Three commodities, 36 months

The joint $F$-test $p$-values in 10.13.4 are 0.2103 (gold), 0.6897 (WTI) and 0.0232 (gas). (a) Apply Bonferroni and Holm at 5% across the three commodities. Give Holm-adjusted $p$-values. (b) The smallest of the 36 month-versus-rest $p$-values is 0.0247. What is its Bonferroni-adjusted value? (c) Three of the 36 raw $p$-values are below 0.05. If all 36 nulls were true and the tests independent, what is the probability of at least three rejections? (d) The out-of-sample sign rule on gold earns $t = 1.94$ on 154 months. Compute the two-sided $p$-value (use $t_{153}$). Holm-adjust it together with the WTI ($t = 1.28$) and gas ($t = 1.61$) rules. What would you need to see before trading it?