Description: This figure shows the Monte Carlo simulation-based rejection probability of finding a significant coefficient of relative risk aversion (γ) for GMM estimation with the non-linear moment condition

Results are based on 1,000 draws of multivariate normally distributed data calibrated to the market excess return as the single test asset (N = 1) and a hypothetical consumption factor with T = 55 years of time-series observations. The far left in the figure shows 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-AR test as reported in Kleibergen and Zhan (2020). The second line in the legend corresponds to inference based on the t-statistic using GMM standard errors (textbook approach). The third line in the legend reports results when using 95% confidence intervals based on a pairwise bootstrap (similar to Burnside, 2011). Figure A.2 in the Appendix shows the results for T = 200.

Interpretation: GMM standard errors over-reject γ estimates for “useless” factors when testing the non-linear pricing equation. Results have to be carefully interpreted, e.g., by asking whether the estimated parameter γ is economically reasonable or whether the factor correlation allows identifying the parameter of interest (e.g., by using the powerful tests shown in Figure 1). The GMM-AR test, as proposed by Kleibergen and Zhan (2020), does not over-reject “useless” factors but has only limited power to detect “useful” factors. Bootstrap confidence intervals also do not over-reject “useless” factors but have relatively high power to detect “useful” factors.