Point Optimal Tests for Testing the Order of Differencing in ARIMA Models

Deciding the order of differencing is an important part in the specification of an autoregressive integrated moving average (ARIMA) mode. In most, though not all, cases this means deciding whether to use the original observations or their first differences. Common test procedures used in this context are some variants of autoregressive unit root tests. In these tests, one tests the null hypothesis that the order of differencing is one against the alternative that it is zero. The null hypothesis thus states that the original series is nonstationary and integrated of order one, whereas the alternative assumes that it is stationary. In this paper the situation is reversed so that our null hypothesis states that the original series is stationary, whereas the alternative states that it is integrated of order one. In our approach the use of a differenced series thus means overdifferencing and, consequently, a model with a moving average unit root. Testing for this moving average unit root is the topic of this paper. As discussed by Saikkonen and Luukkonen [26] and Tanaka [31], test procedures obtained for this null hypothesis can also be used to test the null hypothesis that a multivariate time series is cointegrated with a given theoretical cointegrating vector. Since the null hypothesis of cointegration is often of interest and cannot be naturally tested by autoregressive unit root tests, this connection provides an important motivation for the test procedures of this paper.