Consider the data on Apple stock prices available on Canvas (apple_stock_price.csv). The data is at daily frequency and covers the period between January 3, 2007 and November 8, 2024. The following figure displays the time-series of apple stock price (in log).

Define the training sample as all the information available up to August 1, 2024. Assume the apple stock returns follows a GARCH(1,1) process. More specifically,

rt =εt
εt∣Ωt−1 ∼N(0,σ
2
t
)
σ
2
t
=ω+αε
2
t−1
+βσ
2
t−1

where rt=Δlog(Pt). What is the 2−step ahead forecast of the conditional volatility of rt?

[Remember: We defined the conditional volatility as the square root of the conditional variance]

Question

Consider the data on Apple stock prices available on Canvas (apple_stock_price.csv). The data is at daily frequency and covers the period between January 3, 2007 and November 8, 2024. The following figure displays the time-series of apple stock price (in log).

Define the training sample as all the information available up to August 1, 2024. Assume the apple stock returns follows a GARCH(1,1) process. More specifically,

rt	=εt
εt∣Ωt−1	∼N(0,σ
2
t
)
σ
2
t
	=ω+αε
2
t−1
+βσ
2
t−1

where rt=Δlog(Pt). What is the 2−step ahead forecast of the conditional volatility of rt?

[Remember:  We defined the conditional volatility as the square root of the conditional variance]

BlackTom AI · Accepted Answer

We need to outline how to obtain the 2-step ahead forecast of the conditional volatility for rt under a GARCH(1,1) specification, given the model and the training data cut-off.

First, restate the model in plain terms. The daily return rt is modeled as rt = εt, where εt conditional on information up to t−1, Ωt−1, is normally distributed with mean 0 and variance σt^2. The conditional variance follows a GARCH(1,1) recursion: σt^2 = ω + α εt−1^2 + β σt−1^2. The conditional volatility is defined as the square root of the conditional variance: √σt^2.

To get the 2-step ahead forecast at time t (i.e., using information up to t, with the training sample ending on Aug 1, 2024), we propagate the GARCH recursion forward two steps.

Step 1: Compute the one-step-ahead forecast at time t for σt+1^2, denoted σt+1|t^2:
- By the GARCH(1,1) update, σt+1|t^2 = ω + α εt^2 + β σt^2.
- Here, εt^2 is the squared innovation at time t, which you have observed from the data (since εt = rt under the model). 
- σt^2 is the conditional variance at time t, also observed (or estimated) from the filter at time t.

Step 2: Compute the two-step-ahead forecast for σt+2^2, denoted σt+2|t^2, using the standard identity that the conditional expectation of a future squared innovation given Ωt is its conditional variance:
- σt+2|t^2 = ω + α E[εt+1^2 | Ωt] + β E[σt+1^2 | Ωt].
- Since E[εt+1^2 | Ωt] = σt+1|t^2 and E[σt+1^2 | Ωt] = σt+1|t^2 (the variance is known at t for the next period given Ωt), this collapses to:
  σt+2|t^2 = ω + (α + β) σt+1|t^2.
- Therefore, the two-step-ahead conditional variance is simply the one-step-ahead variance scaled by (α + β) and added to ω, using the σt+1|t^2 value you computed in Step 1.

Putting it together, the double-step forecast for the conditional volatility (the square root of the two-step ahead variance) is:
- σt+2|t = sqrt( σt+2|t^2 ) = sqrt( ω + (α + β) σt+1|t^2 ).
- And σt+1|t^2 itself equals sqrt( σt+1|t^2 ), i.e., the one-step-ahead volatility, if you want the immediate next-period volatility before taking the square root.

Important practical notes:
- You must have estimated the GARCH parameters ω, α, β from the training sample up to Aug 1, 2024 (the problem states the model and training sample; in practice you would fit via maximum likelihood or a similar method).
- You also need the actual observed εt (i.e., rt) and the current σt^2 (the filtered conditional variance) at the cut-off date (Aug 1, 2024).
- If you do not carry forward exact numeric estimates for ω, α, β and the current εt^2 and σt^2, you cannot produce a numeric value like 0.0145. In that case, you would plug in your fitted values and the observed data to compute σt+1|t^2 and then σt+2|t^2, finally taking the square root for the two-step-ahead conditional volatility.

If you provide the estimated (ω, α, β) and the values of εt^2 and σt^2 at Aug 1, 2024, I can perform the numeric computation to yield the 2-step-ahead forecast.

类似问题

What was the world's first widely adopted biodiversity policy?

The aim of the Paris Agreement is to ensure that the earth’s preindustrial temperature is not exceeded by more than º Celsius.

The IPCC's mission is to Blank ______.

Why have most international agreements for environmental protection been based on the honor system?

As was exhibited with the Montreal Protocol and the encouragement of reduced CFC production, trade can be an effective tool to compel compliance with international treaties.

The Stockholm Declaration and UNEP were the result of the Blank ______ first conference on the Human Environment in 1972.

In addition to the direct mission of the Convention on International Trade in Endangered Species (CITES), why was CITES so important?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单