Consider the following model for the mean and volatility of asset returns rt:
rt=βht+εt
εt=
√
ht
ut
ht=μ*+ϕ
*
1
ε
2
t−1
+ϕ
*
2
ε
2
t−2
+ϕ
*
3
ε
2
t−3
𝔼t−1(ut)=0
𝔼t−1(u
2
t
)=1
What is the conditional variance 𝕍t−1(rt)?

Question

BlackTom AI · Accepted Answer

Question restatement: We are given a model for asset returns r_t = β h_t + ε_t with ε_t = sqrt(h_t) u_t, where h_t follows an ARCH(3) process h_t = μ* + φ*1 ε^2_{t-1} + φ*2 ε^2_{t-2} + φ*3 ε^2_{t-3}, and with E_{t-1}(u_t) = 0, E_{t-1}(u_t^2) = 1. The task is to find the conditional variance V_{t-1}(r_t).

Option 1: V_{t-1}(r_t) = 1
- This would imply unit variance regardless of past information, which contradicts the model where volatility depends on h_t. Since h_t is the scale of ε_t, the conditional variance should reflect h_t, not a constant. Therefore this option is incorrect.

Option 2: V_{t-1}(r_t) = h_t
- Under the information set F_{t-1}, h_t is a known (measurable) quantity determined by past ε^2 terms, so the term β h_t is a constant given F_{t-1} and does not contribute to conditional variance. The stochastic part is ε_t = sqrt(h_t) u_t, whose conditional variance given F_{t-1} is Var_{t-1}(sqrt(h_t) u_t) = h_t Var(u_t | F_{t-1}). Since E_{t-1}(u_t^2) = 1, Var(u_t | F_{t-1}) = 1. Therefore Var_{t-1}(r_t) = h_t. This is the correct structure for the conditional variance in an ARCH-type model.

Option 3: V_{t-1}(r_t) = h_t u_t^2
- This mixes the random variable u_t^2 into the variance expression itself. Conditional variance should be a function of information up to t-1, not depend on the realized u_t. Since u_t^2 is not F_{t-1}-measurable (it involves the current shock), this cannot be the conditional variance. Hence incorrect.

Option 4: V_{t-1}(r_t) = β^2 h_t^2
- This would imply the variance grows with the square of h_t and the square of β, but the variance comes from the innovation ε_t, not from the deterministic part β h_t. The term β h_t is a mean component, and the stochastic part ε_t has variance h_t, not β^2 h_t^2. Therefore this overstates the variance and is incorrect.

Option 5: V_{t-1}(r_t) = β h_t
- This mixes a linear term in h_t with the variance. Variance is a measure of dispersion, which in this setup comes from the innovation ε_t and scales with h_t, but not with an extra factor of β. The β term is part of the conditional mean, not the variance. Thus this option is incorrect.

BU.232.630.W6.SP25 Quiz 3 solutions

Consider the following model for the mean and volatility of asset returns rt: rt=βht+εt εt= √ ht ut ht=μ+ϕ 1 ε 2 t−1 +ϕ * 2 ε 2 t−2 +ϕ * 3 ε 2 t−3 𝔼t−1(ut)=0 𝔼t−1(u 2 t )=1 What is the conditional variance 𝕍t−1(rt)?

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BU.232.630.W6.SP25 Quiz 3 solutions

Consider the following model for the mean and volatility of asset returns rt: rt=βht+εt εt= √ ht ut ht=μ*+ϕ * 1 ε 2 t−1 +ϕ * 2 ε 2 t−2 +ϕ * 3 ε 2 t−3 𝔼t−1(ut)=0 𝔼t−1(u 2 t )=1 What is the conditional variance 𝕍t−1(rt)?

View Explanation

Log in for full answers

Similar Questions

It is Friday evening now, you want to know the conditional variance of the weekly SPY return from now to the end of next Friday. Which of the following statements makes sense? The conditional variance cannot be 0 or negative.

It is Friday evening now, you want to know the conditional variance of the weekly SPY return from now to the end of next Friday. Which of the following statements makes sense? The conditional variance cannot be 0 or negative.

It is Friday evening now, you want to know the conditional variance of the weekly SPY return from now to the end of next Friday. Which of the following statements makes sense? The conditional variance cannot be 0 or negative.

If the conditional variance for tomorrow's return is 0.0004, 𝑉 𝑎 𝑅 𝑡 ( 𝑅 𝑡 + 1 ) = ( 0.02 ) 2 = 0.0004 , then the conditional expectation for tomorrow's return is 2% or -2%.

It is Friday evening now, you want to know the conditional variance of the weekly SPY return from now to the end of next Friday. Which of the following statements makes sense? The conditional variance cannot be 0 or negative.

More Practical Tools for Students Powered by AI Study Helper

Making Your Study Simpler

Consider the following model for the mean and volatility of asset returns rt: rt=βht+εt εt= √ ht ut ht=μ+ϕ 1 ε 2 t−1 +ϕ * 2 ε 2 t−2 +ϕ * 3 ε 2 t−3 𝔼t−1(ut)=0 𝔼t−1(u 2 t )=1 What is the conditional variance 𝕍t−1(rt)?