IOC event study and the trades needed to validate the rule

For Indian Oil (IOC), the market model (12.1) estimated against the Nifty on $\tau\in[-250,-11]$ before 5 Nov 2026 ($L_1 = 240$ daily log returns) gave $\hat\alpha = -0.00034$, $\hat\beta = 1.117$, $\hat\sigma = 0.01337$. Over the same window the Nifty's mean log return was $\bar R_m = -0.00055$ and $\sum_s (R_{ms} - \bar R_m)^2 = 0.01828$. Log returns in the event window were: 5 Nov ($\tau = 0$), IOC $+0.01724$, Nifty $+0.02376$; 6 Nov ($\tau = 1$), IOC $+0.01204$, Nifty $+0.01992$.

(a) Compute $AR_0$ and its $t$-statistic using the variance (12.7). Compare with the $t$-statistic that ignores estimation error ($AR_0/\hat\sigma$). (b) Compute $CAR[0,1]$ and its $t$-statistic from (12.8). Show that for the market model $\gamma'X^*(X'X)^{-1}X^{*\prime}\gamma = L_2^2/L_1 + \big(\sum_{\tau}R_{m\tau} - L_2\bar R_m\big)^2/\sum_s(R_{ms}-\bar R_m)^2$. (c) The Nifty itself rose on 5 Nov partly because crude fell. Explain what the IOC abnormal return measures under this benchmark, and name one benchmark that would measure the total effect of crude on IOC, with its own drawback. (d) The EIA-Wednesday rule fired 36 times in about 15.8 years with an R standard deviation of $0.65R$. If its true net expectancy were $+0.10R$, how many trades are needed for the expected $t$-statistic to reach 2 M03 (3.2a), and for 80% power at $t \ge 2$ (3.2b)? How many years is the second number at the rule's signal rate?