Consider the following model for the mean and volatility of asset returns
𝑟
𝑡
:
𝑟
𝑡
=
𝛽
ℎ
𝑡
+
𝜀
𝑡
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 expectation
𝔼
𝑡
−
1
(
𝑟
𝑡
)
?

Question

Consider the following model for the mean and volatility of asset returns 
𝑟
𝑡
:
𝑟
𝑡
=
𝛽
ℎ
𝑡
+
𝜀
𝑡
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 expectation 
𝔼
𝑡
−
1
(
𝑟
𝑡
)
?

BlackTom AI · Accepted Answer

We start by restating the question in our own words to ensure understanding: we have a model where the return r_t is driven by a latent volatility term h_t and a noise term ε_t, with r_t = β h_t + ε_t. The question asks for the conditional expectation E_{t-1}(r_t) given information up to time t-1. The key is to separate the predictable part from the innovation part.

Option 1: E_{t-1}(r_t) = h_t u_t^2. This form suggests the expectation would depend on h_t and a squared innovation term u_t^2, but by construction ε_t is the shock at time t and, conditional on information up to t-1, its mean is zero. There is no reason the conditional expectation would be proportional to h_t times ε_t^2, and the expression contradicts the given structure r_t = β h_t + ε_t. Therefore this option is not correct.

Option 2: E_{t-1}(r_t) = β r_t − 1. This would imply the conditional expectation depends on the realized r_t itself, which cannot be the case when conditioning on information up to t−1 since r_t includes ε_t, the new shock unobserved at t−1. Thus this is inconsistent with the law of iterated expectations in a time-series setting and is incorrect.

Option 3: E_{t-1}(r_t) = 0. If the model had zero-mean innovation and h_t were also unpredictable from t−1, one could entertain a zero conditional mean. But here r_t contains β h_t, and h_t is generally a function of past shocks ε_{t-1}, which are known at t−1. Since ε_t has mean zero but h_t is not necessarily zero, the conditional expectation is not simply zero. Hence this option is false.

Option 4: E_{t-1}(r_t) = β. This would say the predictable part is a constant β regardless of h_t, which ignores the stateful impact of h_t on expected return. Because r_t = β h_t + ε_t and h_t is generally stochastic given I_{t-1}, the conditional expectation should depend on the current (known) level of h_t, not be just a constant. Therefore this is not correct.

Option 5: E_{t-1}(r_t) = β h_t. This aligns with the model structure: taking conditional expectation of r_t given information up to t−1, E_{t-1}(r_t) = E_{t-1}(β h_t + ε_t) = β E_{t-1}(h_t) + E_{t-1}(ε_t). Since ε_t is a new shock uncorrelated with past information and has mean zero, E_{t-1}(ε_t) = 0. What about E_{t-1}(h_t)? In many formulations, h_t is a function of past ε_{t-1}, which are known at t−1, so h_t is actually known at t−1 (i.e., h_t is I_{t-1}-measurable). If h_t is indeed determined by past information, then E_{t-1}(h_t) = h_t, making E_{t-1}(r_t) = β h_t. This matches the given option and is consistent with the stated dynamics, provided the dependence of h_t on past shocks is as described.

In summary, the reasoning shows that the only option that coherently combines r_t = β h_t + ε_t with E[ε_t | I_{t-1}] = 0 and a past-determined h_t is E_{t-1}(r_t) = β h_t. The other options either involve inappropriate dependence on the realized shock, misstate the role of past information, or incorrectly treat the predictable part of the return.

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