Stationarity, Autocorrelation, White Noise, and Linear Time Series

This tutorial introduces basic concepts about stationarity, autocorrelation, white noise, and linear time series.

1  Stationarity

1.1  Strict stationarity

A time series {\(r_t\)} is said to be strictly stationary if the joint distribution of \((t_{t_1},\cdots,r_{t_k})\,\) is identical to  \((t_{t_1+l},\cdots,r_{t_k+l})\,\) for all t, where k is an arbitrary positive integer and (\(t_1,\cdots,r_k\)) is a collection of k positive integers.

In other words, strict stationarity requires that the joint distribution of (\(r_{t_1},\cdots,r_{t_k}\)) is invariant under time shift. This is a very hard condition that is hard to verify empirically. Read more Stationarity, Autocorrelation, White Noise, and Linear Time Series

R in Time Series: White Noise and Random Walk

This tutorial introduces white noise and random walk.

1  White Noise

1.1  Motivation

When we fit mathematical models to time series data, if the model captured most of the deterministic features of the time series, the residual error series should appear to be a realization of independent random variable from some probability distribution. Due to this criteria of judging how good a model is in fitting given data, it seems natural to build models up from a model of independent randon variation, known as discrete white noise.

Read more R in Time Series: White Noise and Random Walk