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. References Alanko-Huotari K, Mursula K, Usoskin IG, Kovaltsov GA. 2006. Global heliospheric parameters and cosmic-ray modulation: an empirical relation for the last decades. Solar Physics 238: 391–404. Ashley R, Granger CWJ, Schmalansee R. 1980. Advertising and aggregate consumption: an analysis of causality. Econometrica 48: 1149–1167. Attanasio A, Triacca U. 2011. Detecting human influence on climate using neural networks based Granger causality. Theoretical and Applied Climatology 103: 103–107. Brohan P, Kennedy JJ, Harris I, Tett SFB, Jones PD. 2006. Uncertainty estimates in regional and global observed temperature changes: a new dataset from 1850. Journal of Geophysical Research 111: D12106. DOI: 10.1029/2005JD006548. Clark T. 2004. Can out-of-sample forecast comparisons help prevent overfitting? Journal of Forecasting 23: 115–139. Crabtree RH. 1995. Aspects of methane chemistry. Chemical Review 95: 987–1007. Diebold FX, Mariano RS. 1995. Comparing predictive accuracy. Journal of Business and Economical Statistics 13: 253–265. Diks C, Mudelsee M. 2000. Redundancies in the Earth’s climatological time series. Physics Letters A 275: 407–414. Elsner JB. 2006. Evidence in support of the climate change – Atlantic hurricane hypothesis. Geophysical Research Letters 33: L16705. DOI: 10.1029/2006GL026869. Copyright 2011 Royal Meteorological Society 71 Elsner JB. 2007. Granger causality and Atlantic hurricanes. Tellus 59A: 476–485. Gelper S, Croux C. 2007. Multivariate out-of-sample tests for Granger causality. Computational Statistics and Data Analysis 51: 3319–3329. Granger CWJ. 1969. Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37: 424–438. Granger CWJ, Newbold P. 1997. Forecasting Economic Time Series. Academic Press: New York, NY. Hansen J, Sato M, Ruedy R, Kharecha P, Lacis A, Miller R, Nazarenko L, Lo K, Schmidt GA, Russell G, Aleinov I, Bauer S, Baum E, Cairns B, Canuto V, Chandler M, Cheng Y, Cohen A, Del Genio A, Faluvegi G, Fleming E, Friend A, Hall T, Jackman C, Jonas J, Kelley M, Kiang NY, Koch D, Labow G, Lerner J, Menon S, Novakov T, Oinas V, Perlwitz Ja, Perlwitz Ju, Rind D, Romanou A, Schmunk R, Shindell D, Stone P, Sun S, Streets D, Tausnev N, Thresher D, Unger N, Yao M, Zhang S. 2007. Climate simulations for 1880–2003 with GISS modelE. Climate Dynamics 29: 661–696. Hegerl GC, Zwiers FW, Braconnot P, Gillett NP, Luo Y, Marengo Orsini JA, Nicholls N, Penner JE, Stott PA. 2007. Understanding and attributing climate change. In Climate Change 2007: The Physical Science Basis, Solomon S, Qin D, Manning M, Chen Z, Marquis M, Averyt KB, Tignor M, Miller HL. (eds). Cambridge University Press: Cambridge, 663–745. Kaufmann RK, Stern DI. 1997. Evidence for human influence on climate from hemispheric temperature relations. Nature 388: 39–44. Kaufmann RK, Zhou L, Myneni RB, Tucker CJ, Slayback D, Shabanov NV, Pinzon J. 2003. The effect of vegetation on surface temperature: A statistical analysis of NDVI and climate data. Geophysical Research Letters 30: 2147. DOI: 10.1029/2003GL018251. Kaufmann RK, Seto KC, Schneider A, Liu Z, Zhou L, Wang W. 2007. Climate response to rapid urban growth: evidence of a humaninduced precipitation deficit. Journal of Climate 20: 2299–2306. Kodra E, Chatterjee S, Ganguly AR. 2011. Exploring Granger causality between global average observed time series of carbon dioxide and temperature. Theoretical and Applied Climatology 104: 325–335. Lean JL, Rind DH. 2008. How natural and anthropogenic influences alter global and regional surface temperatures: 1889 to 2006. Geophyical. Research Letters 35: L18701. DOI: 10.1029/ 2008GL034864. McCracken MW. 2007. Asymptotics for out-of-sample tests of Granger causality. Journal of Econometrics 140: 719–752. Mohkov II, Smirnov DA. 2008. Diagnostics of cause-effect relation between solar activity and the Earth’s global surface temperature. Izvestya Atmospheric and Oceanic Physics 44(3): 263–272. Mohkov II, Smirnov DA, Nakonechny PI, Kozlenko SS, Seleznev EP, Kurths J. 2011. Alternating mutual influence of El Niño/Southern Oscillation and Indian monsoon. Geophysical Research Letters 38: L00F04. DOI: 10.1029/2010GL045932. Mosedale TJ, Stephenson DB, Collins M, Mills TC. 2006. Granger causality of coupled climate processes: ocean feedback on the North Atlantic Oscillation. Journal of Climate 19: 1182–1194. Park JY, Phillips PCB. 1989. Statistical inference in regressions with integrated processes: Part 2. Econometric Theory 5: 95–131. Pasini A, Lorè M, Ameli F. 2006. Neural network modelling for the analysis of forcings/temperatures relationships at different scales in the climate system. Ecological Modelling 191: 58–67. Ramaswamy V, Boucher O, Haigh J, Hauglustaine D, Haywood J, Myhre G, Nakajima T, Shi GY, Solomon S. 2001. Radiative forcing of climate change. In Climate Change 2001: The Scientific Basis, Houghton JT, Ding Y, Griggs DJ, Noguer M, van der Linden PJ, Dai X, Maskell K, Johnson CA. (eds). Cambridge University Press: Cambridge. 349–416. Reichel R, Thejll P, Lassen K. 2001. The cause-and-effect relationship of solar cycle length and the Northern Hemisphere air surface temperature. Journal of Geophysical Research 106(A8): 15635–15641. Sato M, Hansen JE, McCormick MP, Pollack J. 1993. Stratospheric aerosol optical depths (1850–1990). Journal of Geophysical Research 98: 22987–22994. Atmos. Sci. Let. 13: 67–72 (2012) 72 Sims CA, Stock JH, Watson MW. 1990. Inference in linear time series models with some unit roots. Econometrica 58: 113–144. Stock JH, Watson MW. 1989. Interpreting the evidence on moneyincome causality. Journal of Econometrics 40: 161–181. Sun L, Wang M. 1996. Global warming and global dioxide emission: an empirical study. Journal of Environmental Management 46: 327–343. Triacca U. 2001. On the use of Granger causality to investigate the human influence on climate. Theoretical and Applied Climatology 69: 137–138. Triacca U. 2005. Is Granger causality analysis appropriate to investigate the relationship between atmospheric concentration of carbon dioxide and global surface air temperature? Theoretical and Applied Climatology 81: 133–135. Usoskin IG, Mursula K, Solanki SK, Schüssler M, Kovaltsov GA. Copyright 2011 Royal Meteorological Society A. Attanasio, A. Pasini and U. Triacca 2002. A physical reconstruction of cosmic ray intensity since 1610. Journal of Geophysical Research 107(A11): 1374. DOI: 10.1029/2002JA009343. Usoskin IG, Alanko-Huotari K, Kovaltsov GA, Mursula K. 2005. Heliospheric modulation of cosmic rays: monthly reconstruction for 1951–2004. Journal of Geophysical Research 110: A12108. DOI: 10.1029/2005JA011250. Wang Y-M, Lean JL, Sheeley NR Jr. 2005. Modeling the Sun’s magnetic field and irradiance since 1713. Astrophysical Journal 625: 522–538. West KD. 1996. Asymptotic inference about predictive ability. Econometrica 64: 1067–1084. Yuval A, Hsieh WW. 2002. The impact of time-averaging on the detectability of nonlinear empirical relations. Quarterly Journal of the Royal Meteorological Society 128: 1609–1622. Atmos. Sci. Let. 13: 67–72 (2012)

1/--страниц