Chapter 8 Time series
In this chapter, we provide a brief introduction to performing inference in time series models using a Bayesian framework. There is a large literature on time series in statistics and econometrics, making it impossible to present a thorough treatment in just a few pages of an introductory book. However, there are excellent books on Bayesian inference in time series; see, for instance, West and Harrison (2006), Petris, Petrone, and Campagnoli (2009), and Pole, West, and Harrison (2018).
A time series is a sequence of observations collected in chronological order, allowing us to track how variables change over time. However, it also introduces technical challenges, as we must account for statistical features such as autocorrelation and stationarity. Since time series data is time-dependent, we adjust our notation. Specifically, we use t and T instead of i and N to explicitly indicate time.
Our starting point in this chapter is the state-space representation of time series models. Much of the Bayesian inference literature in time series adopts this approach, as it allows dynamic systems to be modeled in a structured way. This representation provides modularity, flexibility, efficiency, and interpretability in complex models where the state evolves over time. It also enables the use of recursive estimation methods, such as the Kalman filter for dynamic Gaussian linear models and the particle filter (also known as sequential Monte Carlo) for non-Gaussian and nonlinear state-space models. The latter method is especially useful for online predictions or when there are data storage limitations. These inferential tools are based on the sequential updating process of Bayes’ rule, where the posterior at time t becomes the prior at time t+1.
Remember that we can run our GUI typing shiny::runGitHub("besmarter/BSTApp", launch.browser=T) in the R console or any R code editor and execute it. However, users should see Chapter 5 for details.