The long-run relationship between market risk and return
Bibliographic record
Abstract
Many finance applications require an annual measure of the market premium for equity. Using a long sample combined with a very parsimonious conditional variance function, we find a positive relationship between market risk and expected excess returns. Unlike traditional exponential-smoothing filters, our specification has a well-defined unconditional variance and allows for mean reverting volatility forecasts. Although total volatility is significantly priced, the smooth long-run component in volatility is more important for capturing the dynamics of the premium. This parameterization produces realistic time-varying market equity premium estimates over the entire 1840-2003 period. For example, our results show that the premium was relatively low in the mid-1990s but has recently increased. Results are robust to univariate specifications that condition on either levels or logs of past realized volatility (RV), as well as to a new bivariate risk-return model of returns and RV for which the conditional variance of excess returns is the conditional expectation of the realized volatility process.
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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 | 0.002 | 0.019 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".