Journal Article

Models as Approximations I

Pitkin, Emil, Brown, Lawrence, Zhao, Linda, Berk, Richard, Buja, Andreas, Traskin, Mikhail, Zhang, Kai, George, Edward

2019

Overview

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-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/√𝑁; (5) inference needs to be based on model-robust standard errors, including sandwich estimators or the 𝑥-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.

• Publisher: Institute of Mathematical Statistics
• Subject: Approximation / Diagnostic systems / Estimating techniques / Mathematical models / Nonlinearity / Parameters / Regression analysis / Robustness (mathematics) / Standard error / Statistical analysis / Statistical inference