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

Let e_t be white noise with variance 9. A process is defined by Y_t = e_t + 0.4 e_{t-1}. Using the autocorrelation at lag 1 of this process, which statement is correct?

The MA(1) process is stationary with variance 9 x 1.16 = 10.44 and lag 1 autocovariance 0.4 x 9 = 3.6, so the lag 1 autocorrelation is about 0.345. Autocorrelations are zero beyond lag 1.

  1. AY_t is non-stationary because it depends on the past noise term
  2. BAutocovariance at lag 1 is 3.6, variance is 9, so rho_1 = 0.4
  3. CAutocovariance at lag 1 is 3.6, variance is 10.44, so rho_1 is about 0.345Correct
  4. DAutocovariance at lag 1 is 1.44, variance is 10.44, so rho_1 is about 0.138
  5. Autocovariance at lag 1 is 3.6 and rho_k = 0.345 for every k >= 1

Explanation

MA(1) is always stationary. Var = 9(1+0.16) = 10.44. Cov lag 1 = 0.4 x 9 = 3.6. So rho_1 = 3.6/10.44 = 0.345. Option 1 forgets the theta^2 term in the variance. Option 4 is wrong since MA(1) autocorrelation is zero beyond lag 1.

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