# Trend in exogenous variable in time series

I have a time series of a variable V1 with seasonality and a strong trend. The trend however seems to be closely related to (and caused by) the trend of another time varying variable (V2). As V2 grows i.e shows a positive trend, the trend on V1 declines (see chart).

I want to use V2 as an exogenous variable in the time series forecast of V1. I am using prophet for this where I forecast V2 and then use it as an exogenous variable to forecast V1. But, since V2 has a trend that affects the trend of V1 does it make sense to use V2 as an exogenous variable at all? I can simply run a forecast with and without the exogenous variable to see what difference it makes. It might not be much, but I am more interested in knowing if there is a fundamental flaw in using exogenous variables with trends.

The question is similar to this one, but it was not very clear to me.

Lets say you want to predict $$y_t$$ based on its past $$y_{0},y_{1},...,y_{t-1}$$, as well as based on exogenous variable $$x_t$$. Models such as the additive Prophet model treat $$x_t$$ as independent over time. Hence, Prophet will not notice a trend in the process $$\{x_0,x_1,...,x_t\}$$ because it only looks at $$x_t$$ and then immediately forgets $$x_t$$ (see here)
However, this does not mean that you cannot use $$x_t$$ to predict $$y_t$$. If $$x_t$$ is a good external regressor for $$y_t$$, then there must exist some function $$f$$ such that $$f(x_t)\approx y_t$$. This does not change if there is a trend in $$\{x_0,x_1,...,x_t\}$$. For example, if your exogenous variable is actually a straight line with slope $$k_1$$ and y-intercept $$d_1$$, and your target variable is a straight line with slope $$k_2$$ and y-intercept $$d_2$$, then the solution is $$f(x_t)=\frac{k_2}{k_1}(x_t-d_1)+d_2=y_t$$. Since Prophet treats all external variables as linear regressors, it can solve this particular problem.
For the data you posted, I do not think that there is a linear function from the exogenous variable to the target variable. So Prophet should not be able to learn this exactly - in fact, it cannot be more exact than the best linear function from $$x_0,...,x_t$$ to $$y_0,...,y_t$$.