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

For a sample from an AR(2) process, which pattern of sample autocorrelation function (ACF) and partial autocorrelation function (PACF) is most consistent with this model?

The ACF decays gradually while the PACF cuts off after lag 2. This is the signature of an AR(2) process, because partial autocorrelations vanish beyond the autoregressive order. The reverse pattern, with ACF cutting off, indicates a moving average process instead.

  1. AACF cuts off after lag 2; PACF decays gradually
  2. BBoth ACF and PACF cut off after lag 2
  3. CACF decays gradually; PACF cuts off after lag 2Correct
  4. DACF cuts off after lag 1; PACF cuts off after lag 1
  5. Both ACF and PACF are insignificant at all lags

Explanation

An AR(p) process has an ACF that decays (geometrically or as damped oscillation) and a PACF that is zero beyond lag p. For AR(2) the PACF therefore cuts off after lag 2. The first option describes an MA(2) process.

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