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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. What does this mean?
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When \(X_t\) and \(Y_t\) are \(I(1)\) (integrated of order 1), it means that both time series are non-stationary but their first differences are stationary. In other words, the series themselves exhibit trends or random walks, but the changes from one period to the next are stationary.

Cointegration refers to a specific relationship between these two non-stationary series. If there exists a parameter \(\theta\) such that the linear combination \(Y_t - \theta X_t\) is \(I(0)\), it means that this linear combination is stationary. In other words, even though \(X_t\) and \(Y_t\) individually are non-stationary, there is a specific linear relationship between them that is stationary.

This implies that \(X_t\) and \(Y_t\) share a common stochastic trend, and they move together over time in such a way that their long-term equilibrium relationship is maintained. The parameter \(\theta\) is often referred to as the cointegrating coefficient, and it represents the equilibrium relationship between the two series.

In practical terms, cointegration suggests that any deviation from this equilibrium relationship is temporary and will be corrected over time, meaning that \(X_t\) and \(Y_t\) are bound together by some long-term relationship despite short-term fluctuations.

Cointegration is a key concept in time series econometrics, especially in the analysis of financial and economic data, as it allows for modeling and understanding the long-term relationships between non-stationary variables.
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