The paper by C. W. J. Granger explores the concept of cointegration in economic variables, which refers to the long-term equilibrium relationship between non-stationary time series. The author introduces the idea that certain economic variables, such as interest rates, commodity prices, and income, should not diverge too much in the long run. This concept is crucial for understanding the dynamics of economic systems and is particularly relevant in macroeconomics. Granger defines the properties of integrated time series, distinguishing between $I(0)$ (stationary) and $I(1)$ (non-stationary) series. He explains that $I(1)$ series, often seen in macroeconomic data, require differencing to become stationary. The paper then delves into the concept of cointegration, where a linear combination of two $I(1)$ series is $I(0)$, indicating a long-term equilibrium relationship. This relationship is captured in an error-correction model, which helps in understanding how deviations from equilibrium are corrected over time. The paper also discusses the testing of cointegration, including the Dickey-Fuller test for stationarity and the Johansen test for multiple cointegrating vectors. It highlights the importance of cointegration in econometric models, suggesting that it can improve short-run forecasts and long-run predictions. Granger emphasizes that cointegration allows economic theories to be incorporated into time-series models, enhancing their predictive power. Finally, the paper touches on more advanced topics, such as time-varying parameters and non-linear cointegration, which extend the applicability of cointegration in economic analysis. The conclusion underscores the growing acceptance of cointegration in empirical research and its potential to bridge the gap between theoretical and empirical economic analysis.The paper by C. W. J. Granger explores the concept of cointegration in economic variables, which refers to the long-term equilibrium relationship between non-stationary time series. The author introduces the idea that certain economic variables, such as interest rates, commodity prices, and income, should not diverge too much in the long run. This concept is crucial for understanding the dynamics of economic systems and is particularly relevant in macroeconomics. Granger defines the properties of integrated time series, distinguishing between $I(0)$ (stationary) and $I(1)$ (non-stationary) series. He explains that $I(1)$ series, often seen in macroeconomic data, require differencing to become stationary. The paper then delves into the concept of cointegration, where a linear combination of two $I(1)$ series is $I(0)$, indicating a long-term equilibrium relationship. This relationship is captured in an error-correction model, which helps in understanding how deviations from equilibrium are corrected over time. The paper also discusses the testing of cointegration, including the Dickey-Fuller test for stationarity and the Johansen test for multiple cointegrating vectors. It highlights the importance of cointegration in econometric models, suggesting that it can improve short-run forecasts and long-run predictions. Granger emphasizes that cointegration allows economic theories to be incorporated into time-series models, enhancing their predictive power. Finally, the paper touches on more advanced topics, such as time-varying parameters and non-linear cointegration, which extend the applicability of cointegration in economic analysis. The conclusion underscores the growing acceptance of cointegration in empirical research and its potential to bridge the gap between theoretical and empirical economic analysis.