The hidden cost of forcing a regression through zero

Question

Tempted by a theory that says "y must be zero when x is zero," you drop the intercept and fit y = bx + \varepsilon (regression through the origin). Explain what can go wrong, and when forcing the line through zero is actually justified.

Accepted Answer

What the constraint does. Removing the intercept forces the fitted line to pass through the origin. The slope becomes \hat b = \frac{\sumi xi yi}{\sumi xi^2}, which differs from the usual centered formula, it uses raw products, not deviations from means. The dangers. 1. Bias if the true intercept is nonzero. If the real relationship has any offset, forcing zero pivots the line to compensate, distorting the slope, potentially a lot, and in either direction. You have imposed a false constraint, and misspecification bias follows. 2. Residuals no longer sum to zero. With an intercept, OLS guarant…

The hidden cost of forcing a regression through zero

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