19.6 Random Walk

A time series with no autocorrelation is called White Noise. Consider the following plot of White Noise:

autoplot(ts(rnorm(100)))+ggtitle("White Noise")+theme_bw()

Let \(\epsilon_t\) be white noise then the random walk without drift is \[y_t = y_{t-1} + \epsilon_t\]

This is called an autoregressive model of order 1 or AR(1). Example: \[y_1 = y_0 + \epsilon_1\] Consider a different model written as \[y_2 = y_1 + \epsilon_2 = y_0 + \epsilon_1 + \epsilon_2\] This is not a stationary process and it can be shown that \(E(y_t) = y_0\) and \(Var(y_t) = t \cdot \sigma^2\). However \[y_t - y_{t-1} = \Delta y_t = \epsilon_t\]

Let \(\epsilon_t\) be white noise then the random walk with drift is \[y_t = c+y_{t-1} + \epsilon_t\] where \(c\) is the drift parameter. It can be shown that \(E(y_t) = y_0 + t \cdot c\) and \(Var(y_t) = t \cdot \sigma^2\). An autoregressive model AR(p) can be written as \[y_t = c+ \sum^p_{i=1} \phi_p y_{t-p} + \epsilon_t\]