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: enter image description here

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?


1 Answer 1


Just regress Y on X:

$$Y=b_0+b_1X+ e$$

and you will likely find some negative significant $b_1$ coefficient even though both series are just unrelated random walks.

You can also see that as one series increases other one decrease so you would expect they are correlated in negative way in this case.

  • 1
    $\begingroup$ Thank you, once again exactly what I looked for. $\endgroup$
    – Meowdog
    Jul 29, 2021 at 21:47

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