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3091Long-Run Variance of a Quiet GARCH ProcessFor a GARCH(1,1) model h t=\omega+ r t-1 2+ h t-1 with \omega= 1 10 , = 1 5 , and = 3 5 , assume + <1. Compute the unconditional variance E[h t].统计中等derivation未尝试面试订阅3093Steady Variance from Daily GARCH ParametersFor a GARCH(1,1) model h t=\omega+ r t-1 2+ h t-1 with \omega=1, = 1 10 , and = 4 5 , assume + <1. Compute the unconditional variance E[h t].统计中等derivation未尝试面试订阅3096Tomorrow Variance After a Large ShockIn a GARCH(1,1) model with \omega= 1 10 , = 1 5 , and = 7 10 , suppose the current squared return is r t 2=4 and the current conditional variance is h t=2. Compute h t+1 .统计简单derivation未尝试面试订阅3099Volatility Update from a Moderate ReturnIn a GARCH(1,1) model with \omega=1, = 3 20 , and = 3 5 , suppose the current squared return is r t 2=4 and the current conditional variance is h t=5. Compute h t+1 .统计简单derivation未尝试面试订阅3101Two-Step Forecast from Today’s VarianceFor a GARCH(1,1) process with \omega= 1 10 , = 1 5 , = 3 5 , suppose you already know the one-step-ahead conditional variance h t+1 =2. Compute E t[h t+2 ] and E t[h t+3 ].统计中等derivation未尝试面试订阅3103Two-Day Ahead Variance MeanFor a GARCH(1,1) process with \omega=1, = 1 10 , = 4 5 , suppose you already know the one-step-ahead conditional variance h t+1 =5. Compute E t[h t+2 ] and E t[h t+3 ].统计中等derivation未尝试面试订阅3106Half-Life When Alpha Plus Beta Equals 0.8In a GARCH-style volatility recursion, a deviation from long-run variance decays approximately by the factor = + = 4 5 each step. What is the half-life of the deviation?统计中等derivation未尝试面试订阅3111Does This GARCH Have a Finite Long-Run Variance?For a GARCH(1,1) model with \omega= 1 5 , = 1 4 , and = 3 4 , decide whether the model has a finite unconditional variance. If it does, compute it.统计中等derivation未尝试面试订阅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6024Persistence and the Covariance-Stationarity VerdictA GARCH(1,1) has =0.20, =0.75. Compute the persistence + and state whether the process is covariance-stationary (i.e. has a finite, time-invariant unconditional variance). Give the persistence as a decimal.统计简单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6027When GARCH(1,1) Becomes EWMARiskMetrics EWMA updates variance as h t=(1- )r t-1 2+ h t-1 . State the constraints on (\omega, , ) that make GARCH(1,1) coincide exactly with EWMA, and give the implied when =0.06. Report as a decimal.统计中等数值题未尝试面试订阅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6029News-Impact Update from a Signed ReturnA GARCH(1,1) has \omega=0.00001, =0.08, =0.90. Today's conditional variance is h t=0.0004 and today's return is r t=-0.03. Compute tomorrow's conditional variance h t+1 as a decimal.统计中等derivation未尝试面试订阅