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ATMOSPHERIC SCIENCE LETTERS
Atmos. Sci. Let. 13: 67–72 (2012)
Published online 7 November 2011 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/asl.365
A contribution to attribution of recent global warming by
out-of-sample Granger causality analysis
Alessandro Attanasio,1 Antonello Pasini2 * and Umberto Triacca3
1 Department of Pure and Applied Mathematics, University of L’Aquila, L’Aquila, Italy
2 CNR, Institute of Atmospheric Pollution Research, Monterotondo Stazione, Rome, Italy
3 Department of Economics, University of L’Aquila, L’Aquila, Italy
*Correspondence to:
A. Pasini, CNR, Institute of
Atmospheric Pollution Research,
via Salaria km 29.300, I-00015
Monterotondo Stazione,
Rome, Italy.
E-mail: [email protected]
Received: 15 September 2011
Revised: 4 October 2011
Accepted: 13 October 2011
Abstract
The topic of attribution of recent global warming is usually faced by studies performed
through global climate models (GCMs). Even simpler econometric models have been applied
to this problem, but they led to contrasting results. In this article, we show that a genuine
predictive approach of Granger analysis leads to overcome problems shown by these models
and to obtain a clear signal of linear Granger causality from greenhouse gases (GHGs) to
the global temperature of the second half of the 20th century. In contrast, Granger causality
is not evident using time series of natural forcing. Copyright  2011 Royal Meteorological
Society
Keywords:
attribution; global warming; Granger causality; time series
1. Introduction
2. Data
Global climate models (GCMs) are the standard
dynamical tools for catching the complexity of the
climate system. Their application to attribution studies has shown that the global warming of the second half of the 20th century can be attributed mainly
to anthropogenic greenhouse gases (GHGs) (Hegerl
et al., 2007).
In the meantime, even simpler empirical models
(adopting artificial intelligence or econometrics techniques) have been applied to the problem of temperature attribution and were considered in Hegerl et al.
(2007). Recent studies (Pasini et al., 2006; Attanasio
and Triacca, 2011) show that nonlinear models generally corroborate the attribution results obtained by
GCMs. Otherwise, using econometric linear models of
Granger causality leads to obtain contrasting results
(Sun and Wang, 1996; Kaufmann and Stern, 1997;
Reichel et al., 2001; Triacca, 2001, 2005; Mohkov
and Smirnov, 2008; Kodra et al., 2011). Recently,
moreover, these Granger models showed their usefulness when applied to specific causality problems
in the climate system (Diks and Mudelsee, 2000;
Kaufmann et al., 2003; Elsner, 2006, 2007; Mosedale
et al., 2006; Kaufmann et al., 2007; Mohkov et al.,
2011).
In this framework, we reconsider a linear Granger
analysis for temperature attribution by an out-ofsample approach, able to overcome problems that can
rise from data analysis and overfitting in previous insample ones. In doing so, we analyze the causality
relationships (in the sense of Granger) between natural/anthropogenic forcings and global temperature and
find clear and non-ambiguous results.
Time series of mean annual data for the period
1850–2007 are considered for the following variables:
Copyright  2011 Royal Meteorological Society
• HadCRUT3 combined global land and marine
surface temperature anomalies (T) (Brohan et al.,
2006). Data available at http://www.cru.uea.ac.uk/
cru/data/;
• Total solar irradiance (TSI) (Lean and Rind, 2008),
with background from Wang et al. (2005). Data
available at www.geo-fu.berlin.de;
• Cosmic ray intensity (CRI) from count rates of
a polar neutron monitor (Usoskin et al., 2002,
2005; Alanko-Huotari et al., 2006). Data available
at ftp.ncdc.noaa.gov;
• Stratospheric aerosol optical thickness (SAOT) at
550 nm as compiled by Sato et al. (1993), updated
by giss.nasa.gov in 1999 and extended to 2007 with
zero values. Data available at http://data.giss.nasa.
gov;
• CO2 , CH4 and N2 O concentrations (Hansen et al.,
2007). Data available at http://data.giss.nasa.gov;
we calculate radiative forcings (RFs) as in
Ramaswamy et al. (2001).
Here, we consider TSI, CRI or SAOT as natural
forcings, and CO2 RF, CH4 RF, N2 O RF or GHG-total
RF (CO2 + CH4 + N2 O) as anthropogenic forcings
which can Granger cause the variable y = T.
3. Methods
The climate system is highly nonlinear and applying
linear methods for climatic attribution could seem generally inappropriate. Actually, if we consider annual
68
A. Attanasio, A. Pasini and U. Triacca
averages, it is quite reasonable that as a consequence
of the central limit theorem (Yuval and Hsieh, 2002),
averaging can produce near-linear climate relations
among variables of the climate system, even if we have
to do with highly nonlinear relations at shorter spacetime scales. Thus, we are confident that also a linear
model can provide a ‘first-order’ estimate of existence
or non-existence of causal links in the climate system.
In a time series approach, we say that a variable x Granger causes another variable y if future
values of y can be better predicted using the past
values of x and y rather than only past values of y. Thus, we can build an autoregressive
(AR) model of order k – AR(k ) – using just y values, a vector autoregressive (VAR) model of the
same order – VAR(k ) – using both y and x values and evaluate their forecast performance on a
test set in the future, in terms of mean square
error (MSE). We say that x Granger causes y if
MSE[VAR(k )]<MSE[AR(k )] in a statistical significant way.
Usually the presence of Granger causality is tested
using in-sample tests. However, the reliability of this
form of Granger causality tests seems to depend on
many data characteristics. It has been shown, for
instance, that the use of non-stationary data can yield
to spurious causality results (Park and Phillips, 1989;
Stock and Watson, 1989; Sims et al., 1990). In insample cases, in fact, it is crucial to establish the
stochastic properties of the time series involved, by
analyzing whether these series are stationary, nonstationary or co-integrated. Of course, the weakness
of this approach is that incorrect conclusions drawn
by this preliminary analysis may affect the results of
causality tests and their reliability.
In order to overcome these problems, according to
the analysis in Ashley et al. (1980), we use a technique
that relies on the out-of-sample comparison of the
forecasting performance of two linear models. This is
not based on assumptions coming from a preliminary
data analysis, so it may be more robust in terms of
model selection biases and overfitting (Clark, 2004;
Gelper and Croux, 2007). Furthermore, according to
Granger’s definition, Granger causality builds upon the
notion of incremental predictability, so that our outof-sample approach is more keeping the spirit of the
original definition by Granger (1969).
Adopting this approach, for the bivariate time series
zi ,t = (yt , xi ,t ), where i = 1, . . . , 7 identify our forcings, we consider a VAR model – VAR(k ), as follows:
zi ,t =
TD(it )
+
k
j(i ) zi ,t−j + wt(i )
(1)
we assume that det (i ) (B ) has zeroes lying on
or
the unit circle, where (i ) (B ) = I2 −
k outside
(i ) j
j =1 j B .
Since
of yt+1 based on
the optimal linear predictors
Iyxi = yt , xi ,t , yt−1 , xi ,t−1 , . . . and Iy = {yt , yt−1 , . . .}
are
k
)
(i )
P yt+1 | Iyxi = TD(i1,t+1
+
φ11,j
yt+1−j
j =1
+
k
(i )
φ12,j
xi ,t+1−j
(2)
j =1
k
(i )
(i )
φ11,j
yt+1−j
P yt+1 | Iy = TD1,t+1 +
j =1
+
k
(i )
φ12,j
P xi ,t+1−j | Iy
(3)
j =1
we have that
MSE P yt+1 | Iyxi = MSE P yt+1 | Iy
(4)
if and only if
(i )
(i )
(i )
φ12,1
= φ12,2
= . . . = φ12,k
= 0.
(5)
Thus, in our application we compare the predictive
ability one step ahead of the two following nested
regression models in terms of MSE:
yt = δ1(i ) +
VAR :
k
αj(i ) yt−j
j =1
+
k
βj(i ) xi ,t−j + vt(i )
(6)
j =1
AR :
yt = δ2 +
k
γj yt−j + ut
(7)
j =1
Here, δ1(i ) and δ2 are constants included as deterministic terms, xi is our forcings, αj(i ) , βj(i ) and γj are
coefficients of our regressions, vt(i ) and ut are univariate white noises. The order k of the models is kept
low (k = 1, . . . , 4) – in doing so the models are parsimonious and the residuals are uncorrelated – and the
models finally selected are those endowed with the
best predictive performance on each test set.
j =1
where TD(it ) = TD(i1,t) , TD(i2,t) is the deterministic
(i )
component, j(i ) = φnm,j
are 2 × 2 coefficient matri
(i )
(i )
(i )
ces and the process wt = w1,t
, w2,t
is a bivariate white noise. Adopting B as the lag operator,
Copyright  2011 Royal Meteorological Society
4. Application and results
In this framework, we analyze which external natural and anthropogenic forcings are able to Granger
cause global temperature (y = T), by testing the
null hypotheses of non-Granger causality. As cited
Atmos. Sci. Let. 13: 67–72 (2012)
A contribution to attribution of recent global warming
above, we consider TSI, CRI or SAOT as natural
forcings, CO2 RF, CH4 RF, N2 O RF or GHG-total
RF (CO2 + CH4 + N2 O) as anthropogenic forcings.
Data sets refer to the period 1850–2007. The outof-sample tests are performed on five test sets which
span the following periods: 1941–2007, 1951–2007,
1961–2007, 1971–2007 and 1981–2007. Fixed and
recursive schemes are adopted for predictions and the
statistical significance of results is evaluated by MSE-t
and MSE-REG tests.
For each test set, the correspondent training set is
composed by data patterns since 1850 till the year
before the beginning of the test set itself. Under the
recursive scheme, we use the training set for the first
estimate and forecast out-of-sample one step ahead;
then, we add an annual pattern to our training set,
obtain a second estimate and forecast for the next
year and so on, iteratively. Under the fixed scheme,
the parameters are estimated only once on the original
training set and every one-step ahead forecast is
obtained using just these fixed parameters.
The null hypothesis is tested using two tests
described in McCracken (2007): the MSE-t test, a similar t-type test commonly attributed to Diebold and
Mariano (1995) or West (1996), and the MSE-REG
test, originally suggested by Granger and Newbold
(1997). Critical values of the test statistics are provided by McCracken (2007) for stationary series and
are computed for the cases in which parameters are
estimated by ordinary least squares (OLS), nonlinear
least squares or Gaussian maximum likelihood.
In particular, we test the null hypothesis of equal
forecast accuracy
H0 : E (ut )2 = E (vt(i ) )2
(8)
versus
H1 = E (ut )2 > E (vt(i ) )2
(9)
Thus, let us consider the residuals v̂t(i ) and ût of the
2
Equations (6) and (7) and define dt = (ût )2 − v̂t(i ) .
We regress dt on a constant a on the test set, obtaining
MSE-t =
â
se(â)
(10)
where â is the OLS estimate of a and se(â) is the
â’s standard error. Furthermore, in order to calculate
MSE-REG statistics, we consider the following regression model:
ût − v̂t(i ) = c ût + v̂t(i ) + et
(11)
on the test set, where et is a white noise. The MSEREG statistics can be thus evaluated by use of the
t-statistics associated with the coefficient c, i.e.
MSE-REG =
ĉ
se(ĉ)
Copyright  2011 Royal Meteorological Society
(12)
69
where ĉ is the OLS estimate of c and se(ĉ) is the ĉ’s
standard error.
In our case, we do not use the critical values
described in McCracken (2007) because our time
series are not stationary. So we perform the following
bootstrap procedure in order to calculate our critical
values:
1. Calculate forecasts of the models (6) and (7) using
fixed and recursive schemes.
2. Evaluate MSE-t and MSE-REG statistics.
3. Under the null hypothesis of non-causality, estimate
the restricted model (7) employing the full sample
and extract the estimates δ̂2 , γ̂1 , . . . , γ̂k and the
residuals ût .
4. Apply bootstrap procedure (resampling with
replacement) on ût and obtain the pseudo-residuals
ut∗ .
5. Generate the pseudo-data yt∗ under the null hypothk
∗
γ̂j yt−j
+
esis of non-causality using yt∗ = δ̂2 +
j =1
ut∗
6. Considering the pseudo-data yt∗ , repeat steps 1
and 2 calculating MSE-t and MSE-REG bootstrap
statistics.
7. Repeat steps from 4 to 6 for N times.
8. Calculate the bootstrap p-value.
The first result obtained by our analysis is very
clear. If we take TSI, CRI or SAOT as x variable,
in every case (any natural forcing, scheme and test
set considered) the null hypothesis of non-Granger
causality on y = T is never rejected (with only
two exceptions), even just at a 10% significance
level (Table I). Vice versa, there is a clear general
evidence of Granger causality from anthropogenic
forcings to global temperature (refer Table II for
detailed results).
In particular, GHG-total RF has a good ‘linear explanatory power’ for the global temperature’s
behavior in every period since 1941. For instance,
in Figure 1 a comparison of performance of the
AR(3) and VAR models (using TSI or GHG-total
RF) in forecasting T values since 1971 is shown.
It is clear that TSI gives no ‘surplus value’ to the
forecast of temperature by the AR model AR(3)
(green and blue lines are quite superimposed), while
GHG-total RF permits to better predict temperature
anomalies.
In more details, we have considered as natural forcings some well-known influence factors on temperature (the indices of sun radiation and volcanic dusts),
but also a more controversial factor like the index of
CRI used here. In every case, no sign of Granger
causality has been ‘detected’ by our analysis since
1941, if we except the rejection of the null hypothesis
just at the 10% significance level for the influence of
TSI on the first test set and CRI on the second one, in
the cases of fixed schemes adopted.
Atmos. Sci. Let. 13: 67–72 (2012)
70
A. Attanasio, A. Pasini and U. Triacca
Table I. Results of Granger causality analysis for natural forcings.
Fixed scheme
Forcing
Test set
VAR order
MSE-t
SAOT
1941–2007
1951–2007
1961–2007
1971–2007
1981–2007
4
4
4
4
4
−1.69(ns)
−1.91(ns)
−2.31(ns)
−0.84(ns)
−0.78(ns)
TSI
1941–2007
1951–2007
1961–2007
1971–2007
1981–2007
4
4
4
3
4
CRI
1941–2007
1951–2007
1961–2007
1971–2007
1981–2007
4
2
4
4
4
Recursive scheme
MSE-REG
VAR order
MSE-t
MSE-REG
−1.93(ns)
−2.10(ns)
−2.59(ns)
−0.85(ns)
−0.78(ns)
4
4
4
4
4
−0.27(ns)
−0.22(ns)
−0.31(ns)
−0.38(ns)
−0.72(ns)
−0.31(ns)
−0.24(ns)
−0.34(ns)
−0.40(ns)
−0.80(ns)
2.36∗∗∗
0.85 (ns)
0.49 (ns)
−0.22(ns)
−1.93(ns)
2.10∗∗∗
0.75 (ns)
0.45 (ns)
−0.23(ns)
−2.15(ns)
4
4
4
4
4
−0.69(ns)
−0.58(ns)
−0.92(ns)
−1.10(ns)
−1.86(ns)
−0.68(ns)
−0.57(ns)
−0.96(ns)
−1.24(ns)
−2.35(ns)
−1.37(ns)
1.54∗∗∗
−1.25(ns)
−1.78(ns)
−2.01(ns)
−1.33(ns)
1.52∗∗∗
−1.31(ns)
−2.11(ns)
−2.21(ns)
3
2
4
4
4
−1.00(ns)
−0.25(ns)
−1.52(ns)
−1.41(ns)
−1.62(ns)
−0.98(ns)
−0.26(ns)
−1.48(ns)
−1.47(ns)
−1.89(ns)
Here a non-significant (ns) statistics implies that the null hypothesis of non-Granger causality cannot be rejected at 10% significance level, while
∗∗∗ indicates that the null hypothesis is rejected at 10% significance level. VAR orders are referred to the models with the best predictive ability.
Table II. Results of Granger causality analysis for anthropogenic forcings.
Fixed scheme
Forcing
CO2 RF
Test set
VAR order
MSE-t
Recursive scheme
MSE-REG
MSE-REG
4.43∗
4.84∗
2.61∗∗
2.62∗
2.56∗
1.48∗∗
2.54∗
3.09∗
2.93∗
2.90∗
1.75∗
3.96∗
3.72∗
3.42∗
3.54∗
3.96∗
4.23∗
3.94∗
3.53∗
3.60∗
4.23∗
1
1
1
1
1
2.85∗
2.94∗
2.83∗
2.72∗
2.85∗
3.00∗
2.95∗
2.87∗
2.79∗
3.00∗
1
1
1
1
1
4.74∗
3.86∗
3.63∗
3.81∗
3.73∗
4.20∗
4.10∗
3.73∗
3.82∗
3.70∗
2
2
1
2
1
3
1
3
3
3
3.36∗
4.43∗
4.07∗
4.13∗
2.85∗
3.46∗
4.91∗
4.65∗
4.83∗
3.18∗
3
3
3
3
3
N2 O RF
1941–2007
1951–2007
1961–2007
1971–2007
1981–2007
1
1
1
1
1
CH4 RF
1941–2007
1951–2007
1961–2007
1971–2007
1981–2007
Total RF
1941–2007
1951–2007
1961–2007
1971–2007
1981–2007
4.14∗
4.15∗
2.16∗∗
MSE-t
3
3
3
3
4
3
1
1
3
4
3.83∗
VAR order
2.39∗
1941–2007
1951–2007
1961–2007
1971–2007
1981–2007
3.58∗
4.62∗
5.19∗
2.82∗
1.25∗∗
1.37∗
2.29∗
0.95 (NS)
2.39∗
2.36∗
2.79∗
2.56∗
2.57∗
1.92∗
1.08∗∗
1.17∗
2.23∗
0.81 (NS)
2.36∗
2.36∗
2.86∗
2.69∗
2.74∗
2.01∗
Here non-significant (NS) statistics implies that the null hypothesis of non-Granger causality cannot be rejected at 5% significance level, while ∗ and
∗∗ indicate that the null hypothesis is rejected at 1 and 5% significance levels, respectively. VAR orders are referred to the models with the best
predictive ability.
Otherwise, the insertion of GHG-total RF in a
VAR model always permits to reject the null hypothesis of non-Granger causality (in both fixed and
recursive schemes) at 1% significance level. VARs
with CO2 RF and N2 O RF supply very similar
results, so that the Granger causality from these forcings to global temperature can be considered very
robust.
Finally, if we consider CH4 RF, in some cases
Granger causality is less strong, the results seem
Copyright  2011 Royal Meteorological Society
to partially depend on the scheme used and cannot
be considered very robust (refer Table II for further
details). The well-known strongly nonlinear physicochemical interactions of methane in the climate system
(Crabtree, 1995) could explain the difficulty of this
linear method to weight CH4 influences on T.
Here, all our VARs are bivariate. Of course, models
of higher dimension can be used in order to further
test the robustness of results. This extention will be
considered in future investigations.
Atmos. Sci. Let. 13: 67–72 (2012)
A contribution to attribution of recent global warming
0.6
Temperature anomalies [K]
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
1975
1980
1985
1990
1995
2000
2005
Time [years]
Figure 1. Sketch of the out-of-sample model performance on
the test set 1971–2007 by fixed scheme. The different lines
show data series of global temperature anomalies (black line),
AR(3) prediction (blue line), VAR(3) prediction using TSI (green
line) and VAR(3) prediction using GHG-total RF (red line).
5. Conclusions
In this article, an analysis has been performed for
understanding which external forcings are able to
Granger cause the recent behavior of global temperature. Differently from previous in-sample approaches
to this problem, in our out-of-sample Granger causality framework we show that a simple linear predictive
method can give a clear contribution to the assessment of temperature attribution. In particular, a strong
evidence of a causal link from anthropogenic forcings
to temperature behavior has been found through our
method during the recent period of recognised global
warming.
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