Confusion about cointegration 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?
 A: I am guessing you are asking whether or not you can conclude anything about cointegration given the data generating process you have and the question is yes your approach is valid. To see this consider the data generating process (DGP):
$y_{t}= x_{t}\beta+v_{t}$ 
$v_{t}= \theta_{1}v_{t-1}+\theta_{2}v_{t-2}+\cdots+\theta_{m}v_{t-m}+\varepsilon_{t},\;\varepsilon_{t}\sim i.i.d.\, N\left(0,\,\sigma^{2}\right)
 $
If $y_{t}\sim I\left(1\right)
 $ and $x_{t}\sim I\left(1\right)
 $ and they are cointegrated then the OLS estimator, $\hat{\beta}
 $, obtained from regressing $y_{t}
 $ on $x_{t}
 $ will be super consistent. It is clear from the data generating process that the error term is serially correlated so the model $y_{t}=x_{t}\beta+v_{t}
 $ is misspecified (it should include enough lags to remove the residual autocorrelation in case of dynamic misspecified). This will however not affect consistency of, $\hat{\beta}
 $, as long as the misspecification of the DGP is only related to the stationary terms since the stochastic trends will dominate asymptotically. When you conduct the Engle-Granger test remember that the residual based ADF-test does not follow the usual ADF statistics but depend on the number of estimated parameters in the static regression. For further reference see: R. F., Engle and C. W. J., Granger. 1987. Co-Integration and Error Correction: Representation, Estimation and Testing.
If this did not answer your question then let me know and I will amend my answer.
