FRM Part I · FRM Exam Part I · Stationary Time Series
For the ARMA(1,1) process Y_t = 0.5 Y_{t-1} + e_t + 0.4 e_{t-1}, with e_t white noise of variance 1, the first-order autocorrelation is given by rho(1) = (1+phi*theta)(phi+theta)/(1+2*phi*theta+theta^2). What is rho(1), rounded to three decimals?
Using the ARMA(1,1) formula with phi 0.5 and theta 0.4, the numerator is 1.2 times 0.9, or 1.08, and the denominator is 1.56, giving a first-order autocorrelation of about 0.692.
- A0.667
- B0.900
- C0.776Correct
- D0.714
Explanation
phi=0.5, theta=0.4. 1+phi*theta=1.2. phi+theta=0.9. Numerator=1.08. Denominator=1+2(0.2)+0.16=1.56. rho(1)=1.08/1.56=0.692. Recheck: 1.08/1.56=0.6923.
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