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November 2019 Models as Approximations I: Consequences Illustrated with Linear Regression
Andreas Buja, Lawrence Brown, Richard Berk, Edward George, Emil Pitkin, Mikhail Traskin, Kai Zhang, Linda Zhao
Statist. Sci. 34(4): 523-544 (November 2019). DOI: 10.1214/18-STS693

Abstract

In the early 1980s, Halbert White inaugurated a “model-robust” form of statistical inference based on the “sandwich estimator” of standard error. This estimator is known to be “heteroskedasticity-consistent,” but it is less well known to be “nonlinearity-consistent” as well. Nonlinearity, however, raises fundamental issues because in its presence regressors are not ancillary, hence cannot be treated as fixed. The consequences are deep: (1) population slopes need to be reinterpreted as statistical functionals obtained from OLS fits to largely arbitrary joint ${x\textrm{-}y}$ distributions; (2) the meaning of slope parameters needs to be rethought; (3) the regressor distribution affects the slope parameters; (4) randomness of the regressors becomes a source of sampling variability in slope estimates of order $1/\sqrt{N}$; (5) inference needs to be based on model-robust standard errors, including sandwich estimators or the ${x\textrm{-}y}$ bootstrap. In theory, model-robust and model-trusting standard errors can deviate by arbitrary magnitudes either way. In practice, significant deviations between them can be detected with a diagnostic test.

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Andreas Buja. Lawrence Brown. Richard Berk. Edward George. Emil Pitkin. Mikhail Traskin. Kai Zhang. Linda Zhao. "Models as Approximations I: Consequences Illustrated with Linear Regression." Statist. Sci. 34 (4) 523 - 544, November 2019. https://doi.org/10.1214/18-STS693

Information

Published: November 2019
First available in Project Euclid: 8 January 2020

zbMATH: 07240208
MathSciNet: MR4048582
Digital Object Identifier: 10.1214/18-STS693

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.34 • No. 4 • November 2019
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