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What Is Cointegration Definition Methods And Examples

Leo Migdal
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what is cointegration definition methods and examples

In econometrics, cointegration is a statistical property that describes a long-run equilibrium relationship among two or more time series variables, even if the individual series are non-stationary (i.e., they contain stochastic trends). In such cases, the variables may drift in the short run, but their linear combination is stationary, implying that they move together over time and remain bound by a stable equilibrium. More formally, if several time series are individually integrated of order d (meaning they require d differences to become stationary) but a linear combination of them is integrated of a lower order, then those... That is, if (X,Y,Z) are each integrated of order d, and there exist coefficients a,b,c such that aX + bY + cZ is integrated of order less than d, then X, Y, and Z... Cointegration is a crucial concept in time series analysis, particularly when dealing with variables that exhibit trends, such as macroeconomic data. In an influential paper,[1] Charles Nelson and Charles Plosser (1982) provided statistical evidence that many US macroeconomic time series (like GNP, wages, employment, etc.) have stochastic trends.

If two or more series are individually integrated (in the time series sense) but some linear combination of them has a lower order of integration, then the series are said to be cointegrated. A common example is where the individual series are first-order integrated (⁠ I ( 1 ) {\displaystyle I(1)} ⁠) but some (cointegrating) vector of coefficients exists to form a stationary linear combination of them. The first to introduce and analyse the concept of spurious—or nonsense—regression was Udny Yule in 1926.[2] Before the 1980s, many economists used linear regressions on non-stationary time series data, which Nobel laureate Clive Granger... 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.

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. A test used to establish if there is a correlation between several time series in the long term A cointegration test is used to establish if there is a correlation between several time series in the long term. The concept was first introduced by Nobel laureates Robert Engle and Clive Granger in 1987 after British economist Paul Newbold and Granger published the spurious regression concept. Cointegration tests identify scenarios where two or more non-stationary time series are integrated together in a way that they cannot deviate from equilibrium in the long term. The tests are used to identify the degree of sensitivity of two variables to the same average price over a specified period of time.

Before the introduction of cointegration tests, economists relied on linear regressions to find the relationship between several time series processes. However, Granger and Newbold argued that linear regression was an incorrect approach for analyzing time series due to the possibility of producing a spurious correlation. A spurious correlation occurs when two or more associated variables are deemed causally related due to either a coincidence or an unknown third factor. A possible result is a misleading statistical relationship between several time series variables. 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.

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 o4-mini 5 min read · April 19, 2025 Dive into the fundamentals of cointegration relationships—learn theory, tests, and applications for robust time‑series analysis.

Reference: Johansen, S. (1995). Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. This article explores the emerging trends and opportunities for advocacy in Latin American internati... Fog computing is a paradigm that extends cloud computing to the edge of the network, bringing comput... The Cointegration Method is a powerful statistical tool used in time series analysis to identify relationships between non-stationary time series data.

It helps analysts determine whether two or more series move together over time, despite potential short-term fluctuations. This method is particularly valuable in economics and finance, where understanding long-term relationships can lead to more informed investment decisions. Understanding the Cointegration Method involves a few key components: Non-Stationarity: This refers to a time series that has a mean and variance that change over time. Many financial time series exhibit non-stationary behavior. Stationarity: A stationary time series has constant mean and variance over time.

Cointegration requires that the series be non-stationary but can still have a stable relationship. Cointegrating Equation: This is a linear combination of the non-stationary series that results in a stationary series. Finding this equation is essential for establishing cointegration.

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