# Cointegration

Consider a time series of the inflation rate $$(\text{y}_{\text{t}})$$ regressed on a time series of interest rates $$(\text{x}_{\text{t}})$$:

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

In this case, we have two different time series, $$\text{y}_{\text{t}}$$ and $$\text{x}_{\text{t}}$$. Either one of the time series is subject to non-stationarity. Recall that we test for non-stationarity using the Dickey-fuller test for each time series to check for unit roots.

## Outcomes of the Dickey-Fuller test

The results of a Dickey-Fuller test can either be:

##### a. Both the time series are covariance stationary.

This means that the time series do not have a unit root. In this case, we use linear regression to test the relationship between the two series. Besides, the regression coefficients are statistically reliable.

##### b. Only the dependent variable time series is covariance stationary.

This means that some of the linear regression assumptions are violated. Therefore, the estimated regression coefficients and the standard error would be inconsistent.

##### c. Only the independent variable time series is covariance stationary.

This has a similar implication to outcome two above.

##### d. None of the time series is covariance stationary

If none of the time series is covariance stationary, check whether or not they are cointegrated. Two time-series are said to be cointegrated if an economic or financial relationship exists between them, preventing them from diverging without bound in the long run.

We can test for cointegration using either the Engle-Granger or Dickey-Fuller test. The error term will be stationary, and the hypothesis tests will be valid if both the time series are covariance stationary and cointegrated.

## Testing for Cointegration

We can test whether the two time series are cointegrated through the following steps:

• Regress one variable on the other variable using the model: $$\text{y}_{\text{t}}=\text{b}_{0}+\text{b}_{1}\text{x}_{\text{t}}+\epsilon_{\text{t}}$$
• Test the errors for a unit root using the Dickey-Fuller test. The critical t-values are determined using the Engle and Granger test.

Rejection of the null hypothesis implies that the error terms are covariance stationary, and both series are cointegrated. Recall that the null hypothesis is that the error term has a unit root.

• Use regression to model the relationship between the two-time series if they are cointegrated.

If the (Engler-Granger) Dickey-Fuller test fails to reject the null hypothesis, we conclude that the error term is not covariance stationary. This means that the two series are not cointegrated. Consequently, the linear regression is invalid.

## Question

An analyst tests the two time series errors for a unit root using the Dickey-Fuller test and determines the critical values using the Engle and Granger test. The test fails to reject the null hypothesis. The most accurate conclusion is that:

1. The two series are cointegrated.
2. The two series are not cointegrated.
3. The error terms are covariance stationary.

#### Solution

If the (Engler-Granger) Dickey-Fuller test fails to reject the null hypothesis, we conclude that the error terms are not covariance stationary. This means that the two series are not cointegrated. Consequently, a linear regression cannot be used.

LOS 5 (n) Explain how time-series variables should be analyzed for nonstationary and/or cointegration before use in linear regression.

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