The seasonal $F$-test

(a) Prove the second equality in (10.27): $\frac{(\mathrm{RSS}_R - \mathrm{RSS}_U)/q}{\mathrm{RSS}_U/(n-k)} = \frac{R^2/q}{(1-R^2)/(n-k)}$ with $q = 11$, $k = 12$. (b) The month-dummy regression for NG=F over Sep 2000–Oct 2026 ($n = 314$) has $R^2 = 0.0697$. Compute $F$, its $p$-value and the 5% critical value of $F_{11,302}$. (c) Show that $\hat\beta_m$ in (10.26) is the sample mean of month $m$'s returns, and that with 26 observations and a monthly standard deviation of 16.08% its standard error is about 3.15%. How large would a month's true mean have to be for a 5% two-sided test to detect it with 80% power? (d) Why does the test reject more readily with classical than with HC-robust standard errors for gas? (Think about which months are most volatile.)