Part V · Chapter 15

Forecasting Methods

Stationary AR(1) forecasts decay geometrically to the mean. Random-walk forecasts stay at the last observed value. Drift adds a steady trend. The model dictates the forecast shape.

Learning objectives

Compute h-step forecasts for AR(1), RW, RW + drift.

Distinguish static vs dynamic forecasts.

Read forecast intervals (uncertainty grows with horizon).

Choose the right benchmark for evaluation.

Forecast Horizon

Model φ (AR only) 0.7 Drift δ 0.10 σ 1.0 Forecast horizon h 40

Seed:

Forecast behavior

—

The 95% interval = forecast ± 1.96 × σ_h. Width grows with h.

📐 Forecast formulas

\[ \text{AR(1): } \hat y_{T+h|T} = \phi^h y_T \quad(\to 0 \text{ as } h\to\infty) \]

\[ \text{RW: } \hat y_{T+h|T} = y_T \quad \text{RW + drift: } \hat y_{T+h|T} = y_T + h\delta \]

AR(1) variance σ²(1−φ^{2h})/(1−φ²)

RW variance hσ² (grows linearly)

Static uses actual y_{T+h−1}

Dynamic uses ŷ_{T+h−1}

🔍 What to look for

AR(1) forecast curves toward the mean; intervals saturate.
RW forecast is flat; intervals grow as √h forever.
RW + drift produces a sloped forecast line.

⚠️ Pro Tip: What to Avoid

Student says

"Random walk forecast must be zero."

Why this is wrong

The best forecast for a RW at any horizon is the LAST observed value, not zero. The series has no mean to revert to.

Correct interpretation

ŷ_{T+h|T} = y_T for RW; ŷ = y_T + hδ if drift is present.

📝 Mini-quiz

📋 Key Takeaways

Model	h-step forecast	Forecast variance
AR(1)	φ^h y_T	σ²(1−φ^{2h})/(1−φ²)
RW	y_T	hσ²
RW + drift	y_T + hδ	hσ²