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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Core concepts of time series models

Which condition makes the MA(1) process X_t = e_t + θ e_{t-1} invertible?

The MA(1) process is invertible when |θ| < 1. This puts the root of 1 + θz = 0 outside the unit circle, allowing the noise term to be expressed as a convergent infinite autoregression in past observations.

  1. A|θ| > 1
  2. B|θ| < 1Correct
  3. Cθ > 0 only
  4. DThe roots of 1 + θz = 0 lie inside the unit circle
  5. θ = 1

Explanation

Invertibility needs the root of 1 + θB = 0, namely z = -1/θ, to lie outside the unit circle, which means |θ| < 1. Then e_t can be written as a convergent infinite AR expression in past X values. Roots inside the unit circle fail this.

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