Multivariate Time Series Analysis and Applications

1Fundamental concepts and issues in multivariate time series analysis

With the development of computers and the internet, we have had a data explosion. For example, a study of monthly cancer rates in the United States during the past 10 years can involve 50 or many hundreds or thousands of time series depending on whether we investigate the cancer rates for states, cities, or counties. Multivariate time series analysis methods are needed to properly analyze these data in a study, and these are different from standard statistical theory and methods based on random samples that assume independence. Dependence is the fundamental nature of the time series. The use of highly correlated high‐dimensional time series data introduces many complications and challenges. The methods and theory to solve these issues will make up the content of this book.

1.1 Introduction

In studying a phenomenon, we often encounter many variables, Z_i,t, where i = 1, 2, …, m, and the observations are taken according to the order of time, t. For convenience we use a vector, Z_t = [Z_1,t, Z_2,t, …, Z_m,t]′, to denote the set of these variables, where Z_i,t is the ith component variable at time t and it is a random variable for each i and t. The time t in Z_t can be continuous and any value in an interval, such as the time series of electric signals and voltages, or discrete and be a specific time point, such as the daily closing price of various stocks or the total monthly sales of various products at the end ...