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?

  • 4
    $\begingroup$ Cointegration is a phenomenon when a weighted sum of two integrated time series is stationary. (I leave integration of orders above I(1) and the respective cointegration aside for simplicity.) This does not depend on any model. OLS is an estimation technique, not a model. You have posted a lot of questions, would it be possible to see which ones are the most important? Maybe when we clarify those, the other will be clear automatically. What is your application, or are you wondering about theory? $\endgroup$ Mar 23, 2015 at 20:08
  • $\begingroup$ I simplified my question. For example, I have an AR(1) model, The residuals can be written as below ε_t =y_t - x_t*β + φ_1*v_(t-1) if I can prove x and y are both I(1), ε_t is I(0), then can I conclude that x and y are cointegrated? Do I also need to test if v_(t-1) is I(1) in order to say x and y are cointegrated ? $\endgroup$
    – user71817
    Mar 24, 2015 at 14:17
  • 3
    $\begingroup$ I would need to think about it. But why wouldn't you use the existing standard cointegration tests? Johansen, Engle-Granger and other techniques are well established and their implementations are available in different statistical software packages. If you want to make inference about cointegration, use them. If you want to model your data using an ARIMAX model, the variables being cointegrated may pose a problem. I would rather use a VECM instead. $\endgroup$ Mar 24, 2015 at 14:28
  • $\begingroup$ see stats.stackexchange.com/questions/118296/… or stats.stackexchange.com/questions/171066/… $\endgroup$
    – user83346
    Oct 24, 2015 at 7:47

1 Answer 1


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.

  • $\begingroup$ Thanks Dan for your clarification. I am not sure if the residule based ADF test should be performed on v_t (residuals from OLS model) or residulas ε_t (residuals from AR model)? $\endgroup$
    – user71817
    Mar 25, 2015 at 19:36
  • $\begingroup$ should I consider v_t =θ1* v_t−1 +θ2* v_t−2 +⋯+θm *v_t−m+ ε_t ,ε_t ∼i.i.d.N(0,σ 2 ) as an error correction mechanisam ? $\endgroup$
    – user71817
    Mar 25, 2015 at 19:39
  • $\begingroup$ Your welcome. You'll have to test $v_{t}$! $\endgroup$
    – Plissken
    Mar 25, 2015 at 19:46
  • $\begingroup$ Yes. If $v_{t}$ is stationary then you have cointegration. The second step of the procedure is that you include the error term, $v_{t}$, in a well specified model. $y_{t}$ error corrects if the coefficient on $v_{t}$ is negative and statistically significant. The magnitude of the coefficient indicates the speed of adjustment back to equilibrium. $\endgroup$
    – Plissken
    Mar 25, 2015 at 22:18
  • $\begingroup$ Now the concept of cointegration makes better sense to me. Have you seen any literature formally states that the second equation in the AR model is an error correction mechanism, and hence the 1st equation is the long run equilibrium ? $\endgroup$
    – user71817
    Mar 26, 2015 at 20:04

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