Why does a regression on prices lie?

The first lesson said prices trend and returns do not, and gave the number: two unrelated series, the S&P 500 fund and the lira, each correlated 0.95 with a clock. This lesson names the property, shows the damage, and gives the one test that matters at a desk.

Stationary, in plain words

A series is stationary when its statistical properties — its mean, its spread, its correlation with its own past — do not depend on when you look. Daily returns are close to stationary: the mean return of 2016 and the mean return of 2024 are two draws from roughly the same distribution. Prices are not. The mean price of SPY.US in 2016 was about 200 and in 2026 about 700; a "mean price" is a number about the window, not the market.

A random walk is the simplest non-stationary series: each value is the last one plus a random step. Its steps are stationary; its levels wander without limit, and two independent random walks will drift together or apart for long stretches by nothing but chance.

The damage, measured

Generate two random walks of 2,500 steps each — the length of ten years of trading days — from independent normal draws. Measured on 2026-09-04 with a fixed seed: the correlation of the two levels was −0.42; the correlation of the two steps was −0.014. Nothing connects the series, and a regression of one level on the other would report a strong, highly "significant" relationship, because the t-statistic from the previous lesson assumes stationary residuals and gets non-stationary ones. Granger and Newbold showed this in 1974 and called it spurious regression; the next unit is the demonstration on real data.

The lira and the S&P: two levels with a 0.95 correlation to time and, through it, to each other. Their daily changes over the same decade correlated at −0.11, weak and of the opposite sign (measured 2026-09-28). Any analysis that put the two price series in a regression would find a relationship; any analysis on returns would find none; only one of them is right.

The test at a desk

Formal tests exist — Dickey and Fuller's is the standard — and none is needed to catch the common case. Ask two questions. Does the series have a level it returns to? A return does: zero, roughly. A price does not. Would the mean of the first half and the second half be the same number? For returns, close; for prices, never. If either answer is no, difference the series — take changes or returns — before computing anything that involves a mean, a correlation or a regression.

What survives on levels

Not nothing. A ratio of two prices that should stay near a level — the same share on two exchanges, a fund and its NAV, two bonds of one issuer — can be stationary even though each price is not, and that is what the derivatives domain's spread trades rely on. The test is the same: does the ratio have a level it comes back to, on data that includes a stress?

In the data

Any two /eod/ pulls make the demonstration: /eod/SPY.US?from=2016-09-02&to=2026-09-03&fmt=json and /eod/USDTRY.FOREX?from=2016-09-02&to=2026-09-03&fmt=json, correlated once on adjusted_close and once on the daily changes of it.

Try it now

1. Both series above, joined on the 2,514 dates they share and computed on 28 September 2026:

The first number says the two move together almost perfectly; the second says that, day by day, a rising dollar against the lira came weakly with a falling S&P. Say which of the two describes a relationship a trader could use, and name the property of the first series that manufactured the 0.95. 2. The two, on the same range:

3. A colleague reports that a share's price is 0.9 correlated with an index's price and proposes to trade the gap. Which of the two desk questions do you ask first?