# Cointegration - same thing as stationary residuals?

So I'm aware that cointegration means there is some linear combination of the set of variables that is stationary. So, if you do a regression and find stationary residuals, can you just immediately say "oh, yes these variables are cointegrated" ? Because you have then just found some linear combination that is stationary?

My concern is the oft cited reasons/bad consequences for why you shouldn't do a regression with trending/undifferenced/non-stationary variables. I won't repeat them here, but the one caveat is always "oh but you can regress these non-stationary variables if they are cointegrated". Is this the same as saying "go ahead and try regressions with any non-stationary variables you want, and if the residuals are stationary you can say, well it was ok to do this regression after all because these variables are cointegrated"?

It just seems too simple, or too good to be true.

## 2 Answers

No this is not true. In order to consider a cointegrating relationship your variables need to be at least integrated of order one, $I\left(1\right)$. In order to carry out a cointegration analysis you would first have to conduct a unit root test to see if your time series are in fact $I\left(1\right)$. Then you could conduct a cointegration test on the relevant series, some of the more popular being the Johansen trace test/maximum eigenvalue test (estimated using maximum likelihood) or the more robust Engle-Granger method (estimated using OLS). If you only have two variables or only suspect one cointegrating relationship you could use the Engle-Granger while the Johansen test can accomodate several cointegrating relationship.

Consider an eonomic example: You are interested in testing wheter or not money and output cointegrate. You would first run a/several unit root test/s on the series in order to see wheter or not the were in fact $I(1)$. If they were in fact $I\left(1\right)$ you could test for a cointegrating relationship using the Engle-Granger method since you only have two variables, hence you can maximum have one cointegrating vector.

First you would run the regression: $y_{t}=\beta_{0}+\beta_{1}m_{t}+u_{t}$, where $\beta_{0}$ is a constant, $m_{t}$ is money, $y_{t}$ is output and $u_{t}$ is the error term. After running this regression you would run a unit root test on the residuals to see if they were stationary or $I\left(1\right)$. If they are stationary the series cointegrate! What is crucial for cointegration is that the series share a common stochastic trend and that they are at least integrated of order 1. By just regressing one $I\left(1\right)$ series on another $I\left(1\right)$ series you could end up with a spurious regression if they do not share a common stochastic trend, i.e. cointegrate. You could also deal with the spurious regression problem by including enough lags for each variable of interest when using a dynamic model!

Note that you can have different kinds of non-stationarity. A trend-stationary series which has an upwards trend is non-stationary. By detrending the series or including a time trend you can make these residuals stationary (see the Frisch–Waugh–Lovell theorem) although there is no cointegration present at all. Further, you can have non-stationary series due to level shifts (structural breaks) or sub-samples with differing degree of volatility. You can have an $I\left(1\right)$ series which can be made stationary by differencing it once.

Hopefully this answered your question. I would recommend you to read up on stationarity, integration and cointegration.

• Dan, I think the second sentence of your answer is not strictly true. The single equation generalized error correction model (which is a cointegrating regression) is agnostic with respect to the integration of predictor variables. (as described by Banerjee; de Boef; de Boef & Keele) – Alexis Jul 10 '14 at 16:50

Yes I should have been more precise on that one Alexis. It is possible to have fractional integration and hence fractional cointegration as well but I didn't want to get into that as it complicates things. Most econometrics books discuss cointegration within an $I\left(1\right)$ and $I\left(2\right)$ framework. What is important to remember is: “If two or more series are individually integrated (in the time series sense) but some linear combination of them has a lower order of integration, then the series are said to be cointegrated” (from wiki). Correct me if I am wrong but what I think you are talking about is the autoregressive distributed lag (ADL) model which is a dynamic model with stationary variables. This can also be represented as an ECM model but it doesn't have to have anything to do with cointegration. Try to take a look in Verbeek. M. 2004. A Guide to Modern Econometrics. 2nd ed. on page 310-312.

I tried to comment directly on your comment but unfortunately my answer was too long.

• The ADL has a representation within the generalized error correction model, but the latter is, again, agnostic with respect to the integration of its predictors. Also, per Keele & de Boef "The tight linkage between cointegration and error correction models stems from the Granger representation theorem. According to this theorem, two or more integrated time series that are cointegrated have an error correction representation, and two or more time series that are error correcting are cointegrated (Engle and Granger 1987). In short, the two concepts are isomorphic, as each implies the other." – Alexis Jul 11 '14 at 0:22
• For more on the single equation generalized ECM, see Banerjee, A., Dolado, J. J., Galbraith, J. W., and Hendry, D. F. (1993). Co-integration, error correction, and the econometric analysis of non-stationary data. Oxford University Press, USA. , or De Boef, S. and Keele, L. (2008). Taking time seriously. American Journal of Political Science, 52(1):184–200. – Alexis Jul 11 '14 at 0:24