<p>Just regress Y on X:</p>
<p><span class="math-container">$$Y=b_0+b_1X+ e$$</span></p>
<p>and you will likely find some negative significant <span class="math-container">$b_1$</span> coefficient even though both series are just unrelated random walks.</p>
<p>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.</p>

CC BY-SA 4.0
2021-07-29T21:47:04.027Z
Thank you, once again exactly what I looked for.
Hyperbolic PDE friend
CC BY-SA 4.0
2021-07-29T21:46:07.193Z
2021-07-29T21:46:07.193Z
1muflon1
<p>Assume we are given two independent random walks
<span class="math-container">$$
Y_t = Y_{t-1} + \varepsilon_{1, t}, \quad \varepsilon_{1, t} \sim \mathcal{N}(0, 1) \\
X_t = X_{t-1} + \varepsilon_{2, t}, \quad \varepsilon_{2, t} \sim \mathcal{N}(0, 4) \\
$$</span>
where <span class="math-container">$\mathbb{E}[\varepsilon_{1, t} \varepsilon_{2, s}] = 0$</span> for all <span class="math-container">$s, t$</span>.</p>
<p>I simulated this for <span class="math-container">$t \in \{1, 2, ..., 1000\}$</span> in R and got:
<a href="https://i.sstatic.net/YAmQh.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/YAmQh.png" alt="enter image description here" /></a></p>
<p>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?</p>

<econometrics><regression><time-series><r>
How to recognize correlation in spurious regression case
Hyperbolic PDE friend
# How to recognize correlation in spurious regression case

(Asked by: username_0 on 2021-07-29)

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](https://i.sstatic.net/YAmQh.png)](https://i.sstatic.net/YAmQh.png)

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?

**Tags:** econometrics, regression, time-series, r

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## 1 answers

### Answer by username_1 on 2021-07-29. Score: 5 | Accepted answer

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.

> **Comments:**
> **username_0**: Thank you, once again exactly what I looked for.  

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