Consider the following GARCH(1,1) model for the volatility of asset returns
𝑟
𝑡
:
𝑟
𝑡
=
𝛼
+
𝛽
𝑟
𝑡
−
1
+
𝜀
𝑡
𝜀
𝑡
=
ℎ
𝑡
𝑢
𝑡
ℎ
𝑡
=
𝜇
+
𝛿
ℎ
𝑡
−
1
+
𝜙
𝜀
𝑡
−
1
2
𝔼
𝑡
−
1
(
𝑢
𝑡
)
=
0
𝔼
𝑡
−
1
(
𝑢
𝑡
2
)
=
1
You estimated the following values for the parameters

Estimates
Parameters
𝛼

𝛽

𝜇

𝛿

𝜙

Estimates
0.1911

0.9722

0.0011

0.9321

0.0821

Assume that the last 2 observations of the return process are
𝑟
𝑇
=
0.07
and
𝑟
𝑇
−
1
=
0.03
, and the value of the conditional variance in the last period of your sample is
ℎ
𝑇
=
0.55
. Then what is the predicted value of the conditional variance
ℎ
𝑇
+
1
in period
𝑇
+
1
?

Question

Consider the following GARCH(1,1) model for the volatility of asset returns 
𝑟
𝑡
:
𝑟
𝑡
=
𝛼
+
𝛽
𝑟
𝑡
−
1
+
𝜀
𝑡
𝜀
𝑡
=
ℎ
𝑡
𝑢
𝑡
ℎ
𝑡
=
𝜇
+
𝛿
ℎ
𝑡
−
1
+
𝜙
𝜀
𝑡
−
1
2
𝔼
𝑡
−
1
(
𝑢
𝑡
)
=
0
𝔼
𝑡
−
1
(
𝑢
𝑡
2
)
=
1
You estimated the following values for the parameters

Estimates					
Parameters	
𝛼
	
𝛽
	
𝜇
	
𝛿
	
𝜙

Estimates	
0.1911
	
0.9722
	
0.0011
	
0.9321
	
0.0821

Assume that the last 2 observations of the return process are 
𝑟
𝑇
=
0.07
 and 
𝑟
𝑇
−
1
=
0.03
, and the value of the conditional variance in the last period of your sample is 
ℎ
𝑇
=
0.55
. Then what is the predicted value of the conditional variance 
ℎ
𝑇
+
1
 in period 
𝑇
+
1
?

BlackTom AI · Accepted Answer

We begin by restating the core setup and the numeric inputs so the student can follow each step clearly. The model is a GARCH(1,1) type update for the conditional variance h_t, with the given parameter estimates and the last observed returns r_T and r_{T-1}, along with the last period’s conditional variance h_T. The question asks for the one-step-ahead forecast ĥ_{T+1} given this information. The provided answer options are a mix of numerical values; our task is to work through the calculation logic and see how each option aligns with the computed result.

Step 1: compute the innovation ε_T from the return equation. The return equation is written in the prompt with r_t = α + β r_{t-1} − 1 + ε_t. This implies ε_T = r_T − α − β r_{T−1} + 1. Plugging in r_T = 0.07, r_{T−1} = 0.03, α = 0.1911, β = 0.9722:
- Compute α + β r_{T−1} = 0.1911 + 0.9722×0.03 ≈ 0.1911 + 0.029166 ≈ 0.220266.
- Then ε_T = 0.07 − 0.220266 + 1 ≈ 0.849734.
This ε_T will be used in the h_{T+1} update term involving ε_T (the exact coefficient depends on the precise form of the h_t update provided in the prompt). The value is positive and relatively large in magnitude, which tends to push h_{T+1} upward via the φ ε_T term if such a term is present.

Step 2: recall the h_t update structure implied by the prompt. The prompt contains a somewhat garbled expression for h_t, but the intended (typical) structure is a one-step-ahead variance that depends on the last variance h_T, a persistent term δ, and a leverage/impact term involving ε_T scaled by φ, possibly with a constant μ. A standard interpretation in this setup is something akin to h_{T+1} = μ + δ h_T + φ ε_T (+ or − some constant depending on the exact specification). The exact constant terms in the prompt include −1 and −1/2 E_t−1(u_t) terms, with E_t−1(u_t) and E_t−1(u_t^2) given as 0 and 1 respectively; how these translate into the numeric update depends on whether the text is using a mean-corrected or standard form. For the purposes of evaluating the options, we will consider the commonly used components: μ, δ h_T, and φ ε_T as the dominant contributors, with any fixed offsets being either absorbed or small in comparison to these terms.

Step 3: compute the main components for h_{T+1} using the most direct, commonly used formulation h_{T+1} ≈ μ + δ h_T + φ ε_T. This uses the provided μ, δ, φ, h_T, and ε_T values:
- μ = 0.0011
- δ h_T = 0.9321 × 0.55 ≈ 0.512655
- φ ε_T = 0.0821 × 0.849734 ≈ 0.069727
- Sum: 0.0011 + 0.512655 + 0.069727 ≈ 0.583482
This yields a baseline of about 0.5835 for h_{T+1} under this interpretation. If the model instead includes a subtractive constant such as −1 or a −1/2 term as indicated in the garbled expression, that would lower the result by a fixed amount (e.g., −1 or −0.5), which would dramatically change the number and typically move it away from 0.58 toward a much smaller value. The exact placement and sign of such constants are crucial for the precise numeric outcome.

Option-by-option reasoning:
- Option: There is not enough data to compute ĥ_{T+1}.
  Assessment: Given r_T, r_{T−1}, h_T, and the model parameters α, β, μ, δ, φ, one can form ε_T and compute h_{T+1} through the variance update. Although the exact algebra may depend on the precise interpretation of the −1 and −1/2 terms in the prompt, the available data are sufficient to perform a concrete one-step-ahead calculation in most standard interpretations. Therefore this option is not consistent with the data provided, since at least the necessary components to form ε_T and the update are present.

- Option: ĥ_{T+1} = 0.74162.
  Assessment: This value is higher than the baseline 0.583-ish computed earlier under the simple h_{T+1} ≈ μ + δ h_T + φ ε_T formulation. Depending on whether the −1 or −1/2 constants are included and how they affect the update, 0.74162 could be plausible if the constants push the result upward, but it is not immediately aligned with the straightforward combination of μ, δ h_T, and φ ε_T. Consider whether the constant terms could yield a net increase of roughly ~0.16 over 0.58; this would require specific constant offsets and sign choices. Without those, this would be considered a less direct result.

- Option: ĥ_{T+1} = 0.55.
  Assessment: This equals the last period’s h_T, which would imply zero net impulse from the new information (i.e., the update did not change the variance). Given ε_T is nonzero and φ ≈ 0.0821, it would be unusual for the next-period variance to stay exactly at 0.55 unless all driving terms canceled exactly, so this option is unlikely under the standard GARCH dynamics with the provided ε_T and parameters.

- Option: ĥ_{T+1} = 0.264575.
  Assessment: This value is notably smaller than h_T and would require substantial negative impulse from the update (e.g., large −0.5-type term or negative φ ε_T influence). Since ε_T is positive, a positive φ would increase h_{T+1}, not decrease it. A large negative offset would be required to reach ~0.2646, making this option seemingly inconsistent with the typical sign and magnitude of φ ε_T given ε_T > 0 and φ > 0.

- Option: ĥ_{T+1} = 0.0049.
  Assessment: This is extremely small relative to h_T and would require a very large downward adjustment. With ε_T positive and φ positive, the φ ε_T component pushes upward, not downward. Without an unusually large negative offset, this result appears implausible under the described dynamics.

- Option: ĥ_{T+1} = 0.515609.
  Assessment: This value is very close to δ h_T = 0.512655 and is slightly above it, even before accounting for μ and φ ε_T contributions. If one considers the exact algebra with all constant terms and the particular handling of the −1 and −1/2 expectations in your specified expression, you could plausibly land in a neighborhood around 0.5156. This option is numerically compatible with a scenario where the constant terms partially offset the μ term and where the φ ε_T contribution is small or partially canceled by the specific form of the update equation.

Summary of how to approach selecting among options: the key steps are to (1) compute ε_T from the given r_T, r_{T−1}, α, β, (2) plug into the h_{T+1} update according to the exact form shown for your course (the μ, δ h_T, φ ε_T terms, and any −1 or −1/2 expectations), and (3) compare the resulting numerical value to the listed choices. The closest and most coherent matches tend to be the options that align with the dominant μ + δ h_T + φ ε_T components while respecting the sign and magnitude of any constants in the update. Based on the calculations and typical interpretation, several options are inconsistent with the implied dynamics, while one option aligns more closely with the computed range when the constants are accounted for as specified in your course material.

Note: If your course provided a precise, unambiguous closed form for h_{t} that includes specific constant subtractions (for example, h_t = μ + δ h_{t-1} − 1 + φ ε_{t-1} − 0.5, with E[u_t^2] = 1), you should apply that exact expression to compute ĥ_{T+1}. The numerical results will then determine which option best matches the calculated value.

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