# Can negative relationship between X and Y be spurious

Let's assume I have X and Y and both X and Y have positive relationship. In such case in which both series trend in the same direction, we need to test for cointegration to be sure that relationship is not spurious. This is clear.

But what if we assume that I have X and Y and both X and Y have negative relationship? In my point of view, I should proceed in the same way as when relationship is positive - I must test whether relationship is not spurious. In fact it is written on this block that cointegrating relationship can exist between two negatively correlated variables with deterministic trends. This should imply that also spurious relationship can exist between two negatively related variables. Am I correct?

The things that my colleague mentioned that relationship can be spurious only of series trend in the same direction, which confused me.

• What does "on this block" mean? – Nick Cox Feb 7 '16 at 15:05

To put it shortly, the sign does not change the essence.

You could have the following pair of series that are roughly "mirror images" of each other plus a stationary component:

$$y_{1,t} = \sum_{\tau=1}^t \varepsilon_{\tau} + u_t$$

and

$$y_{2,t} = -\sum_{\tau=1}^t \varepsilon_{\tau} + v_t$$

where $\varepsilon_{\tau}$, $u_t$ and $v_t$ are stationary series. $y_{1,t}$ and $y_{2,t}$ are both integrated because they contain a cumulative sum of a stationary series $\varepsilon_{\tau}$. Their linear combination

$$y_{1,t}+y_{2,t}=u_t+v_t$$

will be a stationary series because it is a sum of two stationary series. Hence, $y_{1,t}$ and $y_{2,t}$ are cointegrated.

cointegrating relationship can exist between two negatively correlated variables with deterministic trends

perhaps requires a little clarification. Cointegration between two series means that the series share a common stochastic trend. If there are deterministic trends involved, you should adjust for them. For example, you could have

$$y_{1,t} = \sum_{\tau=1}^t \varepsilon_{\tau} + u_t$$

and

$$y_{2,t} = -\sum_{\tau=1}^t \varepsilon_{\tau} + t + v_t$$.

Here $y_{1,t}$ and $y_{2,t}$ are cointegrated when the linear trend in $y_{2,t}$ is adjusted for: the linear combination of the two integrated series and the trend

$$y_{1,t}+y_{2,t}-t=u_t+v_t$$

will be a stationary series.

Given two integrated but not cointegrated series, you may find a spurious relationship regardless of whether the series happen to wander in roughly the same direction or in roughly opposite directions. Regressing one series on the other will be problematic in both cases just because of the infinite memory of both processes; you will not be able to interpret the point estimate of the regression coeficient, its standard error or the $R^2$ statistic in the usual way.