Fig. 5 presents various standard time series diagnostic plots of the model fitted to the NCDC temperature anomaly data.
Fig. 5a illustrates that the standardized innovation residuals appear to be random with no evident outliers and constant variance.
The partial autocorrelation plot (Fig. 5b) strongly indicates that the main autocorrelations of the model residuals have
been adequately represented by expression (2). Furthermore, the Ljung-Box test (Ljung and Box, 1978) supports the hypothesis
that all autocorrelations of the residuals up to lag 24 are zero (p-value = 0.26). The Breusch–Pagan test (Maddala, 2001)
was non-significant further supporting the hypothesis that the residuals are homoscedastic. However, a Q-Q plot of the standardized
innovation residuals (Fig. 5c) suggests that the tails of the residual distribution are slightly heavier than that of a
normal distribution. In addition, the Shapiro–Wilk test (Royston, 1982) yields a p-value of less than 0.01 which also indicates
that the residuals are non-normal. These facts together support the need to use a non-parametric bootstrap when simulating
the global temperature anomaly time series even though little difference was observed in the simulation results between
this and when the parametric bootstrap was used. Mild departures from the validated modelling assumptions examined
in this section may occur for a variety of reasons (Brohan et al., 2006), but the simulation results presented below indicate
that, even if they exist, they have only had a minor impact.