I have a question regarding the concept of cointegration. Does the concept of cointegration apply to any model? Or it only applies to OLS?
For example, I fit the following model $$y_t=x_t \beta+v_t$$
$$v_t=-\theta_1 v_{t-1}-…-\theta_m v_{t-m}+\varepsilon_t$$
If the ADF tests prove that both dependent variable $y$ and independent variable $x$ are I(1), and the residuals $\varepsilon_t$ are stationary at I(0), can I conclude $y$ and $x$ are cointegrated? The residuals $\varepsilon_t$ are the prediction errors from the model?
I checked the definition of cointegration, it looks like as long as the residuals from OLS are I(0) and both $x$ and $y$ are I(1), then $x$ and $y$ are cointegrated? Do I need to test stationarity of the residuals from my model in order to prove that $x$ and $y$ are cointegrated?