Summary of key results · M13 Forecasting

Result	Formula
Efficiency (MDS)	$\mathbb E_t[r_{t+1} - r^e_{t+1}] = 0 \Rightarrow \operatorname{Cov}(f_t, \varepsilon_{t+1}) = 0$ for any $\mathcal F_t$ -measurable $f_t$
Grossman–Stiglitz	$e^{ac}\sqrt{\operatorname{Var}(u\mid\theta)/\operatorname{Var}(u\mid P_\lambda)} = 1$ at interior $\lambda^\*$
AR(1)	stationary iff $\lvert\phi\rvert<1$ ; $\gamma_0 = \sigma^2/(1-\phi^2)$ ; $\rho_k = \phi^k$ ; $R^2 = \phi^2$
AR( $p$ )	roots of $\Phi(z)$ outside unit circle ⇔ eigenvalues of companion matrix inside
Yule–Walker	$R\boldsymbol\phi = \boldsymbol\rho$ ; $\sigma^2 = \gamma_0(1 - \boldsymbol\phi'\boldsymbol\rho)$
AR(1) forecast	$\mathbb E_tX_{t+h} = \mu + \phi^h(X_t-\mu)$ ; error var $\sigma^2(1-\phi^{2h})/(1-\phi^2)$ ; half-life $\ln0.5/\ln\phi$
OOS $R^2$	$1 - \sum(r - \hat r)^2/\sum(r - \bar r_t)^2$
Value of $R^2$	$S^{2} = (S^2+R^2)/(1-R^2)$ ; gain $\approx R^2/S^2$ ; pure timer $S^{2} = R^2/(1-R^2)$
Linear pure timer (Gaussian)	$\mathrm{SR}^2 = R^2/(1+R^2)$
Stambaugh bias	$\mathbb E[\hat\beta-\beta] = \gamma\,\mathbb E[\hat\rho-\rho] \approx -\gamma(1+3\rho)/T$ , $\gamma = \sigma_{uv}/\sigma_v^2$
GARCH(1,1)	$\sigma_t^2 = \omega + \alpha\varepsilon_{t-1}^2 + \beta\sigma_{t-1}^2$ ; $\bar\sigma^2 = \omega/(1-\alpha-\beta)$
GARCH ARMA form	$\varepsilon_t^2 = \omega + (\alpha+\beta)\varepsilon_{t-1}^2 + \eta_t - \beta\eta_{t-1}$
GARCH forecast	$\mathbb E_T\sigma^2_{T+k} = \bar\sigma^2 + (\alpha+\beta)^{k-1}(\sigma^2_{T+1} - \bar\sigma^2)$ ; half-life $\ln0.5/\ln(\alpha+\beta)$
GARCH average variance over $H$	$\bar\sigma^2 + (\sigma_{T+1}^2 - \bar\sigma^2)(1-p^H)/(H(1-p))$
GARCH log-likelihood	$-\tfrac12\sum[\ln2\pi + \ln\sigma_t^2 + \varepsilon_t^2/\sigma_t^2]$
GARCH kurtosis	$3(1-p^2)/(1-p^2-2\alpha^2)$
Hamilton filter	$\xi_{t\mid t}\propto\xi_{t\mid t-1}\odot f(y_t)$ ; $\xi_{t+1\mid t} = P^\top\xi_{t\mid t}$ ; $\ell = \sum\ln(\xi_{t\mid t-1}'f(y_t))$
Kim smoother	$\xi_{t\mid n} = \xi_{t\mid t}\odot P(\xi_{t+1\mid n}\oslash\xi_{t+1\mid t})$
Two-state ergodic	$\pi_0 = (1-p_{11})/(2-p_{00}-p_{11})$ ; duration $1/(1-p_{jj})$
Overlapping labels	$\operatorname{corr}(y_i,y_j) = \max(0, 1-\lvert i-j\rvert/h)$
Purge rule	train iff $t_{1,i} < T_{\text{start}}$ or $t_{0,i} > T_{\text{end}}$ ; embargo $\lceil\epsilon N\rceil$ after block
Brier	$\bar S_B(p;q) = (p-q)^2 + q(1-q)$
Log score	$\bar S_L(p;q) = H(q) + D_{KL}(q\|p)$
Murphy	$\mathrm{BS} = \mathrm{REL} - \mathrm{RES} + \mathrm{UNC}$
Bayes (odds)	posterior odds = prior odds × likelihood ratio
Journal sample size	$N\approx(1.96\,s/d)^2$