Omitted Variable Bias

Abstract

Omitted variable bias arises when a feature that affects price is left out of a model and has a nonzero coefficient on an included regressor in the auxiliary regression of the left-out feature on the included ones --- partial, not merely marginal, correlation; the included regressor's coefficient then absorbs part of the left-out feature's effect. Both requirements are necessary, and the widely held belief that any important missing variable biases a model is false as stated: a left-out feature uncorrelated with the regressors costs precision, not accuracy. The entry distinguishes condition, a state of the house that is largely latent, from the condition score that estimates it (the Journal's 0.00–10.00 scale) and from a remodel, an event whose value is the condition and quality it produced; a remodel indicator beside a condition score stands for whatever the remodel brought that the other features do not record. Theil's formula for the bias is developed and verified exactly against a sixteen-property extension of the Pacifica dataset in which the remodel indicator carries $60,000, the price difference the constructed data attach to a remodel at the same GLA, lot, view and condition score. Omitting the indicator inflates the condition coefficient from $20,000 to $34,816 a point and deflates the view coefficient from $100,000 to $79,178, shifts equal to the product of the omitted coefficient and the auxiliary slopes to the cent. Bias is relative to the question asked: the reduced model answers one question correctly and is wrong only when its coefficients are read as adjustments. The example makes three defensibility points: the bias is invisible in fit statistics, with an R-squared of 0.97 concealing a coefficient off by 74 percent; it does not diminish with sample size; and a biased model can predict accurately while yielding adjustments that are unusable, failing hardest on the atypical properties an appraiser is most often retained to value. The constructed market puts condition into the regression, as most appraisers' regressions do; the Journal's method does not, and the lesson carries, not the specification.