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= Cointegration =

'''Cointegration''' is a statistical property of a collection of time series variables. When two or more time series are cointegrated, it implies that there is a long-term equilibrium relationship between them, even though they may be non-stationary in their levels. This concept is particularly important in the field of econometrics and is widely used in the analysis of financial and economic data.

== Background ==

In time series analysis, a series is said to be [[stationary]] if its statistical properties such as mean, variance, and autocorrelation are constant over time. Many economic and financial time series are non-stationary, meaning they exhibit trends or other patterns that change over time. However, even if individual series are non-stationary, a linear combination of them may be stationary. This phenomenon is known as cointegration.

The concept of cointegration was introduced by [[Clive Granger]] and [[Robert Engle]] in the 1980s, for which they were awarded the [[Nobel Memorial Prize in Economic Sciences]] in 2003.

== Mathematical Definition ==

Consider two time series \(X_t\) and \(Y_t\). These series are said to be cointegrated if:

# Both \(X_t\) and \(Y_t\) are integrated of order 1, denoted as \(I(1)\), meaning they become stationary after differencing once.
# There exists a linear combination \(Z_t = Y_t - \beta X_t\) that is stationary, \(I(0)\).

The parameter \(\beta\) is known as the cointegrating coefficient, and \(Z_t\) is the cointegrating residual.

== Testing for Cointegration ==

Several methods exist for testing cointegration among time series:

* '''Engle-Granger Two-Step Method''': This involves estimating the cointegrating regression and then testing the residuals for stationarity using unit root tests such as the [[Augmented Dickey-Fuller test]].

* '''Johansen Test''': A multivariate approach that allows for more than two time series and tests for the presence of multiple cointegrating vectors.

* '''Phillips-Ouliaris Test''': A residual-based test similar to the Engle-Granger method but with different critical values.

== Applications ==

Cointegration is widely used in [[econometrics]] and [[finance]] for modeling long-term relationships between economic variables. Some common applications include:

* '''[[Pairs trading]]''': A strategy that involves trading two cointegrated stocks to exploit deviations from their long-term equilibrium.

* '''[[Error correction model]]''': A model that incorporates cointegration to adjust short-term deviations from the long-term equilibrium.

* '''[[Macroeconomic modeling]]''': Analyzing relationships between macroeconomic variables such as [[GDP]], [[inflation]], and [[interest rates]].

== Limitations ==

While cointegration provides a powerful tool for analyzing long-term relationships, it has limitations:

* '''Assumption of linearity''': Cointegration assumes a linear relationship between variables, which may not hold in all cases.

* '''Sensitivity to structural breaks''': Cointegration tests can be sensitive to structural breaks in the data, which can lead to incorrect conclusions.

== See Also ==

* [[Stationary process]]
* [[Unit root]]
* [[Vector autoregression]]

== References ==

* Engle, R. F., & Granger, C. W. J. (1987). "Co-integration and error correction: Representation, estimation, and testing." Econometrica, 55(2), 251-276.
* Johansen, S. (1991). "Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models." Econometrica, 59(6), 1551-1580.

[[Category:Time series analysis]]
[[Category:Econometrics]]
[[Category:Statistical methods]]

Cointegration - Revision history

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