# How to recognize correlation in spurious regression case

Assume we are given two independent random walks $$Y_t = Y_{t-1} + \varepsilon_{1, t}, \quad \varepsilon_{1, t} \sim \mathcal{N}(0, 1) \\ X_t = X_{t-1} + \varepsilon_{2, t}, \quad \varepsilon_{2, t} \sim \mathcal{N}(0, 4) \\$$ where $$\mathbb{E}[\varepsilon_{1, t} \varepsilon_{2, s}] = 0$$ for all $$s, t$$.

I simulated this for $$t \in \{1, 2, ..., 1000\}$$ in R and got:

My professor says that this constitutes a spurious relationship. But this implies that somwhere in the plot I should be able to recognize some kind of (unexplicable; it is spurious at last) correlation. But how can I see this?

$$Y=b_0+b_1X+ e$$
and you will likely find some negative significant $$b_1$$ coefficient even though both series are just unrelated random walks.