In a cross-sectional environment, we use the phrase “spurious correlation” to describe a situation where two variables are related through their correlation with a third variable. In particular, if we regress y on x, we find a significant relationship. But when we control for another variable, say, z, the partial effect of x on y becomes zero. Naturally, this can also happen in time series contexts with I(0) variables.
As we discussed in Section 10.5, it is possible to find a spurious relationship between time series that have increasing or decreasing trends. Provided the series are weakly dependent about their time trends, the problem is effectively solved by including a time trend in the regression model.
When we are dealing with integrated processes of order one, there is an additional complication.
Even if the two series have means that are not trending, a simple regression involving two independent I(1) series will often result in a significant t statistic.