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FRM Part I · FRM Exam Part I · Stationary Time Series

A risk analyst examines a stationary series. The sample autocorrelation function declines gradually (geometrically) from lag 1 onward, and the sample partial autocorrelation function also declines gradually rather than cutting off. Which model is most consistent with this pattern?

ARMA(1,1) is most consistent. When both the ACF and the PACF decay gradually without cutting off, the series has both autoregressive and moving-average features. A pure AR process would show a PACF that cuts off, and a pure MA process would show an ACF that cuts off.

  1. AAR(1)
  2. BMA(1)
  3. CARMA(1,1)Correct
  4. DWhite noise

Explanation

An AR(1) has a gradually decaying ACF but a PACF that cuts off after lag 1. An MA(1) has an ACF that cuts off after lag 1 and a PACF that decays. White noise has no significant autocorrelations. Both ACF and PACF decaying gradually is characteristic of a mixed ARMA process.

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