Chapter 8

Multivariate Time Series Analysis and Its Applications

Economic globalization and Internet communication have accelerated the integration of world financial markets in recent years. Price movements in one market can spread easily and instantly to another market. For this reason, financial markets are more dependent on each other than ever before, and one must consider them jointly to better understand the dynamic structure of the global finance. One market may lead the other market under some circumstances, yet the relationship may be reversed under other circumstances. Consequently, knowing how the markets are interrelated is of great importance in finance. Similarly, for an investor or a financial institution holding multiple assets, the dynamic relationships between returns of the assets play an important role in decision making. In this and the next two chapters, we introduce econometric models and methods useful for studying jointly multiple return series. In the statistical literature, these models and methods belong to vector or multivariate time series analysis.

A multivariate time series consists of multiple single series referred to as components. As such, concepts of vector and matrix are useful in understanding multivariate time series analysis. We use boldface notation to indicate vectors and matrices. If necessary, readers may consult Appendix A of this chapter for some basic operations and properties of vectors and matrices. Appendix B provides some results of multivariate normal distribution, which is widely used in multivariate statistical analysis (e.g., Johnson and Wichern, 1998).

Let be the log returns of k assets at time t, where denotes the transpose of . For example, an investor holding stocks of IBM, Microsoft, Exxon Mobil, General Motors, and Wal-Mart may consider the five-dimensional daily log returns of these companies. Here r1t denotes the daily log return of IBM stock, r2t is that of Microsoft, and so on. As a second example, an investor who is interested in global investment may consider the return series of the S&P 500 index of the United States, the FTSE 100 index of the United Kingdom, and the Nikkei 225 index of Japan. Here the series is three-dimensional, with r1t denoting the return of the S&P 500 index, r2t the return of the Financial Times Stock Exchange (FTSE) 100 index, and r3t the return of the Nikkei 225. The goals of this chapter are (a) to explore the basic properties of and (b) to study econometric models for analyzing the multivariate data .

Many of the models and methods discussed in previous chapters can be generalized directly to the multivariate case. But there are situations in which the generalization requires some attention. In some situations, one needs new models and methods to handle the complicated relationships between multiple series. In this chapter, we discuss these issues with emphasis on intuition and applications. For statistical theory of multivariate time series analysis, readers are referred to Lütkepohl (2005) and Reinsel (1993).