Consider the following model for the mean of asset returns
𝑟
𝑡
:
𝑟
𝑡
=
𝛼
+
𝛽
𝑧
𝑡
−
1
+
𝜀
𝑡
where
𝑧
𝑡
−
1
is a predictor of the returns. The model for the volatility is
𝜀
𝑡
=
ℎ
𝑡
𝑢
𝑡
ℎ
𝑡
=
𝜇
*
+
𝜙
1
*
𝜀
𝑡
−
1
2
+
𝜙
2
*
𝜀
𝑡
−
2
2
+
𝜙
3
*
𝜀
𝑡
−
3
2
𝔼
𝑡
−
1
(
𝑢
𝑡
)
=
0
𝔼
𝑡
−
1
(
𝑢
𝑡
2
)
=
1
What is the conditional expected value of the returns
𝔼
𝑡
−
1
(
𝑟
𝑡
)
? Choose the best answer below.

Question

Consider the following model for the mean of asset returns 
𝑟
𝑡
:
𝑟
𝑡
=
𝛼
+
𝛽
𝑧
𝑡
−
1
+
𝜀
𝑡
where 
𝑧
𝑡
−
1
 is a predictor of the returns. The model for the volatility is
𝜀
𝑡
=
ℎ
𝑡
𝑢
𝑡
ℎ
𝑡
=
𝜇
*
+
𝜙
1
*
𝜀
𝑡
−
1
2
+
𝜙
2
*
𝜀
𝑡
−
2
2
+
𝜙
3
*
𝜀
𝑡
−
3
2
𝔼
𝑡
−
1
(
𝑢
𝑡
)
=
0
𝔼
𝑡
−
1
(
𝑢
𝑡
2
)
=
1
What is the conditional expected value of the returns 
𝔼
𝑡
−
1
(
𝑟
𝑡
)
? Choose the best answer below.

BlackTom AI · Accepted Answer

We begin by restating the problem setup to be clear: the return at time t is modeled as
r_t = α + β z_{t-1} + ε_t,
with the conditional mean of ε_t given information up to t−1 being zero, i.e., E_{t−1}(ε_t) = 0. The volatility model provides that ε_t = h_t u_t, and E_{t−1}(u_t) = 0, E_{t−1}(u_t^2) = 1, but the key for the conditional mean of r_t is the mean of ε_t conditional on the information set, which is zero by assumption.

Option analysis:
- Option: E_{t−1}(r_t) = α + β z_t − 1
This option attempts to express the conditional expectation as a linear function of a predictor, but with a time index that appears to be z_t rather than the lagged z_{t−1}, and with an explicit subtractive constant of 1. There are two issues here:
  1) The model specification uses z_{t−1} as the predictor of r_t, not z_t, so replacing z_{t−1} with z_t changes the conditioning structure and is not consistent with the given model.
  2) The “−1” term is not present in the mean equation of r_t. Since E_{t−1}(ε_t) = 0, the conditional expectation should be the deterministic part α + β z_{t−1} (no extra constants), not α + β z_{t−1} − 1.

If the intention is to identify the correct conditional expectation under the stated model, the correct expression would be E_{t−1}(r_t) = α + β z_{t−1} (assuming the predictor is z_{t−1} as written). The provided option deviates in both the indexing of z and the presence of an extra −1 term, making it inconsistent with the model.

Common misconceptions this option might reflect include:
- Confusing lag structure: mixing z_t and z_{t−1} can lead to incorrect conditioning information in the expectation.
- Introducing an arbitrary constant: adding −1 to the conditional mean has no basis in the specified mean equation and would imply a bias term not present in the model.
- Thinking that the error term contributes to the mean: since E_{t−1}(ε_t) = 0, ε_t does not shift the conditional mean, so you should not see a nonzero constant arising from ε_t when computing E_{t−1}(r_t).

In summary, the option given does not align with the model r_t = α + β z_{t−1} + ε_t and E_{t−1}(ε_t) = 0. The correct conditional expectation should involve α + β z_{t−1} and no additional constant, reflecting that all randomness from ε_t has mean zero conditional on information up to t−1.

BU.232.630.W6.SP25 sample_quiz_3

View Explanation

Log in for full answers

Similar Questions

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%.

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler