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6023Long-Run Volatility (Not Variance) from GARCH ParametersA GARCH(1,1) model h t=\omega+ r t-1 2+ h t-1 has \omega=0.04, =0.12, =0.80. Here h t is the conditional variance of daily returns. Report the long-run (unconditional) daily volatility h as a decimal.统计中等derivation未尝试面试订阅6025Five-Step Forecast via the Mean-Reversion FormulaFor a GARCH(1,1) with \omega=0.2, =0.1, =0.8, the one-step-ahead conditional variance is h t+1 =3. Using the closed form E t[h t+k ]= h+( + ) \,k-1 (h t+1 - h), compute the 5-step-ahead forecast E t[h t+5 ] as a decimal.统计困难derivation未尝试面试订阅6026ARCH(1) as the Beta-Zero Special CaseA GARCH(1,1) reduces to ARCH(1) when =0: h t=\omega+ r t-1 2. With \omega=0.7 and =0.3, compute the unconditional variance h as a decimal.统计简单derivation未尝试面试订阅6028Fat Tails: Unconditional Kurtosis of GARCH ReturnsLet r t= h t \,z t with z t\sim N(0,1) i.i.d. and GARCH(1,1) variance. The unconditional kurtosis (when finite) is K=\dfrac 3\,[1-( + ) 2] 1-( + ) 2-2 2 . For =0.1, =0.85, compute K and state whether returns are leptokurtic. Give K as a decimal.统计困难derivation未尝试面试订阅6030One-Step Prediction with a Persistence CoefficientA latent state evolves as x t=0.9\,x t-1 +w t with w t\sim N(0,2). At time t-1 the filtered state is N(4,3). Compute the one-step-ahead predicted mean and predicted variance of x t (before any observation at time t).统计简单derivation未尝试免费6033Innovation Variance and a Standardized SurpriseIn the model y t=x t+v t with v t\sim N(0,3), the predicted state at time t is N(5,7). You then observe y t=11. Compute the innovation (one-step forecast error) variance S, and the standardized innovation (y t-m -)/ S .统计中等derivation未尝试面试订阅6034Where Does the Estimate Land After the Print?The predicted state is N(8,6) and you observe y=14 with measurement noise variance R=2 in y=x+\varepsilon. Compute only the updated (posterior) mean of the state.统计简单数值题未尝试免费6041One-Year Conditional Variance of an OU ProcessAn OU process has mean-reversion speed kappa = 0.5 and diffusion coefficient sigma = 0.3. Given X 0, what is the conditional variance Var(X 1 | X 0)?随机过程中等derivation未尝试面试订阅6043Autocorrelation of a Stationary OU Process at a LagA stationary OU process has mean-reversion speed kappa = 0.7. What is the autocorrelation between X t and X t+2 ?随机过程简单derivation未尝试面试订阅6045Expected Value of a GBMA stock follows dS t = 0.1 S t dt + 0.4 S t dW t with S 0 = 50. Compute E[S 3].随机过程简单数值题未尝试免费6047Variance of a Geometric Brownian MotionA GBM satisfies dS t = 0.06 S t dt + 0.25 S t dW t with S 0 = 1. Compute Var(S 2).随机过程困难数值题未尝试面试订阅6048Ito Isometry for a Time-Weighted IntegralLet W t be standard Brownian motion. Using the Ito isometry, compute E[(integral from 0 to 2 of s dW s) 2].随机过程中等数值题未尝试免费