*4.2. Empirical Data Analysis*

A residual analysis was conducted by looking at the PACF plots for the models with significant goodness of fit test results. It was observed that, in some cases, the PACF still demonstrated a rather clear cut-off at a higher order, suggesting that the AR order of the model could be further increased. Thus, it is suggested that, in this case, it is possible that the model was not a good enough fit, as it did not select a high enough order. This was possible, as the model fitting limited the highest AR order to be less than five, for practical concerns, and as the optimization process was sensitive to the selection of initial points and the initial points were evaluated in a sparser set at higher AR orders, thus resulting in a less-than-ideal fit.

Alternative models with non-Gaussian error provide another perspective of improvement. A Buffered Threshold Autoregressive (BAR) model, as described in Li et al. (2015), has been examined, using a fitting methodology similar to that of the TAR model. However, as the goodness of fit did not improve much, and as it is natural to choose a simpler model given similar goodness of fit, the results of BAR model have not ye<sup>t</sup> been reported. However, other models (such as the Autoregressive Moving-Average model (ARMA)) could still be considered.
