Description: This figure shows the Monte Carlo simulation-based rejection probability of the H0: λ= 0 for tests of the linear factor model

Results are based on 10,000 draws of multivariate normally distributed data calibrated to the 31 test assets and unfiltered consumption (as reported in Table 4). The data have T = 55 time-series observations (as in the empirical data). 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 1.0 with the “true” factor (the “true” factor imposes the sample properties of unfiltered consumption as the population properties). These results can be interpreted as the power of the tests for an empirical factor. The first line in the legend corresponds to a Fama-MacBeth/Shanken t-test for the H0: λ = 0 at the 5% significance level (textbook approach). The second line in the legend corresponds to inference based on the GRS-FAR 95% confidence interval, as in Kleibergen and Zhan (2020). The third line in the legend corresponds to inference based on the bootstrap 95% confidence interval, as in Burnside (2011). Figure A.4 in the Appendix shows the results for T = 200. Figure A.5 and Figure A.6 show results with estimation of the intercept.

Interpretation: The Fama-MacBeth/Shanken approach does not over-reject “useless” factors in small samples, and a researcher can correctly conclude that a “useless” factor is “not significantly priced.” This result is in line with Kan and Zhang (1999) and illustrates that Kleibergen and Zhan (2020) overstate the relevance of the over-rejection problem of Fama-MacBeth/Shanken t-statistics in the presence of a “useless” factor in consumption-based asset pricing. The GRS-FAR test is robust to the “useless” factor problem, but it has only low power to detect “useful” factors when the model is misspecified. The GRS-FAR test requires the model to be correctly specified to be expected to allow for inference on the price of risk, which is unrealistic in empirical work. The bootstrap confidence interval does not over-reject “useless” factors and is powerful in detecting “useful” factors.