White noise

Random walks

Biased random walks

A time series \(\{w_t\}\) is discrete white noise if its values are

1. independent

2. identically distributed with a mean of zero

A time series \(\{w_t\}\) is discrete white noise if its values are

1. independent

2. identically distributed with a mean of zero

Note that distributional form for \(\{w_t\}\) is flexible

We often assume so-called Gaussian white noise, whereby

\[ w_t \sim \text{N}(0,\sigma^2) \]

We often assume so-called Gaussian white noise, whereby

\[ w_t \sim \text{N}(0,\sigma^2) \]

and the following apply as well

    autocovariance:  \(\gamma_k = \begin{cases} \sigma^2 & \text{if } k = 0 \\ 0 & \text{if } k \geq 1 \end{cases}\)

    autocorrelation:   \(\rho_k = \begin{cases} 1 & \text{if } k = 0 \\ 0 & \text{if } k \geq 1 \end{cases}\)

A time series \(\{x_t\}\) is a random walk if

1. \(x_t = x_{t-1} + w_t\)

2. \(w_t\) is white noise

A biased random walk (or random walk with drift) is written as

\[ x_t = x_{t-1} + u + w_t \]

where \(u\) is the bias (drift) per time step and \(w_t\) is white noise