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

To begin, restate the problem setup and what is being forecasted: we are assuming a GARCH(1,1) process for stock returns rt with rt = εt and εt | Ωt−1 ∼ N(0, σt^2), where σt^2 = ω + α εt−1^2 + β σt−1^2. The quantity to forecast is the 2-step ahead conditional volatility, i.e., the square root of E[ r_{t+2}^2 | Ωt ]. We denote the conditional variance at time t as σt^2.

Key relationships in a GARCH(1,1) model under conditional expectation:
- The one-step-ahead conditional variance is σ_{t+1}^2 = ω + α εt^2 + β σt^2.
- For a two-step-ahead forecast, we use the law of iterated expectations with the GARCH recursion: E[σ_{t+2}^2 | Ωt] = ω + α E[ε_{t+1}^2 | Ωt] + β E[σ_{t+1}^2 | Ωt].

Evaluate the expectations given information Ωt:
- E[ε_{t+1}^2 | Ωt] = σ_{t+1}^2, because ε_{t+1} conditional on Ωt is Normal with variance σ_{t+1}^2.
- E[σ_{t+1}^2 | Ωt] = σ_{t+1}^2 in the sense of the conditional expectation of the next-period variance given current information; however, σ_{t+1}^2 itself is a random variable from the perspective of time t, so its expectation is the one-step-ahead variance forecast used in the recursion.

Thus the 2-step-ahead forecast for the conditional variance is:
E[σ_{t+2}^2 | Ωt] = ω + (α + β) σ_{t+1|t}^2, where σ_{t+1|t}^2 is the one-step-ahead variance forecast from time t, given by:
σ_{t+1|t}^2 = ω + α εt^2 + β σt^2.

Putting these together, the two-step ahead conditional variance forecast is:
ĥσ_{t+2}^2 = ω + (α + β) [ω + α εt^2 + β σt^2].

Finally, the quantity asked for is the 2-step ahead forecast of the conditional volatility, i.e., the square root of the above: ĥσ_{t+2} = sqrt( ĥσ_{t+2}^2 ).

What is missing to obtain a numeric value in this problem:
- The actual parameter values (ω, α, β) estimated from the training sample up to August 1, 2024.
- The current values of εt^2 and σt^2 (which derive from the most recent observed rt and σt via the GARCH recursion).
Without these estimates, the numerical result cannot be computed from the information given in the prompt alone.

If one had specific estimated parameters and the latest εt^2 and σt^2, you would plug them into: ĥσ_{t+2}^2 = ω + (α + β) [ω + α εt^2 + β σt^2], and then take the square root to obtain ĥσ_{t+2}.

In short, the formula to compute the 2-step ahead conditional volatility is derived from the GARCH(1,1) recursion and the law of iterated expectations, but a numeric value like 0.0145 requires the estimated parameters ω, α, β and the current state (εt^2, σt^2); those are not provided in the question text. Any reported numeric answer without these inputs would be an explicit numerical outcome from a specific estimated model rather than a universal result.

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