Consider the following GARCH(1,1) model for the volatility of asset returns 𝑟 𝑡 : 𝑟 𝑡 = 𝛼 + 𝛽 𝑟 𝑡 − 1 + 𝜀 𝑡 𝜀 𝑡 = ℎ 𝑡 𝑢 𝑡 ℎ 𝑡 = 𝜇 + 𝛿 ℎ 𝑡 − 1 + 𝜙 𝜀 𝑡 − 1 2 𝔼 𝑡 − 1 ( 𝑢 𝑡 ) = 0 𝔼 𝑡 − 1 ( 𝑢 𝑡 2 ) = 1 You estimated the following values for the parameters 𝛼 𝛽 𝜇 𝛿 𝜙 0.5911 0.9222 0.0112 0.9132 0.0611 Assume that the last 2 observations of the return process are 𝑟 𝑇 = 0.04 and 𝑟 𝑇 − 1 = 0.05 , and the value of the conditional variance in the last period of your sample is ℎ 𝑇 = 0.5 . Then what is the predicted value of the conditional variance ℎ 𝑇 + 1 in period 𝑇 + 1 ? 单项选择题

A

There is not enough data to compute ℎ ̂ 𝑇 + 1 .

B

ℎ ̂ 𝑇 + 1 = 0.7071

C

ℎ ̂ 𝑇 + 1 = 0.4896

D

ℎ ̂ 𝑇 + 1 = 0.2

E

ℎ ̂ 𝑇 + 1 = 0.5

F

ℎ ̂ 𝑇 + 1 = 0.0016

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According to the GARCH model σTHURSDAY2=ω+αRBLANK12+βσBLANK22\sigma_{THURSDAY}^2 = \omega + \alpha R_{BLANK1}^2 +\beta \sigma_{BLANK2}^2 (Hint: fill in day of the week like Monday, Tuesday...) BLANK1:[Fill in the blank], BLANK2:[Fill in the blank],

Consider the following GARCH(1,1) model for the volatility of asset returns 𝑟 𝑡 : 𝑟 𝑡 = 𝜀 𝑡 𝜀 𝑡 = ℎ 𝑡 𝑢 𝑡 ℎ 𝑡 = 𝜇 + 𝛿 ℎ 𝑡 − 1 + 𝜙 𝜀 𝑡 − 1 2 𝔼 𝑡 − 1 ( 𝑢 𝑡 ) = 0 𝔼 𝑡 − 1 ( 𝑢 𝑡 2 ) = 1 You estimated the following values for the parameters Parameters Estimates MLE 𝜇 0.0112 𝛿 0.932 𝜙 0.0811 and the variance-covariance matrix is 𝑉 ( 𝜃 ̂ ) = [ 0.0012 − 0.012 0.001 − 0.012 0.102 − 0.003 0.001 − 0.003 0.003 ] Assume the last observation in your sample has ℎ 𝑇 = 1.5056 . What is the value of the conditional variance 𝑉 𝑇 − 1 ( 𝑟 𝑇 ) ?

Consider the following GARCH(1,1) model for the volatility of asset returns 𝑟 𝑡 : 𝑟 𝑡 = 𝛼 + 𝛽 𝑟 𝑡 − 1 + 𝜀 𝑡 𝜀 𝑡 = ℎ 𝑡 𝑢 𝑡 ℎ 𝑡 = 𝜇 + 𝛿 ℎ 𝑡 − 1 + 𝜙 𝜀 𝑡 − 1 2 𝔼 𝑡 − 1 ( 𝑢 𝑡 ) = 0 𝔼 𝑡 − 1 ( 𝑢 𝑡 2 ) = 1 You estimated the following values for the parameters Estimates Parameters 𝛼 𝛽 𝜇 𝛿 𝜙 Estimates 0.1911 0.9722 0.0011 0.9321 0.0821 Assume that the last 2 observations of the return process are 𝑟 𝑇 = 0.07 and 𝑟 𝑇 − 1 = 0.03 , and the value of the conditional variance in the last period of your sample is ℎ 𝑇 = 0.55 . Then what is the predicted value of the conditional variance ℎ 𝑇 + 1 in period 𝑇 + 1 ?

Consider the following GARCH(1,1) model for the volatility of asset returns 𝑟 𝑡 : 𝑟 𝑡 = 𝛼 + 𝛽 𝑟 𝑡 − 1 + 𝜀 𝑡 𝜀 𝑡 = ℎ 𝑡 𝑢 𝑡 ℎ 𝑡 = 𝜇 + 𝛿 ℎ 𝑡 − 1 + 𝜙 𝜀 𝑡 − 1 2 𝔼 𝑡 − 1 ( 𝑢 𝑡 ) = 0 𝔼 𝑡 − 1 ( 𝑢 𝑡 2 ) = 1 You estimated the following values for the parameters 𝛼 𝛽 𝜇 𝛿 𝜙 0.5911 0.9222 0.0112 0.9132 0.0611 Assume that the last 2 observations of the return process are 𝑟 𝑇 = 0.04 and 𝑟 𝑇 − 1 = 0.05 , and the value of the conditional variance in the last period of your sample is ℎ 𝑇 = 0.5 . Then what is the predicted value of the conditional variance ℎ 𝑇 + 1 in period 𝑇 + 1 ?

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