Time Series Modelling

Real‑world systems rarely evolve as isolated random observations. Instead, they unfold over time, with the present influenced by the past through mechanisms that introduce memory, persistence, and dynamic structure. Time series modelling provides a way to represent and study such behaviour within a probabilistic framework.

In this chapter, time series models are introduced from a simulation‑first perspective. Rather than treating models as static statistical objects, we examine them as stochastic processes that can be generated, perturbed, and observed through simulated sample paths.

A time series is a stochastic process indexed by time,

\[ X_t, \quad t = 1, 2, \dots \]

Beginning with processes that exhibit no memory, we gradually introduce dependence, explore how dynamic behaviour emerges, and investigate how shocks propagate through time and across systems.

The emphasis throughout is on understanding how temporal structure arises, what kind of behaviour it produces, and why it matters for interpreting data and anticipating future dynamics. By focusing on simulation, the chapter connects time series modelling naturally to the broader study of stochastic processes and computer simulation, highlighting dynamics, interpretation, and intuition over formal estimation procedures.

By the end of this chapter, you will be able to:

• Understand time series as stochastic processes with memory, and distinguish them from sequences of independent random variables.
• Simulate basic time series models, beginning with white noise and simple autoregressive structures.
• Interpret dynamic behaviour such as persistence, oscillation, stability, and mean reversion through simulated sample paths.
• Analyse the impact of shocks and understand how disturbances evolve and decay over time.
• Diagnose temporal dependence using tools such as the autocorrelation function (ACF) applied to simulated data.
• Extend simulation ideas to interacting systems, using multivariate models to study shock propagation across variables.
• Interpret forecasting as dynamic extrapolation, viewing future outcomes as paths generated by underlying stochastic dynamics.

These skills reinforce time series modelling as a core component of stochastic modelling and computer simulation, emphasising intuition, behaviour, and interpretation over purely analytical or estimation‑based approaches.