Limited Time Offer: Save 10% on all 2021 and 2022 Premium Study Packages with promo code: BLOG10    Select your Premium Package »

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

Featured Study with Us
CFA® Exam and FRM® Exam Prep Platform offered by AnalystPrep

Study Platform

Learn with Us

Subscribe to our newsletter and keep up with the latest and greatest tips for success
Online Tutoring
Our videos feature professional educators presenting in-depth explanations of all topics introduced in the curriculum.

Video Lessons

Daniel Glyn
2021-03-24
I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
michael walshe
2021-03-18
Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.
Nyka Smith
2021-02-18
Every concept is very well explained by Nilay Arun. kudos to you man!
2021-02-13
Agustin Olcese
2021-01-27
Excellent explantions, very clear!
Jaak Jay
2021-01-14
Awesome content, kudos to Prof.James Frojan
sindhushree reddy
2021-01-07
Crisp and short ppt of Frm chapters and great explanation with examples.