Residuals Modeling with AR and ARMA Representations

After some time spent on the transition between the Wold decomposition transfer function and the AR and ARMA model parameterizations, this chapter focuses on AR and ARMA parameter estimation, i.e., describes (i) standard numerical solutions for computing the AR and ARMA coefficients, (ii) the main statistical properties of these estimation techniques. More precisely, in this chapter, it is shown that By assuming that the residuals time series Is a realization of a zero mean weak stationary stochastic sequence which does not contain linearly singular components anymore. Has been generated by a stable, causal, linear, and time invariant system. The modeling of the residuals boils down to the estimation of ARMA model coefficients, i.e., a finite number of constant weights thanks to the Wold decomposition theorem. The computation of AR model parameters from residuals realizations can be carried out by using a linear least squares algorithm. The computation of ARMA model parameters from residuals realizations can be carried out by using a pseudolinear least squares or a nonlinear least squares algorithm. Both AR and ARMA model parameters can be estimated consistently under mild practical conditions. The autocorrelation and partial autocorrelation functions are easy-to-implement tools to determine if an AR or ARMA model structure should be considered for residuals time series modeling. Prediction with confidence intervals can be carried out once reliable AR or ARMA model parameters are available.