Forward cumulative GC test results
The three information criterion tests for fitting a VAR model yield mixed results. Lag 7 was indicated as the maximum lag for the AIC and HQ tests, while the BIC test indicated lag 3 as the maximum lag.
We conduct our test at each of these lags (3 and 7), in both RC-2 to GT-2 and GT-2 to RC-2 directions, using forward cumulative windows as described in Section 4. There are 11 tests in each direction for both lags 3 and 7 (a total of 44 tests), and only two tests have p values less than 0.10. If all of these tests were independent of each other, with a significance level of 0.10, approximately four false positives would be expected if the null hypothesis of no GC is true. Thus, we should not interpret that these two p values of less than 0.10 imply GC. The results for the forward cumulative GC tests are reported in Tables 2 and 3.
Note the sudden dip in the size of RC → GT p values as more data are added to the tests, specifically starting with the 1860–1980 lag 3 test. This, along with the interpolation of the CO2 ice core values, leads us to the RCUMGC tests, since this dip in p values may indicate that a GC from RC → GT exists in more recent times while being less significant in earlier times, which may have been masked in the forward cumulative window testing procedure. Thus, with the RCUMGC tests, we look backward in time rather than forward.
As the AIC and HQ suggests seven maximum lags and the BIC suggests 3, we conduct RCUMGC tests in both directions at lags 1–7 for a total of 224 tests. Figures 3 and 4 suggest that with progressively larger latest-window sizes, there is increasing evidence for GC in RC-2 to GT-2. With short latest windows, we see evidence in some lags for GC from GT-2 to RC-2. However, tests at the latest 15, 20, and 25 years may not be credible since they suffer from small sample problems. These problems are exaggerated in high lag (i.e., 4, 5, 6, and 7) tests, where even more data are lost.
All points in Fig. 4 are calculated as per Eq. 4. The plot of H shows that as more data are added, the case becomes stronger for GC from RC → GT. We also find that the overwhelming majority of RC-2-to-GT-2 models have approximately normally distributed residuals, while most of the GT-2-to-RC-2 models do not. Since the GT-2-to-RC-2 residuals are not distributed normally, the significance of the vectors ω = (ω
)T may actually be over- or understated.
The final set of seven points of Fig. 3 (where all data are used, 147 indices for each variable) show that the ratio (H) of F statistic ranges from slightly less than −1 or more than 1, implying no real difference in F ratios in either direction, to 10, meaning an F ratio for RC → GT is 10 times larger than the one from GT → RC. Five of these final seven points have ratios of more than 2, while the other two have ratios close to |1|. Overall, this shows evidence for RC → GT and no significant evidence for GT → RC.
The residuals from the RC → GT models are all approximately normal, while many of those from the GT → RC models (55/112) are not. Thus, the significance of many of the vectors ω = (ω
)T here may be over- or understated here.
These results are not definitive (refer to Section 6.1 for a discussion of limitations), but the RCUMGC procedure gives us a previously unexplored perspective on the relationship between RC and GT.
The results for the forecasting tests are displayed in Table 4.
The results in Table 4 show no appreciable differences between models 3 and 4. However, model 2 seems to have an appreciably higher R
2 holdout and lower maximum APE than model 1.
The above forecasting test results give more evidence for the RC → GT hypothesis and lend more credibility to the results reported in Section 5.3.
Exploratory correlation analysis
We begin by specifying the motivation for integrating the ENSO index into this part of our work. From the forward cumulative GC tests, we see quite high p values before the 1970s. The ENSO index has a significant relationship with global temperature patterns (Ropelewski and Halpert 1986). We hypothesize that the ENSO index may be one factor that affects GT and also possibly statistically obscures the effects of RC, particularly in earlier years. Thus, perhaps ENSO is, at least partly, a reason why we see no GC in our forward cumulative GC tests in Section 4.
We plot spline curves of the ENSO index and GT along with RC (Fig. 5). This plot indicates a relationship between ENSO and GT, as their temporal patterns appear to be very similar in many aspects. Next, we plot a 30-year forward moving window correlation-significance index between GT-1 and ENSO-1, between GT-2 and RC-2, and finally RC-2 and ENSO-2 (Fig. 6). For example, a p value at year 1991 indicates the significance of the correlation between GT-1 and ENSO-1, GT-2 and RC-2, or RC-2 and ENSO-2 for the years 1962–1991.
From Fig. 6, we see that an abrupt change in significance of the correlation between RC-2 and GT-2 occurs in the 30-year time window ending in 1974. This is the same time frame when there was a sudden drop in the RC → GT p value from the forward cumulative GC tests. We find evidence that this sudden change in significance is not solely due to a change in data source, i.e., where in 1959 CO2 values become Mauna Loa data as opposed to ice core data.
Here, we note that an abrupt climate regime change occurred in the 1970s, and its cause is not well known (Graham 1994). An abrupt climate change is said to occur when a climate system is forced across some threshold (Committee on Abrupt Climate Change, National Research Council 2002). Alley et al. (2005) notes that even a slow forcing could cause an abrupt change. This noted abrupt climate regime change corresponds closely with the time frame in which the RC-2 and GT-2 correlation becomes significant. This is an interesting phenomenon and one that deserves attention; we may report observations on this in a future work.
We have seen that there are interesting temporal correlation patterns between the ENSO and GT and especially between RC and GT. The ENSO index may indeed affect the nature of causality between RC and GT. Several GC tests between GT and ENSO as well as GT and RC show evidence from ENSO → GT only. This GC is statistically more significant than that found from RC → GT, supporting the hypothesis that ENSO could be obscuring our view of some causality or at least correlation between RC-2 and GT-2 in earlier years. Table 5 displays the results for these tests. Note in Table 5 that GC tests indicated feedback between RC-2 and ENSO-2.
With the accumulation of all results, we conceive three competing hypotheses as to why there is a sudden jump in correlation significance between RC-2 and GT-2:
The smoothing of the early CO2 values hides the early dependence structure between the two variables. Thus, we only see it later. We remark several paragraphs earlier that we find evidence (contained in the Electronic supplementary material) that this is at least not completely the case, although it may contribute.
ENSO is a more statistically significant covariate at times and thus sometimes hides the RC-2 and GT-2 correlation. Here, we are not limited to ENSO; other atmospheric circulation variables could play a masking role as well.
The aforementioned abrupt regime change in the 1970s explains this jump (Graham 1994; Alley et al. 2005; Committee on Abrupt Climate Change). This hypothesis is supported by Fig. 6: The change in the slope of the GT regression line coincides remarkably with a sudden significance between RC-2 and GT-2.
Note that in reality, these three hypotheses may not be mutually independent—that is, it could be a combination of more than one that causes this jump in correlation. Note also that we are certainly not limited to these three possible explanations.
We stress the fact that these hypotheses are considerably unrefined and are partially visually derived. Climate oscillators are not the only factors which influence the relationship between CO2 and temperature. Kaufmann et al. (1991) and others have discussed the potential warming or cooling effects of tropospheric aerosol activity, which influences the net RC. Future research efforts need to decompose aerosol-related effects from RC to evaluate its significance for GC results. Recent reports (e.g., Schiermeier 2010) highlight our lack of understanding of the impacts of aerosols as one of four major holes in climate science.