Summary of key results · M12 Geopolitics

Result	Formula
Market model	$R_{i\tau}=\alpha_i+\beta_iR_{m\tau}+\varepsilon_{i\tau}$ (12.1)
Abnormal return	$AR^_i=R^_i-X^\hat\theta_i=\varepsilon^_i-X^*(X'X)^{-1}X'\varepsilon_i$ (12.4, 12.6)
AR covariance	$V_i=\sigma_i^2[I+X^(X'X)^{-1}X^{\prime}]$ (12.5)
Single-day variance	$\sigma^2[1+1/L_1+(R_{m\tau}-\bar R_m)^2/\sum(R_{ms}-\bar R_m)^2]$ (12.7)
CAR variance	$\sigma^2[L_2+\gamma'X^(X'X)^{-1}X^{\prime}\gamma]$ (12.8)
Exact test	$SCAR\sim t_{L_1-2}$ (12.9)
J1	$\overline{CAR}/\sqrt{\sum\widehat{\operatorname{Var}}(CAR_i)/N^2}$ (12.10)
Patell	$\sum SCAR_i/\sqrt{N(L_1-2)/(L_1-4)}$ (12.12)
BMP	$\overline{SCAR}/(s_{SCAR}/\sqrt N)$ (12.13), robust to common variance scaling
Clustering	$\operatorname{Var}(\overline{AR})=\frac{\sigma^2}N[1+(N-1)\bar\rho]$
GPR	$100\,s_t/\bar s_{1985-2019}$ (12.14)
Supply shock	$\Delta\ln P=\phi/(\epsilon_s+\epsilon_d)$ (12.21)
Bayes	$P(s\mid n)\propto P(n\mid s)P(s)$ ; odds × LR; log-odds add (12.16–12.18)
Decision tree	chance = expectation, decision = max (12.19)
EVPI	$\mathbb E_s\max_ax-\max_a\mathbb E_sx\ge0$ (12.20)