Whittle estimation of multivariate exponential volatility models

The aim of this thesis is to offer some insights into two topics of some interest for time-series econometric research. The first chapter derives the rates of convergence and the asymptotic normality of the pooled OLS estimators for linear regression panel models with mixed stationary and non-stationary regressors. This work is prompted by the consideration that many economic models of interest present a mixture of I(1) and I(0) regressors, for example models for analysis of demand system or for assessment of the relationship between growth and inequality. We present results for a model where the regressors and the regressand are cointegrated. We find that the OLS estimator is asymptotically normal with convergence rates T p n and p nT for respectively the non-stationary and the stationary regressors. Phillips and Moon (1990) show that in a cointegrated regression model with non-stationary regressors, the OLS estimator converges at a rate of T p n. We find that the presence of one stationary regressor in the model does not increases the rate of convergence. All the results are derived for sequential limits, with T going to infinity followed by n; and under quite restrictive regularity conditions. Chapters 3-5 focus on parametric multivariate exponential volatility models. It has long been recognized that the volatility of stock returns responds differently to good news and bad news. In particular, while negative shocks tend to increase future volatility, positive ones of the same size will increase it by less or even decrease it. This was in fact one of the chief motivations that led Nelson (1991) to introduce the univariate EGARCH model. More recently empirical studies have found that the asymmetry is a robust feature of multivariate stock returns series as well, and several multivariate volatility models have been developed to capture it. Another important property that characterizes the dynamic evolution of volatilities is that squared returns have significant autocorrelations that decay to zero at a slow rate, consistent with the notion of long memory, where the auto-covariances are not absolutely summable. Univariate long-memory volatility models have received a great deal of attention. However, the generalization to a multivariate long-memory volatility model has not been attempted in the literature. Chapter 3 offers a detailed literature review on multivariate volatility models. Chapter 4 and 5 introduce a new multivariate exponential volatility (MEV) model which captures long-range dependence in the volatilities, while retaining the martingale difference assumption and short-memory dependence in mean. Moreover the model captures cross-assets spillover effects, leverage and asymmetry. The strong consistency and the asymptotic normality of the Whittle estimator of the parameters in the Multivariate Exponential Volatility model is established under a variety of parameterization. The results cover both the case of exponentially and hyperbolically decaying coefficients, allowing for different degrees of persistence of shocks to the conditional variances. It is shown that the rate of convergence and the asymptotic normality of the Whittle estimates do not depend on the degree of persistence implied by the parameterization as the Whittle function automatically compensates for the possible lack of square integrability of the model spectral density.