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.

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