Price of Risk Estimates: Small Cross-Section.
| Short Sample (1960–2014) | Full Sample (1928–2014) | ||||||
|---|---|---|---|---|---|---|---|
| T = 55 and N = 7 | T = 87 and N = 7 | ||||||
| Reported Dec. | Garbage Dec. | P-J Dec. | Q4-Q4 Dec. | Unfiltered T.A. | Reported Dec. | Unfiltered T.A. | |
| λ | 4.01 | 2.17 | 7.20 | 1.96 | 2.63 | 8.32 | 2.91 |
| t(λ)sh | 1.35 | 2.44 | 1.46 | 2.06 | 2.79 | 1.02 | 3.63 |
| Btrp c.i.95% | (−4.5,5.8) | (0.6,4.9) | (−6.7,12.8) | (0.4,5.2) | (1.0,6.1) | (−9.7,11.5) | (1.5,4.7) |
| GRS-FAR c.i.95% | disjointed | (2.1, 15.0) | disjointed | disjointed | (3.6, 6.6) | disjointed | empty |
| implied γ | 230.19 | 26.16 | 69.37 | 90.46 | 38.79 | 173.67 | 17.76 |
| # signif. positive | 0 | 7 | 2 | 6 | 7 | 0 | 7 |
| t(βALL), p-value | 0.41 | 0.00 | 0.11 | 0.01 | 0.00 | 0.27 | 0.00 |
| F(β = 0), p-value | 0.60 | 0.01 | 0.09 | 0.07 | 0.00 | 0.10 | 0.00 |
| Short Sample (1960–2014) | Full Sample (1928–2014) | ||||||
|---|---|---|---|---|---|---|---|
| T = 55 and | T = 87 and | ||||||
| Reported Dec. | Garbage Dec. | P-J Dec. | Q4-Q4 Dec. | Unfiltered T.A. | Reported Dec. | Unfiltered T.A. | |
| 4.01 | 2.17 | 7.20 | 1.96 | 2.63 | 8.32 | 2.91 | |
| 1.35 | 2.44 | 1.46 | 2.06 | 2.79 | 1.02 | 3.63 | |
| Btrp c.i.95% | (−4.5,5.8) | (0.6,4.9) | (−6.7,12.8) | (0.4,5.2) | (1.0,6.1) | (−9.7,11.5) | (1.5,4.7) |
| GRS-FAR c.i.95% | disjointed | (2.1, 15.0) | disjointed | disjointed | (3.6, 6.6) | disjointed | empty |
| implied γ | 230.19 | 26.16 | 69.37 | 90.46 | 38.79 | 173.67 | 17.76 |
| # signif. positive | 0 | 7 | 2 | 6 | 7 | 0 | 7 |
| t(βALL), p-value | 0.41 | 0.00 | 0.11 | 0.01 | 0.00 | 0.27 | 0.00 |
| 0.60 | 0.01 | 0.09 | 0.07 | 0.00 | 0.10 | 0.00 | |
Description: The test assets are six portfolios sorted by value and size (from Ken French’s website), plus the market excess return. The reported statistics are the same as in Table 4. Table A.2 in the Appendix reports results with estimation of the intercept.
Interpretation: The small cross-section further mitigates concerns of low power. Garbage and unfiltered consumption (short and full sample) pass the multivariate rank test. Results based on the univariate rank robustness test now align in most cases. In the short sample, garbage and unfiltered consumption have a significant price of risk according to the GRS-FAR test. In the full sample, the GRS-FAR confidence interval is empty for unfiltered consumption. When comparing these results with the different specifications in the previous tables, I find that the Fama-MacBeth Shanken t-statistics and the bootstrap confidence intervals allow for very similar conclusions across the different specifications. On the contrary, the GRS-FAR test offers unbounded, bounded and empty confidence sets across the different specifications, which illustrates the poor power problem in practice.
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