Underdifferencing versus Overdifferencing Tests

In the ARIMA modeling of business, ̄nancial, and macroeconomic time series, a very important question consists on the determination of the degree of integration of the data under analysis. There are essentially three di®erent approaches to this problem. In the ̄rst approach, the analyst tests for underdi®erencing, i.e., for the existence of one or more autoregressive unit roots, in order to decide whether further di®erencing is necessary. In the second approach, the analyst tests for overdi®erencing, i.e., for the existence of a unit root in the moving average polynomial of the ARMA model. In the third approach, the analyst directly estimates the degree of integration of the time series via spectral or time-domain methods. In this paper, we provide a Monte Carlo comparison of these three di®erent approaches. In particular, we contrast the power of the di®erent tests in fractionally integrated and in nearly nonstationary and nearly noninvertible processes.