A sequence of observations indexed by the time of each observation is called a time series. Time series analysis is the art of specifying the most likely stochastic model that could have generated the observed series. The aim is to understand the relations among the variables in order to make better predictions and decisions. One particularly important application of time series analysis in financial econometrics is the study of how volatilities change over time because financial assets demand returns commensurate with their volatility levels.

Introduction

A time series of prices is called a financial time series: The prices can be those of stocks, bonds, currencies, futures, commodities, and countless others. Other time series of financial interest include those of prepayment speeds of MBSs and various economic indicators.

Models for the time series can be conjectured from studying the data or can be suggested by economic theory. Most models specify a stochastic process. Models should be consistent with past data and amenable to testing for specification error. They should also be simple, containing as few parameters as possible. Model parameters are to be estimated from the time series. However, because an observed series is merely a sample path of the proposed stochastic process, this is possible only if the process possesses a property called ergodicity. Ergodicity roughly means that sample moments converge to the population moments as the sample path lengthens.

The basic steps are illustrated with the maximum likelihood (ML) estimation of stock price volatility.