Models as Approximations, Part I: A Conspiracy of Nonlinearity and Random Regressors in Linear Regression
Year: 2014
Type: preprint
Abstract: In the early 1980s Halbert White inaugurated a model-robust'' form of statistical inference based on the 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 can't be treated as fixed. The consequences are deep: (1)~population slopes need to be re-interpreted as statistical functionals obtained from OLS fits to largely arbitrary joint $\xy$~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; (5)~inference needs to be based on model-robust standard errors, including sandwich estimators or the $\xy$~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.
Source: arXiv (Cornell University)
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2566828246