Cointegration Definition Examples Tests Statistics How To

Leo Migdal

-Dec 11, 2025, 1:57 AM

cointegration definition examples tests statistics how to

You may want to read this article first: What is order of integration? Cointegration tests analyze non-stationary time series— processes that have variances and means that vary over time. In other words, the method allows you to estimate the long-run parameters or equilibrium in systems with unit root variables (Rao, 2007). Two sets of variables are cointegrated if a linear combination of those variables has a lower order of integration. For example, cointegration exists if a set of I(1) variables can be modeled with linear combinations that are I(0). The order of integration here—I(1)— tells you that a single set of differences can transform the non-stationary variables to stationarity.

Although looking at a graph can sometimes tell you if you have an I(1) process, you may need to run a test such as the KPSS test or the Augmented Dickey-Fuller test to figure... In order to analyze time series with classical methods like ordinary least squares, an assumption is made: The variances and means of the series are constants that are independent of time (i.e. the processes are stationary). Non-stationary time series (or unit root variables) don’t meet this assumption, so the results from any hypothesis test will be biased or misleading. These series have to be analyzed with different methods. One of these methods is called cointegration.

More formally, cointegration is where two I(1) time series xt and yt can be described by the stationary process ut = yt − αxt. In time series analysis, many variables show trends over time, meaning they are non-stationary. This non-stationarity can be a problem when building statistical models because it can lead to misleading results. However, sometimes two or more non-stationary time series move together in such a way that their combination becomes stationary. This relationship is called cointegration. Cointegration occurs when two or more non-stationary time series move together in such a way that their linear combination becomes stationary.

This indicates a long-term equilibrium relationship between the variables, even if each one individually trends or drifts over time. Reveals stable, long-run relationships between non-stationary variables. Facilitates the use of Error Correction Models (ECM), which capture: Before diving into cointegration, it’s important to understand stationarity: Step 1: Check Stationarity of Individual Series: Sarah Lee AI generated o3-mini 0 min read · March 13, 2025

In today’s data-driven landscape, the cointegration test has emerged as an indispensable tool for researchers and analysts working in econometrics and time series analysis. With financial markets, macroeconomic indicators, and various other domains generating copious amounts of data, understanding the long-run equilibrium relationships between non-stationary time series has become both challenging and critical. This tutorial offers a comprehensive guide on cointegration tests, detailing each step of the methodology, interpreting results accurately, and providing real-world case studies to bridge theory and practice. Time series data are inherently dynamic, evolving over time as economic, financial, or environmental factors shift. Traditional statistical models assume stationarity, which can fail when the underlying variables exhibit trends or other forms of non-stationarity. Cointegration testing steps in to address such challenges by determining whether a collection of non-stationary series can form a stationary relationship—that is, whether a linear combination of them remains stable over time.

Cointegration refers to the phenomenon where two or more non-stationary time series are linked by a long-run, equilibrium relationship. In formal terms, consider two time series, yt y_t yt and xt x_t xt, integrated of order 1 (denoted I(1) I(1) I(1)). If there exists a coefficient β \beta β such that the linear combination ut=yt−βxt u_t = y_t - \beta x_t ut=yt−βxt by Eric · Published January 28, 2020 · Updated October 19, 2023 Cointegration is an important tool for modeling the long-run relationships in time series data.

If you work with time series data, you will likely find yourself needing to use cointegration at some point. This blog provides an in-depth introduction to cointegration and will cover all the nuts and bolts you need to get started. In particular, we will look at: Though not necessary, you may find it helpful to review the blogs on time series modeling and unit root testing before continuing with this blog. Economic theory suggests that many time series datasets will move together, fluctuating around a long-run equilibrium. In econometrics and statistics, this long-run equilibrium is tested and measured using the concept of cointegration.

Cointegration is a statistical method used to test the correlation between two or more non-stationary time series in the long run or for a specified period. The method helps identify long-run parameters or equilibrium for two or more variables. In addition, it helps determine the scenarios wherein two or more stationary time series are cointegrated so that they cannot depart much from the equilibrium in the long run. This method tests the residuals created based on static regression for the presence of unit roots. For example, suppose two non-stationary time series are cointegrated, and the result confirms the stationary characteristic of residuals. However, there are some limitations to this method.

For example, suppose there are two or more non-stationary variables. The method will reflect two or more cointegrated relationships. Also, the method is a single equation model. Recent tests like Johansen's and Philip-Ouliaris have addressed some of these limitations. Johansen test is for testing cointegration between several time-series data at a time. This test overcomes the limitation of an incorrect test result for more than two time series of the Engle-Granger method.

However, this test is subject to asymptotic properties; i.e., it takes a large sample size because a small sample size would give incorrect or false results. There are two further bifurcations of the Johansen test: the Trace test and the Maximum Eigenvalue test. This test proves that when a residual-based unit root test applies to time series, the cointegrated residuals give asymptotic distribution instead of Dickey-Fuller distribution. The resulting asymptotic distributions are known as Philip-Ouliaris distributions. The cointegration test is based on the logic that more than two-time series variables have similar deterministic trends that one can combine over time. Therefore, it is necessary for all cointegration testing for non-stationary time series variables.

One should integrate them in the same order, or they should have a similar identifiable trend that can define a correlation between them. So, they should not deviate much from the average parameter in the short run. In the long run, they should be reverting to the trend. When \(X_t\) and \(Y_t\) are \(I(1)\) and if there is a \(\theta\) such that \(Y_t - \theta X_t\) is \(I(0)\), \(X_t\) and \(Y_t\) are cointegrated. Put differently, cointegration of \(X_t\) and \(Y_t\) means that \(X_t\) and \(Y_t\) have the same or a common stochastic trend and that this trend can be eliminated by taking a specific difference of the... R functions for cointegration analysis are implemented in the package urca.

As an example, reconsider the relation between short-term and long-term interest rates in the example of U.S. 3-month treasury bills, U.S. 10-years treasury bonds and the spread in their interest rates which have been introduced in Chapter 14.4. The next code chunk shows how to reproduce Figure 16.2 of the book. The plot suggests that long-term and short-term interest rates are cointegrated: both interest series seem to have the same long-run behavior. They share a common stochastic trend.

The term spread, which is obtained by taking the difference between long-term and short-term interest rates, seems to be stationary. In fact, the expectations theory of the term structure suggests the cointegrating coefficient \(\theta\) to be 1. This is consistent with the visual result. Following Key Concept 16.5, it seems natural to construct a test for cointegration of two series in the following manner: if two series \(X_t\) and \(Y_t\) are cointegrated, the series obtained by taking the... If the series are not cointegrated, \(Y_t - \theta X_t\) is nonstationary. This is an assumption that can be tested using a unit root test.

We have to distinguish between two cases: Cointegration is a statistical property of a series of time-series variables which, when analyzed, indicate a long-term relationship or equilibrium amongst them, despite being non-stationary when taken individually. Non-stationary data series are those whose statistical properties such as mean, variance, and autocorrelation are not constant over time. However, if these series are cointegrated, it implies that some linear combination of them is stationary, meaning they move together in the long run even though they may diverge in the short term. Cointegration is a crucial concept in econometrics and financial economics, especially in the analysis of time series data that aim to find and quantify long-term economic and financial relationships. Consider the relationship between consumer spending and household income.

Over time, both variables tend to grow, suggesting they are non-stationary. However, economic theory posits that consumer spending is directly influenced by household income. To examine this relationship through the lens of cointegration, one would analyze long-term historical data on spending and income. If it is found that any deviation between consumer spending and household income is temporary and that these variables move together over time (i.e., the gap between them does not widen endlessly), they are... Cointegration here means that there’s a long-run equilibrium relationship between consumer spending and household income, ensuring that discrepancies between them are corrected over time. Cointegration holds significant value in economic and financial analyses because it helps in understanding and predicting long-term relationships between variables.

For policymakers, recognizing cointegrated relationships enables the formulation of more effective economic policies. For traders and investors, cointegration analysis supports the identification of pairs trading opportunities, where two stocks or assets move together in the long term, allowing for strategic buying and selling. Furthermore, in econometric modeling, the concept of cointegration is vital for ensuring the validity and reliability of regression analyses involving time series. Without acknowledging cointegration, models risk being spurious, implying false correlations that may lead to incorrect conclusions and poor predictive performance. Hence, cointegration analysis not only helps in identifying and modeling long-term relationships but also in avoiding potentially misleading inferences in time series data. Cointegration and correlation often get confused, but they are distinct concepts.

Correlation measures the strength and direction of a linear relationship between two variables, without considering non-stationarity or the long-term equilibrium relationship. Cointegration, on the other hand, specifically addresses long-term equilibrium among non-stationary time series. Two or more series can be highly correlated without being cointegrated if they do not share a common stochastic trend. 7. Practical Applications of Cointegration Analysis 8.

Cointegration Definition Examples Tests Statistics How To

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