Unit Root Test Conflict and Possible Overdifferencing
A monthly demand series has 96 observations. An ADF test on levels gives p=0.12; a KPSS test with level and trend gives p=0.03. After first differencing, ADF gives p<0.01 and KPSS gives p=0.10. The sample ACF of first differences has lag-1 = −0.42, lag-2 = 0.10; the PACF has lag-1 = −0.35. Two candidate models are fitted: ARIMA(0,1,1) with MA(1) coefficient −0.88 and AIC 812.4; ARIMA(1,1,0) with AR(1) coefficient −0.38 and AIC 810.9. A Ljung–Box test at lag 12 gives p=0.04 for the ARIMA(1,1,0) residuals and p=0.21 for the ARIMA(0,1,1) residuals. The analyst prefers ARIMA(1,1,0) because its AIC is lower and proposes a 12-month forecast. A colleague warns that the series may have been over-differenced.
1. Interpret the ADF and KPSS results and state whether d=1 is supported.
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ADF on levels has p=0.12, so it does not reject a unit root at 5%; ADF has low power. KPSS with level and trend has p=0.03, rejecting trend stationarity. After first differencing, ADF p<0.01 and KPSS p=0.10, consistent with a stationary first difference. So d=1 is supported, but the tests are not decisive about trend versus difference; the ADF p on levels is weak evidence, not proof of stationarity. The conflict between tests is typical when a series is persistent or has a break.
2. Diagnose the possible over-differencing and evaluate the AIC-based model choice.
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The first-difference ACF has a large negative lag-1 value (-0.42), and the fitted MA(1) coefficient is -0.88, close to the noninvertible boundary. This is the classic signature of over-differencing: if the level series is stationary around a trend, differencing induces a negative MA unit root. The lower AIC of ARIMA(1,1,0) (810.9 vs 812.4) is not decisive, and its Ljung-Box p=0.04 rejects adequate fit at 5%, while ARIMA(0,1,1) has p=0.21. AIC comparisons between nonstationary and overdifferenced models are not a substitute for residual diagnostics.
3. What model should be considered for forecasting, and how should the forecast be validated?
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Do not accept ARIMA(1,1,0) on AIC alone. Refit an ARIMA(1,0,0) with trend or constant to the levels and compare residual diagnostics and out-of-sample forecasts; also consider a model with a structural break or a damped trend. If d=1 is retained, prefer ARIMA(0,1,1) on residual grounds but report the near-unit MA coefficient and the uncertainty it creates. Forecasts from an overdifferenced model can have inflated long-horizon variance; use rolling-origin validation rather than in-sample AIC.