Holding square footage fixed, more bedrooms means smaller rooms, and smaller rooms sell for less. At the same 1,800 square feet, the 5-bedroom house has cramped rooms and worse flow; the 3-bedroom has spacious ones. Real buyers pay for that. The coefficient isn’t reporting “bedrooms are bad” in isolation — it’s reporting what happens to price as bedroom count moves while square footage stays put, which is a very different, and much narrower, question.
This is the single most important habit for reading any linear model: a coefficient’s meaning depends on what else is in the model. Drop square footage entirely and refit on bedrooms alone, and the coefficient would very likely flip positive — because without square footage to hold fixed, “more bedrooms” mostly just means “bigger house,” and bigger houses cost more. Same data, same target, different coefficient sign, because the question being answered changed. A coefficient is not a fact about a feature; it’s a fact about a feature given the company it keeps.
Why each wrong option fails:
- “Causes a price drop” claims causation from a fitted correlation — the classic trap. Nothing about a linear fit tells you bedrooms cause anything; it reports an association conditional on the other features. (Could adding a genuine extra bedroom, without shrinking the others, ever lower price? Almost certainly not — but that’s not the scenario the coefficient describes.)
- “The model is broken” assumes a sensible feature must always show the sign your intuition expects. A correctly fit model reports the data’s actual conditional structure, which can and regularly does surprise you — that’s often the useful part.
- “Pure noise” would be true only if the coefficient’s uncertainty (a standard error, not covered here) swamped its value — nothing in the setup suggests that; the sign has a clean, legible explanation once you condition correctly.
A related trap worth flagging now, revisited properly under “leakage” in stage 3: multicollinearity — when two features are themselves highly correlated (square footage and bedroom count usually are), the model can struggle to tell which one “deserves” credit, and coefficients can become unstable or swap signs with small data changes. The bedroom coefficient here is a legitimate conditional effect, not an artifact — but the same negative-sign surprise can also, in messier data, be exactly that artifact. Telling the two apart takes more than staring at one sign; it’s why stage 3 exists.
Where this goes: so far, fitting meant “minimize squared error” without saying how the search finds the minimizing weights. Next: the loss surface those weights are searched over, and gradient descent — rolling downhill on it — including what happens when the hill is walked with too large a step.