Linear regression is a fundamental building block of statistical data
analysis. It amounts to estimating the parameters of a linear model that maps
input features to corresponding outputs. In the classical setting where the
precision of each data point is fixed, the famous Aitken/Gauss-Markov theorem
in statistics states that generalized least squares (GLS) is a so-called "Best
Linear Unbiased Estimator" (BLUE). In modern data science, however, one often
faces strategic data sources, namely, individuals who incur a cost for
providing high-precision data.
In this paper, we study a setting in which features are public but
individuals choose the precision of the outputs they reveal to an analyst. We
assume that the analyst performs linear regression on this dataset, and
individuals benefit from the outcome of this estimation. We model this scenario
as a game where individuals minimize a cost comprising two components: (a) an
(agent-specific) disclosure cost for providing high-precision data; and (b) a
(global) estimation cost representing the inaccuracy in the linear model
estimate. In this game, the linear model estimate is a public good that
benefits all individuals. We establish that this game has a unique non-trivial
Nash equilibrium. We study the efficiency of this equilibrium and we prove
tight bounds on the price of stability for a large class of disclosure and
estimation costs. Finally, we study the estimator accuracy achieved at
equilibrium. We show that, in general, Aitken's theorem does not hold under
strategic data sources, though it does hold if individuals have identical
disclosure costs (up to a multiplicative factor). When individuals have
non-identical costs, we derive a bound on the improvement of the equilibrium
estimation cost that can be achieved by deviating from GLS, under mild
assumptions on the disclosure cost functions.Comment: This version (v3) extends the results on the sub-optimality of GLS
(Section 6) and improves writing in multiple places compared to v2. Compared
to the initial version v1, it also fixes an error in Theorem 6 (now Theorem
5), and extended many of the result
