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# Unit Roots for Time-Series Analysis

## The Unit Root Problem

An AR(1) series is said to be covariance stationary if the absolute value of the lag coefficient $$\text{b}_{1}$$ is less than 1. If the absolute value of $$\text{b}_{1}=1$$, the time series is said to have a unit root. All random walks have a unit root since they have $$\text{b}_{1}=1$$. This implies that they are not covariance stationary; hence we cannot apply the standard linear regression to test for $$\text{b}_{1}=1$$.

A Dickey-Fuller test can be used to establish if the time series has a unit root. A time series with unit roots should be transformed by first-differencing it to a covariance stationary time series, which can be effectively analyzed using regression analysis.

First-differencing is a technique that involves subtracting the dependent variable in the immediately preceding period from the current value of the time series to define a new dependent variable, $$y$$. Thus, we model the change in the value of the dependent variable.

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

The first-differenced time series can then be modeled as an autoregressive time series. A properly differenced random walk time series is covariance stationary with a mean reversion level of 0.

## Question

Which of the following is the most appropriate approach to transforming a time series with a unit root problem?

1. Performing a Dickey-Fuller test.
2. Modeling the first differences of the time series.
3. Performing a log-linear transformation.

### Solution

A time series with a unit root can be first differenced to transform it into one that is covariance stationary. This is followed by estimating an autoregressive model for the first-differenced series.

First differencing involves subtracting the value of the time series in the immediately preceding period from the current value of the series to define a new dependent variable, $$y$$.

A is incorrect. The Dickey fuller test is used to determine if the time series has a unit root.

C is incorrect. A log-linear transformation is appropriate when data grows at a constant rate.

LOS 3 (j) Describe implications of unit roots for time-series analysis, explain when unit-roots are likely to occur and how to test for them, and demonstrate how a time series with a unit root can be transformed so it can be analyzed with an AR model.

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