Questions / Quant research, Time-series reasoning
Mean of an autoregression
A stationary AR(1) satisfies X_t = 2 + 0.5X_(t−1) + ε_t with mean-zero innovations. What is its unconditional mean?
Answer it in the app It's free. You get hints, a second try if you miss, and a follow-up question.
Show the answer and worked solution
Answer: 4 units
- Stationarity makes E[X_t] and E[X_(t−1)] equal to the same mean μ.
- Take expectations: μ = 2 + 0.5μ + 0, so 0.5μ = 2.
- Therefore μ = 4. The constant term is an input to the recurrence, not its long-run mean.
Interview shortcut. AR(1) mean = c/(1 − φ) = 2/0.5.
Takeaway. The stationary mean solves μ = c + φμ.
In the app this has a follow-up where one thing about the setup changes. Try the follow-up.