Description: This figure shows the Monte Carlo simulation-based probability of finding a “significant factor correlation” such that the coefficient of relative risk aversion can be identified and GMM standard errors are expected to be reliable. Results are based on 10,000 draws of multivariate normally distributed data calibrated to the market excess return (N = 1) and a hypothetical consumption factor with T = 55 years of time-series observations. Results on the far left show the rejection probability of a consumption factor that is in the population uncorrelated with the market excess return (“useless” factor). These results can be interpreted as the size of a test. Moving from the left to the right increases the population correlation coefficient from zero to 1.00 (“useful” factors). These results can be interpreted as the power of the tests. The vertical lines indicate the sample correlation coefficient of alternative consumption measures (see Table 1 for further details). The first line in the legend corresponds to the GMM-rank test as reported in Kleibergen and Zhan (2020). The second line in the legend corresponds to a modified version of the GMM-rank test and is proposed in this paper. The third line in the legend corresponds to a direct test of the correlation coefficient, as in Savov (2011).

Interpretation: The GMM-rank test as in Kleibergen and Zhan (2020) has no power to detect “useful” factors. The reason is that the GMM objective function of this test is numerically ill-conditioned and usually comes with two arbitrary corner solutions. I propose a corrected version that has a unique solution and avoids the issue. This version of the GMM-rank test has some power to detect “useful” factors. However, in short samples with limited time-series observations, a direct test of the correlation coefficient has the highest power to detect “useful” factors.