DSAS - PhD Public Lecture - Jingjia Chu

Title: Modelling the Common Risk among Equities: A Multivariate Time Series Model with an Additive GARCH Structure

Abstract:
A new additive structure of multivariate GARCH model is proposed where the dynamic changes of the conditional correlation between the stocks are aggregated by the common risk term. The observable sequence is divided into two parts, a common risk term and an individual risk term, both following a GARCH type structure. The conditional volatility of each stock will be the sum of these two conditional variance terms. All the conditional volatility of the stock can shoot up together because a sudden peak of the common volatility is a sign of the system shock.

We provide sufficient conditions for strict stationarity and ergodicity of the model. The ergodicity of the model cannot be studied in the standard way because of the non-linearity. After reforming the original mathematical representation of the model into a complicated Markovian structure, the systematic theory for Markov chain from Meyn and Tweedie (2009) is applied.

All the parameters in the model are identifiable in terms of the second conditional moments under mild assumptions. Then there exists a unique solution of parameters in the domain which maximizes the likelihood function for a sufficiently large sample size. The choice of starting values is unimportant within the parameter space defined by the ergodicity theorem. Under some general assumptions we proposed, without specifying the distribution of the innovation, different initial values will lead to the same estimates asymptotically. Once both assumptions for ergodicity and identifiability are satisfied, the quasi maximum likelihood (QML) has become a reasonable method to estimate parameters in practice. The sufficient conditions for the strong consistency of the QML estimator are proposed.

The Monte Carlo simulation example is included in this thesis to demonstrate how to verify the assumptions in the strict stationarity and asymptotic normality theorems. The numeric issues for the estimating process in practice are addressed with possible solutions.

Supervisors: Hao Yu and Reg Kulperger