Limitations of regression analysis

A regression will fit any two columns you give it and return a slope, an intercept and an R² with total confidence, even when the relationship is meaningless. The green-belt skill is knowing the four ways it lies, and the guard against each one.

Why can a high R² mislead you?

1. R² is not causation. A high R² says two series move together. It never says one causes the other. Ice-cream sales and drowning deaths correlate strongly, since both rise in summer, yet neither causes the other. A lurking third variable, temperature, drives both. Before you read a slope as a lever you can pull, ask what the mechanism would be, and whether a confounder could produce the co-movement on its own. Guard against this by demanding a plausible causal story and ruling out obvious confounders before you act on a correlation.

1. Tiny samples fool you. With only three points and a two-parameter line, the fit has almost no freedom to be wrong. Three random points routinely give an R² near 1. Even eight points is thin. A high R² on small n is weak evidence, because a couple of lucky points can manufacture it. Guard against this by treating R² from small samples with deep suspicion. Look at the standard errors and the number of points, and don't stop at the headline R². Our footfall example's R² of 0.99 is impressive for eight points, but you'd want far more data before betting a store-expansion decision on the exact slope.

How do spurious trends and extrapolation mislead?

1. Non-stationarity and spurious trends. Two unrelated series that both trend upward over time will correlate highly for a purely mechanical reason. They share the common factor "time," and there's no real link behind it. Picture a company's revenue and the number of blog posts it published, both rising over six years. The regression can show R² = 0.996, with a slope claiming each post "adds" ₹7,000 of revenue. It's nonsense. Regress the year-on-year changes instead of the levels and the fake relationship usually evaporates. Guard against this by distrusting any two rising lines in a time series. Difference or de-trend the data before regressing, and never read a level-on-level trend fit as a driver.

1. Extrapolation beyond the data. A line fitted over footfall of 12 to 34 thousand says nothing reliable about footfall of 200 thousand. The linear form is a local approximation that may bend or break outside the observed range. FORECAST.LINEAR will happily compute a prediction there, but it's a guess dressed as arithmetic. Guard against this by forecasting only inside, or barely beyond, the range of the x's you actually observed. Flag any out-of-range prediction as an assumption and never present it as a result.

What is the one habit to take away?

Put the four together and you get one habit. A regression is a hypothesis-sharpening tool, and it was never a truth machine. It tells you the best-fit line given these points. Whether that line means anything is a judgment you make with domain knowledge, an eye on the sample size and a healthy fear of trends. Beginners report the R². Professionals interrogate it.