# Regression with cointegrated regressors

Suppose that we have the regression model

$$Y(t)=\alpha +\beta_1X_1(t)+ \cdots +\beta_nX_n(t)+\epsilon(t)$$

One approach to fitting this model is to use OLS. If the predictor variables $X_1(t),\ldots,X_n(t)$ are correlated then we are subject to statistical colinearity distortion and OLS can give an overfit model. These problems can be avoided by using the LASSO or ridge regression.

My question is, what happens when the predictor variables $X_1(t),\ldots,X_n(t)$ are cointegrated? What kinds of problems are introduced by cointegrated predictors and what are the best methods to use to get a reliable fit?

• Could you look at the comments to the answer and help us find out what your setting is? I.e., is $y$ integrated or stationary? Also, if integrated, is $y$ cointegrated with the $x$'s? – Richard Hardy Dec 24 '15 at 9:40
• @RichardHardy Just saw this, I commented in the answer below. – Wintermute Dec 24 '15 at 13:23
• OK, be aware that the original answer considers a different case (as becomes clear from the comments) and may or may not hold in the case you are interested in (so I am not sure accepting it was a good idea). – Richard Hardy Mar 13 '17 at 19:51

## 1 Answer

If there is a cointegrating relationship, that is a linear combination of I(1) variables that is stationary, then it is safe to run an OLS regression. However, the residuals will probably be serially correlated so you might want to adjust the standard errors. One way to do that is by the Newey-West method. To test for cointegration you can either use the Engle-Granger procedure or better yet the Johansen method, up to you.

Also, with regard to your remark about LASSO and ridge, these two estimators are used in very different contexts and as far as I know the LASSO does not perform particularly well in the presence of multicollinearity.

• The answer sounds kind of intuitive, but I am not sure about the theoretical details. Let $X$ be the regressor matrix. If some of the regressors are I(1), some of the elements of $\mathbf{E}(X'X)$ will be infinite. This matrix is involved in the OLS coefficients: $\beta=(X'X)^{-1}X'y$, and $\mathbf{E}$ is used when deriving consistency. So is there really no problem? What properties does the OLS estimator have? What properties does it not have as compared to the case of all-stationary regressors? – Richard Hardy Dec 24 '15 at 0:35
• @RichardHardy Actually, the OLS estimator is super consistent in this case but its limiting distribution is non-normal. The details are rather involved but here is a nice summary econ.ku.dk/metrics/econometrics2_05_ii/slides/… – JohnK Dec 24 '15 at 0:47
• Does the source cover the case where some of the $x$'s are cointegrated among themselves but not with $y$? I think the OP is about such a case rather than the standard case where $y$ is cointegrated with some of the $x$'s (which is indeed covered in the source). – Richard Hardy Dec 24 '15 at 0:51
• @RichardHardy I am not sure what the OP means, I took the question to mean the case where there is a cointegrating relationship amongst all predictors and the response. If that is not the case, I don't think it makes sense to run an OLS regression, although I have no reference at hand. The reason is that if the response is not stationary, this will be a spurious regression. – JohnK Dec 24 '15 at 0:58
• I read the question as, what happens if $y$ is stationary while $x$'s are cointegrated. Then you may have stationarity on both sides of the equation. An example would be one equation of a vector error correction model where the cointegrating vector is not given (when the vector is given, there is no problem as we efectively consider the stationary combination of $x$'s rather than original $x$'s). – Richard Hardy Dec 24 '15 at 9:37