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

An actuary fits an MA(1) model X_t = e_t + theta e_{t-1}. For the model to be invertible, which condition on theta is required?

The condition is |theta| < 1. Invertibility needs the root of 1 + theta B = 0 to lie outside the unit circle, so that the noise term can be expressed as a convergent infinite autoregression of past observations. The sign of theta does not matter.

  1. A|theta| > 1
  2. B|theta| < 1Correct
  3. Ctheta > 0
  4. Dtheta = 1
  5. 0 < theta < 0.5

Explanation

Invertibility requires that e_t can be written as a convergent sum of past X values. This needs the root of 1 + theta B = 0 outside the unit circle, i.e. |theta| < 1. Positivity of theta is not required.

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