Cointegration in the time series context indicates that at least two time series show similar futures in the long run. However, we might not conclude that they have a causal relationship. Even if there is a causal relationship, we might not determine the path of this relationship. For example, let's assume we have variables x and y. x and y can move together over time, but we might not say if x causes y, y causes x, or if a third variable z causes both. We only know that a linear combination of these two variables is stationary.

For instance, if we have a variable with a unit root in the model, we can't directly implement regression. Instead, we take the first difference of these variables if they are I(1) variables and then make the regression. Let's say x and y are I(1). Then, the regression d(y)=a+bd(x) is suitable. If we implement y=a+bx, it will show spurious regression results. However, if a linear combination of x and y is stationary and we know this linear relationship, we can implement this regression: diff(y)=(a+bx+cy)+k diff(x)+n diff (y(-1)). The linear combination (a+bx+cy) is stationary, making the regression not spurious.

We might not say that just because a combination is stationary, it implies causation. Even if causation exists, we might not determine the direction from x to y. This relationship can be written as y=−a/c−b/cx or x=−a/b−c/by.

while cointegration indicates that two variables share a long-term relationship, it does not imply causality. Establishing causality requires further analysis, such as Granger causality testing, to determine the direction and nature of the relationship between the variables. However, even Granger causality might not indicate true causality; it only shows Granger causality. For instance, consider rent prices and house prices: one can trigger the other, and you can show which one is triggering the other using Granger causality. However, a recession might be the real cause triggering both of them. Additionally, it might be that the combination of the two time series just happened to create a stationary variable during those specific periods, so there is no real causation.

@Huseyin Utku Demir , thank you very much for a detailed information on this matter.