposed that allows more flexible fitting of the variance-of-variance function. It is shown that a CEVGARCH(1,1) process with an integrated or mildly "explosive" deterministic drift component may still be stationary if the variance of variance grows with the variance level at least as fast as in the GARCH(1,1), highlighting the importance of the joint role of the variance-of-variance term and the deterministic drift term in determining the stability properties of a time series.; In Chapter 2, it is empirically found that high elasticity of variance of variance in the CEVGARCH is a common property of stock markets around the world, which suggests that stock market volatilities tend to rise fast in response to a series of volatility-increasing shocks. A new EGARCH-like model is introduced to investigate whether the fast variance-of-variance growth estimates coupled with high volatility persistence are not merely a symptom of model misspecification. Empirical results with the SP 500 index data indicate that the elasticity of variance of variance is still high even when the new flexible-drift, exponential CEVGARCH model is used.; Chapter 3 proposes new ways to construct probability integral transforms of random vectors that "scan" multivariate conditional densities from different angles. An illustrative bivariate normal example is given. An empirical example is also given that applies different probability integral transforms to testing the specification of Engle's (2002) dynamic conditional correlation model for multivariate financial time series.