Order of integration

In statistics, the order of integration, denoted I(d), of a time series is a summary statistic, which reports the minimum number of differences required to obtain a covariance-stationary series. In statistics, the order of integration, denoted I(d), of a time series is a summary statistic, which reports the minimum number of differences required to obtain a covariance-stationary series. A time series is integrated of order 0 if it admits a moving average representation with where b {displaystyle b} is the possibly infinite vector of moving average weights (coefficients or parameters). This implies that the autocovariance is decaying to 0 sufficiently quickly. This is a necessary, but not sufficient condition for a stationary process. Therefore, all stationary processes are I(0), but not all I(0) processes are stationary. A time series is integrated of order d if is a stationary process, where L {displaystyle L} is the lag operator and 1 − L {displaystyle 1-L} is the first difference, i.e. In other words, a process is integrated to order d if taking repeated differences d times yields a stationary process. An I(d) process can be constructed by summing an I(d − 1) process: