This paper describes the procedure of testing the presence of a feedback loop for a scalar discrete dynamic stochastic system.
Box and Jenkins proposed a stochastic model for open-loop systems. They employed a transfer function model to describe the dynamics between the input and the output, and an autoregressive moving average model to characterize the input and the disturbance. They also obtained an identification and estimation method. But their method leads to inappropriate estimates if the generating process is a closed-loop system whose input is written as a linear combination of the output. Therefore, at the first stage of the data analysis, the existence of the closed-loop should be tested.
For that purpose, we use the portmanteaulike statistic, which is constructed from the input and the output prewhitened by an autoregressive model fitting with the FPE criterion. This statistic is shown to be asymptotically distributed according to the χ² distribution of certain degrees of freedom. The test procedure is derived from this result. Two real data are analysed to demonstrate the usefulness of the present method.