ARMA 2,1 model large sample

The ACF shows a decaying cyclic pattern whereas the PACF clearly indicates a significant partial auto-correlation at lag 1. Therefore, an AR(1) model is chosen as the initial model. To incorporate the outlier at index 128, an ARX(l) model is fitted to the original Naphtha time series where a dummy variable xt accounts for the effect of the outlieP  

The corresponding residuals standard deviation is estimated as a = 0.002 and the model s AIC) is calculated as -8.13. An analysis of the residuals reveals no hints for a violation of the underlying assumption of independent and identically distributed residuals. 

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Figure 2.11 ACF of residuals, ACF of squared residuals and QQ-plot of residuals for ARMA(2,0) model (small sample, T = 100) and for ARMA(2,1) model (large sample, T = 1000)...

For the small sample, the EACF values (Table 2.3) show an ambiguous pattern. Probably, an ARMA(1,3) or ARMA(3,1) model viould be chosen as initial choices. In contrast, the large sample (Table 2.).) shows a good fit to the theoretical EACF (Table 2.5) and an ARMA(2,1) model seems an appropriate initial choice. Since the analysis of ACF, PACF and EACF cannot reveal an unambiguous favourite model for both samples, all possible order specifications with p 0,4 and q e 0,4 are estimated for both samples. The corresponding AIC , SIC, and HQIC values are displayed in Table 2.6 where the minimum for each criterion and time series is set in bold font and the true model order is coloured in grey. ... 

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