# Ph.d.-forsvar: Rasmus Søndergaard Pedersen:Inference and Testing in Multivariate GARCH Models

Most financial applications are, by nature, multivariate with estimates and forecasts of conditional covariance matrices as important components as in, for example, the rich asset pricing, portfolio choice, and risk management literature. One way of obtaining such estimates and forecasts is by estimation of multivariate generalized autoregressive conditional heteroskedasticity (GARCH) models - a class of models that, by now, is heavily used within the fields of financial econometrics and empirical finance. My thesis consists of three chapters on estimation of and large-sample inference in multivariate GARCH models.

In the first chapter we consider asymptotic inference in the multivariate BEKK-GARCH model based on (co)variance targeting (VT). By definition the VT estimator is a two-step estimator and the theory presented is based on expansions of the modified likelihood function, or estimating function, corresponding to these two steps. Strong consistency is established under weak moment conditions, while sixth-order moment restrictions are imposed to establish asymptotic normality.

Existing literature on VT estimation of multivariate GARCH models, including the first chapter of this thesis, relies on at least finite fourth-order moments of the data generating process in order to derive the large-sample distribution of the variance targeting estimator. Such moment conditions may not be a realistic assumption as financial return distributions are typically found to be heavy tailed. In the second chapter we consider the large-sample properties of the VT estimator for the multivariate extended constant conditional correlation (ECCC-)GARCH model when the distribution of the data generating process has infinite fourth-order moments. Using non-standard limit theory we derive new results for the estimator stating that, under suitable conditions, its limiting distribution is multivariate stable (different from a Gaussian distribution). The rate of consistency of the estimator is shown to depend on the tail shape of the data generating process.

Lastly, in the third chapter we consider testing for volatility spillovers (or interactions) in ECCC-GARCH models. The proposed tests imply that the parameter vector under the null hypothesis lies on the boundary of the maintained hypothesis, which leads to non-standard limiting distributions of the test statistics. The large-sample properties of the quasi-maximum likelihood estimator are derived together with limiting distributions of the related Lagrange multiplier, Wald, and quasi-likelihood ratio statistics. As an empirical illustration, the proposed tests are applied to test for volatility spillovers between returns on foreign exchange rates.