Reversed Residuals In Autoregressive Time Series Analysis

. Both linear and non‐linear time series can have directional features which can be used to enhance the modelling and investigation of linear or non‐linear autoregressive statistical models. For this purpose, reversed pth‐order residuals are introduced. Cross‐correlations of residuals and squared reversed residuals allow extensions of current model identification ideas. Quadratic types of partial autocorrelation functions are introduced to assess dependence associated with non‐linear models which nevertheless have linear autoregressive correlation structures. The use of these residuals and their cross‐correlation functions is exemplified empirically on some deseasonalized river flow data for which a first‐order autoregressive model is a satisfactory second‐order fit. Parallel theoretical computations are undertaken for the non‐linear first‐order random coefficient autoregressive model and comparisons are made. While the data are shown to be strongly non‐linear, their correlational signatures are found to be convincingly different from those of a first‐order autoregressive model with random coefficients.