Endogeneity

Abstract

Endogeneity is the condition in which a regressor is correlated with the structural error --- the price minus what the causal relationship says it should be, a different quantity from the deviation about the conditional mean, which has zero conditional mean by construction --- so that its coefficient does not recover the causal effect of interest. This entry generalizes the omitted-variable case of entry 008 (Omitted Variable Bias), presenting omitted variables, simultaneity, selection, and measurement error as four mechanisms producing one condition, and distinguishing endogeneity from imprecision by the fact that additional sales do not reduce it. Measurement error pulls a coefficient toward zero only in the classical case; the error of a condition score is not classical, and a score that the known sale price has nudged is outcome-dependent measurement error, and its direction is known. Before the variables are chosen the effect must be named: a confounder, a cause of both the regressor and the price, must be held fixed; a mediator, the channel through which the regressor acts on the price, must not be, or the coefficient loses the part of the effect that runs through it. A remodel is a treatment that is not randomly assigned, and condition and quality are its mediators. The central result is that measuring an omitted variable cures the omission without necessarily curing the endogeneity: a remodel coefficient estimated from observational data combines what the remodel brought that the other features do not record with the extra price, beyond what their recorded features explain, of the houses whose owners chose to remodel, and an appraiser answering a renovation-payback question about a particular house from such a coefficient overstates the return, while one who also measured and held fixed all the condition and quality the remodel produced understates it. Since the ordinary regression cannot itself establish that a regressor is exogenous, the defensible response is an explicit argument about mechanism and direction, and design choices in assembling a comparable set do more for defensibility than any estimator applied afterward.