Conditional Expectation

The conditional expectation of price given a vector of characteristics is the mean of the price distribution at that point in characteristics space; as a function of the characteristics it is the conditional expectation function, and it is the object a hedonic regression approximates: least squares recovers its best linear approximation, and recovers the function itself exactly when it is linear. This entry names the estimand that the inaugural *Foundations* issue used without stating, and establishes the decomposition of price into a conditional mean plus a deviation whose conditional mean is zero by construction --- the identity on which the rest of the issue turns. The sales comparison grid is recast as a small nonparametric estimator of a conditional mean, which makes three of its standing assumptions explicit: that the function is smooth enough for nearby comparables to inform the subject, that its slope is roughly constant across the range an adjustment crosses, and that indications are unbiased draws around a common conditional mean. A worked example shows that the eight-comparable Pacifica dataset of issue 1 is exactly noiseless and recovers its generating function, and that the issue-1 regression and its remodel-augmented counterpart target different conditioning sets, each recovered only up to its best linear approximation: in the eight-property constructed market the four-variable conditional expectation function interpolates every price exactly, and Comp H's $46,455 residual is the gap between that function and its fitted linear projection --- which is precisely where the remodel effect hides.