Testing for a unit root under the alternative hypothesis of ARIMA (0, 2, 1)

Showing a dual relationship between ARIMA (0, 2, 1) with parameter θ = -1 and the random walk, a new alternative hypothesis in the form of ARIMA (0, 2, 1) is established in this article for evaluating unit root tests. The power of four methods of testing for a unit root is investigated under the new alternative, using Monte Carlo simulations. The first method testing θ = -1 in second differences and using a new set of critical values suggested by the two authors in finite samples, is the most appropriate from the integration order point of view. The other three methods refer to tests based on t and φ statistics introduced by Dickey and Fuller, as well as, the nonparametric Phillips-Perron test. Additionally, for cases where for the first method a low power is met, we studied the validity of prediction interval for a future value of ARIMA (0, 2, 1) with θ close but greater of -1, using the prediction equation and the error variance of the random walk. Keeping the forecasting horizon short, the coverage of the interval ranged at expected levels, but its average half-length ranged up to four times more than its true value.