Random Walk Process

A time series is said to follow a random walk process if the predicted value of the series in one period is equivalent to the value of the series in the previous period plus a random error.

A simple random walk process can be expressed as follows:

$$\text{x}_{\text{t}}=\text{x}_{\text{t}-1}+\epsilon_{\text{t}}$$

Where:

• \(\text{x}_{\text{t}}\) = Best prediction tomorrow.
• \(\text{x}_{\text{t}-1}\) = Best value today.
• \(\epsilon_{\text{t}}\) = Random error term.

Characteristics of Random Walk Time Series

• An AR(1) time series with \(\beta_{0}=0\) and \(\beta_{1}=1\) is a random walk. This is because the best prediction for tomorrow is the best value today plus a random error term.
• The expected value of the error term \(\epsilon_{\text{t}}\) is equal to zero.
• The variance of the residuals is constant. 
• Random walk with drift: A time series follows a random walk with drift if it has a non-zero constant intercept term. It is expressed as:

$$\text{x}_{\text{t}}=\text{b}_{0}+\text{b}_{1}\text{x}_{\text{t}-1}+\epsilon_{\text{t}}$$

Where:

• \(\text{b}_{0}\) = Constant drift
• \(\text{b}_{1}\) = 1

Note that a random walk is expressed as:

$$\text{x}_{\text{t}}=\text{b}_{0}+\text{b}_{1}\text{x}_{\text{t}-1}+\epsilon_{\text{t}}$$

• \(\text{b}_{0}=0\) for a random walk without drift;
• \(\text{b}_{0}\neq 0\) for a random walk with drift; and
• \(\text{b}_{1}=1\) for a random walk with or without drift.

A random walk has an undefined mean reversion level. If has a mean-reverting level, i.e., \(\text{x}_{\text{t}}=\text{b}_{0}+\text{b}_{1}\text{x}_{\text{t}},\) then \(\text{x}_{\text{t}}=\frac{\text{b}_{0}}{1-\text{b}_{1}}\). However, in a random walk, \(\text{b}_{0}=0\) and \(\text{b}_{1}=1\), so, \(\text{x}_{\text{t}}=\frac{0}{1-1}=0\).

A random walk is not covariance stationary. The covariance stationary property suggests that the mean and variance terms of a time series remain constant over time. However, the variance of a random walk process does not have an upper bound. As \(t\) increases, the variance grows with no upper bound. This implies that we cannot use standard regression analysis on a time series that appears to be a random walk.

Question

The most accurate statement about a random walk is that it:

1. Has a finite mean-reverting level.
2. Has an undefined mean-reverting level.
3. Is covariance stationary.

Solution

The correct answer is B.

A random walk process has an undefined mean-reverting level, and thus it is not covariance stationary.

Reading 5: Time Series Analysis

LOS 5 (i) Describe characteristics of random walk processes and contrast them to covariance stationary processes.