Random Walks and Unit Roots

Random walk. If a time series follows a random walk process, the predicted value of the series (i.e., the value of the dependent variable) in one period is equal to the value of the series in the previous period plus a random error term.

A time series that follows a simple random walk process is described in equation form as $x_t$ = $x_{t-1}$ + ${\epsilon }_t$, where the best forecast of $x_t$ is $x_{t-1}$ and:

1. E(${\epsilon }_t$) = 0: The expected value of each error term is zero.
2. E(${\epsilon }_t^2$) = ${\sigma }^2$: The variance of the error terms is constant.
3. E(${\epsilon }_i$${\epsilon }_j$) = 0; if i ≠ j: There is no serial correlation in the error terms.

Random walk with a drift. If a time series follows a random walk with a drift, the intercept term is not equal to zero. That is, in addition to a random error term, the time series is expected to increase or decrease by a constant amount each period.

A random walk with a drift can be described as:

Covariance stationarity. Neither a random walk nor a random walk with a drift exhibits covariance stationarity. To show this, let’s start by expressing a random walk as:

In either case (with or without a drift), the mean reverting level is $\frac{b_0}{1-b_1}=\frac{b_0}{0}$ (the division of any number by zero is undefined), and as we stated earlier, a time series must have a finite mean reverting level to be covariance stationary. Thus, a random walk, with or without a drift, is not covariance stationary, and exhibits what is known as a unit root ($b_1$ = 1). For a time series that is not covariance stationary, the least squares regression procedure that we have been using to estimate an AR(1) model will not work without transforming the data.

We discuss unit roots and how they are handled in the next section.

As we discussed in the previous LOS, if the coefficient on the lag variable is 1, the series is not covariance stationary. If the value of the lag coefficient is equal to one, the time series is said to have a unit root and will follow a random walk process. Since a time series that follows a random walk is not covariance stationary, modeling such a time series in an AR model can lead to incorrect inferences.

Unit Root Testing for Nonstationarity

To determine whether a time series is covariance stationary, we can (1) run an AR model and examine autocorrelations, or (2) perform the Dickey Fuller test.

In the first method, an AR model is estimated and the statistical significance of the autocorrelations at various lags is examined. A stationary process will usually have residual autocorrelations insignificantly different from zero at all lags or residual autocorrelations that decay to zero as the number of lags increases.

A more definitive test for unit root is the Dickey Fuller test. For statistical reasons, you cannot directly test whether the coefficient on the independent variable in an AR time series is equal to

1. To compensate, Dickey and Fuller created a rather ingenious test for a unit root. Remember, if an AR(1) model has a coefficient of 1, it has a unit root and no finite mean reverting level (i.e., it is not covariance stationary). Dickey and Fuller (DF) transform the AR(1) model to run a simple regression.

To transform the model, they (1) start with the basic form of the AR(1) model and (2) subtract $x_{t-1}$ from both sides:

Then, rather than directly testing whether the original coefficient is different from 1, they test whether the new, transformed coefficient ($b_1$ − 1) is different from zero using a modified t test. If ($b_1$ − 1) is not significantly different from zero, they say that $b_1$ must be equal to 1.0 and, therefore, the series must have a unit root.

First Differencing

If we believe a time series is a random walk (i.e., has a unit root), we can transform the data to a covariance stationary time series using a procedure called first differencing. The first differencing process involves subtracting the value of the time series (i.e., the dependent variable) in the immediately preceding period from the current value of the time series to define a new dependent variable, y. Note that by taking first differences, you model the change in the value of the dependent variable

So, if the original time series of x has a unit root, the change in x, $x_t$ − $x_{t-1}$ = ${\epsilon }_t$, is just the error term. This means we can define $y_t$ as:

Then, stating y in the form of an AR(1) model:

This transformed time series has a finite mean reverting level of $\frac{0}{1-0}=0$ and is, therefore, covariance stationary.