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

A risk analyst fits an ARMA(1,1) model to a stationary series: Y_t = 0.6 Y_{t-1} + e_t + 0.3 e_{t-1}, where e_t is white noise. Which statement best describes why the analyst might prefer this model over a pure AR or pure MA model?

ARMA models are preferred because they combine autoregressive persistence and moving-average shock effects, capturing both decaying autocorrelation and short-lived shock dynamics parsimoniously. They apply to stationary series and still assume white noise errors, unlike a pure MA(1) whose autocorrelation cuts off after lag 1.

  1. AARMA models require the series to contain a unit root, which pure AR models cannot handle
  2. BARMA models can capture both gradually decaying autocorrelation and short-lived shocks with fewer parametersCorrect
  3. CARMA models always produce a zero autocorrelation beyond lag 1
  4. DARMA models remove the need for the white noise assumption on the errors

Explanation

An ARMA model combines AR persistence (geometrically decaying autocorrelation) with MA terms that capture short-term shock effects, often achieving a good fit with fewer parameters than a high-order AR or MA. Option A is wrong because ARMA is for stationary series. Option C describes an MA(1) cutoff, not ARMA(1,1).

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