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

For the MA(1) process X_t = e_t + 0.5 e_{t-1}, where e_t is white noise with variance sigma^2, what is the autocorrelation at lag 1?

The lag-1 autocorrelation is 0.4. For an MA(1) process, rho_1 equals theta divided by (1 + theta squared), which is 0.5 divided by 1.25. The value is not equal to theta itself, and all autocorrelations beyond lag 1 are zero.

  1. A0.5
  2. B0.4Correct
  3. C0.25
  4. D0.8
  5. 0

Explanation

For MA(1) with parameter theta, rho_1 = theta/(1+theta^2). Here this is 0.5/(1+0.25) = 0.5/1.25 = 0.4. Choosing 0.5 uses theta itself, which is the common mistake, and the autocorrelation is zero beyond lag 1.

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