Exact Skewness–Kurtosis Tests for Multivariate Normality and Goodness‐of‐Fit in Multivariate Regressions with Application to Asset Pricing Models*
Bibliographic record
Abstract
Abstract We study the problem of testing the error distribution in a multivariate linear regression (MLR) model. The tests are functions of appropriately standardized multivariate least squares residuals whose distribution is invariant to the unknown cross‐equation error covariance matrix. Empirical multivariate skewness and kurtosis criteria are then compared with a simulation‐based estimate of their expected value under the hypothesized distribution. Special cases considered include testing multivariate normal and stable error distributions. In the Gaussian case, finite‐sample versions of the standard multivariate skewness and kurtosis tests are derived. To do this, we exploit simple, double and multi‐stage Monte Carlo test methods. For non‐Gaussian distribution families involving nuisance parameters, confidence sets are derived for the nuisance parameters and the error distribution. The tests are applied to an asset pricing model with observable risk‐free rates, using monthly returns on New York Stock Exchange (NYSE) portfolios over 5‐year subperiods from 1926 to 1995.
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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 | 0.038 | 0.299 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.005 | 0.004 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 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".