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

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

The lag 1 autocorrelation is 0.4, from theta/(1+theta^2) = 0.5/1.25, and the lag 2 autocorrelation is zero, because an MA(1) process has autocorrelations that cut off after lag 1.

  1. A0.4 and 0Correct
  2. B0.5 and 0.25
  3. C0.5 and 0
  4. D0.8 and 0
  5. 0.4 and 0.16

Explanation

For MA(1) with coefficient theta, rho_1 = theta/(1+theta^2) = 0.5/1.25 = 0.4. The autocorrelation is zero for all lags beyond 1. The option 0.5 forgets the denominator; 0.5 and 0.25 treats it like an AR(1).

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