Database Reference
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comparing the appropriateness of one fitted model against another fitted model.
These measures follow:
• AIC (Akaike Information Criterion)
• AICc (Akaike Information Criterion, corrected)
• BIC (Bayesian Information Criterion)
Because these criteria impose a penalty based on the number of parameters
included in the models, the preferred model is the fitted model with the smallest
AIC, AICc, or BIC value.
Table 8.1
provides the information criteria measures for
the ARIMA models already fitted as well as a few additional fitted models. The
highlighted row corresponds to the fitted ARIMA model obtained previously by
examining the ACF and PACF plots.
Table 8.1
Information Criteria to Measure Goodness of Fit
ARIMA Model (p,d,q) × (P,Q,D)
S
AIC
AICc BIC
(0,1,0) × (1,0,0)
12
1561.38 1561.43 1568.33
(0,1,1) × (1,0,0)
12
1472.43 1472.53 1482.86
(0,1,2) × (1,0,0
)12
1474.25 1474.42 1488.16
(1,1,0) × (1,0,0)
12
1504.29 1504.39 1514.72
(1,1,1) × (1,0,0)
12
1474.22 1474.39 1488.12
In this dataset, the (0,1,1) × (1,0,0)
12
model does have the lowest AIC, AICc, and
BIC values compared to the same criterion measures for the other ARIMA models.
Normality and Constant Variance
The last model validation step is to examine the normality assumption of the
residuals in
Equation 8.15
.
Figure 8.19
indicates residuals with a mean near zero
and a constant variance over time. The histogram in
Figure 8.20
and the Q-Q
plot in
Figure 8.21
support the assumption that the error terms are normally
distributed. Q-Q plots were presented in Chapter 6, “Advanced Analytical Theory
and Methods: Regression.”
plot(arima_2$residuals, ylab = "Residuals")
abline(a=0, b=0)
hist(arima_2$residuals, xlab="Residuals", xlim=c(-20,20))
