FRM Exam Part I · Nonstationary Time Series

Spurious Regression and Differencing for Nonstationary Series

Updated 11 October 2026 · Fact-checked

Spurious regression happens when you regress one nonstationary series on another and get a high R² and a significant slope even though the series are unrelated. First differencing, or using log returns for prices, removes a unit root and restores stationarity, so the regression becomes valid.

Understand Spurious Regression and Differencing

A stationary series has a stable mean, variance and autocovariance over time. A nonstationary series does not. The classic case is a random walk: Yt = Yt-1 + εt. It has a unit root, and shocks never fade. Its variance grows with time.

Now take two independent random walks and regress one on the other. They have no true link. But each one wanders, and over any sample the two paths will look like they move together or apart. OLS picks up that chance co-movement. You get a high R², a large t-statistic and a tiny p-value. This is a spurious regression.

The standard tests break down because the OLS assumptions fail. The residuals are themselves nonstationary and strongly autocorrelated. The usual t-distribution no longer applies, so the t-statistic is far too large. A common warning sign is a high R² together with a very low Durbin-Watson statistic (close to 0).

The fix for a unit-root series is differencing. Take ΔYt = Yt - Yt-1. If Yt is a random walk, ΔYt = εt, which is white noise and stationary. For prices, the log return ln(Pt ÷ Pt-1) is the first difference of the log price, so it does the same job. After differencing, regress changes on changes.

Do not confuse this with detrending. Detrending subtracts a deterministic trend (for example a fitted line a + bt). It works when the series is trend-stationary. It does not remove a stochastic trend from a unit root. Differencing does. Differencing also costs one observation and discards long-run level information.

Key formulas to remember

Random walk (with drift): Yt = δ + Yt-1 + εt; Has a unit root. Variance of Yt grows with t. δ = 0 gives a pure random walk.
First difference: ΔYt = Yt - Yt-1; For a random walk with drift, ΔYt = δ + εt, which is stationary. Lose one observation.
Log return: rt = ln(Pt) - ln(Pt-1) = ln(Pt ÷ Pt-1); First difference of log price. Usually stationary even when the price is not.
Deterministic trend model: Yt = a + b·t + εt; Trend-stationary. Remove the trend by detrending (use the residuals), not by differencing.
Spurious regression signature: High R² + significant t-statistics + very low Durbin-Watson; A warning sign only. Confirm with a unit root test such as Dickey-Fuller.
Second difference: Δ²Yt = ΔYt - ΔYt-1; Used only if the first difference is still nonstationary (series integrated of order 2).

How to solve Spurious Regression and Differencing questions

Use this sequence for any question on spurious regression or making a series stationary.

1Identify what each series is: a price level, an index, a rate, or a return. Levels of prices and indices are usually nonstationary. Returns usually are not.
2Check for a unit root or trend. Look at the stem: a random walk, a coefficient of 1 on the lagged value, or a failed Dickey-Fuller test all point to a unit root.
3Decide the type of nonstationarity. A stochastic trend (unit root) needs differencing. A deterministic trend needs detrending.
4If the question regresses two nonstationary series, check for the spurious signs: high R², large t-statistics, very low Durbin-Watson.
5Apply the fix: compute ΔYt = Yt - Yt-1, or ln(Pt ÷ Pt-1) for prices. Remember you lose one observation.
6Re-state the regression in differences or returns and interpret the slope as a relationship between changes.
7If asked for a value, compute the difference or log return carefully and keep sign and units consistent.

Quickest way: Three-question shortcut

When to use it: Use it when a multiple-choice question asks which remedy or conclusion is correct and you have under two minutes.

Is the series a level of a price, index or other random walk? If yes, assume a unit root.
Unit root means difference it (or use log returns). A fitted time trend means detrend.
High R² with very low Durbin-Watson on levels means suspect spurious regression and do not trust the t-statistics.

Common mistakes in Spurious Regression and Differencing

Trusting a high R² and large t-statistic from a regression of two trending price series.
Students treat the usual OLS test statistics as valid without checking stationarity.
Fix: Check for unit roots first. If both series have them, the t-statistics are unreliable. Difference before regressing.
Detrending a random walk and calling it stationary.
Detrending and differencing both sound like 'removing the trend'.
Fix: Detrending only removes a deterministic trend. A random walk has a stochastic trend, so you must difference it.
Differencing a series that is already stationary.
Students think differencing is always safe.
Fix: Over-differencing adds a moving-average term, removes useful information and lowers precision. Difference only when a unit root is present.
Forgetting that differencing loses one observation.
The sample size is rarely checked after the transformation.
Fix: With n price observations you get n - 1 differences or returns. Use n - 1 in any degrees-of-freedom calculation.
Computing a log return with the wrong ratio or log base.
Speed under time pressure.
Fix: Use ln(Pt ÷ Pt-1), the natural log, with the latest price on top. A price rise gives a positive return.
Believing the high R² proves the series are related over the long run.
Students mix up spurious regression with cointegration.
Fix: Co-movement in levels is only meaningful if the series are cointegrated. Otherwise it is chance.

Worked examples

Example 1

An analyst regresses the level of one equity index on the level of an unrelated commodity price index, both modelled as random walks. The regression gives R² = 0.91, a slope t-statistic of 14, and a Durbin-Watson statistic of 0.15. What is the most likely explanation and the appropriate remedy?

Show the solution

Both series are random walks, so each has a unit root and is nonstationary.
A high R² and large t-statistic with a Durbin-Watson near 0 is the classic sign of spurious regression.
Residuals are highly autocorrelated, so the standard t-distribution does not apply and the t-statistic is overstated.
The remedy for a stochastic trend is first differencing: regress ΔY on ΔX, or use returns.

Answer: The result is most likely spurious. Difference both series (or use log returns) and re-run the regression.

Example 2

An index closes at 4,000 on day 1 and 4,200 on day 2. Compute the first difference and the log return for day 2. Then state how many returns you get from a sample of 251 daily closes.

Show the solution

First difference = 4,200 - 4,000 = 200 index points.
Log return = ln(4,200 ÷ 4,000) = ln(1.05).
ln(1.05) ≈ 0.04879, which is 4.88%.
Each return needs two consecutive prices, so 251 closes give 251 - 1 = 250 returns.

Answer: First difference = 200 points; log return ≈ 4.88%; the sample yields 250 returns.

Exam tips

If the stem says 'random walk', 'unit root' or 'prices', expect the answer to involve differencing or returns.
A high R² with a very low Durbin-Watson on levels is the usual cue for spurious regression.
Separate detrending (deterministic trend) from differencing (unit root). Exam options often swap them.
Count observations: n prices give n - 1 differences.
Do not choose 'the regression is reliable because R² is high' for nonstationary levels.

Practice questions from Nonstationary Time Series

Spurious Regression and Differencing: frequently asked questions

What is spurious regression in time series?

It is a regression of one nonstationary series on another that shows a strong, significant relationship when none exists. The high R² and t-statistics come from shared random wandering, not from a true link. The standard test statistics are invalid in this case.

How do you make a time series stationary by differencing?

Subtract the previous observation from the current one: ΔYt = Yt - Yt-1. For a random walk this gives white noise, which is stationary. For prices, use log returns, ln(Pt ÷ Pt-1), instead.

What is the difference between differencing and detrending?

Detrending subtracts a fitted deterministic trend, such as a + bt, and keeps the residuals. Differencing subtracts the prior value and removes a stochastic trend caused by a unit root. Use detrending for trend-stationary series and differencing for unit-root series.

Does a high R² always mean spurious regression?

No. A high R² is only a warning sign when the variables are nonstationary and residuals are strongly autocorrelated. Test for unit roots, for example with the Dickey-Fuller test, before you decide.