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Intermediate Proof

Derivation of the Explicit Formula for the l-step Ahead Forecast \( \hat{Z}_t(l) \) for a Stationary AR(1) Model

Students often confuse E[ZtFt]=Zt E[Z_t | \mathcal{F}_t] = Z_t with a general forecast. The main pitfall in derivation is incorrect index management during recursive unrolling, leading to errors in the geometric series bounds or the final power of ϕ \phi multiplying Zt Z_t . Careful handling of l l vs. l1 l-1 is key.
Institutional Reference: Time Series Analysis
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