Spurious Regressions with Time-Series Data: Further Asymptotic Results
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Bibliographic record
Abstract
Abstract A "spurious regression" is one in which the time-series variables are non stationary and independent. It is well known that in this context the OLS parameter estimates and the R 2 converge to functionals of Brownian motions, the "t-ratios" diverge in distribution, and the Durbin–Watson statistic converges in probability to zero. We derive corresponding results for some common tests for the normality and homoskedasticity of the errors in a spurious regression. Keywords: Asymptotic theoryHomoskedasticityNormalitySpurious regressionUnit rootsMathematics Subject Classification: Primary 62F05, 62J05, 91B84Secondary 62M10, 62P20 Acknowledgment The author is most grateful to Lauren Dong, Mike Veall, and an anonymous referee for their helpful comments on earlier versions of this article. Notes 1The 10% critical value for the Chi Square distribution with 2 degrees of freedom is 4.60517. 'RR' denotes the 'rejection rate', i.e., the percentage of the 5,000 simulated values for JB that exceeded this critical value. 1The 10% critical value for the Chi Square distribution with 1 degree of freedom is 2.70554. 'RR' denotes the 'rejection rate', i.e., the percentage of the 5,000 simulated values for BPG that exceeded this critical value. 1The 10% critical value for the Chi Square distribution with 1 degree of freedom is 2.70554. 'RR' denotes the 'rejection rate', i.e., the percentage of the 5,000 simulated values for BPG that exceeded this critical value.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.011 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it