Exogeneity tests, weak identification, incomplete models and non-Gaussian distributions: Invariance and finite-sample distributional theory
Bibliographic record
Abstract
We study the distribution of Durbin-Wu-Hausman (DWH) tests for exogeneity from a finite-sample viewpoint, under the null and alternative hypotheses. We consider linear structural models with possibly non-Gaussian errors, where structural parameters may not be identified and where reduced forms can be incompletely specified (or nonparametric). On level control, we characterize the null distributions of all the test statistics. Through conditioning and invariance arguments, we show that these distributions do not involve nuisance parameters. In particular, this applies to several test statistics for which no finite-sample distributional theory is yet available, such as the standard statistic proposed by Hausman (1978). The distributions of the test statistics may be non-standard – so corrections to usual asymptotic critical values are needed – but the characterizations are sufficiently explicit to yield finite-sample (Monte-Carlo) tests of the exogeneity hypothesis. The procedures so obtained are robust to weak identification, missing instruments or misspecified reduced forms, and can easily be adapted to allow for parametric non-Gaussian error distributions. We give a general invariance result (block triangular invariance) for exogeneity test statistics. This property yields a convenient exogeneity canonical form and a parsimonious reduction of the parameters on which power depends. In the extreme case where no structural parameter is identified, the distributions under the alternative hypothesis and the null hypothesis are identical, so the power function is flat, for all the exogeneity statistics. However, as soon as identification does not fail completely, this phenomenon typically disappears. We present simulation evidence which confirms the finite-sample theory. The theoretical results are illustrated with two empirical examples: the relation between trade and economic growth, and the widely studied problem of the return of education to earnings.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.025 | 0.135 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.004 | 0.004 |
| Science and technology studies | 0.001 | 0.009 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".