Essays on Volatility Modeling

Time series of squared returns are typically highly autocorrelated. This has led to a voluminous literature on volatility modeling. Broadly speaking, there are two main classes of volatility models: Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models and stochastic volatility models.
In the first three papers of this thesis, my co-authors and I propose and examine an empirically motivated extension of the GARCH model. We augment the GARCH conditional variance by a time-varying intercept. This specification entails mainly two theoretical challenges. Firstly, the time-variation in the intercept makes the model nonstationary. This complicates the proofs of the two results that typically function as cornerstones of inference in time series models: consistency and asymptotic normality of the (quasi-) maximum likelihood estimator. In the stationary case, these results are known to hold under mild conditions. In the first paper, we use the theory of locally stationary processes to prove that the results continue to hold for our model under slightly stronger but empirically reasonable assumptions. The second challenge is that the parametric function that we use to model the time-variation in the intercept is unidentified if the intercept is constant. This makes it necessary to test for a type of additive misspecification before attempting to fit our proposed model. However, the identification problem complicates this type of inference. The second and third papers propose solutions.
In the last paper, I consider a stochastic volatility model that is motivated by financial theory. A commonly accepted and well motivated prediction of financial theory is that financial leverage makes equity more risky. The theory is based around a structural model of the firm in which equity is modeled as a call option on the firm's assets. A recent contribution in the GARCH literature has exploited this to propose a so-called structural GARCH model. I use the same reasoning to propose a parameterization in the stochastic volatility class: a structural stochastic volatility model. As the model is of the stochastic volatility type, inference is more complicated than in the GARCH case. By a close study of the literature on estimating stochastic volatility, I find that the methods of quasi maximum likelihood and Monte Carlo maximum likelihood work well for the proposed model. I use the methods to develop a framework for estimation and misspecification testing. I provide two empirical examples that highlight the utility and relevance of the model.