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FRM Part I · FRM Exam Part I · Nonstationary Time Series

An analyst runs an augmented Dickey-Fuller test on the level of a series and obtains a statistic of -1.50 against a 5% critical value of -2.86, failing to reject. She then tests the first difference and obtains -5.20 against the same critical value. Later she estimates an AR(1) on the level: Y_t = 0.97Y_{t-1} + ε_t. Which conclusion is best supported?

The series is best viewed as I(1): the unit root is not rejected in levels but is rejected in first differences, so the differenced series is stationary and should be modeled. An AR coefficient of 0.97 does not prove stationarity because it is hard to distinguish from one.

  1. AThe series is stationary in levels because the AR coefficient 0.97 is below 1
  2. BThe series is integrated of order one, I(1), and modeling the first difference is appropriateCorrect
  3. CThe series is integrated of order two, I(2), because differencing was needed
  4. DThe tests contradict each other, so no conclusion about stationarity is possible

Explanation

Failing to reject in levels suggests a unit root; rejecting in first differences suggests the differenced series is stationary. That pattern defines an I(1) series. A coefficient of 0.97 is close to 1 and is statistically hard to distinguish from it; it does not prove stationarity. I(2) would require the first difference to also have a unit root.

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