Description: This table shows estimates of the coefficient of relative risk aversion (γ) using the non-linear GMM moment condition:

The market excess return is the single test asset. Below the GMM estimate of γ is the GMM standard error, and the 95% confidence interval of γ according to a pairwise bootstrap or the GMM-AR test. J_T is the value of the objective function, p(rαnk), KZ, is the p-value to the GMM-rank test as in Kleibergen and Zhan (2020), which leads to an ill-conditioned objective function (Figure 2). p(rαnk), corrected, is the p-value for an alternative version of the GMM-rank test where the objective function is well-behaved (also shown in Figure 2). p(cor), is the p-value for a direct test of the correlation coefficient. The short sample is considered by Kleibergen and Zhan (2020) but not the full sample period.

Interpretation: According to the GMM-rank test, as implemented by Kleibergen and Zhan (2020), no consumption factor is significantly correlated with the market excess return. The unbounded/disjointed GMM-AR confidence intervals do not allow to conclude that consumption has explanatory power for the equity premium. However, the direct test of the correlation coefficient indicates that garbage, Q4-Q4, and unfiltered consumption are significandy correlated with the market excess return. The bootstrap confidence intervals do not include zero and indicate that garbage, Q4-Q4 and unfiltered consumption help to explain the equity premium.

The difference in the test outcomes are in line with the power curves shown in Figures 1 and 3. The GMM-AR and the GMM-rank tests suffer from the “GMM trap” problem and have low or no power to detect “useful” factors. The modified GMM-rank test, or testing the correlation coefficient directly, is considerably more powerful in detecting “useful” factors (Figure 1). Bootstrap confidence intervals do not over-reject “useless” factors and are powerful in detecting “useful” factors (Figure 3). Kleibergen and Zhan (2020) attribute the difference between GMM-AR test and GMM standard errors to an incorrect size of the textbook approach in the presence of a “useless” factor. Except for reported consumption, the extended results in this paper show that the difference in the two tests can be attributed to the low power of the GMM-AR test.