This paper deals with an identification of the class of nonstationary stochastic processes known as Auto-Regressive Integrated (ARI) processes. The basic idea of this paper is to formulate the ARI process as a linear regression model and estimate the parameters of the process by the least squares. With this formulation, the determination of the numbers of parameters is usually difficult. This difficulty is overcome by introducing the criterion PSS (Prediction Sum of Squares) developed in the multivariate analysis. Moreover, the minimal realization algorithm of Mayne-Kalman is applied to determine the degree of the backward difference and to calculate the parameters which represent the stationary part of the process. Some numerical examples are tested and discussed at the end of this paper.