# 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

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

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

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