Granger causality and global warming : the causal links between oxygen, carbon and solar insolation over the past 1.8 million years

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Faculteit Economie en Bedrijfskunde, Amsterdam School of Economics Bachelorscriptie en afstudeerseminar Econometrie - Final version of thesis

Semester 2 - Blok: 5 & 6

Supervisors: H. Li and M. Hennequin

Granger Causality and Global Warming:

The Causal Links between Oxygen, Carbon and

Solar Insolation over the past 1.8 Million Years

David Kranenburg (10574697)

June 29

th

- 2016

Abstract

This thesis explores the linear and nonlinear Granger causality relationships between δ18_O,

δ13_{C and solar insolation over the past 1.8 million years. The extent to which nonlinearities}

play a role in the causality relationships among these variables and how the causal linkages have changed over time are investigated. In order to discover this, the linear Granger causality test and the nonparametric Diks-Panchenko test are used on the ODP site 1146 dataset. The dataset contains values of δ18_{O, δ}13_{C and solar insolation at 15}o_{N latitude and is split into}

three climatic periods (period II, Ia and Ib). For the purpose of research, the Granger tests

are performed on both the raw data and VAR-ltered residuals. The following conclusions are derived from the results of the research. A linear Granger causality relationship exists from δ18_{O to δ}13_{C in all three periods. The causal relationship from solar insolation to δ}18_O

is linear for the two ancient periods and nonlinear for the most recent period. Solar insolation is a linear Granger cause for δ13_{C in period I}

a only. The reverse of this linkage is also found

to be signicant for the same period, but that is most likely due to chance. The unexpected Granger causality linkage from δ18_{O to insolation is discovered in every period.}

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Statement of Originality

This document is written by Student David Kranenburg who declares to take full respon-sibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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1 Introduction

As temperature has constantly been increasing, new records in global temperature have been recorded recently. Pollution has aected most individuals throughout the world. Conse-quently, global warming is a major problem the contemporary world is faced with. Given its increase in importance over the past century, controversy has arisen surrounding the causes of climate change. Therefore, research has to be conducted in order to determine the rela-tionships between climatological variables, as an attempt to explain the causes of impending global warming.

The Granger causality test (Granger, 1969) is often used to research causal relation-ships between climatological variables. By using this test, one can determine which variable inuences the other. The classical linear Granger causality test relies on the assumption of a linear model (Granger, 1969). However, new methods have been developed to test for Granger causality in the case of nonlinear models.

The climate system is highly nonlinear, therefore a linear model for testing Granger causality between climatological variables seems inappropriate (Attanasio, Pasini & Triacca, 2012). Nonetheless, the aforementioned researchers used a linear model to test for Granger causality between climatological variables and the temperature. They motivated the use of a linear model by arguing that the annual averaging of climatological variables produces nearly linear relations between these variables. Diks and Mudelsee (2000) tested for nonlin-ear Granger causality between oxygen, carbon isotopes and solar insolation. Their resnonlin-earch concludes that the couplings between these variables are nonlinear and increase over time.

The purpose of this thesis is to test for the existence of both linear and nonlinear Granger causality relationships among three climatological variables, over the past 1.8 mil-lion years. The extent to which nonlinearities play a role in the causal relations among the climatological variables and how these relations have changed over time are to be investi-gated. The variables under investigation are oxygen, carbon and solar insolation at 15o_N

latitude. Oxygen isotopes are a proxy for the global ice volume and carbon isotopes are a proxy for deep water circulation. The data on these isotopes comes from Ocean Drilling Program (ODP) site 1146, a drilling project which studies sedimentation layers located on

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the seaoor of the northern South China Sea. This data provides the benthic oxygen and carbon isotope measurements for the past 1.8 million years. Solar insolation is considered to be the main driving force behind the amount of oxygen and carbon on earth. Values of the solar insolation at 15o_{N latitude over the past 1.8 million years are calculated by Berger and}

Loutre (1991). The dataset is split into two climatic periods (period II, between 1822-892 ka ago and period I, starting from 892 ka ago), to take the Mid-Pleistocene Climate Transition into account. After analysis of the age plot of the total dataset, it became clear that the sedimentation process in the northern South China Sea has been slowing down since 318 ka ago. Therefore, period I is divided into period Ia, between 892-318 ka ago, and period Ib,

starting from 318 ka ago.

The research of this thesis consists of a two-stage empirical framework aimed to explore the nature of the Granger causality among the climatological variables. Firstly, in the pre-lter stage, a bivariate vector autoregressive (VAR) model is constructed for each pair of the climatological variables. Linear Granger causality between the variables is then tested using a classical linear Granger test (Granger, 1969). Nonlinear Granger causality relationships are tested using the nonparametric Diks-Panchenko test (Diks & Panchenko, 2006). Secondly, in the lter stage, both the linear and nonlinear Granger tests are applied to the residuals of the bivariate VAR model. This stage ensures that the linear causal relationships are captured by the model and that any remaining causality is nonlinear in nature. Furthermore, the residuals of the complete trivariate VAR model are considered to take possible eects of the other variables into account. As before, both the linear and nonlinear Granger causality tests are applied on these residuals, in order to test each pair of variables for Granger causality.

The remainder of this thesis is organized as follows. Section 2 reviews the classi-cal linear Granger causality test and describes the Diks-Panchenko test for nonparametric Granger causality. Other methods to explore Granger causality on the climate system are mentioned afterwards. Moreover, Section 2 presents known Granger causality relationships between climatological variables and comments on the use of Granger causality tests on the climate system. Section 3 describes the dataset used for research, provides an analysis of the data and explains the empirical methodology. The results of the Granger causality tests are presented and analysed in Section 4. Section 5 gives a conclusion of the thesis.

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2 Granger causality testing on the climate system

Over the past twenty years, much research has been conducted in order to explore the cou-plings among climatological variables. The concept of Granger causality has shown its utility in the study of these couplings. Both linear and nonlinear Granger causality tests have al-ready been applied by others to several causality problems in the climate system. This thesis applies the linear Granger causality test and the Diks-Panchenko test for nonlinear Granger causality.

The concept of Granger causality is the rst to be explained in this section. Next, the conventional Granger test for linear causality is reviewed and known linear causality re-lationships in the climate system are analysed. The nonparametric Diks-Panchenko test is explained afterwards. Moreover, other methods to identify Granger causality are mentioned, together with the results of their application to the climate system. Then, some comments on the use of Granger causality tests on climatological variables are taken into consideration. Finally, a conclusion to this section is given.

2.1 The concept of Granger causality

Granger (1969) introduced the concept of Granger causality. This concept has proven to be useful in the detection of causal relationships among sets of two or more time series. The Granger causality technique has been applied in dierent elds of study, for example climate studies, even though it was rst developed for application in econometrics. The linear Granger test (Granger, 1969) is an important test which identies linear causality between time series. Recently, there has been an increasing interest in methods which detect nonlinear causality. The Diks-Panchenko test (Diks & Panchenko, 2006) is one of the methods for detection of nonlinear Granger causality relationships.

Granger (1969) dened causality between variables in terms of predictability. Granger causality is dened as follows: let {Xt} and {Yt} be two strictly stationary time series. Then,

intuitively, {Xt} is said to be a Granger cause of {Yt} if current and future values of Y are

better predicted using past values of both X and Y , rather than only using past values of Y. More formally, let IX,t and IY,t be the information sets of X and Y , consisting of past

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values of Xt and Yt respectively, up to and including time t. Let `∼' dene equivalence in

distribution. Then {Xt} is a Granger cause of {Yt} if, ∀k ≥ 1, k ∈ N :

(Yt+1, ..., Yt+k) | (IX,t, IY,t) (Yt+1, ..., Yt+k) | IY,t. (1)

This denition is generic and involves no model assumptions. In practice one often uses k=1; testing for Granger non-causality then amounts to comparing the one-step-forward conditional distribution of {Yt} with and without past and present values of {Xt}.

The null hypothesis of the Granger causality test is as follows:

H0 : X is not a Grange cause of Y . (2)

One should note that the null hypothesis of a Granger causality test always states the ab-sence of Granger causality from one variable to another. Therefore, a rejection of the null hypothesis (2) means that there is evidence of Granger causality from X to Y , which, in other words, means that a causal link from X to Y exists. However, it is possible for a causal link to be found between X and Y , while they are, in fact, uncoupled. This may happen when both X and Y are driven by a third variable, say Z (Diks & Mudelsee, 2000).

2.2 Linear Granger causality testing on the climate system

This subsection describes the linear Granger causality test. The results found through the application of linear Granger causality tests on the climate system are then analysed.

2.2.1 Test for linear Granger causality

The Granger test for linear causality (Granger, 1969) tests for bivariate linear causality. This test assumes stationary time series and involves the estimation of a linear reduced-form VAR model. Suppose {Xt} and {Yt} are stationary time series given by the bivariate VAR(L)

model:

Xt = c1+ A(L)Xt+ B(L)Yt+ X,t

Yt= c2+ C(L)Xt+ D(L)Yt+ Y,t, t = 1, 2, ..., N

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where c1 and c2 represent constants for each equation, L is the lag operator and the coecient

matrices A(L), B(L), C(L) and D(L) are polynomials in L with all roots outside the unit circle. The regression error terms X,t and Y,t are assumed to be white noise: mutually

independent and individually distributed (i.i.d.) with zero mean, constant variance and no correlation across time (particularly, no autocorrelation).

In order to test whether lagged values in Y have a signicant linear predictive power for the present value of X (i.e., whether Y is a linear Granger cause for X), a test of joint exclusion restrictions (F - or χ2_{-test) has to be applied. The null hypothesis that Y is not}

a linear Granger cause of X is rejected if the coecients in B(L) are jointly signicantly dierent from zero. Likewise, to test if X is a Granger cause for Y , one has to test whether the coecients in C(L) are jointly signicantly dierent from zero. When both tests reject the null hypothesis, then Granger causality exists in both directions (feedback) between the two series.

2.2.2 Linear Granger causality in climatological variables

The bivariate linear Granger causality test, explained in the previous paragraph, and variants of this test are applied earlier to explore causality in the climate system. This paragraph describes the methods used to explore linear Granger causality linkages and the results of their application in earlier studies to the climate system. The references in this paragraph are also used in Attanasio, Pasini and Triacca (2012) and Hennequin (2012).

Reichel, Thejll and Lassen (2001) nd that solar activity inuences the mean land air temperature of the Northern Hemisphere, as a result of the application of the bivariate linear Granger causality test (with the F -test of joint exclusion restrictions).

Mosedale et al. (2006) applied a bivariate linear Granger test (using a χ2_{-test of joint}

exclusion restrictions) to examine the causal links between sea surface temperatures (SST) and the North Atlantic Oscillation (NAO). They nd that SST is a linear Granger cause for the NAO.

Elsner (2006, 2007) used the bivariate linear Granger causality test (with the F -test of joint exclusion restrictions), to explore causality between the global mean near-surface air temperature (GT) and the Atlantic sea surface temperature (SST). He concludes that GT is

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a Granger cause for SST, but the reverse of this linkage is incorrect.

Kodra, Chatterjee and Ganguly (2011) applied a variant of the bivariate linear Granger causality test, namely the reverse cumulative Granger causality test (RCUMGC test), on CO2

and the globally averaged land surface temperature. Their research concludes that an increase in CO2 emission causes an increase in temperature. However, these researchers mention that

their data violated some necessary assumptions on which the RCUMGC test relies, which must be taken into account when interpreting their results.

Sun and Wang (1996) used the classic Granger causality partial F -test to investigate the causality between CO2 and global temperature. They conclude that CO2 Granger causes

the temperature, as do Kodra, Chatterjee and Ganguly (2011).

Kaufmann and Stern (1997) nd evidence that greenhouse gases inuence the climate, through the application of linear Granger causality tests. However, Triacca (2001) criti-cizes this conclusion. He brings forth other possible interpretations of this research, thereby achieving dierent results.

Triacca (2005) also re-examined the linear Granger causality between CO2 and global

mean temperature. He applied the methodology from Toda and Yamamoto (1995). This methodology is robust to (co-)integration properties of data. Triacca concludes that there is no detectable linear Granger causality from CO2 to the temperature and states that it is

in-appropriate to use a linear Granger test to explore a causal link between these two variables. The aforementioned studies show that linear Granger causality tests should be applied cautiously because spurious results may be obtained when underlying assumptions of the linear Granger test are violated (Hennequin, 2012).

2.3 Diks-Panchenko test for nonlinear Granger causality

The basis of the Diks-Panchenko test (Diks & Panchenko, 2006) is the same as the Hiemstra-Jones test (Hiemstra & Hiemstra-Jones, 1994). The aim of the nonparametric DP-test is to check whether past values of a variable inuence current and future values of another variable. In a nonparametric setting, it is impossible to condition on an innite number of past observations (i.e. Xt, Xt−1,... and Yt, Yt−1,...); therefore in practice, tests are restricted to nite orders in

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YlY

t = (Yt−lY+1,...,Yt), are dened. When past observations of X

lX

t do not include information

about future values, then, in the case of k=1, it follows from (1) and (2) that testing for Granger non-causality comes down to testing the following null hypothesis:

H0 : Yt+1| (XtlX, Y lY

t ) ∼ Yt+1 | YtlY. (4)

For stationary bivariate time series, this null hypothesis is a statement about the invariant distribution of the (lX + lY +1)-dimensional vector Wt = (XtlX,Y

lY

t ,Zt), with Zt = Yt+1.

We now drop the time index t to keep the notation compact. Furthermore, lX = lY =1 is

assumed. Thus, under the null hypothesis, the conditional distribution of Z given (X,Y ) = (x,y) is equal to that of Z given Y = y. Hence, the joint probability density function fX,Y,Z(x,y,z) and its marginals must satisfy the following relationship, which states that X

and Z are independent conditionally on Y = y, for each xed value of y: fX,Y,Z(x, y, z) fY(y) = fX,Y(x, y) fY(y) · fY,Z(y, z) fY(y) . (5)

Diks and Panchenko (2006) show that this reformulated null hypothesis indicates:

q ≡ E[fX,Y,Z(X, Y, Z)fY(Y ) − fX,Y(X, Y )fY,Z(Y, Z)] = 0. (6)

Let ˆfW(Wi)denote a local density estimator of a dW-variate random vector W at Wi dened

by: ˆ fW(Wi) = (2n)−dW n − 1 · X j,j6=i I_ijW (7) where IW

ij = I(|| Wi - Wj|| ≤ n), with I(·) the indicator function and n the bandwidth,

depending on the sample size n. Given this estimator, the test statistic is a scaled sample version of q in Equation 6: Tn(n) = n − 1 n(n − 2) · X i

( ˆfX,Y,Z(Xi, Yi, Zi) ˆfY(Yi) − ˆfX,Y(Xi, Yi) ˆfY,Z(Yi, Zi)) (8)

Suppose that, for lX = lY =1, n = Cn−β, (C ≥ 0, 1₄ ≤ β ≤1₃). Then Diks and Panchenko

(2006) prove that (8) satises √

n(Tn(n) − q) Sn

d

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where d

→ denotes convergence in distribution and Sn is an estimator of the asymptotic

vari-ance of Tn(·). The aforementioned researchers recommended the implementation of a

one-tailed version of this test, leading to the rejection of the null hypothesis if the left hand side of Equation 9 is too large, because it is usually more powerful than a two-tailed test. This recommendation is followed for all the DP-tests applied in this thesis.

2.4 Other methods to explore Granger causality

Apart from the linear Granger causality test and the Diks-Panchenko test, other methods have been used to examine the causal links among climatological variables. Kodra, Chatter-jee and Ganguly (2011) state that a nonparametric HJ-test for Granger causality (Hiemstra & Jones, 1994) is a potential method to research these causal relationships. However, this thesis applies an improved version of this HJ-test, namely the DP-test for Granger causality (Diks & Panchenko, 2006).

Another method to analyse the nonlinear causal links among variables is by applying neural networks. Attanasio and Triacca (2011) used this approach and nd that CO2Granger

causes the temperature.

Attanasio, Pasini and Triacca (2012) applied a linear Granger analysis, but their method diers from the methods described in paragraph 2.2.2, in the way that these re-searchers took annual averages of climatological data and used a linear out-of-sample ap-proach to test for Granger causality. They performed a bootstrap procedure, because their time series were not stationary. These researchers motivated that, even though the climate system is highly nonlinear, the use of a linear model is justied when one takes annual aver-ages of climatological variables into account, because the annual averaging produces nearly linear relations among these variables. Their analysis concludes that greenhouse gases are a linear Granger cause for the temperature while no linear Granger causality was found for natural forcings on the temperature.

Mokhov and Smirnov (2009) introduce the concept of long term Granger causality and research the causal links between CO2 and the temperature. The method used by these

re-searchers involves empirical modelling, which extends the Granger causality concept so that it deals with longer-term behaviour. They conclude that the rise in temperature during the

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last decades can only be explained if CO2 is present in the model.

Diks and Mudelsee (2000) used statistical techniques of mutual information and redun-dancy to examine the nonlinear coupling among δ18_{O, δ}13_{C and solar insolation. They nd}

that the relationships among these variables are nonlinear. The couplings vary over four dif-ferent climatological periods. Their test for nonlinear Granger causality concludes that δ18_O

Granger causes δ13_{C, but the reverse is inapplicable. Solar insolation is a Granger cause for}

δ13C in the rst period and a Granger cause for δ18O in the second and third period.

2.5 Comments on the use of Granger causality tests on the climate

system

Although Granger causality tests have been widely used to explore couplings in the climate system, one has to be aware of the several problems which can arise when applying them.

Firstly, tests for Granger causality depend on which additional variables are included or excluded from the test equation. Omitting an important causal variable from the test equation leads to an omitted variable bias. This bias can lead to false conclusions about Granger causality (Lütkepohl, 1982). Particularly, assuming that one variable is a Granger cause for another variable only implies a real causal relationship if other possible causes are accounted for (Granger, 1988). This thesis explores the causal linkages among oxygen, carbon and solar insolation. Besides the aforementioned three, other variables might have inuenced the climate system in the northern South China Sea for the past 1.8 million years. An example of an important missing climatological variable is the dust ux (Tian, Wang & Cheng, 2004; Hennequin, 2012). Therefore, an omitted variable bias is likely to occur when testing for Granger causality in this thesis, but the extent to which this bias inuences the results and conclusions in this thesis is unclear.

Secondly, there are some shortcomings of using Granger causality tests on the climate system. Granger causality tests only detect directional couplings and only quantify their short-term eects (Mokhov & Smirnov, 2009). It is often more important to learn the long-term behaviour of a process. Moreover, a Granger causality test does not quantify the strength or importance of causal inuences. However, these long-term causal inuences or

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the strength of causal inuences are not part of the topic of research in this thesis.

2.6 Conclusion

Dierent methods have been used so as to study the causal relationships among climatolog-ical variables. Still, some uncertainty exists in regard to the exact causal relationships and whether these relationships are linear or nonlinear. Consequently, more research has to be conducted on these relationships. The analysis of the ODP 1146 dataset might help to give insight into these issues. Some researchers found a linear causal relationship between clima-tological variables, so a linear causal link between the three climaclima-tological variables under investigation might exist. However, Diks and Mudelsee (2000) found that δ18_{O is a nonlinear}

Granger cause for δ13_{C, whereas the reverse is inapplicable. By applying the nonparametric}

DP-test to the dataset, I expect this result to be found. Solar insolation is expected to be a main driving force behind δ18_{O and δ}13_{C, which is in line with results of Reichel, Thejll and}

Lassen (2001) and Diks and Mudelsee (2000).

The problem of an omitted variable bias is likely to occur when applying the Granger causality tests, since besides the three variables under investigation, other variables, for ex-ample the dust ux, might impact the climate system on the northern South China Sea. However, it is not clear to what extent this bias inuences the Granger causality test results and conclusions from research. Another limitation to the use of Granger causality tests on the climate system is that it does not take the long-term eects and strength of causal inu-ences into account.

This section explained the concept of Granger causality and described the dierent Granger causality tests that can be applied to the climate system. Moreover, some comments on the use of Granger causality tests on the climate system were taken into consideration. The next section describes the data used for this thesis, provides a data analysis and explains the empirical methodology used for research.

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3 Data and empirical methodology

This section rst describes the data used for research and explains the relationships between climatological variables and the climate. A preliminary data analysis is then provided in order to gain insight into the characteristics of the data. Finally, the empirical methodology for research is explained.

3.1 Data

The data under consideration consists of benthic oxygen (δ18O) and carbon (δ13C) isotope

records from Ocean Drilling Program (ODP) site 1146, located in the northern South China Sea (Clemens & Prell, 2003). The records cover the past 1800 ka (1ka = 1000 years). Fur-thermore, a time series of solar insolation at 15o_{N latitude (which nearly matches the latitude}

of the northern South China Sea: 19oN latitude) over the past 1800 ka will be included as

a third variable. The solar insolation values at 15o_{N latitude are calculated by Berger and}

Loutre (1991). Linear interpolation has been applied to these insolation values in order to match them with the times of the observations from ODP site 1146. This subsection explains the three variables and describes their relationships to the climate.

Solar insolation is an important leading variable in the context of the climate system (see Section 2). It is a measure of the amount of solar radiation energy received by the Earth, on a given latitude. Insolation depends on several astronomical parameters of the Earth (Berger and Loutre, 1991), such as the Earth's orbit (the ecliptic), its obliquity (the tilt of the equator on the ecliptic), its eccentricity (a measure of the shape of the Earth`s orbit around the sun) and its precession (the gradual shift in orientation of the Earth's axis of rotation). The aforementioned parameters are called Milankovitch parameters and they vary cyclically over periods of 20-100 ka. According to the Milankovitch theory, these variations in the Earth's orbital elements cause the insolation, and thus the Earth's climate, to change. The delta notation in δ18_{O and δ}13_{C represents the relative deviation of the isotope}

ratio from a reference standard (VPDB), for example:

δ18O = ( 18_O 16_O)sample− ( 18_O 16_O)V P DB (1816O_O)V P DB (10)

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and analogously for δ13_{C, but then with}13_{C and} 12_{C. The δ}18_{O and δ}13_{C isotope values are}

measured on benthic foraminifera. Foraminifera can preserve their isotope ratio for millions of years and are widely used in the study of climate change (Pearson, 2012).

The δ18_{O values are a proxy of global ice volume (Shackleton, 1987). A positive}

inuence from δ18_{O to temperature has been found by Yao et al. (1996). The δ}13_{C values}

are a proxy for deep and bottom water circulation (Mackensen & Bickert, 1999). Minimal values in δ13_{C are a sign of starting deglaciation (Spero & Lea, 2002). A negative relation}

between δ13_{C and temperature is found by Skrzypek et al. (2007).}

3.2 Data analysis

The data analysis begins by presenting and discussing the graphs of time series of the vari-ables. Then, the time intervals problem is discussed and the descriptive statistics of the variables are shown. Lastly, a check for stationary time series is conducted.

3.2.1 Analysis of the graphs of time series

The ODP 1146 data consists of 1045 observations of benthic δ18O and δ13C over the past

1.8 million years, in the northern South China Sea. The timespan of this data is part of a climatic period called the Pleistocene. Some major climate changes occurred during the Pleistocene, which are also reected in the ODP 1146 data.

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From circa 1.8 to 1.6 million years ago, one can observe an increasing trend in the δ13_C

time series. However, starting from 1.6 million years ago, there seems to be a decreasing trend in this time series. The explanation for this change is that 1.6 million years ago a reorganization of the oceanic carbon reservoir occurred, which is associated with the restruc-turing of the Southern and tropical Pacic Ocean (Wang, Li & Tian, 2014). This decreasing trend in δ13_{C changes into an increasing trend at around 900 ka. The time series of δ}18_O

also show an increase after this time. Around 900 ka, the Mid-Pleistocene Climate Transi-tion (MPT) occurred, a relatively abrupt increase in global ice volume (Mudelsee & Schulz, 1997). The MPT led to late Pleistocene ice ages, with large glacial-interglacial amplitudes. The largest amplitude changes in δ18_{O start at circa 400 ka. At this time the Mid-Brunhes}

Event (MBE) occurred (Wang, Li & Tian, 2014). The MBE signied a further development of the glacial-interglacial contrast in climate.

The MPT is considered to be the most inuential from the three aforementioned events. Having taken this event into account, the dataset (solar insolation, δ18_{O and δ}13_{C) is divided}

into two climatic periods for the remainder of this thesis. As per Diks and Mudelsee (2000), the dataset is split at 892 ka. The ancient period (period II) is between 1822.7 to 892.64 ka ago and the more recent period (period I) starts from 891.82 ka ago. Period II consists of 578 observations and period I consists of 467 observations.

3.2.2 Inequality in time intervals

When inspecting the ODP 1146 dataset, one notices that the time intervals between obser-vations are not of equal length. The reason for this is that the obserobser-vations are measured on sediment cores and sedimentation is not a constant process. This inequality in time intervals might lead to a bias in estimates and can inuence the results obtained from the empirical research. Therefore it is important to analyse this problem. In order to do this, an age plot of the total dataset and a histogram of the time intervals are presented and discussed.

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Figure 2: Age plot of the total dataset

In gure 2 one can observe roughly "constant" time intervals for the rst 952 observations (1822.70-318.87 ka ago). But, at 318.87 ka, we see a bend in the age plot. After 318.87 ka ago, the time intervals between observations increase. This indicates that the sedimentation process slowed down after 318.87 ka ago. This change in sedimentation process might be related to the MBE. If ignored, it might lead to incorrect ndings in the empirical research. As an attempt to overcome this problem, period I is divided into two periods: period Ia, between 891.82 to 318.87 ka ago and period Ib, starting from 318.87 ka

ago. Period Ia consists of 374 observations and period Ib of 93 observations.

Figure 3: Histogram of the time intervals

Next, a histogram of the time intervals of the total dataset (including all periods) is discussed. Figure 3 shows that most of the time intervals are between 0 and 4 ka, which for

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the most part corresponds to the time intervals in period II and Ia. However, there are some

larger time intervals, in a range from 4 to 10.9 ka. These are almost all part of period Ib.

Consequently, this histogram supports the division of the dataset. The mean time interval is 1.738 ka, with a standard deviation of 1.065.

In conclusion, the analysis of the time intervals made it clear that the dataset can be divided into two parts with roughly "constant" time intervals (an ancient part, consisting of observations from before 318.87 ka ago (period II and Ia), and a more recent part, consisting

of observations starting from 318.87 ka ago (period Ib)). As an attempt to overcome the

problem of inequality in time intervals, period I is divided into period Iaand Ib. Therefore, in

order to take both the MPT and the time intervals problem into account, the dataset (solar insolation, δ18_{O and δ}13_{C) is divided into three periods: period II (1822.70-892 ka ago, 578}

observations), period Ia (892-318.87 ka ago, 374 observations) and period Ib (starting from

318.87 ka ago, 93 observations).

3.2.3 Descriptive statistics

The descriptive statistics for the three periods are shown in Table 1, 2 and 3. For δ18_{O, the}

mean increases from period II to period I (period Ia and Ib almost have the same mean in

δ18O). Its variance also increases over the periods. These results are in line with the climatic events mentioned earlier. For δ13C we see a decrease in mean from period II to I

a, and an

increase from period Ia to Ib. Its variance increases from period II to Ia and decreases from

period Ia to Ib. The solar insolation has an almost steady mean over the periods, but its

variance largely decreases from period II to Ia and largely increases from period Ia to Ib.

No clear upward or downward pattern is observed in the skewness and kurtosis of the three variables. The correlation matrices show that all correlations between the variables (except the correlations between insolation and δ13C) are signicant at the 5% level. The correlations

become stronger over time (with the exception of the correlation between δ13_{C and δ}18_{O from}

period Ia to Ib), which suggests that the couplings between the variables might have become

stronger over time, as concluded by Diks and Mudelsee (2000). However, correlation is not the same as and does not imply causation, so these results should be interpreted cautiously. A causality analysis is of more utility in the investigation of relationships among the variables.

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Table 1: Descriptive statistics period II Period II δ13C δ18O Insolation Mean -1.017 3.837 456.028 Variance 0.082 0.089 465.516 Skewness -0.271 -0.303 0.150 Kurtosis 2.880 2.467 2.514 Correlation matrix δ13C 1.000 δ18O -0.574** 1.000 Insolation 0.002 -0.183** 1.000

`**' denotes statistically signicant correlation at the 5% level

Table 2: Descriptive statistics period Ia

Period Ia δ13C δ18O Insolation Mean -1.268 4.105 455.921 Variance 0.106 0.155 277.589 Skewness 0.288 -0.436 0.172 Kurtosis 2.467 2.897 2.514 Correlation matrix δ13C 1.000 δ18O -0.685** 1.000 Insolation 0.028 -0.239** 1.000

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Table 3: Descriptive statistics period Ib Period Ib δ13C δ18O Insolation Mean -1.115 4.088 457.336 Variance 0.088 0.171 414.168 Skewness 0.118 -0.473 0.019 Kurtosis 2.449 2.554 2.262 Correlation matrix δ13C 1.000 δ18O -0.454** 1.000 Insolation 0.140 -0.397** 1.000

`**' denotes statistically signicant correlation at the 5% level

3.2.4 Check for stationary time series

It is of importance to research whether the time series of δ18_{O, δ}13_{C and solar insolation are}

stationary, since stationarity is required for the application of VAR models and the Granger causality tests in Section 4. To research this, an Augmented Dickey-Fuller (ADF) test is applied on every variable, for each period. The null hypothesis of an ADF test states non-stationarity.

For each period, a trend is observed in the graphic of δ13C, therefore a trend term

is included in the test equation of this variable. The graphics of δ18_{O and solar insolation}

variables did not show a trend, so no trend term is included for these two variables. Every variable includes a constant in the test equation. The lag lengths are selected by the SIC.

After application of the ADF tests on the δ13C time series, it becomes clear that, for

each period, the trend term is insignicant (or close to zero when signicant) in the test equation of δ13C. Thus, the trend term will be excluded for every variable (insolation, δ18O

and δ13_{C) in the Granger causality tests in Section 4.}

The p-values of the ADF tests are presented in Table 4. All the ADF tests rejected the null hypothesis of a unit root (non-stationarity) at 1% level, except for δ18_{O in period}

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hypothesis of non-stationarity still gets rejected at 5% signicance level. Therefore, the time series are all assumed to be stationary.

Table 4: p-values of the ADF tests

Period δ13_C _δ18_O _Insolation

II 0.000*** 0.000*** 0.000*** Ia 0.009*** 0.000*** 0.000***

Ib 0.000*** 0.012** 0.000***

`**', `***' denote statistically signicant stationarity at the 5%, 1% level

3.3 Empirical methodology

The research consists of a two-stage empirical framework, repeated for each period. This two-stage framework not only makes it possible to nd linear and nonlinear Granger causal-ity among pairs of variables before VAR-ltering, but also after VAR-ltering and when the eects of all variables are accounted for. By applying this method, a broader insight into the causality among the climatological variables is obtained. A similar method was applied earlier to explore causality in other elds, for example to explore the causal linkages among stock markets (De Gooijer & Sivarajasingham, 2008), exchange rates (Bekiros & Diks, 2008a) and oil prices (Bekiros & Diks, 2008b).

Firstly, in the pre-lter stage, the linear and nonlinear causal linkages are explored on the raw time series of the three variables. To do this, a bivariate VAR model is constructed for each pair of variables. The Schwarz information criterion (SIC) is used to determine the amount of lags that should be included for every estimated VAR model throughout this thesis. The VAR models include a constant, but no trend term. In order to explore linear Granger causality among pairs of variables, a linear Granger test is applied. The Diks-Panchenko test is then applied in order to test for nonlinear causality linkages among the variables.

Secondly, during the lter stage, the data is ltered by both bivariate and trivariate VAR models. The residual series of these models are investigated for linear and nonlinear Granger causality. In order to verify that any remaining causality is nonlinear in nature, a Diks-Panchenko test is applied on each pair of residual series. Lastly, a linear Granger test is applied, using a VAR model specication for the residual series so as to check if linear

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causality linkages after the VAR-ltering still exist.

This section explained the dataset used for research, provided a preliminary data analy-sis and described the empirical methodology for research. The following section analyses the results of the Granger causality tests on the dataset.

4 Results of the Granger causality tests

This section discusses the results of the Granger causality tests on the ODP 1146 dataset. The results are presented in Table 5, 6 and 7. The outcomes of the linear Granger test and the Diks-Panchenko test on the raw data will be analysed rst. Then, the results of the tests on the bivariate and trivariate VAR-ltered residuals will be discussed. This section ends with a comparison and conclusion of the outcomes.

4.1 Granger causality tests on raw data

In order to apply a linear Granger test on the dataset, a bivariate VAR model was constructed for each pair of variables, for each period. The SIC selected the lag lengths of the variables: for every pair of variables the lag length in each period was two, except for δ13C-δ18O in

period II, which had a lag length of three. Every model included a constant.

For the Diks-Panchenko test, the time series were rescaled to zero mean and unit variance. The lag lengths were set at lX = lY =1 and the bandwidth nwas set at 1, which is

within the common range (0.5-1.5) used in practice (Diks & Panchenko, 2006). The results of the causality tests for period II, Iaand Ib are shown in Table 5, 6 and 7 respectively. When

interpreting the results, one should be aware of the fact that a signicant result might be a coincidence. For example, when testing at a 1% level, one out of a hundred tests rejects a null hypothesis, while the null hypothesis is actually true (type I error).

4.1.1 Linear Granger causality test

The Granger causality tests on the raw data provided some notable results. The linear Granger test detected the relationship δ18_{O → δ}13_{C for all three periods, at a 1% signicance}

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level. The relationship insolation → δ18_{O was found at 1% signicance level for period II}

and Ia, and at 5% signicance level for period Ib. Insolation → δ13C is found to be signicant

for period Ia. These results are in line with what was expected. However, some results are

contrary to our expectations. In period II and period Ib the causal link δ18O → insolation

is found, at a 10% and 5% level respectively. A possible explanation for this result is given by Mitrovica, Forte and Pan (1997). These researchers state that this variable can inuence insolation, via changes in the Earth's moment of inertia. However, this linkage is unexpected, because δ18_{O is not considered of great signicance on insolation.}

4.1.2 Diks-Panchenko test

The Diks-Panchenko test detected the relationship δ18O → δ13C for period II and period

Ia, at 1% signicance level. The relationship insolation → δ18O was found for period II and

period Ia, at 5% signicance level. The relationship insolation → δ13C was found at 10%

signicance level in period Ia.

Some results are unexpected. For period Ia, the relationships δ18O → insolation and

δ13_{C → insolation were signicant. The DP-test on period I}

b did not result in any signicant

relationships in the raw data. This may be caused by the low amount of observations in period Ib (93 observations). Diks and Panchenko (2006) showed that applying the DP-test

on a dataset with a low amount of observations might cause under-rejection.

4.2 Granger causality tests on VAR-ltered residuals

The nonparametric DP-tests on raw data indicated that some nonlinear Granger causality might exist between the three variables. A VAR-lter has to be applied to ensure that the causality relationships found are nonlinear in nature. For the bivariate VAR-ltering, the residual series of the three bivariate VAR models of the rst stage are investigated for linear and nonlinear causality, by applying the same tests as before. Furthermore, a trivariate VAR model, containing the three variables, was estimated for every period. The SIC selected two lags for every trivariate VAR model. The trivariate VAR models included a constant. Each pair of residual series of the trivariate VAR models was tested for causality.

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4.2.1 Linear Granger causality test

For the linear Granger causality test, a bivariate VAR model was constructed for each pair of residual series of the VAR models. All the bivariate VAR models of the residual series contained a constant. For the bivariate VAR-ltering, the selected lag length was zero by the SIC. The same result is obtained for the trivariate VAR-ltering. This suggests that there is no linear Granger causality in the residuals. To ensure that this is the case, a linear Granger causality test is applied on the bivariate VAR model of the residual time series, including a constant and one lag. The tests resulted in no signicant linear causality between the pairs of residuals. Therefore, the linear causal linkages disappeared by applying the VAR lter, implying that the linear relationships are captured by the VAR model.

4.2.2 Diks-Panchenko test

For the Diks-Panchenko test, the same lag lengths and bandwidth are used as in the pre-lter stage. The Diks-Panchenko test on the VAR-pre-ltered residuals resulted in a few nonlinear causality linkages between the variables. The same signicant linkages were found for both the bivariate and the trivariate VAR-ltering. For period Ib, the nonlinear link insolation →

δ18O is found at a 10% signicance level. The unexpected causality link δ18O → insolation is found for both period II and period Ia, at a 10% signicance level.

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Table 5: p-values of the causality tests period II

5a: Linear Granger causality test

Residuals Residuals

Variables Raw data (bivariate (trivariate

VAR-ltering) ltering)

X Y X→Y Y→X X→Y Y→X X→Y Y→X

δ13C δ18O 0.957 0.000*** 0.715 0.812 0.717 0.717

δ18O Insolation 0.072* 0.000*** 0.953 0.993 0.985 0.917

δ13C Insolation 0.173 0.171 0.870 0.664 0.916 0.134

`*',`**', `***' denote statistically signicant correlation at the 10%, 5%, 1% level

5b: DP-test for nonlinear Granger causality

Residuals Residuals

δ13C δ18O 0.155 0.000*** 0.694 0.447 0.915 0.693

δ18O Insolation 0.142 0.041** 0.056* 0.657 0.095* 0.686

δ13C Insolation 0.559 0.859 0.142 0.355 0.142 0.312

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Table 6: p-values of the causality tests period Ia

Residuals Residuals

δ13C δ18O 0.597 0.000*** 0.824 0.666 0.841 0.595

δ18O Insolation 0.562 0.000*** 0.916 0.568 0.891 0.587 δ13C Insolation 0.876 0.012** 0.682 0.352 0.795 0.454 `*',`**', `***' denote statistically signicant correlation at the 10%, 5%, 1% level

Residuals Residuals

δ13C δ18O 0.152 0.000*** 0.418 0.312 0.363 0.523

δ18O Insolation 0.016** 0.046** 0.056* 0.115 0.068* 0.196 δ13C Insolation 0.004*** 0.055* 0.690 0.583 0.726 0.366 `*',`**', `***' denote statistically signicant correlation at the 10%, 5%, 1% level

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Table 7: p- values of the causality tests period Ib

Residuals Residuals

δ13C δ18O 0.216 0.001*** 0.739 0.521 0.699 0.731

δ18O Insolation 0.018** 0.029** 0.808 0.846 0.918 0.865

δ13C Insolation 0.333 0.105 0.722 0.864 0.425 0.475

Residuals Residuals

Variables Raw data (bivariate VAR- (trivariate

δ13C δ18O 0.213 0.313 0.808 0.447 0.911 0.592

δ18O Insolation 0.256 0.211 0.719 0.080* 0.737 0.090*

δ13C Insolation 0.771 0.487 0.833 0.168 0.599 0.426

4.3 Comparison of the results

The Granger causality tests provided some noteworthy results. In this subsection the results are compared in order to come to the following conclusions about the causality relationships between the climatological variables.

4.3.1 Causal linkages between δ18O and δ13C

Both the linear Granger causality test and the DP-test on raw data detected a highly sig-nicant causal link from δ18_{O to δ}13_{C, for periods II and I}

a. For period Ib, this relationship

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disappeared in every period after the VAR-ltering, which indicates that there is a linear causal relationship from δ18_{O to δ}13_{C. This is in conict with the result found by Diks and}

Mudelsee (2000), their research concludes in that this link is nonlinear in nature. No signi-cant causality has been found for δ13_{C on δ}18_O.

4.3.2 Causal linkages between insolation and δ18O

The linear Granger test showed that insolation is a signicant Granger cause for δ18O in

period II, Ia and Ib, at a 1% level for the rst two periods, and 5% level for the last. This

relationship is also found by the DP-test on raw data, for period II and Ia at 5% signicance

level. For period Ib, this relationship only counts after the VAR-lter, and not on the raw

data. That this result is not found by the DP-test on raw data of period Ib might be because

of the under-rejection of the DP-test for a small amount of observations. However, the link insolation → δ18O is not signicant after VAR-ltering on period II and I

a, so it is not clear

whether this linkage is nonlinear or not. Insolation might be a linear Granger cause for δ18_O

in period II and Ia, and a nonlinear Granger cause for δ18O in period Ib.

The unexpected link δ18_{O → insolation is found through the linear Granger test to be}

on a 10% signicance level for period II, and 5% level for period Ib. This link was also found

signicant by the DP-test for period Iaon the raw data (5% signicance level), and on a 10%

signicance level for period II and Ia after VAR- ltering. This suggests that the causal link

δ18O → insolation changed its nature over time, from nonlinear for period II and Ia to linear

for period Ib. A possible explanation for this unexpected causal link is given by Mitrovica,

Forte and Pan (1997). These researchers state that this climatological variable can inuence insolation, via changes in the Earth's moment of inertia. However, δ18O is not considered of

great signicance on insolation, so this is an odd result.

4.3.3 Causal linkages between insolation and δ13_C

Insolation has been found as a signicant Granger cause for δ13_{C in period I}

aonly. Both the

linear Granger test and the DP-test on raw data found this linkage at a 5% and 10% signif-icance level. This causal relationship disappeared after VAR-ltering, so the link insolation → δ13_{C is expected to be linear for period I}

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is only found signicant at a 1% level at the Diks-Panchenko test on raw data for period Ia.

This is an unexpected causality relationship and most likely due to chance.

This section discussed and compared the results of the Granger causality tests on the dataset. The next section is a conclusion of the thesis.

5 Conclusion

The research on the causality linkages among climatological variables is helpful in the debate on impending global warming. An important question in the study of the recently observed climate change is which climatological variable is driving which. Granger causality tests are often applied on the climate system as an attempt to answer this question. The purpose of this thesis was to test for the existence of linear and nonlinear Granger causality relation-ships between three climatological variables (δ13C, δ18O and solar insolation), over the past

1.8 million years. The extent to which nonlinearities play a role in the causal relationships among the climatological variables and how these relations have changed over time were in-vestigated.

The analysis of earlier studies on Granger causality in the climate system led to the following expectations about the causality linkages among the three aforementioned clima-tological variables. The causality linkage from δ18O to δ13C was previously found to be

nonlinear and there was no causality linkage from δ13_{C to δ}18_{O in earlier studies. Solar}

in-solation was expected to be a main driving force behind δ18O and δ13C, but it was not clear

whether this linkage is linear or nonlinear. The variables δ18_{O and δ}13_{C were not expected}

to be a Granger cause for insolation.

Two dierent Granger causality tests were applied in this thesis so as to research the nature of the causality linkages among the aforementioned climatological variables. The raw data as well as the ltered residuals of bivariate and trivariate VAR models were tested for linear and nonlinear causality using the linear Granger causality test and the nonparametric Diks-Panchenko test.

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dataset (in the northern South China Sea) were explored in this thesis. The ODP 1146 dataset provides proxies for the global ice volume (reected by δ18_{O) and the deep water}

circulation (reected by δ13_{C), over the past 1.8 million years. Interpolated solar insolation}

values at 15o_{N latitude were also included to this dataset.}

The dataset was split into three periods to investigate how the causality relationships have changed over time. The rst split was at 892 ka ago to take the Mid-Pleistocene Climate Transition into account. The second separation in the dataset was at 318 ka ago because the sedimentation process in the northern South China Sea slowed down after 318 ka ago. The aim of this second split was to overcome the unequal time intervals problem. The ADF test for stationarity resulted in the conclusion that the time series of the climatological variables were stationary for every period, which was required for the application of the VAR models and the Granger causality tests in this thesis.

The results of the Granger causality tests on the raw data and VAR-ltered residuals led to the following conclusions about the causality linkages between δ18_{O, δ}13_{C and solar}

insolation. For every period, δ18_{O is a linear Granger cause for δ}13_{C, which is in conict with}

the expectation that this linkage is nonlinear. There is no signicant causality linkage from δ13C to δ18O, which is in line with what was expected. Solar insolation is a signicant linear Granger cause for δ18_{O in period II and I}

a and a nonlinear Granger cause for δ18O in period

Ib, which suggests that this linkage changed its nature of causality over time. The unexpected

causality linkage δ18_{O → insolation is nonlinear for period II and I}

a, and linear for period Ib.

This result is odd, because δ18_{O is not considered of great signicance for insolation. There is}

a signicant linear Granger causality linkage from insolation to δ13_{C in period I}

a only. This

is in line with what was expected. However, the unexpected link δ13_{C → insolation is also}

found to be signicant in period Ia. This unexpected causality relationship is most likely due

to chance.

There are some limitations to the research of this thesis. An omitted variable bias likely occurred, since the dust ux is an important climatological variable which is not included in the dataset of this thesis. However, it is not clear to what extent this bias inuenced the results and conclusions of this thesis. Whether more variables, besides the dust ux and the three variables in this thesis, have an inuence on the climate system at this location,

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can be subject to further research. Another limitation is that the Granger causality tests do not take the long-term eects and strength of causal inuences into account. These are also important to consider.

Analysis of the ODP 1146 dataset made it clear that the time intervals between ob-servations are unequal in length. When ignored, this leads to biased results. Even though the separation of the dataset into climatological periods helped to overcome this problem for a large part, it might still be possible that the results of the Granger causality tests were aected by unequal time intervals. Moreover, the splitting of the dataset left period Ib with

93 observations only. This might have resulted in under-rejection of the null hypothesis when the Diks-Panchenko test was applied on period Ib.

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Granger causality and global warming : the causal links between oxygen, carbon and solar insolation over the past 1.8 million years

Granger Causality and Global Warming:

The Causal Links between Oxygen, Carbon and

Solar Insolation over the past 1.8 Million Years

David Kranenburg (10574697)

June 29

th

- 2016

Contents

1 Introduction

2 Granger causality testing on the climate system

2.1 The concept of Granger causality

2.2 Linear Granger causality testing on the climate system

2.3 Diks-Panchenko test for nonlinear Granger causality

2.4 Other methods to explore Granger causality

2.5 Comments on the use of Granger causality tests on the climate

system

2.6 Conclusion

3 Data and empirical methodology

3.1 Data

3.2 Data analysis

3.3 Empirical methodology

4 Results of the Granger causality tests

4.1 Granger causality tests on raw data

4.2 Granger causality tests on VAR-ltered residuals

4.3 Comparison of the results

5 Conclusion

References

4.2 Granger causality tests on VAR-ltered residuals