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

An analyst fits an AR(2) model Y_t = 0.5 Y_{t-1} + 0.3 Y_{t-2} + e_t. Which statement about its stationarity is correct?

The AR(2) is stationary because all roots of its characteristic polynomial lie outside the unit circle; here the roots are about 1.60 and -2.51. The sum of coefficients, 0.8, is below one, consistent with this. The intercept and the sign of phi2 are irrelevant.

  1. AIt is nonstationary because phi2 is positive
  2. BIt is stationary because the sum of coefficients, 0.8, is less than 1 and the roots lie outside the unit circleCorrect
  3. CIt is nonstationary because the sum of the coefficients exceeds 0.5
  4. DIt is stationary only if the intercept is zero

Explanation

Characteristic equation 1 - 0.5z - 0.3z^2 = 0 has roots z = [-0.5 ± sqrt(0.25+1.2)]/(0.6) giving 1.60 and -2.51 in magnitude, both greater than 1. So the process is stationary. The sign of phi2 and the intercept do not determine stationarity.

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