Table 4. Price of Risk Estimates:...

Table 4.

Price of Risk Estimates: Estimation Without Intercept.

	Short Sample (1960–2014)					Full Sample (1928–2014)
	T = 55 and N = 31					T = 87 and N = 21
	Reported Dec.	Garbage Dec.	P-J Dec.	Q4-Q4 Dec.	Unfiltered T.A.	Reported Dec.	Unfiltered T.A.
λ	0.31	2.09	10.14	2.28	2.44	7.87	2.66
t(λ)_sh	0.57	2.51	1.05	1.87	2.59	1.00	3.32
Btrp c.i._95%	(−4.8,4.9)	(0.6,4.6)	(−10.8,13.6)	(−1.8,5.4)	(0.9,5.4)	(-9.2,9.3)	(1.2,4.3)
GRS-FAR c.i._95%	unbounded	(−0.8, 7.8)	unbounded	unbounded	disjointed	empty	(0.9, 5.5)
implied γ	17.64	25.25	97.75	104.84	35.95	164.24	16.26
# signif. positive	0	31	0	25	31	1	21
t(β_ALL) p-value	0.50	0.00	0.21	0.04	0.00	0.28	0.00
F(β = 0), p-value	0.27	0.01	0.53	0.27	0.21	0.00	0.00

	Short Sample (1960–2014)					Full Sample (1928–2014)
	T = 55 and N = 31					T = 87 and N = 21
	Reported Dec.	Garbage Dec.	P-J Dec.	Q4-Q4 Dec.	Unfiltered T.A.	Reported Dec.	Unfiltered T.A.
λ	0.31	2.09	10.14	2.28	2.44	7.87	2.66
t(λ)_sh	0.57	2.51	1.05	1.87	2.59	1.00	3.32
Btrp c.i._95%	(−4.8,4.9)	(0.6,4.6)	(−10.8,13.6)	(−1.8,5.4)	(0.9,5.4)	(-9.2,9.3)	(1.2,4.3)
GRS-FAR c.i._95%	unbounded	(−0.8, 7.8)	unbounded	unbounded	disjointed	empty	(0.9, 5.5)
implied γ	17.64	25.25	97.75	104.84	35.95	164.24	16.26
# signif. positive	0	31	0	25	31	1	21
t(β_ALL) p-value	0.50	0.00	0.21	0.04	0.00	0.28	0.00
F(β = 0), p-value	0.27	0.01	0.53	0.27	0.21	0.00	0.00

Description: The test assets are 31 portfolios in the short sample (ten size, value, and investment decile portfolios, plus the market excess return) and 21 test assets in the full sample (size and value decile portfolios, plus the market excess return), λ is the cross-sectional Fama-MacBeth estimate of the price of risk when the cross-sectional intercept is restricted to zero

E (R_{i, t}^{e}) = β_{C, i} \times λ .

t(λ)_sh is the Shanken (1992)-corrected t-statistic for the price of risk. Angle brackets report the 95% bootstrap confidence interval for the price of risk, as in Burnside (2011). Brackets report the 95% confidence interval for the price of risk according to the GRS-FAR test, as in Kleibergen and Zhan (2020). γ is the coefficient of relative risk aversion implied by $\hat{λ} / v a r (Δ C_{t})$ ⁠. The bottom of the table reports number of significant individual betas, the univariate rank robustness test t(β_ALL) and the multivariate rank test F(β = 0). Table A.l in the Appendix reports results with estimation of the intercept.

Interpretation: Short sample: according to the GRS-FAR test, no consumption measure has a significant price of risk. However, the bootstrap confidence intervals indicate that garbage and unfiltered consumption have a significant price of risk. Full sample: The GRS-FAR test and the bootstrap confidence interval suggest that unfiltered consumption has a significant and positive price of risk. The full sample evidence is not shown in Kleibergen and Zhan (2020).

The difference in the test outcomes in the small sample are in line with the power curves shown in Figure 5. The bootstrap approach and the GRS-FAR test do not over-reject “useless” factors. The Fama-MacBeth t-statistics over-reject “useless” factors in large samples and not in small samples. Fama-MacBeth t-statistic and the bootstrap approach are considerably more powerful than the GRS-FAR test in detecting “useful” factors. Kleibergen and Zhan (2020) attribute the difference between Fama-MacBeth t-statistics and the GRS-FAR test to an incorrect size of the textbook approach in the presence of a “useless” factor. Instead, the extended results in this paper show that the difference in the two tests can be attributed to a low power of the GRS-FAR test.

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