Level 4 · Can the line forecast a bigger budget?

Prediction inside and outside the data

A fitted line is a summary of the points it was fitted on. Inside the range of x those points cover, it is a reasonable description. Outside that range it is an assumption: that the relationship keeps going in a straight line where you have never looked. Using a model outside the range of its data is called extrapolation, and it is the most common way a good-looking regression produces a bad forecast.

Harbor & Pine’s monthly spend runs from $23,861 to $70,675. The line’s prediction at $110,000:

5.503 * 110_000 - 40_288   # ≈ 565,000

$565,000 is 77% more than the best month the store has ever had. Maybe. Or demand runs out: there are only so many households shopping for a sofa in a given month, and returns on ad spend almost always flatten as budgets grow. The data cannot say, because it has no months like that.

What R² does and does not measure

R² = 0.845 says the line follows the ups and downs of the 36 months it was fitted on closely. It is not:

• a forecast error. “R² 0.85, so within 15%” is a common and wrong reading;
• evidence of cause. The holiday season lifts both spend and revenue (level 2), and that shared driver is a big part of the 0.845;
• valid outside the range. It was computed on the fitted months only.

The intercept shows the same problem at the other end: at zero spend the line predicts -$40,288 of revenue, which is impossible. Zero is as far outside the data on the left as $110,000 is on the right.

What to tell Jordan

“In past months, higher spend went with higher revenue. The line is fitted on budgets up to about $71,000 and includes the holiday effect, so it can’t forecast $110,000. If we want to know, we should raise spend in some regions first and compare them with the rest.”

Common mistakes

• Forecasting beyond the data because the formula happily returns a number.
• Reading R² as accuracy or as a percentage of correct predictions.
• Fixing the model instead of the question. More rows, daily data or a fancier curve still have no observations at $110,000 a month.