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

A time series follows the process X_t = μ + e_t + θ e_{t-1}, where e_t is white noise with variance σ². Which statement about the autocorrelation function (ACF) of this MA(1) process is correct?

The MA(1) ACF is non-zero at lag 1 and zero for all higher lags, because the process depends on only one previous noise term. Its partial ACF tails off, and the lag-1 autocorrelation is θ/(1+θ²) rather than θ.

  1. AThe ACF is non-zero at all lags and decays geometrically
  2. BThe ACF is non-zero at lag 1 and equal to zero for all lags greater than 1Correct
  3. CThe ACF cuts off after lag 1 and the partial ACF also cuts off after lag 1
  4. DThe ACF is zero at lag 1 and non-zero at lag 2
  5. The ACF equals θ at lag 1 for every value of σ²

Explanation

For an MA(q) process, autocovariances vanish beyond lag q. So the MA(1) ACF is non-zero at lag 1 and zero afterwards. The PACF of an MA process tails off rather than cutting off, which rules out the third option. The lag-1 ACF is θ/(1+θ²), not θ.

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