Chapter 4 Time Series Concepts

Updated: April 29, 2021

Financial variables such as asset prices and returns are naturally ordered by time. That is, these variables are time series variables. When we construct returns, the time index or data frequency becomes the investment horizon associated with the return. Typical data frequencies are daily, weekly, monthly and annual. In building probability models for time series variables, the time ordering of the data matters because we may think there are important temporal dependencies among the variables. For example, we might have reason to believe that the return on an asset this month is correlated with the return on the same asset in the previous month. This autocorrelation can only be defined if the time ordering of the data is preserved. A major complication in analyzing time series data is that the usual assumption of random sampling from a common population is not appropriate because it does not allow for any kind of time dependence in the time series variables. We would like to retain the notion that the observed data come from some population model, perhaps with time-invariant parameters, but we would like to allow the variables to have time dependencies. Fortunately, we can do this if the time series data come from a stationary time series process.

This chapter reviews some basic times series concepts that are essential for describing and modeling financial time series. Section 4.1 defines univariate time series processes and introduces the important concepts of stationarity and ergodicity. Covariance stationary time series processes are defined, which gives meaning to measuring linear time dependence using autocorrelation. The benchmark Gaussian White Noise process and related processes are introduced and illustrated using R. Some common non-stationary time series processes are also discussed including the famous random walk model. Section 4.2 introduces covariance stationary multivariate time series process. Such processes allow for dynamic interactions among groups of time series variables. Section 4.3 discusses time series model building and introduces the class of univariate autoregressive-moving average time series models and multivariate vector autoregression models. The properties of some simple models are derived and it is shown how to simulate observations from these models using R. The chapter concludes with a brief discussion of forecasting from time series models.

The R packages used in this chapter are mvtnorm and vars. Make sure these packages are installed and loaded before running the R examples in this chapter.