Exploring Granger causality between global average observed time series of carbon dioxide and temperature
- 1.6k Downloads
- 28 Citations
Abstract
Detection and attribution methodologies have been developed over the years to delineate anthropogenic from natural drivers of climate change and impacts. A majority of prior attribution studies, which have used climate model simulations and observations or reanalysis datasets, have found evidence for human-induced climate change. This papers tests the hypothesis that Granger causality can be extracted from the bivariate series of globally averaged land surface temperature (GT) observations and observed CO_{2} in the atmosphere using a reverse cumulative Granger causality test. This proposed extension of the classic Granger causality test is better suited to handle the multisource nature of the data and provides further statistical rigor. The results from this modified test show evidence for Granger causality from a proxy of total radiative forcing (RC), which in this case is a transformation of atmospheric CO_{2}, to GT. Prior literature failed to extract these results via the standard Granger causality test. A forecasting test shows that a holdout set of GT can be better predicted with the addition of lagged RC as a predictor, lending further credibility to the Granger test results. However, since second-order-differenced RC is neither normally distributed nor variance stationary, caution should be exercised in the interpretation of our results.
1 Introduction
The Fourth Assessment Report (AR4) of the Intergovernmental Panel on Climate Change notes that it is “very likely” (90%) that “most” (greater than 50%) of the increase in global average temperature in the second half of the twentieth century can be attributed to anthropogenic greenhouse gases (Solomon et al., IPCC WG1 AR4 Report, Summary for Policymakers 2007). Substantial advances have been made in climate change detection and attribution through the analysis of climate model outputs (e.g., Hasselmann 1979; Santer et al. 1991; Santer et al. 1993; Hegerl et al. 1996; Hegerl et al. 1997; Meehl et al. 2007; Barnett et al. 2008). A few detection and attribution studies have also relied solely on observations (e.g., Tol and De Vos 1993; Sun and Wang 1996; Tol and De Vos 1998; Triacca 2005). Past work attempting to statistically assess Granger causality from observed emissions to globally averaged temperature observations has varied in both method and results (e.g., Sun and Wang 1996; Triacca 2005).
We hypothesize that Granger causality (GC) may serve as a tool for attribution in the multivariate case of anthropogenic emissions, natural cycles, and temperature. As a step in testing this hypothesis, this work introduces a variant of the classic bivariate GC test, reverse cumulative Granger causality (RCUMGC) testing, applied to two variables, a proxy for radiative forcing (RC), which in this work is a transformation of CO_{2}, and global land surface temperature anomalies (GT). The results presented in this work seem to indicate GC from RC to GT, at least in relatively large datasets. However, the appropriateness of the classical GC tests we employ are subject to several probabilistic assumptions being valid, many of which can be verified from data. These assumptions are not all met, and so our results have a degree of uncertainty beyond the usual uncertainty quantified in statistical testing. In addition, we find that the effects of RC might be overshadowed by other more statistically significant causal variables, such as the El Niño Southern Oscillation Index (ENSO).
Section 2 briefly discusses GC in its general form and in the specific form employed here, as well as previous applications of GC in climatology, several specifically between CO_{2} and temperature. Section 3 describes the specific data used as well as how it is preprocessed. Section 4 describes the testing procedures selected and discusses why they were used. In addition, we discuss the limitations of the chosen methods. Section 5 reveals the results and discusses their implications. Section 6 discusses the limitations of this work and briefly proposes its possible future extensions.
The Electronic supplementary material contains details on selected procedures from this work.
2 Granger causality overview
GC is a technique first developed for use in econometrics. It attempts to identify causal relationships between sets of two or more time series (Granger 1969).
We present GC for two variables, which we employ in this work. To say one variable X Granger causes another variable Y is to say that, by using past values of both variables X and Y, we can better predict future values of Y than by using only past values of Y. That is, past observations of X contain information useful for predicting Y, beyond what is available from past observations of Y itself.
If β = (β _{1},...β _{ n })^{ T } is not the zero vector (0,..0)^{ T } and ω = (ω _{1},...ω _{ n })^{ T } is the zero vector (0,..0)^{ T }, then X is said to Granger cause Y. If ω is not the zero vector and β is the zero vector, then Y Granger causes X. If neither β nor ω is the zero vector, then there is dependence in both directions, i.e., feedback between X and Y. If both β and ω are the zero vectors, there is no GC. The terms ϵ represent the white noise innovation at each instance of time t and are assumed to be independently and identically distributed with a bivariate normal law. The terms ϕ _{0} and λ _{0} represent intercepts for each equation.
GC has been applied numerous times in climate studies. Elsner (2006, 2007) applied a GC analysis to sea surface temperature anomalies and global surface temperature anomalies for Atlantic hurricanes. GC has been used to assess the “feedback of daily sea surface temperatures (SSTs) on daily values of the North Atlantic as simulated by a realistic coupled general circulation model (GCM)” (Mosedale et al. 2006). They find that SST Granger causes the North Atlantic Oscillation. Kaufmann and Stern (1997) use GC tests to show evidence that Southern Hemisphere leads the Northern Hemisphere in regards to temperature, which suggests that humans have contributed to climate change. Salvucci et al. (2002) uses GC to investigate soil moisture feedbacks from precipitation in Illinois. This work attempts to show evidence for causality going from soil moisture to precipitation, using a form of GC involving a Markov model (Salvucci et al. 2002). Sun and Wang (1996), using a classic GC partial F test, suggest that CO_{2} Granger causes global temperature. Smirnov and Mokhov (2009) introduce the idea of long-term Granger causality and apply it to temperature and CO_{2}; their method involves empirical modeling, extending the concept of GC to deal with longer-term behavior. They conclude that the rise in temperature over the last several decades can only be explained with the presence of CO_{2}. However, Triacca (2005) suggests that there is no significant GC from CO_{2} to global temperature, and that GC does not appear to be an appropriate tool for studying the causal relationship between these two variables. This work uses methodology from Toda and Yamamoto (1995), which is robust to the integration/co-integration properties of the data, but might require large sample size to obtain reliable results. Attanasio and Triacca (2010), a follow-up on Triacca (2005), find that a neural network based (and hence non-linear) approach suggests that GC exists from radiative forcing to global temperature.
We follow the classical F test methodology for our analysis in this paper. In case of potential violation of the classical statistical assumptions that are needed to justify the F test, Hacker and Hatemi-J (2006) suggests using a bootstrap distribution instead. Another potential method involves using a non-parametric test for GC (Hiemstra and Jones 1994).
Recently, grouped Granger causality models have appeared in literature. This extends the notion of temporal lags in GC to spatial lags, meaning neighboring spatial points can be tested for GC in addition to past values as discussed earlier. This method has been applied to observations in a study of climate change (Lozano et al. 2009a, b).
3 Data and preliminary data analysis
3.1 Data
The CO_{2} atmospheric concentration data in annual parts per million (ppm) are from the Mauna Loa observatory starting at 1959 (C.D. Keeling, T.P. Whorf, and the Carbon Dioxide Research Group Scripps Institute of Oceanography (SIO) University of California La Jolla, CA, USA). The CO_{2} concentration from several months in 1964 was originally missing, but the open-source statistical software R has a base package (“datasets”) that contains the monthly PPM data from 1959 to 1997, with those missing months in 1964 estimated by linear interpolation, which gives us the 1964 annual average. The annual 1964 value from R is added to the existing data, giving us observations from 1959 to 2008.
For annual CO_{2} PPM values from 1860 to 1958, we use the 20-year smoothed values estimated from the Law Dome DE08, DE08-2, and DSS ice cores (Etheridge et al., Division of Atmospheric Research, CSIRO, Aspendale, Victoria, Australia; J-.M. Barnola, Laboratoire of Glaciologie et Geophysique de l’Environnement, Saint Martin d’Heres-Cedex, France and V.I. Morgan Antarctic Division, Hobart, Tasmania, Australia).
GT for 1860–2008 are obtained from the Climate Research Unit at the University of East Anglia. These global land surface temperature values are expressed as anomalies from the average of the base years 1961–1990.
We obtain ENSO annual indices for 1860–2008 from http://jisao.washington.edu/data/globalsstenso/ (Todd Mitchell, Joint Institute for the Study of the Atmosphere and Ocean, University of Washington, Seattle, WA, USA). The December 2008 value was not yet available at the time of analysis, so 2008 is an average of January through November. The anomalies are expressed in hundredths of degrees Celsius as deviations from the period 1950–1979.
3.2 Preliminary data analysis
Inference based on ordinary least squares parameter estimates and normality-driven white noise process is justifiable when the bivariate time series is second-order stationary (Kwiatkowski et al. 1992). Consequently, it is necessary to find a level of differencing at which each variable is approximately normal and stationary.
All variables used in this study, shown with respective definitions
Variable name | Variable definition |
---|---|
GT | Raw, undifferenced annual GT series, 1860–2008. GT also refers to GT in general |
GT-1 | First-differenced GT series |
GT-2 | Second-differenced GT series |
RC | Transformation of CO_{2}, undifferenced. 1860–2008. RC also refers to RC in general |
RC-1 | First-differenced RC series |
RC-2 | Second-differenced RC series |
ENSO | Raw, undifferenced ENSO series from 1860 to 2008. ENSO also refers to ENSO in general |
ENSO-1 | First-differenced ENSO series |
ENSO-2 | Second-differenced ENSO series |
For example, GT-k represents the k-differenced GT series.
Figure 1 suggests that neither the RC nor the GT series are stationary or have normal innovations. To test this formally, we conduct the Shapiro–Wilks test (Shapiro and Wilks 1965) and the KPSS test (Kwiatkowski et al. 1992), respectively.
The Shapiro–Wilks tests reveal that RC is not approximately normal at any level of differencing, likely due to the interpolated values of the ice core CO_{2} data. This will have implications which will be addressed later in our analysis. GT is sufficiently normal at GT-1 and GT-2. However, note that the Shapiro–Wilks test relies on the assumption that data are independent. Autocorrelation plots of all six variables reveal that none of the variables may be deemed independent.
The KPSS test is used in many applications to test for stationarity of a series (Kwiatkowski et al. 1992). Based on the result of this test, the RC series may be considered stationary only at RC-2, while the GT series may be considered stationary at both GT-1 and GT-2. These tests partially confirm the results of Stern and Kaufmann (1999). We conduct GC tests, then, with the series GT-2 and RC-2.
The ENSO index, which is integrated further into our analysis, is found to be approximately normal and stationary at ENSO-1. Thus, when conducting GC tests on GT and ENSO, we use GT-1 and ENSO-1.
4 Methods
In order to estimate the maximum lags at which our data will be tested for GC, we use vector autoregression (VAR), an extension of univariate autoregression (Zivot and Wang 2002). We look at three different criteria in order to find an optimal maximum lag: the Akaike information criterion (AIC), the Bayesian information criterion (BIC), and the Hannan–Quinn (HQ) statistic. These criteria attempt to find a lag with sufficient information content on the two variables without over fitting (Zivot and Wang 2002).
Once we have identified a set of lags at which to test for GC, we conduct the forward cumulative windows GC test (Triacca 2005). Specifically, we perform classic GC partial F tests, in both directions, GT to RC and RC to GT, on range of lags, using forward cumulative windows (years 1860–1900, 1860–1910, 1860–1920... 1860–2008, adding 10 years for each test, except for the last, where only 8 years are added). Section 5.1 discusses the results from this test.
Subsequently, we introduce an alternative test of causality between these two variables: RCUMGC testing. This test is motivated by the results of the forward cumulative tests as well as the interpolation present in the ice core CO_{2} data. The motivation from the forward cumulative tests is presented in Section 5.1. The interpolation of many annual ice core CO_{2} values may mask some of the dependence structure between RC and GT. Thus, the RCUMGC conducts some tests without the presence of any ice core CO_{2} data.
In this test, we apply GC F tests with additive “latest windows,” meaning that a GC test is conducted, for example, corresponding to the last 30 years (1979–2008) of RC-2 and GT-2. The test is conducted for the last 15, 20, 25, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, and 147 years. These tests are conducted for seven lag values, in both RC-2-to-GT-2 and GT-2-to-RC-2 directions, so there are 224 total RCUMGC tests. The years 1959–2008 (50 years) contain no ice core CO_{2}-derived RC values.
We also conduct a forecasting test. We consider four forecasting scenarios: predicting GT using past values of GT, GT using past values of RC and GT, RC using past values of RC, and RC using past values of RC and GT. All the four forecasting problems are studied using lag values 1, 2, and 3 and using the GT-2 and RC-2 series. We use the first 137 time points as a training set to construct the forecasting procedure, and use the last 10 time points as the test set for evaluating each forecasting scheme. We evaluate the forecasting schemes using the coefficient of determination (R ^{2}), the mean absolute percentage error (MAPE), and the maximum absolute percentage error (Max. APE) of the 10-length validation set. The results of this forecast test are discussed in Section 5.3.
Finally, an exploratory analysis of the dependence structure between RC and GT shows that, at times, RC-2 is negatively correlated with GT-2. One possible explanation for this periodic negative correlation is the ENSO index. Hence, we integrate ENSO into our dependence structure analysis. Specifically, we test for correlation significance between GT-2 and RC-2, between GT-1 and ENSO-1, and between RC-2 and ENSO-2, using 30-year moving windows. We also use the same GC F tests to check for GC between GT-1 and ENSO-1. The results and of this analysis are discussed in Section 5.4.
5 Results
5.1 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.
Causality tests with forward cumulative windows, lag = 3
Period | Test statistic (RC → GT) | p value (RC → GT) | Test statistic (GT → RC) | p value (GT → RC) |
---|---|---|---|---|
1860–1900 | 0.537 | 0.661 | 1.023 | 0.396 |
1860–1910 | 1.040 | 0.385 | 0.569 | 0.638 |
1860–1920 | 1.037 | 0.384 | 0.572 | 0.636 |
1860–1930 | 0.941 | 0.426 | 0.603 | 0.615 |
1860–1940 | 0.636 | 0.595 | 0.339 | 0.797 |
1860–1950 | 0.709 | 0.549 | 0.375 | 0.771 |
1860–1960 | 0.805 | 0.494 | 0.255 | 0.857 |
1860–1970 | 0.119 | 0.948 | 0.298 | 0.827 |
1860–1980 | 1.329 | 0.269 | 0.402 | 0.752 |
1860–1990 | 2.220 | 0.089^{a} | 0.239 | 0.869 |
1860–2008 | 1.983 | 0.119 | 0.624 | 0.601 |
Causality tests with forward cumulative windows, lag = 7
Period | Test statistic (RC → GT) | p value (RC → GT) | Test statistic (GT → RC) | p value (GT → RC) |
---|---|---|---|---|
1860–1900 | 0.577 | 0.766 | 2.424 | 0.059^{a} |
1860–1910 | 0.637 | 0.721 | 0.637 | 0.721 |
1860–1920 | 0.622 | 0.735 | 0.777 | 0.611 |
1860–1930 | 0.478 | 0.846 | 0.516 | 0.818 |
1860–1940 | 0.252 | 0.969 | 0.716 | 0.659 |
1860–1950 | 0.198 | 0.985 | 0.447 | 0.869 |
1860–1960 | 0.309 | 0.948 | 0.303 | 0.951 |
1860–1970 | 0.583 | 0.768 | 0.677 | 0.691 |
1860–1980 | 0.804 | 0.586 | 0.469 | 0.855 |
1860–1990 | 0.868 | 0.535 | 1.226 | 0.295 |
1860–2008 | 1.482 | 0.179 | 1.634 | 0.132 |
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 CO_{2} 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.
5.2 RCUMGC results
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 _{1},...ω _{ n })^{ 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 _{1},...ω _{ n })^{ 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.
5.3 Forecast results
Two forecasting models are built in each direction (RC-2 → GT-2 and vice versa) using a lag 3 GC model
Model | R^{2} holdout | MAPE | Maximum APE |
---|---|---|---|
1. \( {{G\hat{T}}} = {\hbox{PastGT}} \) | 0.722 | 0.215 | 1.431 |
2. \( {{G\hat{T}}} = {\hbox{PastGT}} + {\hbox{PastRC}} \) | 0.767 | 0.186 | 0.987 |
3. \( {{R\hat{C}}} = {\hbox{PastRC}} \) | 0.495 | 0.005 | 0.012 |
4. \( {{R\hat{C}}} = {\hbox{PastRC}} + {\hbox{PastGT}} \) | 0.514 | 0.005 | 0.011 |
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.
5.4 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.
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 CO_{2} 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.
ENSO, GT, and RC Granger tests
GT-1 → ENSO-1 | ENSO-1 → GT-1 | RC-2 → ENSO-2 | ENSO-2 → RC-2 | |||||
---|---|---|---|---|---|---|---|---|
F test | p value | F test | p value | F test | p value | F test | p value | |
Lag1 | 2.436 | 0.121 | 20.253 | 1.39e−05^{a} | 13.224 | 3.8e−04^{a} | 18.804 | 2.71e−05^{a} |
Lag2 | 0.600 | 0.550 | 10.620 | 5.06e−05^{a} | 4.611 | 0.012^{b} | 4.560 | 0.012^{b} |
Lag3 | 0.531 | 0.662 | 6.167 | 0.001^{a} | 2.810 | 0.042^{b} | 3.079 | 0.030^{b} |
Lag4 | 0.654 | 0.625 | 4.892 | 0.001^{a} | 2.195 | 0.073^{c} | 4.936 | 0.001^{a} |
Lag5 | 0.543 | 0.743 | 3.435 | 0.006^{a} | 2.395 | 0.041^{b} | 2.763 | 0.021^{b} |
Lag6 | 0.332 | 0.919 | 2.735 | 0.016^{b} | 1.249 | 0.286 | 1.910 | 0.084^{c} |
Lag7 | 0.328 | 0.940 | 2.437 | 0.022^{b} | 0.932 | 0.484 | 1.675 | 0.121 |
- 1.
The smoothing of the early CO_{2} 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.
- 2.
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.
- 3.
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 CO_{2} 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.
6 Concluding remarks
6.1 Limitations
One caveat in general with GC lies in the outside variable factor. If X seems to cause Y but is simply highly correlated with Z, which actually causes Y, then it is possible to incorrectly designate X as the causal influence. In climate applications, this type of error might be complex and hard to detect.
Another possible limitation in using classic bivariate GC F test, more specific to this work, is that it follows the assumption that the predictand should be normally distributed. We do not meet the normality assumption with RC at any level of differencing. This could potentially affect our p values, especially those obtained in GT → RC models, where RC is the predictand.
The conditional heteroskedasticity found in RC-2 may also pose a problem to our GC testing. This non-constant variance is a result of smoothing in the early ice core CO_{2} data, where many annual values are interpolated. The change in data source may have some effect on our GC tests and/or our results in Fig. 6. We do not study the effect of using smoothed averages between 1860 and 1959. In general, such imputation or smoothing has the effect of reducing noise variance; hence, p values of hypothesis tests tend to be recorded as lower than their true values.
In fact, if we look at Fig. 4, we note that after a latest-window size of 50, there is a sudden jump in most of the H trend lines. Perhaps not coincidentally, the jump occurs when the latest 10 ice core-derived RC values are added to the test (1949–1958). As implied in the previous paragraph, the interpolation of many of the ice core values could have the effect of artificially increasing the significance of the F tests. Thus, perhaps the larger F _{1} values at latest-window sizes 60 and above are not a result of GC but a result of the interpolated ice core values. These two effects, then, may be confounded.
One final limitation concerns the number of RCUMGC tests we conduct. Since we are conducting 224 such tests and 57 p values significant at 0.05 (39 at 0.01), the standard problems of false positives and false negatives associated with multiple testing will be present. These can be controlled using standard procedures; see for example Benjamini and Hochberg (1995) for a method of controlling the false discovery rate. This same limitation applies to the forward cumulative tests.
6.2 Implications
- 1.
RC does seem to Granger cause GT, as demonstrated by the RCUMGC test. The forecast procedure adds support to this.
- 2.
There is a sudden jump in correlation significance between RC and GT beginning in the 1970s—in Section 5.4, we propose several competing hypotheses for this phenomenon. It is possible that no one of these hypotheses stands alone.
- 3.
There may be one or more variables that Granger cause GT besides RC, including but not limited to ENSO. An investigation of such variables and their relation to GT might be pursued in future research.
- 4.
The data (particularly RC-2) do not meet the assumptions necessary to apply the chosen GC F tests, and so to further substantiate our results, we may have to investigate a test that takes our data limitations into consideration.
Notes
Acknowledgements
The authors are grateful to Umberto Triacca for sharing data from a previous study, and to Karsten Steinhaeuser and Shih-Chieh Kao for their valuable input. All computations were done using R and Microsoft Excel. This research funded through the Summer Undergraduate Laboratory Internship (SULI) program of the United Stated Department of Energy (US DOE). The research was performed at the Oak Ridge National Laboratory (ORNL), which in turn is managed by UT-Battelle, LLC, for the US Department of Energy under Contract DE-AC05-00OR22725. The United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allows others to do so, for United States Government purposes. The research of E.K. and A.R.G. was partially supported by the Laboratory Directed Research & Development (LDRD) Program of the Oak Ridge National Laboratory (ORNL) as part of a project on climate extremes and uncertainty led by A.R.G. at ORNL.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Supplementary material
References
- Alley RB, Marotzke J, Nordhaus WD, Overpeck JT, Peteet DM, Pielke RA Jr, Pierrehumbert RT, Rhines PB, Stocker TF, Talley LD, Wallace JM (2005) Abrupt climate change. Science 299(5615):2005–2010CrossRefGoogle Scholar
- Attanasio A, Triacca U (2010) Detecting human influence on climate using neural networks based Granger causality. Theor Appl Climatol. doi: 10.1007/s00704-010-0285-8 Google Scholar
- Barnett TP, Pierce DW, Hidalgo HG, Bonfils C, Santer BD, Das T, Bala G, Wood AW, Nozawa T, Mirin AA, Cayan DR, Dettinger MD (2008) Human-induced changes in the hydrology of the Western United States. Science 319(5866):1080–1083CrossRefGoogle Scholar
- Benjamini Y, Hochberg Y (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. J R Stat Soc B 57(1):289–300Google Scholar
- Committee on Abrupt Climate Change, National Research Council (2002) Abrupt climate change. National Academy, WashingtonGoogle Scholar
- Elsner J (2006) Evidence in support of the climate change—Atlantic hurricane hypothesis. Geophysical Research Letters 33:L16705CrossRefGoogle Scholar
- Elsner J (2007) Granger causality and Atlantic hurricanes. Tellus 59(4):476–485CrossRefGoogle Scholar
- Graham NE (1994) Decadal-scale climate variability in the tropical and North Pacific during the 1970s and 1980s: observations and model results. Clim Dyn 10(3):135–162CrossRefGoogle Scholar
- Granger CWJ (1969) Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37(3):424–438CrossRefGoogle Scholar
- Hacker RS, Hatemi-J A (2006) Tests for causality between integrated variables using asymptotic and bootstrap distributions: theory and application. Appl Econ 38(13):1489–1500CrossRefGoogle Scholar
- Hasselmann K (1979) On the signal-to-noise problem in atmospheric response studies. In: Shaw DB (ed) Meteorology over the Tropical Oceans. Roy al Meteorological Society, London, pp 251–259Google Scholar
- Hegerl GC, von Storch H, Hasselmann K, Santer BD, Cubasch U, Jones PD (1996) Detecting greenhouse-gas-induced climate change with an optimal fingerprint method. J Clim 9:2281–2306CrossRefGoogle Scholar
- Hegerl GC, Hasselmann K, Cubasch U, Mitchell JFB, Roeckner E, Voss R, Waszkewitz J (1997) Multi-fingerprint detection and attribution analysis of greenhouse gas, greenhouse gas-plus-aerosol and solar forced climate change. Climate Dyn 13:613–634CrossRefGoogle Scholar
- Hiemstra C, Jones JD (1994) Testing for linear and non-linear Granger causality in the stock price-volume relation. J Finance 49(5):1639–1664CrossRefGoogle Scholar
- Kaufmann RK, Stern DI (1997) Evidence for human influence on climate from hemispheric temperature relations. Nature 388:39–44CrossRefGoogle Scholar
- Kaufmann YJ, Fraser RS, Mahoney RL (1991) Fossil fuel and biomass burning effect on climate—heating or cooling? J Clim 4:578–588CrossRefGoogle Scholar
- Kutner M, Nachtsheim C, Neter J, Li W (2004) Applied linear statistical models. McGraw-Hill, New YorkGoogle Scholar
- Kwiatkowski D, Phillips PCB, Schmidt P, Shin Y (1992) Testing the null hypothesis of stationarity against the alternative of a unit root. J Econom 54:159–178CrossRefGoogle Scholar
- Lozano A, Naoki A, Liu Y, Rosset S (2009) Grouped graphical Granger modeling methods for temporal causal modeling. The 15th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, Paris, France.Google Scholar
- Lozano A, Li H, Niculescu-Mizil A, Liu Y, Perlich C, Hosking J, Abe N (2009) Spatial-temporal causal modeling for climate change attribution. The 15th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, Paris, France.Google Scholar
- Meehl GA, Arblaster JM, Tebaldi C (2007) Contributions to natural and anthropogenic forcing to changes in temperature extremes over the United States. Geophys Res Lett 34:L19709CrossRefGoogle Scholar
- Mosedale TJ, Stephenson DB, Collins M, Mills TC (2006) Granger causality of coupled climate processes: ocean feedback on the North Atlantic Oscillation. J Climate 19(7):1182–1194CrossRefGoogle Scholar
- Myhre G, Highwood EJ, Shine K, Stordal F (1998) New estimates of radiative forcing due to well mixed greenhouse-gases. Geophys Res Lett 25(14):2715–2718CrossRefGoogle Scholar
- Ropelewski CF, Halpert MS (1986) North American precipitation and temperature patterns associated with the El Nino/Southern Oscillation. Mon Weather Rev 114:2352–2362CrossRefGoogle Scholar
- Salvucci GD, Saleem JA, Kaufmann RK (2002) Investigating soil moisture feedbacks on precipitation with tests of Granger causality. Adv Water Resour 25(8–12):1305–1312CrossRefGoogle Scholar
- Santer BD, Wigley TML, Jones PD, Schlesinger ME (1991) Multivariate methods for the detection of greenhouse-gas-induced climatic change. In: Schlesinger ME (ed) Greenhouse-gas-induced climatic change: a critical appraisal of simulations and observations. Elsevier, Amsterdam, pp 511–536Google Scholar
- Santer BD, Wigley TML, Jones PD (1993) Correlation methods in fingerprint detection studies. Climate Dyn 8:265–276CrossRefGoogle Scholar
- Schiermeier Q (2010) The real holes in climate science. Nature 463:284–287CrossRefGoogle Scholar
- Shapiro SS, Wilk MB (1965) An analysis of variance test for normality (complete samples). Biometrika 52(3/4):591–611CrossRefGoogle Scholar
- Smirnov DA, Mokhov II (2009) From Granger causality to long-term causality: application to climactic data. Phys Rev E. 80(1) doi: 10.1103/PhysRevE.80.016208
- Solomon S, Qin D, Manning M, Chen Z, Marquis M, Averyt KB, Tignor M, Miller HL, IPCC (2007) Summary for policymakers. In: Climate change 2007: the physical science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA.Google Scholar
- Stern DI, Kaufmann RK (1999) Econometric analysis of global climate change. Environ Modell Softw 14(6):597–605CrossRefGoogle Scholar
- Stigler S (2005) Correlation and causation: a comment. Perspect Biol Med 48:S88–S94Google Scholar
- Sun L, Wang M (1996) Global warming and global dioxide emission: an empirical study. J Environ Manage 46:327–343CrossRefGoogle Scholar
- Toda HY, Yamamoto T (1995) Statistical influence in vector autoregressions with possibly integrated processes. J Econom 66(1–2):225–250CrossRefGoogle Scholar
- Tol RSJ, De Vos AF (1993) Greenhouse statistics—time series analysis. Theor Appl Climatol 48:63–74CrossRefGoogle Scholar
- Tol RSJ, De Vos AF (1998) A Bayesian statistical analysis of the enhanced greenhouse effect. Clim Change 38:87–112CrossRefGoogle Scholar
- Triacca U (2005) Is Granger causality analysis appropriate to investigate the relationship between atmospheric concentration of carbon dioxide and global surface air temperature? Theor Appl Climatol 81:133–135CrossRefGoogle Scholar
- Zivot E, Wang J (2002) Modeling financial time series with S-PLUS. Insightful, SeattleGoogle Scholar