A Graphical Procedure for Determining Nonstationarity in Time Series

Abstract

Integrated processes as models for time series data have proved to be an important component of the highly flexible class of ARIMA(p, d, q) models. Determining the amount of differencing, d, has been a difficult task: too little and the process is not yet second-order stationary; too much and the process is more variable than it need be. It is shown that by introducing the notion of generalized covariances, developed by Matheron (1973) for spatial processes, the amount of differencing needed can be read easily from a sequence of graphs showing averages of squares of primary data increments. Formal inference to determine if the last difference really is necessary can then be carried out. Time series data are analyzed in this way and compared with the hypothesis-testing approach illustrated by Dickey, Bell, and Miller (1986). Once the order of differencing has been diagnosed, either the differenced time series can be analyzed or the generalized covariance of the undifferenced series can be estimated.

Key Words:

• Generalized covariance
• Generalized increment
• Integrated process
• Intrinsic random function of order k; Primary increment
• Spatial process
• Unit-root test