The state space model is widely used to handle time series data driven by related latent processes in many fields. In this article, we suggest a framework to examine the relationship between state space models and autoregressive integrated moving average (ARIMA) models by examining the existence and positive-definiteness conditions implied by auto-covariance structures. This study covers broad types of state space models frequently used in previous studies. We also suggest a simple statistical test to check whether a certain state space model is appropriate for the specific data. For illustration, we apply the suggested procedure in the analysis of the United States real gross domestic product data. Copyright © 2011 John Wiley & Sons, Ltd.