 # The Unit Root Test for Nonstationary

Unit root testing involves checking whether the time series is covariance stationary. We can either form an AR model and check for autocorrelations or perform a Dickey and Fuller test.

A t-test is performed to examine the statistical significance of autocorrelations at various lags. The autocorrelations of a stationary process will be insignificantly different from zero at all lags or decrease rapidly to zero as the number of lags becomes large.

Recall that an AR (1) model is said to be covariance stationary if the absolute value of  is less than 1. Additionally, the time series will have a unit root if the lag coefficient equals 1. However, random walks with drift are NOT covariance stationary. Therefore, we cannot use standard linear regression to test for covariance stationary Dickey-Fuller unit root test is therefore used to test for unit roots.

## Dickey-Fuller Test

The Dickey-Fuller test uses the transformed AR(1) model by substracting  from both sides of the equation. i.e.,

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

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

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

Let the transformed coefficient, $$\text{b}_{1}-1=\text{g}$$. Then,

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

We then perform a t-test with the null hypothesis $$\text{H}_{0}:\text{g}_{1}=0$$ (a test of $$\text{b}_1={1}$$, which implies the time-series has a unit root), against the alternative hypothesis $$\text{H}_{\text{a}}:\text{g}_{1}<0$$ (a test of $$\text{b}_1={1}$$, which implies the time series has no unit root).

If $$\text{g}_{1}$$ is not significantly different from 0, then $$\text{b}_{1}= 0$$, the series must have a unit root. The null hypothesis will be rejected if the time series does not have a unit root. On the other hand, failure to reject the null hypothesis means that the time series has a unit root.

A time series with a unit root problem can be modeled by modeling the first differenced series with an autoregressive time series.

## Question

If the null hypothesis $$\text{H}_{0}:\text{g}_1=0$$ under the Dickey-Fuller test cannot be rejected; the most accurate conclusion is that:

1. The time series is covariance stationary.
2. The time series does not have a unit root.
3. The time series has a unit root.

#### Solution

If the null hypothesis cannot be rejected, then $$\text{b}_1-1=0$$ and $$\text{b}_1=1$$. This implies that the time series has a unit root. If the null hypothesis is rejected, the time series does not have a unit root.

A is incorrect. Rejection of the null hypothesis implies that $$\text{b}_1-1=0$$ and so the time series has a unit root. However, a time series with a unit root is not covariance stationary.

LOS 5 (k) Describe the steps of the unit root test for nonstationary and explain the relation of the test to autoregressive time-series models.

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