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Spurious regression and how to avoid it

Today let's understand spurious regression: how two series with nothing real connecting them can still move together so closely that a regression between them looks like undeniable evidence, and how to catch it before it fools you.

Thistle and Vine is a small bakery chain. Nimbus is a fitness app. They share no supplier, no customers, no season, nothing that would tie one to the other.

But watch their numbers over the same 60 months, January 2021 to December 2025: Thistle and Vine's monthly revenue climbs from $40,181 to $50,120, and Nimbus's monthly paying subscribers climb from 1,492 to 2,183. Below, both series are indexed to start at 100, so a dollar figure and a headcount can share one chart.

Look at how closely the two lines track each other, climbing in the same wobbly, drifting way, even though a bakery chain and a fitness app have nothing to do with one another.

Two businesses with nothing in common

Let's build both series exactly as they happened. Thistle and Vine's revenue starts at $40,181 in January 2021. Every month after that, revenue moves by a random amount, on average $160 up, but with swings of about $300 either way, so plenty of months dip instead of climb. Nimbus's subscriber count is built the same way, starting at 1,492, moving by a random amount every month too, on average up by 9 subscribers, with swings of about 25 either way.

Neither series knows anything about the other. Whatever moves revenue in a given month, a slow Tuesday or a marketing push, has nothing to do with whatever moves Nimbus's subscriber count that same month. The two are built from separate random number streams, and a bakery chain and a fitness app share no supplier, no customer base and no season that could quietly link them.

RInteractive R
# Build Thistle and Vine's revenue and Nimbus's subscribers as two separate random walks, then combine them into one tsibble library(tsibble) set.seed(348) revenue <- round((cumsum(rnorm(60, mean = 1.6, sd = 3)) + 400) * 100) set.seed(10348) subs <- round((cumsum(rnorm(60, mean = 0.9, sd = 2.5)) + 150) * 10) df <- tsibble( month = yearmonth("2021 Jan") + 0:59, revenue = revenue, subs = subs, index = month ) head(df, 3) #> # A tsibble: 3 x 3 [1M] #> month revenue subs #> <mth> <dbl> <dbl> #> 1 2021 Jan 40181 1492 #> 2 2021 Feb 40636 1484 #> 3 2021 Mar 41032 1478 tail(df, 3) #> # A tsibble: 3 x 3 [1M] #> month revenue subs #> <mth> <dbl> <dbl> #> 1 2025 Oct 49278 2141 #> 2 2025 Nov 49714 2146 #> 3 2025 Dec 50120 2183

  

  





By December 2025, revenue has reached $50,120 and subscribers have reached 2,183. Notice the first three rows though: subscribers actually dip slightly, from 1,492 down to 1,478, before turning around and climbing for the rest of the 60 months. That is exactly what a random monthly step looks like. It does not march straight up, it wanders, sometimes backward, before the upward drift wins out over the long run.