Cointegration test; model with different number of explanatory variables I have run an ADF test on the residuals of an ARDL and a DOLS model to test for cointegration. I have 3 explanatory variables and 1 response variable. When I run the ADF test on the residuals on both models with all 3 explanatory variables included I get the result that they are cointegrated.
However, when I run the ADF test on the residuals of the DOLS model where only 1 explanatory variable is included I get the result that they are not cointegrated.
What could a possible explanation be of this?
Thank you!

Let $y$ be the response variable and $x_1,x_2,x_3$ be the explanatory variables. It could be that the pair $(y,x_1)$ is not cointegrated while the 4 variables $(y,x_1,x_2,x_3)$ are cointegrated. This would be the case if $y$ were a linear combination of the three $x$s plus a stationary variable, e.g. $y=x_1+x_2+x_3+\varepsilon$, where the $x$s were independent random walks.
