题目3111 · 统计
Does This GARCH Have a Finite Long-Run Variance?
For a GARCH(1,1) model with $\omega=\frac{1}{5}$, $\alpha=\frac{1}{4}$, and $\beta=\frac{3}{4}$, decide whether the model has a finite unconditional variance. If it does, compute it.
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中文题目题目6028 · 统计
Fat Tails: Unconditional Kurtosis of GARCH Returns
Let $r_t=\sqrt{h_t}\,z_t$ with $z_t\sim N(0,1)$ i.i.d. and GARCH(1,1) variance. The unconditional kurtosis (when finite) is $K=\dfrac{3\,[1-(\alpha+\beta)^2]}{1-(\alpha+\beta)^2-2\alpha^2}$. For $\alpha=0.1$, $\beta=0.85$, compute $K$ and state whether returns are leptokurtic. Gi
打开 →题目3091 · 统计
Long-Run Variance of a Quiet GARCH Process
For a GARCH(1,1) model $h_t=\omega+\alpha r_{t-1}^2+\beta h_{t-1}$ with $\omega=\frac{1}{10}$, $\alpha=\frac{1}{5}$, and $\beta=\frac{3}{5}$, assume $\alpha+\beta<1$. Compute the unconditional variance $E[h_t]$.
打开 →题目6023 · 统计
Long-Run Volatility (Not Variance) from GARCH Parameters
A GARCH(1,1) model $h_t=\omega+\alpha r_{t-1}^2+\beta h_{t-1}$ has $\omega=0.04$, $\alpha=0.12$, $\beta=0.80$. Here $h_t$ is the conditional variance of daily returns. Report the long-run (unconditional) daily volatility $\sqrt{\bar h}$ as a decimal.
打开 →题目3093 · 统计
Steady Variance from Daily GARCH Parameters
For a GARCH(1,1) model $h_t=\omega+\alpha r_{t-1}^2+\beta h_{t-1}$ with $\omega=1$, $\alpha=\frac{1}{10}$, and $\beta=\frac{4}{5}$, assume $\alpha+\beta<1$. Compute the unconditional variance $E[h_t]$.
打开 →题目6027 · 统计
When GARCH(1,1) Becomes EWMA
RiskMetrics EWMA updates variance as $h_t=(1-\lambda)r_{t-1}^2+\lambda h_{t-1}$. State the constraints on $(\omega,\alpha,\beta)$ that make GARCH(1,1) coincide exactly with EWMA, and give the implied $\lambda$ when $\alpha=0.06$. Report $\lambda$ as a decimal.
打开 →题目6026 · 统计
ARCH(1) as the Beta-Zero Special Case
A GARCH(1,1) reduces to ARCH(1) when $\beta=0$: $h_t=\omega+\alpha r_{t-1}^2$. With $\omega=0.7$ and $\alpha=0.3$, compute the unconditional variance $\bar h$ as a decimal.
打开 →题目6025 · 统计
Five-Step Forecast via the Mean-Reversion Formula
For a GARCH(1,1) with $\omega=0.2$, $\alpha=0.1$, $\beta=0.8$, the one-step-ahead conditional variance is $h_{t+1}=3$. Using the closed form $E_t[h_{t+k}]=\bar h+(\alpha+\beta)^{\,k-1}(h_{t+1}-\bar h)$, compute the 5-step-ahead forecast $E_t[h_{t+5}]$ as a decimal.
打开 →题目3106 · 统计
Half-Life When Alpha Plus Beta Equals 0.8
In a GARCH-style volatility recursion, a deviation from long-run variance decays approximately by the factor $\rho=\alpha+\beta=\frac{4}{5}$ each step. What is the half-life of the deviation?
打开 →题目6029 · 统计
News-Impact Update from a Signed Return
A GARCH(1,1) has $\omega=0.00001$, $\alpha=0.08$, $\beta=0.90$. 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.
打开 →题目6024 · 统计
Persistence and the Covariance-Stationarity Verdict
A GARCH(1,1) has $\alpha=0.20$, $\beta=0.75$. Compute the persistence $\alpha+\beta$ and state whether the process is covariance-stationary (i.e. has a finite, time-invariant unconditional variance). Give the persistence as a decimal.
打开 →题目3096 · 统计
Tomorrow Variance After a Large Shock
In a GARCH(1,1) model with $\omega=\frac{1}{10}$, $\alpha=\frac{1}{5}$, and $\beta=\frac{7}{10}$, suppose the current squared return is $r_t^2=4$ and the current conditional variance is $h_t=2$. Compute $h_{t+1}$.
打开 →题目3103 · 统计
Two-Day Ahead Variance Mean
For a GARCH(1,1) process with $\omega=1$, $\alpha=\frac{1}{10}$, $\beta=\frac{4}{5}$, suppose you already know the one-step-ahead conditional variance $h_{t+1}=5$. Compute $E_t[h_{t+2}]$ and $E_t[h_{t+3}]$.
打开 →题目3101 · 统计
Two-Step Forecast from Today’s Variance
For a GARCH(1,1) process with $\omega=\frac{1}{10}$, $\alpha=\frac{1}{5}$, $\beta=\frac{3}{5}$, suppose you already know the one-step-ahead conditional variance $h_{t+1}=2$. Compute $E_t[h_{t+2}]$ and $E_t[h_{t+3}]$.
打开 →题目3099 · 统计
Volatility Update from a Moderate Return
In a GARCH(1,1) model with $\omega=1$, $\alpha=\frac{3}{20}$, and $\beta=\frac{3}{5}$, suppose the current squared return is $r_t^2=4$ and the current conditional variance is $h_t=5$. Compute $h_{t+1}$.
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