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FRM Part I · FRM Exam Part I · Stationary Time Series

Consider ARMA(1,1) processes of the form Y_t = δ + φY_{t-1} + θε_{t-1} + ε_t, where ε_t is white noise. Which parameter pair gives a covariance-stationary process?

The process with φ = 0.5 and θ = 1.8 is stationary because stationarity in an ARMA(1,1) requires only that the absolute value of the autoregressive coefficient be below one. The moving-average coefficient influences invertibility, not stationarity, so a large θ does not matter.

  1. Aφ = 0.5, θ = 1.8Correct
  2. Bφ = 1.2, θ = 0.3
  3. Cφ = 1.0, θ = 0.2
  4. Dφ = -1.05, θ = 0.1

Explanation

Covariance stationarity of an ARMA process depends only on the autoregressive part: the AR root must lie inside the unit circle, which for ARMA(1,1) means |φ| < 1. Only φ = 0.5 meets this. θ = 1.8 affects invertibility, not stationarity. The other options have |φ| ≥ 1 and so are nonstationary.

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