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

To begin, let’s lay out what the model provides and what the conditioning means.
- The return is r_t = β h_t + ε_t.
- The innovation is ε_t = h_t u_t, with the given conditional facts E_{t-1}(u_t) = 0 and E_{t-1}(u_t^2) = 1.
- The conditioning, denoted E_{t-1}(.), refers to information available up to time t−1.

Now evaluate E_{t-1}(r_t) by conditioning on the information up to t−1. Since r_t = β h_t + ε_t, we have:
- E_{t-1}(r_t) = E_{t-1}(β h_t) + E_{t-1}(ε_t).
- If h_t is known or independent of the fresh shock u_t given the t−1 information (i.e., h_t is F_{t-1}-measurable), then E_{t-1}(β h_t) = β h_t.
- For the remaining term, ε_t = h_t u_t, so E_{t-1}(ε_t) = E_{t-1}(h_t u_t) = E_{t-1}(h_t) E_{t-1}(u_t) if h_t is independent of u_t conditional on t−1. Given E_{t-1}(u_t) = 0, we obtain E_{t-1}(ε_t) = 0.

Putting these pieces together, the conditional expectation reduces to E_{t-1}(r_t) = β h_t + 0 = β h_t, under the usual assumption that h_t is known at time t−1 (or at least uncorrelated with u_t conditional on F_{t-1}).

Why the other potential interpretations don’t change the result here: if h_t were random and not F_{t-1}-measurable but still independent of u_t conditional on F_{t-1}, the cross-term would still vanish due to E_{t-1}(u_t) = 0, leaving E_{t-1}(r_t) = E_{t-1}(β h_t) which, in many formulations, collapses to β h_t when h_t is F_{t-1}-measurable. The key point is that the innovation component ε_t has zero mean conditional on the information up to t−1, so it does not contribute to the conditional mean of r_t.

Therefore, the stated conditional expectation E_{t−1}(r_t) = β h_t follows from the given relations and the zero-mean property of u_t at the conditioning time.

BU.232.630.W6.SP25 Quiz 3 solutions

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