Abstract

At present, the role of natural variability in influencing climate behaviour is widely discussed. The generally accepted view is that atmosphere-ocean coupled circulation patterns are able to amplify or reduce temperature increase from interannual to multidecadal time ranges, leaving the principal driving role to anthropogenic forcings. In this framework, the influence of these circulation patterns is considered synchronous with global temperature changes. Here, we would like to investigate if there exists a lagged influence of these indices on temperature. In doing so, an extension of the Granger causality technique, which permits to test both direct and indirect causal influences, is applied. A lagged influence of natural variability is not evident in our analysis, if we except weak influences of some peculiar circulation indices in specific periods.

1. Introduction

The search for the fundamental causes of the recent global warming has been extensively performed in the last decade by attribution studies through a dynamical approach using Global Climate Models (GCMs): see, for instance, Hegerl and Zwiers (2011). The results clearly indicate the necessity to consider anthropogenic forcings (and especially greenhouse gases radiative forcing—GHGRF) if one wants to recover the increase in global temperature (T) detected in the last half-century (Bindoff et al., 2013).

However, several studies show how circulation patterns of the coupled atmosphere-ocean system can influence the evolution of temperature behaviour, possibly amplifying or reducing temperature increase over interannual or multidecadal time ranges (Hoerling et al., 2008; DelSole et al., 2011). Generally, in these studies, the influences of these circulation patterns on global temperature appears to be synchronous (see, for instance, Hurrell, 1996; Comrie and McCabe, 2013).

Here, we would like to investigate if there exists a lagged influence of these indices on global temperature. The motivation of our study relies on the evidence that these single oscillation patterns [here synthesized by indices based on Sea Surface Temperatures (SSTs)], usually show teleconnections with delayed effects on temperatures elsewhere, on very wide regions: see, for instance, previous studies about physically based relationships between Atlantic and Pacific SSTs (Saenko et al., 2004; Wang et al., 2011). Furthermore, the origins for the delayed increases in global surface temperature accompanying El Niño events have been explored, too (Trenberth et al., 2002). Finally, in other empirical attribution studies, the values of circulation patterns have been also used as influence factors with a delayed effect on global temperature (Lean and Rind, 2008; Foster and Rahmstorf, 2011) even for indices here not considered, such as the North Atlantic Oscillation (Li et al., 2013).

In order to do so, we adopt a well-known technique for testing causality links, the so-called Granger causality analysis (Granger, 1969), which represents the appropriate statistical tool for identifying lagged influences. Recently, this analysis method has been applied to several problems in climate science and extensively to the attribution topic (Kaufmann and Stern, 1997; Diks and Mudelsee, 2000; Wang et al., 2005; Triacca, 2005; Mosedale et al., 2006; Elsner, 2007; Kodra et al., 2011; Attanasio et al., 2012; Pasini et al., 2012; Triacca et al., 2013; Stern and Kaufmann, 2014; Pasini et al., 2015).

In particular, for the first time, we investigate both direct influences and indirect chains of causality, using a generalization of the notion of Granger causality proposed by Dufour and Renault (1998): the so-called multi-horizon causality. The results give new insight in the role of circulation patterns for temperature determination.

The paper is organized as follows. In Section 2, we introduce the notion of the multi-horizon causality. In Section 3, we describe the data and the used testing procedures. Section 4 presents the obtained results. Section 5 provides some concluding remarks.

2. Multi-horizon Granger causality

The notion of Granger causality was first introduced by Norbert Wiener (1956) and later reformulated and formalized by Clive Granger (1969). Conceptually, the idea of Granger causality is quite simple. Suppose that we have two variables, x, y, and a vector z of m auxiliary variables, and that we first attempt to forecast yt+1 using past terms of y and z. We then try to forecast yt+1 using past terms of y, x and z. We say that x Granger causes y, if the second forecast is found to be more successful, according to standard cost functions. If the second prediction is better, then the past of x contains a useful information for forecasting yt+1 that is not in the past of y and z. Clearly, Granger causality is based on precedence and predictability.

If z is empty, we deal with a bivariate Granger causality, otherwise with a multivariate Granger causality. The causal relationship between the variables x and y has often been investigated in a bivariate system (m = 0). However, it is well known that in a bivariate framework problems of spurious causality and of non-causality due to omission of a relevant variable can arise. These problems can be solved if a vector z of auxiliary variables is considered in the analysis.

It is important to note that, when m > 0, the original definition of Granger could be insufficient to capture indirect causal links since it only deals with one-step-ahead prediction. If a causal chain from x to y via z (xzy) exists, it is possible that x fails to cause y for horizon 1 and causes y at horizon h > 1. Thus, an extension of Granger causality is necessary in multivariate models. Following Dufour and Renault (1998), we will use the following definitions of non-causality.

Let (yt,xt,zt) be a (3 × 1) vector time series. Consider the following information sets:  

Iyxz(t)={yt,xt,zt,yt1,xt1,zt1,}
 
Iyx(t)={yt,xt,yt1,xt1,}
 
Iyz(t)={yt,zt,yt1,zt1,}
 
Iy(t)={yt,yt1,}

For any positive integer h (the prediction horizon), we denote with P(yt+h|I(t)) the optimal linear forecast of the variable yt+h based on the information set I(t).

We say that:

  • (i) x does not cause y at horizon h given Iyx(t) (denoted xhy|Iyx(t)) if  

    P(yt+h|Iy(t))=P(yt+h|Iyx(t))t

  • (i.bis) x does not cause y at horizon h given Iyxz(t) (denoted xhy|Iyxz(t)) if  

    P(yt+h|Iyz(t))=P(yt+h|Iyxz(t))t

  • (ii) x does not cause y up to horizon h given Iyx(t) (denoted x(h)y|Iyx(t)) if  

    xky|Iyx(t)for k=1,,h

  • (ii.bis) x does not cause y up to horizon h given Iyxz(t) (denoted x(h)y|Iyxz(t)) if  

    xky|Iyxz(t)for k=1,,h

We observe that the conditions (i) and (ii) are equivalent [see Dufour and Renault (1998, Proposition 2.3)].

It is important to underline that the causal relationship between two variables x and y is not guaranteed to be conserved when a third variable z is considered in the analysis. In particular, we can have the following situations:

  • Spurious causality. x1y|Iyx(t) and x1y|Iyxz(t)

  • Spurious non-causality. x1y|Iyx(t) and x1y|Iyxz(t)

Another interesting pattern of causality is the following:

  • Indirect causality. x1y|Iyxz(t) and x2y|Iyxz(t)

It is possible to show that if x1y|Iyxz(t) and x2y|Iyxz(t), then there exists a causal chain from x to y via z, that is x1z|Iyxz(t) and z1y|Iyxz(t) [see Propositions 2.3 and 2.4 of Dufour and Renault (1998)]. Thus, we have called this pattern indirect causality.

3. Data and methodology

Here, we deal with the annual time series for the period 1866–2011:

3.1. Bivariate analysis

We are interested to study, separately, Granger causality from AMO, PDO or SOI to global temperature anomalies. First of all, a bivariate Granger causality analysis is performed by means of the following unrestricted VAR model:  

(1)
[ytxt]=[μ1μ2]+j=1k[α11,jα12,jα21,jα22,j][ytjxtj]+[ε1tε2t]

where μ=(μ1,μ2) is a vector of constants, αil,j are fixed coefficients and εt=(ε1t,ε2t) is a bivariate white noise process. In this framework, the variable x does not cause y at horizon 1 given Iyx(t) if and only if α12,j=0, for j=1,,k. Therefore, the null hypothesis of non-causality is given by  

H0:α12,j=0j=1,,k.

In order to test this hypothesis, we use an out-of-sample approach (see Ashley et al., 1980). Our sample of observations (yt,xt)t=1N is divided into a training set and a test set. The test set is composed by the last P observations, while the training sample consists of all previous R=NP observations. In particular, we consider the unrestricted model (1) and the following restricted model  

(2)
[ytxt]=[γ1γ2]+j=1k[β11,j0β21,jβ22,j][ytjxtj]+[w1tw2t]

Considering the Mean Squared Prediction Errors (MSPEs)  

MSPE(ε1t)=1Pt=R+1R+Pε1t2
 
MSPE(w1t)=1Pt=R+1R+Pw1t2

the null hypothesis of Granger non-causality becomes  

H0:E[MSPE(w1t)]E[MSPE(ε1t)]=0

where E is the expectation value operator.

The alternative hypothesis is that the restricted model provides a bigger MSPE than the unrestricted model.

Using the training set, the parameters of the models (1) and (2) are estimated by Ordinary Least Squared (OLS) and the P one-step-ahead forecast errors, for t=R+1,,R+P, are calculated as follows:  

ε^1t=ytμ^1j=1kα^11,jytjj=1kα^12,jxtj
 
w^1t=ytγ^1j=1kβ^11,jytj

Then, we calculate the mean square prediction errors  

MSPE(ε^1t)=1Pt=R+1R+Pε^1t2
 
MSPE(w^1t)=1Pt=R+1R+Pw^1t2

In order to test the null hypothesis, we use two tests described in McCracken (2007): the MSE-t, commonly attributed to Diebold and Mariano (1995) or West (1996), and the MSE-REG tests, suggested by Granger and Newbold (1977).

Defining dt=ε^1t2w^1t2, the MSE-t statistics is obtained by regressing dt on a constant a on the test set, thus obtaining  

MSE-t=a^se(a^)

where a^ is the OLS estimate of a and se(a^) is the a^’s standard error. Furthermore, in order to calculate MSE-REG statistics, we consider the following regression model:  

(ε^1tw^1t)=c(ε^1t+w^1t)+et

on the test set, where et is a white noise. The MSE-REG statistics can be thus evaluated by use of the t-statistics associated with the coefficient c, i.e.  

MSE-REG=c^se(c^)

where c^ is the OLS estimate of c and se(c^) is the c^’s standard error.

In our case, we do not use the critical values described in McCracken (2007) because several time series are not stationary (Kaufmann and Stern, 1997; Attanasio, 2012; Triacca et al., 2013). The critical values of the tests are calculated by means of the following bootstrap method (bootstrap based on residuals):

  1. Calculate forecasts of the models (1) and (2) for the time series yt using a forecast schemes.

  2. Evaluate MSE-t and MSE-REG statistics.

  3. Under the null hypothesis of non-causality, estimate the restricted model (2) employing the full sample and extract the estimates γ^j, β^lm,j and the residuals w^t.

  4. Apply bootstrap procedure (resampling with replacement) on w^t and obtain the pseudo-residuals wt*.

  5. Create the pseudo-data given by  

    [yt*xt*]=[γ^1γ^2]+j=1k[γ^11,j0γ^21,jγ^22,j][ytj*xtj*]+[w1t*w2t*]

  6. Using the pseudo-data, repeat the steps 1 and 2 calculating MSE-t and MSE-REG bootstrap statistics.

  7. Repeat steps from 4 to 6 for M times (in our application we use a high value for M (10000), which should lead to avoid problems with the convergence of P-values).

  8. Calculate the bootstrap P-values which is the proportion of the MSE-t (or MSE-REG) estimated bootstrap statistics that exceed the same statistic evaluated on the observed data.

The model order k, with k{1,2,3,4}, of the unrestricted model (1) is selected by means of Akaike information criteria (AIC) on the training set.

We close this subsection remembering that the problem of having to test for multi-horizon non-causality does not emerge in bivariate models. In fact, it is possible to show that x does not cause y at horizon 1 given Iyx(t) if and only if x does not cause y at any horizon h given Iyx(t). The reason is that in a bivariate system any causal effect of x on y must flow directly from x to y: a causal chain cannot exist.

3.2. Trivariate analysis

In this subsection, in order to investigate the various patterns of causality (spurious causality, spurious non-causality, indirect causality), we introduce the formalism for a trivariate analysis. In this case, the unrestricted VAR(k) model becomes  

(3)
[ytxtzt]=[c1c2c3]+j=1k[ϕ11,jϕ12,jϕ13,jϕ21,jϕ22,jϕ23,jϕ31,jϕ32,jϕ33,j][ytjxtjztj]+[u1tu2tu3t]

The variable x does not cause y at horizon 1 given Iyxz(t) if and only if ϕ12,j=0, for j=1,,k. However, in this situation, the causality may be indirect through z if ϕ13,j and ϕ32,j are not zero for some j. In this case, when x does not cause y at horizon 1 given Iyxz(t), the existence of an indirect causality through z implies that x causes y at horizon 2 given Iyxz(t). Giles (2002) proposed a sequential procedure that provides information on the horizon at which the causality, if any, arises. Here, we use this procedure. First we test the null hypothesis  

H0(1):ϕ12,1==ϕ12,k=0

If this hypothesis is rejected, then x1y|Iyxz(t): we call this ‘direct causality’. Otherwise, the following conditional null hypotheses are tested  

H0(2):ϕ32,1==ϕ32,k=0|ϕ12,1==ϕ12,k=0
 
H0(3):ϕ13,1==ϕ13,k=0|ϕ12,1==ϕ12,k=0

If H0(2) and H0(3) are rejected, then x1z|Iyxz(t), z1y|Iyxz(t) and x1y|Iyxz(t). Thus, we conclude that x2y|Iyxz(t). In particular, x2y|Iyxz(t) when we accept one or both of the hypotheses H0(2) and H0(3).

Even in this case, in order to test the null hypotheses H0(1), H0(2) and H0(3), we use the out-of-sample approach. The hypothesis H0(1) is tested using the unrestricted model (3) and the following restricted model  

(4)
[ytxtzt]=[a1a2a3]+j=1k[θ11,j0θ13,jθ21,jθ22,jθ23,jθ31,jθ32,jθ33,j][ytjxtjztj]+[v1tv2tv3t]

Estimating the parameters of these two models by means of OLS, we can obtain the P one-step-ahead forecast errors of the first equation of the two models and the mean square prediction errors MSPE(u^1t) and MSPE(v^1t), respectively. If MSPE(v^1t)>MSPE(u^1t) and this difference is statistical significant, then the null hypothesis H0(1) is rejected and we conclude that x1y|Iyxz(t).

Otherwise, indirect causality must be investigated. In this case, we examine indirect Granger causality by means of the chain  

x1z|Iyxz(t)andz1y|Iyxz(t)

Thus, we consider the VAR model (3) imposing ϕ12,j=0, for j=1,,k 

(5)
[ytxtzt]=[c1c2c3]+j=1k[ϕ11,j0ϕ13,jϕ21,jϕ22,jϕ23,jϕ31,jϕ32,jϕ33,j][ytjxtjztj]+[u1tu2tu3t]

The null hypothesis H0(2) can be tested considering the previous model (5) and the following model  

(6)
[ytxtzt]=[a1a2a3]+j=1k[θ11,j0θ13,jθ21,jθ22,jθ23,jθ31,j0θ33,j][ytjxtjztj]+[v1tv2tv3t]

Estimating the model parameters via OLS, we obtain the P one-step-ahead forecast errors  

u^3t=ztc^3j=1kϕ^31,jxtjj=1kϕ^32,jytjj=1kϕ^33,jztj
 
v^3t=zta^3j=1kθ^31,jxtjj=1kθ^33,jztj

We calculate the mean square prediction errors MSPE(u^3t) and MSPE(v^3t) and test the null hypothesis H0(2) employing the MSE-t and MSE-REG tests. The critical values are always calculated by means of previous bootstrap method. Finally, the null hypothesis H0(3) can be tested using the same procedure described for H0(2).

It is important to underline that, assuming a 100α1% significance level for a test of H0(1) and a 100α2% significance level for a test of H0(2) and H0(3), the overall size is bounded by α=α1+2α2.

The model order k, with k{1,2,3,4}, of the unrestricted VAR model in equation (3) is selected, considering the training set, by means of AIC.

Finally, it is worthwhile to stress that in our study the forecasts are calculated by means of the fixed scheme. Under this scheme, each one-step-ahead forecast is generated using parameters that are estimated only once using data from 1 to R.

4. Results

In this framework, we analyse which pattern of variability (among those ones considered here) are able to have a lagged influence on global temperature T, which is our y variable, by means of Granger causality. Following the same approach used in previous papers (Attanasio et al., 2012; Pasini et al., 2012), the out-of-sample tests are performed on five test sets which span the following periods: 1941–2011, 1951–2011, 1961–2011, 1971–2011, 1981–2011. The bivariate results obtained by our analysis are very clear: if we take PDO as x variable in equation (1), the null hypothesis of Granger non-causality on T is often rejected (with only two exceptions) at a 5% significance level (Table 1). Otherwise, there is a clear general evidence of Granger causality from AMO or SOI to global temperature. In fact, the null hypothesis of non-causality is always rejected at 5% level (Tables 2 and 3).

Table 1.

Results of bivariate Granger non-causality tests from PDO to T

Test set Model order P-value 
[1941–2011] 0.0071a 
[1951–2011] 0.0093a 
[1961–2011] 0.1383 
[1971–2011] 0.3560 
[1981–2011] nc 
Test set Model order P-value 
[1941–2011] 0.0071a 
[1951–2011] 0.0093a 
[1961–2011] 0.1383 
[1971–2011] 0.3560 
[1981–2011] nc 

nc indicates that the MSPE of the unrestricted model (1) is bigger than the MSPE of the restricted model (2), so that the MSE-t test is not calculated.

aIndicates that the null hypothesis is rejected at 5% significance level.

Table 2.

Results of bivariate Granger non-causality tests from AMO to T

Test set Model order P-value 
[1941–2011] 0.0001a 
[1951–2011] 0.0020a 
[1961–2011] 0.0017a 
[1971–2011] 0.0010a 
[1981–2011] 0.0043a 
Test set Model order P-value 
[1941–2011] 0.0001a 
[1951–2011] 0.0020a 
[1961–2011] 0.0017a 
[1971–2011] 0.0010a 
[1981–2011] 0.0043a 

aIndicates that the null hypothesis is rejected at 5% significance level.

Table 3.

Results of bivariate Granger non-causality tests from SOI to T

Test set Model order P-value 
[1941–2011] 0.0019a 
[1951–2011] 0.0015a 
[1961–2011] 0.0010a 
[1971–2011] 0.0005a 
[1981–2011] 0.0010a 
Test set Model order P-value 
[1941–2011] 0.0019a 
[1951–2011] 0.0015a 
[1961–2011] 0.0010a 
[1971–2011] 0.0005a 
[1981–2011] 0.0010a 

aIndicates that the null hypothesis is rejected at 5% significance level.

In the trivariate case, we consider z=GHGRF in order to study the robustness of the bivariate results. Previous studies (Attanasio et al. 2012; Pasini et al. 2013; Stern and Kaufmann 2014) have shown that GHGRF Granger causes global temperature.

Here, we test both direct and indirect causality of circulation patterns on T. In the first case, we test the MSPEs coming from the unrestricted and restricted models (3) and (4). If no direct causality is found, the possibility of an indirect causality through a causality chain is explored. The chains are formed by a first causal link between the oscillation pattern considered and GHGRF: a causal link between these variables is physically possible because of the influence of an oceanic release of GHGs in the atmosphere. The second link is obviously between GHGRF and T. If these two causal links should be both verified in a statistical significant manner by means of Granger tests, then an indirect causality arises even if no direct causality exists between a circulation pattern and T.

When PDO is considered as x in equation (3), we never find direct or indirect Granger causality (Tables 4 and 5).

Table 4.

Results of direct Granger non-causality tests from PDO to T in the trivariate system given by PDO, T and GHGRF

Test set Model order P-value 
[1941–2011] 0.2564 
[1951–2011] 0.0987 
[1961–2011] nc 
[1971–2011] nc 
[1981–2011] nc 
Test set Model order P-value 
[1941–2011] 0.2564 
[1951–2011] 0.0987 
[1961–2011] nc 
[1971–2011] nc 
[1981–2011] nc 

nc indicates that the MSPE, for the equation of T, of the unrestricted model (3) is bigger than the MSPE of the restricted model (4), so that the MSE-t test is not calculated.

Table 5.

Results of indirect Granger non-causality tests from PDO to T via GHGRF, in the trivariate system given by PDO, T and GHGRF

Test set Model order PDO1GHGRF (P-value) GHGRF1T (P-value) 
[1941–2011] nc 0.0035a 
[1951–2011] 0.1460 0.0065a 
[1961–2011] nc 0.0070a 
[1971–2011] nc 0.0082a 
[1981–2011] 0.3577 0.0088a 
Test set Model order PDO1GHGRF (P-value) GHGRF1T (P-value) 
[1941–2011] nc 0.0035a 
[1951–2011] 0.1460 0.0065a 
[1961–2011] nc 0.0070a 
[1971–2011] nc 0.0082a 
[1981–2011] 0.3577 0.0088a 

nc indicates that the MSPE, for the equation of GHGRF, of the unrestricted model (5) is bigger than the MSPE of the restricted model (6), so that the MSE-t test is not calculated.

aIndicates that each null hypotheses is rejected at 2.5% significance level.

The other results are very impressive. In particular, the bivariate outcomes of SOI and AMO are not statistically robust. In fact, there is no direct Granger causality (with only one exception), at 5% significant level, from SOI to T in our test sets (Table 6). Even the causal chains are always interrupted (Table 7). However, it is worthwhile to note that in the fourth test set the empirical evidence that does not support the null hypothesis H0(1) is not too strong. In fact, the P-value is just slightly smaller than 0.05.

Table 6.

Results of direct Granger causality from SOI to T in the trivariate system given by SOI, T and GHGRF

Test set Model order P-value 
[1941–2011] 0.2429 
[1951–2011] 0.3208 
[1961–2011] 0.2191 
[1971–2011] 0.0428a 
[1981–2011] 0.1110 
Test set Model order P-value 
[1941–2011] 0.2429 
[1951–2011] 0.3208 
[1961–2011] 0.2191 
[1971–2011] 0.0428a 
[1981–2011] 0.1110 

aIndicates that the null hypothesis is rejected at 5% significance level.

Table 7.

Results of indirect Granger causality from SOI to T via GHGRF, in the trivariate system given by SOI, T and GHGRF

Test set Model order SOI1GHGRF (P-value) GHGRF1T (P-value) 
[1941–2011] 0.1651 0.0045a 
[1951–2011] nc 0.0071a 
[1961–2011] nc 0.0066a 
[1971–2011] – – 
[1981–2011] nc 0.0289 
Test set Model order SOI1GHGRF (P-value) GHGRF1T (P-value) 
[1941–2011] 0.1651 0.0045a 
[1951–2011] nc 0.0071a 
[1961–2011] nc 0.0066a 
[1971–2011] – – 
[1981–2011] nc 0.0289 

nc indicates that the MSPE, for the equation of GHGRF, of the unrestricted model (5) is bigger than the MSPE of the restricted model (6), so that the MSE-t test is not calculated.

aIndicates that each null hypotheses is rejected at 2.5% significance level.

When x=AMO, the null hypothesis of non-causality is never rejected and we do not find a direct causality link (Table 8). In all test sets, the causal chains are not completed (Table 9).

Table 8.

Results of direct Granger causality from AMO to T in the trivariate system given by AMO, T and GHGRF

Test set Model order P-value 
[1941–2011] 0.1720 
[1951–2011] 0.2400 
[1961–2011] nc 
[1971–2011] 0.1344 
[1981–2011] 0.3461 
Test set Model order P-value 
[1941–2011] 0.1720 
[1951–2011] 0.2400 
[1961–2011] nc 
[1971–2011] 0.1344 
[1981–2011] 0.3461 

nc indicates that the MSE, for the equation of T, of the unrestricted model (3) is bigger than the MSE of the restricted model (4), so the MSE-t test is not calculated.

Table 9.

Results of indirect Granger causality from AMO to T via GHGRF, in the trivariate system given by AMO, T and GHGRF

Test set Model order AMO1GHGRF (P-value) GHGRF1T (P-value) 
[1941–2011] nc 0.0047 
[1951–2011] nc 0.0081a 
[1961–2011] nc 0.0075a 
[1971–2011] 0.9622 0.0097a 
[1981–2011] 0.0754 0.0309 
Test set Model order AMO1GHGRF (P-value) GHGRF1T (P-value) 
[1941–2011] nc 0.0047 
[1951–2011] nc 0.0081a 
[1961–2011] nc 0.0075a 
[1971–2011] 0.9622 0.0097a 
[1981–2011] 0.0754 0.0309 

nc indicates that the MSPE, for the equation of GHGRF, of the unrestricted model (5) is bigger than the MSPE of the restricted model (6), so that the MSE-t test is not calculated.

aIndicates that each null hypotheses is rejected at 2.5% significance level.

In summary, we find direct Granger causality just for SOI on the fourth test set. Thus, the apparent causality from AMO and SOI to T, which we found in a bivariate framework, generally disappears when the most influent context causal variable—GHGRF—is inserted in the information set, even considering a possible causal chain through this variable. Nevertheless, weak signals of natural variability influence can be still recognized in single time intervals, namely the weaker influence of SOI in the more recent decades.

Finally, we should point out:

  • the P-values of the MSE-REG test are very similar to those of the MSE-t test;

  • using BIC (Bayesian information criterion) to select the VAR orders, the results of very low causality found employing AIC are further strengthened.

These results are available from the authors upon request.

5. Conclusions

In this paper, we have analysed the causal role of the climate natural variability, here exemplified by three circulation patterns, on the behaviour of global temperature. In particular, once accepted the idea that a synchronous relationship exists between these patterns and global temperature, we have investigated the presence of possible lagged influences.

After a first evidence of strong causality for AMO and SOI, this has revealed itself as a spurious causality due to omission of variables in the information set considered. Once completed this set with data about greenhouse gases, the causality between natural variability and global temperature disappeared almost completely, even in the framework of the original analysis performed here about the role of possible indirect links.

In general, a lagged causal link from the indices of natural variability considered here to global temperature is not evident in our analysis, if we exclude some cases of weak influences in specific periods.

References

Allan
RJ
Nicholls
N
Jones
PD
Butterworth
IJ
.
A further extension of the Tahiti-Darwin SOI, early SOI results and Darwin pressure
.
J Climate
 
1991
;
4
:
743
9
.
Ashley
R
Granger
CWJ
Schmalansee
R
.
Advertising and aggregate consumption: an analysis of causality
.
Econometrica
 
1980
;
48
:
1149
67
.
Attanasio
A
.
Testing for linear Granger causality from natural/anthropogenic forcings to global temperature anomalies
.
Theor Appl Climatol
 
2012
;
110
:
281
9
.
Attanasio
A
Pasini
A
Triacca
U
.
A contribution to attribution of recent global warming by out-of-sample Granger causality analysis
.
Atmos Sci Lett
 
2012
;
13
:
67
72
.
Bindoff
NL
Stott
PA
AchutaRao
KM
et al
.
Detection and attribution of climate change: from global to regional
. In:
Stocker
TF
Qin
D
Plattner
GK
et al.  (eds).
Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change
 .
Cambridge
:
Cambridge University Press
,
2013
, pp.
867
952
.
Comrie
AC
McCabe
GJ
.
Global air temperature variability independent of sea-surface temperature influences
.
Prog Phys Geog
 
2013
;
37
:
29
35
.
DelSole
T
Tippett
MK
Shukla
JA
.
A significant component of unforced multidecadal variability in the recent acceleration of global warming
.
J Climate
 
2011
;
24
:
909
26
.
Diebold
FX
Mariano
RS
.
Comparing predictive accuracy
.
J Bus Econ Stat
 
1995
;
13
:
253
65
.
Diks
C
Mudelsee
M
.
Redundancies in the Earth’s climatological time series
.
Phys Lett A
 
2000
;
275
:
407
14
.
Dufour
JM
Renault
E
.
Short run and long run causality in time series: theory
.
Econometrica
 
1998
;
66
:
1099
125
.
Elsner
JB
.
Granger causality and Atlantic hurricanes
.
Tellus
 
2007
;
59
:
476
85
.
Enfield
DB
Mestas-Nunez
AM
Trimble
PJ
.
The Atlantic multidecadal oscillation and its relation to rainfall and river flows in the continental US
.
Geophys Res Lett
 
2001
;
28
:
2077
80
.
Foster
G
Rahmstorf
S
.
Global temperature evolution 1979–2010
.
Environ Res Lett
 
2011
;
6
:
044022
.
Giles
J
.
Testing for two-step Granger noncausality in trivariate VAR models
. In:
Ullah
A
Wan
ALT
Chaturvedu
A
(eds).
Handbook of Applied Econometrics and Statistical Inference
 .
New York, NY
:
CRC Press
,
2002
, pp.
371
99
.
Granger
CWJ
.
Investigating causal relations by econometric methods and cross-spectral methods
.
Econometrica
 
1969
;
37
:
424
38
.
Granger
CWJ
Newbold
P.
Forecasting Economic Time Series
 .
New York, NY
:
Academic Press
,
1977
.
Hansen
J
Sato
M
Ruedy
R
et al
.
2007
.
Climate simulations for 1880–2003 with GISS model E
.
Clim Dynam
 
29
:
661
96
.
Hegerl
GC
Zwiers
FW
.
Use of models in detection and attribution of climate change
.
WIREs Clim Change
 
2011
;
2
:
570
91
.
Hoerling
M
Kumar
A
Eischeid
J
Jha
B
.
What is causing the variability in global mean land temperature?
Geophys Res Lett
 
2008
;
35
:
L23712
.
Hurrell
JW
.
Influence of variations in extratropical wintertime teleconnections on northern hemisphere temperature
.
Geophys Res Lett
 
1996
;
23
:
665
8
.
Kaufmann
RK
Stern
DI
.
Evidence for human influence on climate from hemispheric temperature relations
.
Nature
 
1997
;
388
:
39
44
.
Kodra
E
Chatterjee
S
Ganguly
AR
.
Exploring Granger causality between global average observed time series of carbon dioxide and temperature
.
Theor Appl Climatol
 
2011
;
104
:
325
35
.
Können
GP
Jones
PD
Kaltofen
MH
Allan
RJ
.
Pre-1866 extensions of the southern oscillation index using early Indonesian and Tahitian meteorological readings
.
J Climate
 
1998
;
11
:
2325–39
.
Lean
JL
Rind
DH
.
How natural and anthropogenic influences alter global and regional surface temperatures: 1889 to 2006
.
Geophys Res Lett
 
2008
;
35
:
L18701
.
Li
J
Sun
C
Jin
F-F
.
NAO implicated as a predictor of Northern Hemisphere mean temperature multidecadal variability
.
Geophys Res Lett
 
2013
;
40
:
5497
502
.
McCracken
MW
.
Asymptotics for out-of-sample tests of Granger causality
.
J Econometrics
 
2007
;
140
:
719
52
.
Morice
CP
Kennedy
JJ
Rayner
NA
Jones
PD
.
Quantifying uncertainties in global and regional temperature change using an ensemble of observational estimates: the HadCRUT4 data set
.
J Geophys Res
 
2012
;
117
:
D08101
.
Mosedale
TJ
Stephenson
DB
Collins
M
Mills
TC
.
Granger causality of coupled climate processes: ocean feedback on the North Atlantic Oscillation
.
J Climate
 
2006
;
19
:
1182
94
.
Pasini
A
Triacca
U
Attanasio
A
.
Evidence of recent causal decoupling between solar radiation and global temperature
.
Environ Res Lett
 
2012
;
7
:
034020
.
Pasini
A
Triacca
U
Attanasio
A
.
On the role of sulfates in recent global warming: a Granger causality analysis
.
Int J Climatol
 
2015
;
35
:
3701
6
.
Ramaswamy
V
Boucher
O
Haigh
J
, et al
.
Radiative forcing of climate change
. In: Houghton JT, Ding Y, Griggs DJ, et al. (eds).
Climate Change 2001: The Scientific Basis
 .
Cambridge, UK
:
Cambridge University Press
,
2001
, pp.
349
416
.
Ropelewski
CF
Jones
PD
.
An extension of the Tahiti-Darwin southern oscillation index
.
Mon Weather Rev
 
1987
;
115
:
2161
5
.
Saenko
OA
Schmittner
A
Weaver
AJ
.
The Atlantic-Pacific seesaw
.
J Climate
 
2004
;
17
:
2033
8
.
Smith
TM
Reynolds
RW
.
Improved extended reconstruction of SST (1854–1997)
.
J Climate
 
2004
;
17
:
2466
77
.
Stern
DI
Kaufmann
RK
.
Anthropogenic and natural causes of climate change
.
Clim Change
 
2014
;
122
:
257
69
.
Trenberth
KE
Caron
JM
Stepaniak
DP
Worley
S
.
Evolution of El Niño-Southern Oscillation and atmospheric surface temperatures
.
J Geophys Res
 
2002
;
107
:
D8
.
Triacca
U
.
Is Granger causality analysis appropriate to investigate the relationship between atmospheric concentration of carbon dioxide and global surface air temperature?
Theor Appl Climatol
 
2005
;
81
:
133
5
.
Triacca
U
Attanasio
A
Pasini
A
.
Anthropogenic global warming hypothesis: testing its robustness by Granger causality analysis
.
Environmetrics
 
2013
;
24
:
260
8
.
Wang
YM
Lean
JL
Sheeley
NR
Jr
.
Modeling the Sun’s magnetic field and irradiance since 1713
.
Astrophys J
 
2005
;
625
:
522
38
.
Wang
X
Wang
C
Zhou
W
Wang
D
Song
J
.
2011
.
Teleconnected influence of North Atlantic sea surface temperature on El Niño onset
.
Clim Dynam
 
37
:
663
76
.
West
KD
.
1996
.
Asymptotic inference about predictive ability
.
Econometrica
 
64
:
1067
84
.
Wiener
N
.
The theory of prediction
. In:
Beckenback
EF
(ed).
The Theory of Prediction
 .
New York, NY
:
McGraw-Hill
,
1956
.

Author notes

*Correspondence Alessandro Attanasio, Department of Computer Engineering, Computer Science and Mathematics, University of L’Aquila, L’Aquila 67100, Italy; E-mail: alessandro_attanasio@yahoo.it
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact journals.permissions@oup.com