Conditional Expectation

Abstract

The conditional expectation of price given a vector of characteristics is the mean of the price distribution at that point in characteristics space. Viewed as a function of the characteristics, it is called the conditional expectation function (CEF), and it is the function a hedonic regression approximates. Least squares gives the best linear approximation to the CEF, and it reproduces the CEF exactly when the CEF itself 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. That decomposition always holds; what the later entries ask is whether the equation a regression actually fits, the features it conditions on, and the sales it has allow its rates to be read as the market effects wanted. The sales comparison grid, when its adjustments are extracted from the sales, can be viewed as a small nonparametric estimator of a conditional mean, which makes three of its standing assumptions explicit. A worked example shows that the eight-comparable Pacifica dataset of Issue 1 is exactly noiseless and returns the equation its prices were built from, with condition on the Journal's 0.00–10.00 score (average 5.00, good 7.00, excellent 9.00; $20,000 a point) and a remodel indicator whose $60,000 is the conditional price difference associated with the attributes the remodel produced but the conditioning variables do not represent. The example puts condition into the regression, as most appraisers' regressions do; the Journal's own method, the Residual Constraint Approach, does not, and the entry says how it differs. The four-variable regression of Issue 1 and its remodel-augmented counterpart target different conditioning sets, and Comp H's $46,455 residual is the gap between the four-variable conditional expectation function and its fitted linear projection, which is where the remodel's effect hides.