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

The question presents a model for asset returns: r_t = α + β z_{t−1} + ε_t, where z_{t−1} is a predictor of the returns and ε_t is the error term with mean zero given information up to time t−1. The task is to identify the conditional expected value E_{t−1}(r_t).

Option 1: E_{t−1}(r_t) = α + 1 − β. This form is incorrect. It treats the expectation as a constant offset of α and subtracts β, but it ignores the actual predictor z_{t−1} and its coefficient, which are essential to the conditional mean. There is no basis in the model for adding a constant 1 minus β; the predictor z_{t−1} must appear in the conditional expectation.

Option 2: E_{t−1}(r_t) = β z_t−1 + α. At first glance, this is numerically the same as α + β z_{t−1} if we interpret z_t−1 as z_{t−1}. However, the typical notation in the model uses z_{t−1} as the predictor with a minus one subscript, not z_t−1 as a product term. If we interpret z_t−1 as z_{t−1}, this option matches the model. The issue is that the option’s formatting suggests a potential ambiguity about the indexing. In principle, with the standard interpretation, this is equivalent to the correct form, but the safer, explicit expression is α + β z_{t−1}.

Option 3: E_{t−1}(r_t) = α + β z_t − 1. This mirrors the model using the same structure but with the predictor written as z_t instead of z_{t−1}, and with the “−1” understood as a subscript rather than a literal minus one. If the intended meaning is α + β z_{t−1}, then this option is the correct expression, provided z_t−1 is interpreted as z_{t−1}. Given the model uses z_{t−1} as the predictor, this option aligns with the stated equation when the minus one is treated as a subscript, not a subtraction.

Option 4: E_{t−1}(r_t) = β z_t − 1. This omits the intercept α entirely, which is inconsistent with the model r_t = α + β z_{t−1} + ε_t. Dropping α leads to an incorrect conditional mean, and also uses z_t rather than the required z_{t−1} (if we interpret the indexing strictly). Therefore this is not correct.

Option 5: E_{t−1}(r_t) = z_t − 1. This option completely misrepresents the structure of the model by replacing the linear combination α + β z_{t−1} with a simple transformation of z_t. It ignores both α and β, and uses the wrong form of the predictor entirely. This is thus incorrect.

In summary, the correct conditional mean must reflect the linear relationship with the predictor z_{t−1} and include the intercept α. Among the choices, the expression that matches the model’s form, when interpreted with the standard indexing (z_{t−1}) and the minus-one in the subscript rather than as a literal minus one, is α + β z_t − 1, i.e., option 3. The other options fail because they either omit α, misplace the predictor’s index, or misinterpret the subscript notation, leading to incorrect conditional expectations.

BU.232.630.W4.SP25 sample_quiz_3

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