The random walk (RW) model

Time Series Analysis in R

David S. Matteson

Associate Professor at Cornell University

Random walk

Random Walk (RW) is a simple example of a non-stationary process.

A random walk has:

  • No specified mean or variance.
  • Strong dependence over time.
  • Its changes or increments are white noise (WN).
Time Series Analysis in R

Random walk

Time series plots of Random Walk:

Time Series Analysis in R

Random walk

The random walk recursion:

$$Today = Yesterday + Noise$$

More formally:

$$ Y_t = Y_{t-1} + \epsilon_t$$

where $\epsilon_t$ is mean zero white noise (WN).

  • Simulation requires an initial point $Y_0$.

  • Only one parameter, the WN variance $\sigma^{2}_{\epsilon}$.

Time Series Analysis in R

Random walk - I

The random walk process:

$$Y_t = Y_{t-1} + \epsilon_t$$

where $\epsilon_t$ is mean zero WN

As $Y_t - Y_{t-1} = \epsilon_t \rightarrow $ diff(Y) is WN

Time Series Analysis in R

Random walk - II

Time Series Analysis in R

Random walk with drift - I

The random walk with a drift:

$$Today = Constant + Yesterday + Noise$$

More formally: $$Y_t = c + Y_{t-1} + \epsilon_t$$

where $\epsilon_t$ is mean zero white noise (WN).

  • Two parameters, the constant $c$ , and the WN variance $\sigma_{\epsilon}^2$.
  • $Y_t - Y_{t-1} = $ ? $\rightarrow$ WN with mean $c$!
Time Series Analysis in R

Random walk with drift - II

Time series plots of Random Walk with drift:

Time Series Analysis in R

Let's practice!

Time Series Analysis in R

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