Chapter 1

Financial Time Series and Their Characteristics

Financial time series analysis is concerned with the theory and practice of asset valuation over time. It is a highly empirical discipline, but like other scientific fields theory forms the foundation for making inference. There is, however, a key feature that distinguishes financial time series analysis from other time series analysis. Both financial theory and its empirical time series contain an element of uncertainty. For example, there are various definitions of asset volatility, and for a stock return series, the volatility is not directly observable. As a result of the added uncertainty, statistical theory and methods play an important role in financial time series analysis.

The objective of this book is to provide some knowledge of financial time series, introduce some statistical tools useful for analyzing these series, and gain experience in financial applications of various econometric methods. We begin with the basic concepts of asset returns and a brief introduction to the processes to be discussed throughout the book. Chapter 2 reviews basic concepts of linear time series analysis such as stationarity and autocorrelation function, introduces simple linear models for handling serial dependence of the series, and discusses regression models with time series errors, seasonality, unit-root nonstationarity, and long-memory processes. The chapter also provides methods for consistent estimation of the covariance matrix in the presence of conditional heteroscedasticity and serial correlations. Chapter 3 focuses on modeling conditional heteroscedasticity (i.e., the conditional variance of an asset return). It discusses various econometric models developed recently to describe the evolution of volatility of an asset return over time. The chapter also discusses alternative methods to volatility modeling, including use of high-frequency transactions data and daily high and low prices of an asset. In Chapter 4, we address nonlinearity in financial time series, introduce test statistics that can discriminate nonlinear series from linear ones, and discuss several nonlinear models. The chapter also introduces nonparametric estimation methods and neural networks and shows various applications of nonlinear models in finance. Chapter 5 is concerned with analysis of high-frequency financial data, the effects of market microstructure, and some applications of high-frequency finance. It shows that nonsynchronous trading and bid–ask bounce can introduce serial correlations in a stock return. It also studies the dynamic of time duration between trades and some econometric models for analyzing transactions data. In Chapter 6, we introduce continuous-time diffusion models and Ito's lemma. Black–Scholes option pricing formulas are derived, and a simple jump diffusion model is used to capture some characteristics commonly observed in options markets. Chapter 7 discusses extreme value theory, heavy-tailed distributions, and their application to financial risk management. In particular, it discusses various methods for calculating value at risk and expected shortfall of a financial position. Chapter 8 focuses on multivariate time series analysis and simple multivariate models with emphasis on the lead–lag relationship between time series. The chapter also introduces cointegration, some cointegration tests, and threshold cointegration and applies the concept of cointegration to investigate arbitrage opportunity in financial markets, including pairs trading. Chapter 9 discusses ways to simplify the dynamic structure of a multivariate series and methods to reduce the dimension. It introduces and demonstrates three types of factor model to analyze returns of multiple assets. In Chapter 10, we introduce multivariate volatility models, including those with time-varying correlations, and discuss methods that can be used to reparameterize a conditional covariance matrix to satisfy the positiveness constraint and reduce the complexity in volatility modeling. Chapter 11 introduces state-space models and the Kalman filter and discusses the relationship between state-space models and other econometric models discussed in the book. It also gives several examples of financial applications. Finally, in Chapter 12, we introduce some Markov chain Monte Carlo (MCMC) methods developed in the statistical literature and apply these methods to various financial research problems, such as the estimation of stochastic volatility and Markov switching models.

The book places great emphasis on application and empirical data analysis. Every chapter contains real examples and, in many occasions, empirical characteristics of financial time series are used to motivate the development of econometric models. Computer programs and commands used in data analysis are provided when needed. In some cases, the programs are given in an appendix. Many real data sets are also used in the exercises of each chapter.