Random Walk

Basic Concepts

A random walk time series y₁, y₂, …, y_n takes the form

where

If δ = 0, then the random walk is said to be without drift, while if δ ≠ 0, then the random walk is with drift (i.e. with drift equal to δ).

It is easy to see that for i > 0

It then follows that E[y_i] = y₀ + δi, var(y_i) = σ²i and cov(y_i, y_j) = 0 for i ≠ j. The variance values are not constants but vary with time i, and so this type of time series is not stationary. Also, the mean values are constant only for a random walk without drift.

Note too that since cov(ε_i,ε_j) = 0 for i ≠ j, it follows that

Note that the first difference z_i = y_i – y_i-1 of a random walk is stationary since it takes the form

which is a purely random time series.

Plot

Example 1: Graph the random walk with drift y_i = y_i-1 + ε_i where the ε_i ∼ N(0,.5).

The graph is shown in Figure 1. All the cells in column B contain the formula =NORM.INV(RAND(),0,.5), cell C4 contains the formula =1+B4 and cell C5 contains the formula =1+B5+C4.

As we can see, the graph shows a clear upward trend and the ACF shows a slow descent.

Figure 1 – Random Walk

First differences are taken between the y values as shown in Figure 2. E.g. cell C5 contains the formula = B5-B4 (where column B replicates the values in column C from Figure 1). We see from the chart that the trend has been eliminated. We also see from the Ljung-Box test (cell F13) that the ACF values for the first 7 lags are statistically equal to zero, consistent with a purely random process.

Figure 2 – First differences of a random walk