Narrow Results

In this chapter we study the relationships among random variables, which will be characterized by the joint probability distribution of random variables. Most insights into multivariate distributions can be gained by focusing on bivariate distributions. We first introduce the joint probability distribution of a bivariate random vector (X, Y ) via the characterization of the joint cumulative distribution function, the joint probability mass function (when (X, Y ) are discrete), and the joint probability density function (when (X, Y ) are continuous) respectively. We then characterize various aspects of the relationship between X and Y using the conditional distributions, correlation, and conditional expectations. The concept of independence and its implications on the joint distributions, conditional distributions and correlation are also discussed. We also introduce a class of bivariate normal distributions.

In this chapter, we shall build on the fundamental notions of probability distribution and statistics in the last chapter, and extend consideration to a sequence of random variables. In financial application, it is mostly the case that the sequence is indexed by time, hence a stochastic process. Interesting statistical laws or mathematical theories result when we look at the relationships within a stochastic process. We introduce an application of the Central Limit Theorem to the study of stock return distributions.

The landscape of financial econometrics was forever changed and augmented greatly when Professor Engle introduced the Autoregressive Conditonal Heteroskedasticity model in 1982. It was an ingeniously embedded tool in specifying the dynamics of volatility coupled with the underlying asset price process. This greatly extended the space of stochastic processes including those in the Box Jenkins approach. ARCH was very successfully extended to GARCH by Prof. Bollerslev, and to-date there continues to be a huge number of variations of processes building on the embedded tooling idea. We consider such process modelling to be particularly relevant in estimating risks in market prices and in risk management in today's market of high and persistent volatilities. The topic of maximum likelihood is given more discussion here, as well as ideas of asymptotic efficiency.