Multivariate Time Series, Linear Systems and Kalman Filtering

Abstract

This chapter is devoted to the analysis of the time evolution of random vectors. The first section presents the generalization to the multivariate case of the univariate time series models studied in the previous chapter. Modern accounts of time series analysis increasingly rely on the formalism and the techniques developed for the analysis of general stochastic systems. Even though financial applications have remained mostly immune to this evolution, because of its increased popularity and its tremendous potential, we decided to include this alternative approach in this chapter. The tone of the chapter will have to change slightly as we discuss concepts and theories which were introduced and developed in engineering fields far remote from financial applications. The practical algorithms were developed mostly for military applications. They led to many civil and technological breakthroughs. Here, we present the main features of the filtering algorithms, and we use financial examples as illustrations, restricting ourselves to linear systems.