Testing hypotheses about glacial cycles against the observational record


Corresponding author: Robert K. Kaufmann, Department of Earth and Environment, Boston University, Boston, MA, USA. (kaufmann@bu.edu)


[1] We estimate an identified cointegrated vector autoregression model of the climate system to test hypotheses about the physical mechanisms that may drive glacial cycles during the late Pleistocene. Results indicate that a permanent doubling of CO2 generates a 11.1°C rise in Antarctic temperature. Large variations in atmospheric CO2 over glacial cycles are driven by changes in sea ice and sea surface temperature in southern oceans and marine biological activity. The latter can be represented by a two-step process in which iron dust increases biological activity and the increase in biological activity reduces CO2 concentrations. Glacial variations in ice volume, as proxied by δ18O are driven by changes in CO2 concentrations, global and high latitude solar insolation, latitudinal gradients in solar insolation, and the atmospheric concentration of CO2. The model is able to quantify the effects of ice volume and temperature on sea level, such that in the long-run, sea level rises 14 m per 0.11‰ δ18O and about 17 m/°C of sea surface temperature in southern oceans. Beyond these specific results, the multivariate model suggests omitted variables may bias bivariate analyses of these mechanisms.

1 Introduction

[2] Referring to climate variables that represent various aspects of glacial cycles, Ruddiman [2006] quotes Laurent Labeyrie “Everything is correlated to everything.” At first glance, strong collinearity seems an insurmountable obstacle to efforts that use statistical techniques to understand the physical mechanisms that drive glacial cycles.

[3] Strong correlations among trending variables also pose obstacles to a better understanding of economic systems. To surmount this obstacle, economists now use statistical techniques that are based on the idea of cointegration and error correction. These concepts allow economists to evaluate whether correlations among trending variables correspond to statistically meaningful long-run relationships and to trace the dynamics by which variables adjust back to equilibrium after having been pushed away by exogenous shocks. The power of this statistical methodology is the basis for Clive Granger's 2003 Nobel Prize in economics.

[4] Using the ideas of cointegration and equilibrium error correlation, Kaufmann and Juselius (in review) (Herein KJ) use a cointegrated vector autoregressive (CVAR) model to evaluate the role that orbital, seasonal, and spatial variations in solar insolation play in glacial cycles. Results indicate that glacial cycles in temperature, CO2, and ice volume can be simulated accurately using orbital parameters, with lesser contributions from seasonal variations at specific latitudes.

[5] These results are based on a CVAR model whose ability to simulate glacial cycles is determined purely by statistical criteria. This narrow criterion raises the question whether these results reflect the physical and/or biological mechanisms that are hypothesized to generate glacial cycles or whether they are a result of pure statistical curve fitting.

[6] The aim of this paper is, therefore, to go beyond the purely statistical description of the CVAR and test whether the cointegrating relations in the CVAR are consistent with a basic understanding of how the climate system works. This is done by identifying long-run relationships among 10 climate variables and four solar variables and assessing whether the results are consistent with physical and/or biological mechanisms. Identification allows us to discriminate among mechanisms currently described in the literature and quantify the physical and/or biological mechanisms embodied in the CVAR, such as the large (80–100 ppm) changes in atmospheric CO2.

[7] Results indicate that the statistical relations embodied in the CVAR are consistent with the physical and/or biological mechanisms described in the literature. A permanent doubling of CO2 generates 11.1°C rise in Antarctic temperature. Large variations in atmospheric CO2 over glacial cycles are driven by changes in sea ice and sea surface temperature in southern oceans and marine biological activity, which is represented by a two-step process in which iron dust increases biological activity and the increase in biological activity reduces CO2 concentrations. Glacial variations in ice volume, as proxied by δ18O, are driven by changes in global and high latitude solar insolation, latitudinal gradients in solar insolation, and the atmospheric concentration of CO2. Finally, the model is able to quantify the effects of ice volume and ocean temperature on sea level, such that after complete adjustment, sea level rises 14 m per 0.11‰ δ18O and about 17 m/°C of sea surface temperature in southern oceans.

[8] These results, and the methods used to obtain them, are described in four sections and a set of Supporting Information. Section 2 describes the time series that are used to proxy various aspects of the climate and how the CVAR model can be used to test hypotheses about the mechanisms that may drive glacial cycles. Section 3 describes diagnostic statistics that are used to evaluate the identified CVAR model. Section 4 interprets the statistical results relative to hypotheses for the physical mechanisms that are thought to play an important role in glacial cycles. Section 5 concludes by describing the general importance of a multivariate approach, as opposed to the bivariate approach that is used by many analyses.

2 Methodology

[9] We expand the three variable CVAR model described by KJ by including the following six climate variables, land surface temperature (Temp), atmospheric carbon dioxide (CO2), atmospheric methane (CH4), ice volume (Ice), sea surface temperature (SST), and sea level (Level) and study how they are linked to four physical variables, iron (Fe), sea-salt sodium (Na), non sea-salt calcium (Ca), and non-sea-salt sulfate (math formula). This ten-dimensional dynamic adjustment model allows us to test hypotheses about the physical and/or biological mechanisms that create glacial cycles in the late Pleistocene (previous 391 Kyr), the so called “Vostok period.” The Vostok period is chosen because it contains four glacial cycles and reduces data aggregation across cores. Subsequent efforts will expand the sample period to include the previous 800 Kyr and test whether the relationships among variables during the “Vostok period” are stable.

2.1 Data

[10] Data for temperature, carbon dioxide, and methane are obtained from cores drilled into the Antarctic ice sheet. Carbon dioxide and methane are well-mixed gasses and so measurements from the Antarctic ice sheet proxy global concentrations. The temperature proxy represents local conditions, but can be converted to global values by assuming a scaling derived from a limited set of observations can be applied across all observations [Masson-Delmotte et al., 2010; Masson-Delmotte et al., 2006]. The data that are used to proxy ice volume are derived from 57 cores drilled by the Deep-Sea Drilling Project and Ocean Drilling Program across the globe [Lisiecki and Raymo, 2005]. The proxy for sea surface temperature is constructed using alkenones from site PS2489-2/ODP1090 in the subantarctic Atlantic. Data for sea level are reconstructed using oxygen isotope records from Red Sea sediments. The sources for these data (and those described below), the number of observations, units of measure, and their original time scale are described in Table 1.

Table 1. Time Series Included in the CVAR
VariableSourceUnitTime ScaleObs

Jouzel et al. 2007

∆ avg. last 1 kyrEDC3710

Leithi et al. 2008


Loulergue et al. [2008]


Lisiecki and Raymo [2005]


Wolff et al. [2006]

µg m−2yr−1EDC2187

Wolff et al. [2006]

µg m−2yr−1EDC2195

Wolff et al. [2006]

µg m−2yr−1EDC2195

Wolff et al. [2006]

µg m−2yr−1EDC2195
Sea level

Siddall et al. [2003]

Sea surface temp

Martinez-Garcia et al. [2009]

Degrees CEDC3121

Paillard et al. [1996]

Dimensionless index-391

Paillard et al. [1996]


Paillard et al. [1996]

Dimensionless index-391
Seasonal insolation

Paillard et al. [1996]


[11] Surface temperature, atmospheric CO2, and ice volume are among the most commonly used representatives for glacial cycles and are thought to be directly related to one another, and so their inclusion requires little justification. Methane is included because it may be related to surface temperature; variations in methane concentrations generate the second largest change in radiative forcing due to greenhouse gasses [Kohler et al., 2010]. Sea surface temperature is included because it is related to air surface temperature and it may affect rates of marine biological activity and sea ice. Finally, sea level is included because models based on physical processes cannot simulate sea level changes accurately [Vermeer and Rahmstorf, 2009] and this inability leads to a plethora of semi-empirical models that are estimated from synthetic and real data [e.g., Rahmstorf, 2007; Horton et al., 2008; von Storch et al., 2008; Vermeer and Rahmstorf, 2009], which can be compared to results generated here.

[12] Physical and biological mechanisms that may link the six climate variables are represented by the four proxy variables mentioned above. As described below, we interpret these proxies as they are used in the literature. The basis for these interpretations and the sources of uncertainty that are associated with their values are described in section I of the Supporting Information.

[13] Fe is derived almost entirely from terrestrial sources and is included to proxy a so-called iron fertilization effect, which may enhance the biotic uptake of CO2 [Martin, 1990]. Sulfate originates mainly from marine biogenic emissions of dimethylsulphide (after removing sea-salt sources using the Na data), and so proxies marine biological activity [Cosme et al., 2005]. It is included to represent the possible effect of iron-containing dust on biological activity and/or the effect of biological activity on atmospheric CO2.

[14] Sea salt sodium is derived from the sea-ice surface and proxies the extent of winter sea-ice [Wolff et al., 2003]. Sea-salt sodium is included to represent the possible effect of sea ice on the flow of CO2 from the ocean to the atmosphere [Stephens and Keeling, 2000]. Non-sea-salt calcium has a terrestrial origin (mainly Patagonia) and may represent changes in temperature, moisture, vegetation, wind strength, glacial coverage, or changes in sea level in and around Patagonia [Basile et al., 1997]. It is included to represent climate conditions at a locale thought to play an important role in glacial cycles.

[15] To make these data amenable to a statistical analysis, we convert them to a common time scale (EOS Data Center 3, EDC3) using conversions from Parrenin et al. [2007] and Ruddiman and Raymo [2003]. Unevenly spaced observations are interpolated (linearly) to generate a data set in which each series has a time step of 1 Kyr.

[16] Solar insolation is exogenous in the statistical model and is represented by four time series. Orbital variations are represented by precession, obliquity, and eccentricity. These changes generate spatial and temporal variations that are represented by summer-time insolation at 65°S (SunSum). This series is chosen based on results reported by KJ. They simulate 12 CVAR models that specify summer-time insolation at 5° intervals between 60° and 85° north and south and compare the accuracy of in-sample simulations using statistical criteria. They find that a CVAR that uses summer-time insolation at 65°S generates the most accurate in-sample simulation for Antarctic temperature. As summer-time insolation is related to the orbital parameters, the question of multicollinearity and its effect on the results is an important issue. As explained in the Supporting Information section II, multicollinearity has little effect on findings of cointegration and therefore should have little effect on our results. Consistent with this, KJ shows that seasonal measures of solar insolation have information about glacial cycles beyond that in orbital variations.

[17] To eliminate the effects on inverting matrices with elements that differ greatly in size (due to different units of measurement), each of the 14 time series is standardized as follows:

display math(1)

where yt is the value (in original units), math formula is the average value over the sample period, and Var(y) is the variance over the sample period.

2.2 The Statistical Methodology of the Cointegrated VAR

[18] The cointegrated VAR (CVAR) model is developed for the statistical analysis of nonstationary processes. For example, a random walk variable, xt = xt − 1 + εt, is a simple example of a first order nonstationary variable. It is characterized by a tendency of xt to drift persistently away from its long-run mean in a nonstationary manner, whereas its increment, xt − xt − 1 = εt, is a stationary error process. A random walk variable also can be formulated as the cumulation of these increments, math formula where math formula is called a stochastic trend. A stochastic trend differs from a linear deterministic time trend by having stochastic, rather than deterministic, increments over time and represents, therefore, a more realistic description of many climate variables. Stochastic trends can be eliminated partly by differencing the series and partly by cointegration (linear combinations of several series that cancel the stochastic trends). But differencing removes all long-run information in the data, whereas cointegration ensures that it is preserved. For further details, see section III of the Supporting Information. The CVAR model builds on this insight by combining both differences and cointegration to describe short-run adjustment and long-run relations in the same model. Previous climate-related applications of the CVAR model include analyses of the relation between surface temperature and radiative forcing during the instrumental temperature record [e.g., Kaufmann et al., 2011; Kaufmann and Stern, 2002]. These uses and its use here are supported by a result that indicates the cointegration error correction model is consistent with a zero dimension climate energy balance model [Kaufmann et al., 2013].

[19] When applying the CVAR model to the data described in Table 1, we assume that all climate and physical variables are endogenous, whereas all solar variables are exogenous. Based on this distinction, the CVAR model is defined as:

display math(2)

where xt is a vector of climate and physical variables whose behavior is being modeled endogenously, wt is a vector of exogenous solar variables, μ0 is a vector of constant terms, A0, A1, Γ11 and Π are matrices of regression coefficients, Δ is the first difference operator (Δxt = xt − xt − 1), and εt is Niid(0,Ω).

[20] When xt is nonstationary, the long-run matrix Π is either zero or of reduced rank r. This is formulated as:

display math(3)

where α is a matrix of adjustment coefficients and β is matrix of cointegration coefficients that define stationary deviations from long-run equilibrium relationships, math formula. Equation ((2)) subject to condition ((3)) can be solved by reduced rank regression. See Johansen [1996] for further details.

[21] Geometrically, the long-run relationships, math formula, can be thought of as an attractor set towards which the climate system moves after having been pushed away from its equilibrium by an exogenous shock, such as a change in Earth's orbital position. The speed of adjustment at which the system returns to equilibrium is described by the elements of α. If Π = αβ' = 0, then the model does not contain any information about the long-run properties of the climate system. The remaining components of the model (A0Δwt, A1Δwt − 1, Γ11Δxt − 1) describe period-to-period transitory changes in climate. But, if the VAR model is estimated for nonstationary climate data in levels ignoring cointegration, then the presence of unit roots in the variables will invalidate some of the statistical inferences of the model. By including both the long-run relations and the first difference terms, the CVAR model preserves all long-run and short-run information in the data, enables the correct use of standard inference based on (X2, F, t), and ensures the validity of R2. (see section IV of the Supporting Information.) Thus, when data are highly persistent, the CVAR model generally produce more reliable results than a regression model based on ordinary least squares. (see section IV of the Supporting Information.) This means that the CVAR model can be used to evaluate whether climate variables are causally associated well beyond the efforts to determine system dynamics from potentially spurious correlations between lags and leads [e.g., Caillon et al., 2003; Shakun et al., 2012]. (see section V of the Supporting Information).

2.3 Illustrating the Methodology

[22] To give the physical intuition for the CVAR model, we use a simple model that includes only three of our 10 endogenous variables, Temp, CO2, Ice, and one of the four exogenous variables, Precession. We assume that the rank is two, i.e., there are two cointegrating relations among the four variables. For simplicity, the short-run effects,A0Δwt, A1Δwt − 1, Γ1Δxt − 1, are set to zero, and the CVAR model becomes:

display math(4)

[23] The β coefficients in ((4)) correspond to estimated eigenvectors obtained by solving the reduced rank problem and are uniquely defined based on the ordering of the eigenvalues. Such ordering is not likely to have a meaningful physical interpretation, and the unrestricted β relations in ((4)) should be considered statistical regularities which only exceptionally may have a physical interpretation. To give them a physical interpretation, we need to impose identifying restrictions on each cointegration relation. Generally, the structure of cointegration relations is statistically identified when it is not possible to take a linear combination of the two cointegration relations without violating any of the imposed restrictions [Johansen and Juselius, 1994, Juselis, 2006, Chapter 12]. It is physically identified when the resulting relations can be given a meaningful physical interpretation. Therefore, identifying restrictions should preferably be consistent with extant hypotheses about the physical and biological mechanisms postulated to drive glacial cycles. For example, the hypothesis that the level of surface temperature in the long-run is associated with the level of CO2, and Precession can be formulated as β13 = 0 in the first cointegrating relation which becomes:

display math(5)

where u1,t is an equilibrium error. The relation tells us that temperature, CO2, and precession tend to move together over time in a way that cancels their glacial cycle movements. The nature of the relation between variables (positive or negative) is given by the signs of the β coefficients. The estimates of the identified β coefficients are super consistent and, therefore, generally accurate. They are asymptotically normal and can be tested by a Student's t statistics [Johansen, 1996].

[24] Two points are worth mentioning:

  1. The relation ((5)) is just-identified as we can always impose r − 1 = 1 restriction on each of the cointegration relations without changing the value of the likelihood function. Hence, u1,t is by construction stationary.
  2. The fact that the variables in ((5)) are cointegrated does not say anything about how they are causally related. Any inference about causality is associated with the adjustment coefficients α.

[25] The α coefficients provide information about how the system reacts when the equilibrium error u1,t is different from zero. If ((5)) correctly describes a relation for Temp, then we would expect an adjustment in the equation for Temp to take place when it has been hit by an exogenous shock, a process called equilibrium correction. The condition for equilibrium correction is that α11 be significantly different from zero and α11β11 < 0. If only the equation for Temp equilibrium corrects to the relation ((5)), we say that the latter is specifically a cointegrating relation for surface temperature. But if α21 is significant and α21β12 < 0, then CO2 also is equilibrium correcting to relation ((5)) implying that it can also be considered a cointegrating relation for CO2. In this case, both temperature and CO2 simultaneously (within the time unit) adjust towards equilibrium, so that each of them eliminates part of the disequilibrium. Thus, an exogenous shock can have feedback effects on several variables in the system and normalizing on one specific variable does not necessarily limit a causality interpretation of the other variables.

[26] If Temp significant equilibrium corrects to ((5)), then the equilibrium value for surface temperature, Temp * can be found by normalizing on Temp in ((5)) and solving:

display math(6)

[27] In equilibrium, u1t = 0. The size of the adjustment coefficient (after normalization) measures the speed with which the system adjusts when it has been pushed away from equilibrium. For example, a value of α11 = − 0.6 (for a normalization on Temp) implies that 60% of a shock that has moved the surface temperature away from its previous equilibrium value is eliminated in this period, and the remaining 40% is gradually eliminated over the following periods. The average time it takes to eliminate the effect of an exogenous shock is ln(2)/α, so if α is very small, an exogenous shock can have a long lasting effect on the climate system.

[28] To identify the full structure, we also need to identify the second relation. The hypothesis here is that in the long run Ice is only related to CO2, formulated as β21 = β24 = 0, which gives rise to the relation:

display math(7)

where u2,t is stationary provided that CO2 and Ice form a cointegrating relation. One of the two zero restrictions is over-identifying and can be tested with a Likelihood Ratio test distributed as a χ2 with one degree of freedom. If the test rejects, then u2,t is not stationary implying that either Precession or Temp would have to be included in relation ((7)) to make it stationary. The hypothesis that ((7)) describes a relation for Ice would be checked against a significant value of α32 and α32β23 < 0.

[29] Taken together, the two sets of restrictions satisfy generic identification (it is not possible to take a linear combination of the two relations without violating some of the zero restrictions). We say that the structure is both generically and physically identified when the over identifying restrictions are not statistically rejected and the relations define physical and/or biological relationships that are postulated to drive glacial cycles. In this simple example, we would expect β11 and β12 to have opposite signs so that Temp and CO2 have a positive relation in ((6)), and β22 and β23 have the same sign so that there is a negative relation between Ice and CO2 when ((7)) is normalized on Ice or CO2.

3 Results

[30] Several diagnostic tools can be used to determine the cointegration rank of Π (i.e., the number of cointegrating relations). The most common is the likelihood based trace test [Johansen, 1996] which tests the null of p-r unit roots in the process. If the test fails to reject zero unit roots, then the process can be considered stationary, if it fails to reject P unit roots, then there is no cointegration and hence no long-run equilibrium relations. Any value between zero (full rank) and P (no cointegration) implies the existence of both equilibrium relations and stochastic drivers.

[31] Based on this test, the hypothesis of one unit root, i.e., at least nine cointegration relations, barely fails to be rejected (p < 0.066), which suggests that a tenth cointegration relation may be present. This interpretation is bolstered by the small (in absolute terms), but statistically significant estimates of the α coefficients that load the tenth cointegrating relation into the equations for CO2 and Ice. Taken together, this suggests that the tenth cointegrating relation is very persistent and, if included in the model, would add to the internal climate dynamics. Understanding this effect is the focus of a future effort; for this analysis, the matrix is assigned full rank, (i.e., ten cointegrating relations).

[32] To identify the long-run structure, we impose altogether 26 over-identifying restrictions on the 10 cointegrating relations χ2(26)  =  11.14,  p > 0.99. The corresponding estimates of α and β are reported in Tables 2 and 3.

Table 2. The Cointegrating Vectors as Indicated by the Elements of the β Matrix
  • Test statistics are statistically significantly different from zero at the:

  • **


  • *

    5%, +10% level.

CR#2 -1.000**0.891**------0.178**----
Table 3. Rates of Adjustment to Disequilibrium in the Cointegrating Relations, as Given by Elements of the α Matrix
  • Test statistics are statistically significantly different from zero at the:

  • **


  • *

    5%, +10% level.


[33] In-sample simulations indicate that the model can reproduce the main features of the glacial cycles for each of the eight endogenous variables of interest based on the four exogenous solar insolation variables (Figures 1a–1h). Section VI of the Supporting Information provides the formal conditions on the CVAR model under which the simulations are done. The ability of the CVAR to simulate glacial cycles allows us to use the model to quantify the relationship among endogenous and/or exogenous variables by (1) simulating the model to equilibrium (by holding solar insolation constant), and (2) changing a given variable (e.g., doubling atmospheric CO2) and simulating the model to a new equilibrium. This is done for either a single endogenous variable, which represents the direct effect, or all endogenous variables, which represents the system-wide climate effect.

Figure 1.

To evaluate the degree to which the CVAR can simulate the endogenous variables based on changes in the four exogenous variables for solar insolation, we simulate the model and compare the results to the observed values. For each panel, the observed values are given by the black line where the simulated values are given by the red line. Assessing the match between the black and red lines is given in section VI of the Supporting Information. (a) temperature, (b) atmospheric carbon dioxide, (c) atmospheric methane, (d) δ18O ice volume, (e) Na sea ice, (f) sea level, (g) sea surface temperature, and (h) math formula biological activity.

4 Discussion

[34] Here we interpret the 10 cointegrating relations relative to hypotheses regarding the physical and biological mechanisms that are postulated to drive long-run changes in land surface temperature, CO2, CH4, ice volume, sea level, and sea surface temperature. As described in section VII of the Supporting Information, these results are likely to be relatively unaffected by errors in chronology and/or tuning climate variables to solar insolation.

4.1 Land Surface Temperature

[35] The equilibrium relation for land surface temperature is given by the first cointegrating relation, which includes Temp and CO2 (Table 2). The radiative forcing of CO2 is a log function of its concentration. Nonetheless, CO2 is specified as a linear function (equation (1)) of its concentration because a linear specification probably is more appropriate when CO2 is on the left-hand side of the CVAR model (as described in section VIII of the Supporting Information, the linear specification has little effect on the results described below).

[36] As indicated in Table 4, CO2 is positively associated with surface temperature. Furthermore, the value for α11 (−0.50) indicates that land surface temperature adjusts towards the equilibrium value for temperature implied by CR #1 (Table 4). Similarly, the value for α21 is statistically different from zero and α21β12 < 0, which implies that CO2 also adjusts to disequilibrium in the relation between Temp and CO2 (Table 4). This simultaneous adjustment is inconsistent with results that changes in atmospheric CO2 lead changes in surface temperature [Shakun et al., 2012]. This difference may be caused by the higher frequency of the observations, the lag/lead methodology, and/or the omission of other climate variables by Shakun et al. [2012].

Table 4. Long-run Equilibrium Relations Implied by the Cointegrating Relations
 Cointegrating Relations Normalized for Equilibrium Adjusting VariableRate of Equilibrium Correction
  1. Each of the 10 cointegrating relations is normalized (e.g., equation (5)) by the variable in the cointegrating relation that equilibrium error corrects to the long-run value implied by the cointegrating relation (e.g., equation (6)). The rate at which the variable adjusts towards the equilibrium implied by the equation is given in the last column and corresponds to the element of α that loads the cointegrating relation into the equation for the equilibrium error-correcting variable.

CR#1Tempt = 0.83CO2t1t−0.495
CO2t = 1.027Tempt1t−0.137
CR#2CH4t = –0.89Icet +0.18Ecct2t−0.495
CR#3Icet = −0.97CO2t − 1 −0.10Ecct − 13t−0.105
CO2t = −1.03Icet − 1 −0.10Ecct − 13t−0.204
CR#4Fet = 0.16Cat −0.85Levelt −2.02*Oblt +0.08*Prect4t−0.447
CR#5SSTt = −0.25Fet −1.00Nat −0.24SunSumt5t−0.189
CR#6SO4t = 0.72Fet6t−0.156
Fet = 1.04SO4t6t−0.090
CR#7Cat = −0.71*Levelt −0.25Ecct7t−0.166
Levelt = −1.41Cat −0.0.35Ecct7t−0.028
CR#8Levelt = −2.5*CO2t −2.01Icet +1.29SSTt8t−0.300
CR#9CO2t = −2.02Nat −0.43SO4t −0.77SSTt9t−0.229
Nat = −0.49CO2t −0.21SO4t −0.38SSTt9t−0.330
CR#10Icet = −0.76*Ecct +4.46Oblt +2.88SunSumt10t−0.037

[37] The direction of adjustment and the inclusion of other variables that affect CO2 and temperature allows the CVAR to quantify the long-run effect of CO2 on Temp. A permanent 180 ppm increase in atmospheric CO2 increases the long-run Antarctic temperature by about 11.1°C, which corresponds to a global value of about 5.6°C [Masson-Delmotte et al., 2006, 2010]. This increase represents the total temperature adjustment brought about by the direct effect of CO2 on temperature and the indirect effects, by which CO2 affects one or more of the other eight endogenous variables and changes in one or more of these eight endogenous variables affect temperature. Conversely, if only land surface temperature is allowed to adjust to the increase in CO2, Antarctic temperature rises 3.4°C, which corresponds to a global value of 1.7°F. To better understand these direct and indirect effects, future efforts will test competing hypotheses about the role that CO2 plays in glacial cycles [e.g., Shakun et al., 2012; Soon, 2007; Shackleton, 2000] and quantify the mechanisms and rates of adjustment by which elevated concentrations of CO2 affect surface temperature.

4.2 Atmospheric Carbon Dioxide

[38] The CVAR is used to test competing hypotheses about the physical and/or biological mechanisms that cause atmospheric CO2 to vary 80–100 ppm over the course of a glacial cycle during the late Pleistocene. Empirical tests against the observational record are important because extant climate models cannot reproduce the observed changes in CO2 [Archer et al., 2000].

[39] Despite this inability, there is general agreement that the ocean plays a critical role. But the mechanism(s) that transfers large quantities of CO2 between the atmosphere and ocean are the focus of considerable debate [Webb et al., 1997; Broecker and Henderson, 1998; Archer et al., 2000; Sigman and Boyle, 2000. Many explanations fall into three categories: (1) ocean alkalinity, (2) marine biological productivity and the rate at which this carbon sinks (i.e., the marine biological pump), and (3) physical processes that influence the rate at which CO2 flows between the atmosphere and ocean. Sediment data are not consistent with the hypothesis that increased weathering or coral reef formation move large quantities of CO2 from the atmosphere to the ocean during glacial periods [e.g., Archer and Maier-Raimer, 1994], and so we do not explore this hypothesis.

[40] Explanations based on the marine biological pump focus on changes in the availability of nutrients and/or changes in phytoplankton taxa. According to the iron fertilization hypothesis [Martin, 1990], ocean deposition of iron-rich dust from terrestrial sources alleviates nutrient constraints and enhances biological productivity. This effect need not be direct, as described by the “silica-leakage” hypothesis [Brzezinski et al., 2002; Hutchins and Bruland, 1998].

[41] Statistical results are consistent with the general outline of the iron hypothesis. CR #6, which can be interpreted as a cointegrating relation for math formula, shows a positive relation between math formula and Fe (Table 4). This positive relation is consistent with the hypothesis that ocean deposition of iron-rich dust from terrestrial sources enhances biological productivity. This statistical relationship also is consistent with experimental results that indicate adding iron to ocean surface waters increases biological activity [Coale et al., 1996, 2004; Boyd et al. 2000; Tsuda et al. 2003]. Conversely, this result is inconsistent with those generated by Kaufmann et al. [2010] who conclude that there is only a weak bivariate correlation between sulfate and dust flux over the last 150 Kyr.

[42] The second step of the iron fertilization hypothesis is consistent with CR #9, which can be interpreted as a cointegrating relation for CO2. As indicated in Table 4, increased levels of math formula are associated with reduced concentrations of CO2, which may be caused by increased net primary production. This statistical result is consistent with experiments that suggest a large role for iron fertilization in glacial cycles [Blain et al., 2007], but contradicts experimental results that indicate that adding iron to ocean surface waters does not increase the rate at which carbon sinks [Coale et al., 1996; 2004; Boyd et al., 2000; Tsuda et al., 2003].

[43] We quantify the effect of iron fertilization on CO2 by allowing the model to come to equilibrium, increasing Fe by 218 µg m- 2year- 1 which is the large increase from 372 to 355 Kyr before present, and allowing math formula and CO2 only to come to a new equilibrium. This direct effect reduces CO2 by 8.5 ppm. This estimate is similar to the 20 ppm reduction associated with dust deposition in the Southern Ocean [Indermuhle et al., 2000; Rothlisberger et al., 2004; Fischer et al., 2010]. Conversely, if all endogenous variables are allowed to adjust, and Fe is maintained at the higher level, the equilibrium reduction in CO2 is 177 ppm, which is larger than the maximum value of 80 ppm attributed to previous deglaciations [Parekh et al., 2004; Ridgewell, 2003]. The climatic feedbacks that induce this large increase are different from the biogeochemical feedbacks described by Parekh et al. [2006].

[44] The ninth cointegrating relation also is consistent with one of the physical mechanisms postulated to affect the flow of CO2 between the atmosphere and the ocean; sea ice in the high latitudes of Southern oceans retards the flow of carbon from the ocean to the atmosphere and thereby lowers the atmospheric concentration of CO2 [Stephens and Keeling, 2000]. As indicated in Table 4, winter sea ice (as proxied by Na) is negatively related to CO2 in the long-run, either directly, as proposed by Stephens and Keeling [2000], or indirectly via increased stratification due to denser bottom water caused by intense sea ice formation near Antarctica [Watson et al., 2006]. Finally, the negative relation between SST and CO2 in CR #9 is consistent with the hypothesis that Antarctic temperatures and CO2 concentrations are closely linked to changes in Southern ocean surface temperatures [De Boer et al., 2007; Sigman et al., 2004; Stephens and Keeling, 2000; Toggweiler et al., 2006].

[45] We evaluate the effect of changes in sea ice by allowing the CVAR to equilibrate to solar insolation at the time of the last glacial maximum and then reducing Na by 358 µg m- 2year- 1, which is the change over the last 20 Kyr. This reduction generates a 39.6 ppm increase in CO2 if all endogenous variables (other than Na) are allowed to adjust to the reduced value of Na. This effect is smaller than the 67 ppm estimated by Stephens and Keeling [2000] but larger than estimates generated by Morales Maqueda and Rahmstorf [2002] and Kurahashi-Nakamura et al. [2007].

[46] But if feedbacks are turned off, such that only CO2 is allowed to adjust to the 358 µg m- 2year- 1 reduction in Na, CO2 increases 8.2 ppm. This small increase is consistent with the small effect simulated by physical models [e.g., Morales Maqueda and Rahmstorf, 2002]. Indeed, the reduced direct effect simulated by the single equation adjustment is consistent with arguments that the small effect of sea ice, which is simulated by physical models, indicates that these models are relatively insensitive to changes in climate [Kohfeld et al., 2005].

4.3 Atmospheric Methane

[47] The mechanism(s) that drives methane concentrations is uncertain [Fluckiger et al., 2004], but many hypotheses focus on obliquity, precession, and temperature [Jouzel et al., 2007; Loulergue et al., 2008]. The third cointegrating relation, which includes Ice, CH4, and Eccentricity, can be interpreted as the equilibrium relation for methane. As indicated in Table 4, the negative relation with Ice is consistent with hypotheses that ice sheets affect atmospheric CH4 via the deposition of peat, the freeze/thaw cycle of the active soil layer, and the seasonal extent of snow cover [Loulergue et al., 2008; Scmidt et al., 2004].

4.4 Ice Volume

[48] The long-run determinants of ice volume are given by the third and tenth cointegrating relations. As indicated by Table 4, CR #3 shows a negative long-run relation between CO2 and Ice. This relationship extends previous efforts, which simulate glacial terminations based on orbital forcings and ice volume only [e.g., Imbrie et al., 2011; Tziperman et al., 2006; Parrenin and Paillard, 2003].

[49] The tenth cointegrating relation in Table 4 represents the relation between ice volume and various aspects of solar insolation. The negative relation between eccentricity and global insolation (also in CR#3) is consistent with the notion that an increase in global solar insolation (albeit small) reduces ice volume. The positive relation between ice and summer insolation in the Southern Hemisphere is consistent with the well known effect of Northern Hemisphere high latitude summer-time insolation on ice volume. That is, an increase in Southern Hemisphere summer time insolation at 65° is associated with a reduction in Northern Hemisphere summer insolation at 65°, which is associated with an increase in ice volume.

[50] The tenth cointegrating relation also indicates a positive relation between Ice and Obliquity, which seems to contradict a basic understanding of glacial/interglacial cycles. But the positive effect of obliquity represents its effect on ice volume beyond the effect of summer-time insolation. As such, the positive effect of obliquity on Ice may be caused by latitudinal differences in insolation that are correlated with obliquity. Specifically, there is a phase reversal in the relation between obliquity and total solar insolation such that subtracting insolation on different sides of 43°–44° creates a 41 Kyr cycle that is strongly correlated with obliquity. For example, the difference between daily insolation on the June solstice at 65°N and 30°N is dominated largely by obliquity [Loutre et al., 2004]. Furthermore, the maximum gradient is associated with maximum values in mean annual insolation at low latitudes, which correspond to maximum values of obliquity.

[51] Latitudinal gradients in solar insolation (as proxied by obliquity) may affect ice volume via atmospheric circulation [e.g., Raymo and Nisancioglu, 2003]. Large gradients may increase the poleward transport of water, which would increase precipitation at high latitudes. And the increased precipitation would add to ice volume. For example, Johnson [1991] explains the transition from isotopic stage 6 to 5 using summer insolation gradients. Similarly, Masson-Delmotte et al. [2006] relates rapid changes in Greenland to large-scale changes in atmospheric circulation.

4.5 Sea Level

[52] The long-run relationship for sea level is given by the eighth cointegrating relation, which includes Sea level, Ice, CO2, and SST (Table 4). This result allows us to untangle the effects of ice volume and ocean temperature on sea level, which Wright et al. [2009] describe as the holy grail of Pleistocene paleoceanography. CR#8 indicates that a change of 0.11‰ in δ18O changes sea level by about 14 m at equilibrium. This value is slightly larger than the widely used 10 m per 0.11‰ in δ18O [Fairbanks and Matthews, 1978]. CR#8 also indicates that a 1°C rise in sea surface temperature (in the sub Antarctic Atlantic) raises sea level by about 17 m at equilibrium. This estimate probably understates the effect of a 1°C rise in ocean temperature because the time series for sea surface temperature used here changes by about 12°C over the sample period, compared to about 5°C for bottom water temperature [e.g., Elderfield et al., 2010, Sosdian and Rosenthal, 2009]. Note that both of these values represent the full equilibrium response of sea level and so cannot be used to estimate effects due to changes in temperature or ice volume as the climate system moves from one partial equilibrium to another partial equilibrium (i.e., it cannot be used to compute changes since the last glacial maximum).

[53] Despite the potential for improvements by using a more integrated (over space and depth) measure for SST, the results reported here probably are more reliable than semi-empirical efforts to model sea level. These models use ordinary least squares to estimate a statistical relationship for the rate of sea level rise based on the increase in temperature relative to a base temperature at which sea level is in (presumed) equilibrium [e.g., Rahmstorf, 2007] and a rate at which temperature rises [Vermeer and Rahmstorf, 2009]. Because these efforts ignore the highly persistent nature of climate variables, statistical estimates for the relation between SST and sea level that are generated by ordinary least squares have a small sample bias [Stock, 1987], overstate the models' explanatory power, and overstate the ability of tests to reject the null hypothesis that the regression coefficients are statistically different from zero [Schmith et al., 2007].

4.6 Sea Surface Temperature

[54] The equilibrium relation for sea surface temperature is given by the fifth cointegrating relation (Table 4), which includes SST, Fe, Na, and SunSum. The negative relation with Na suggests that sea ice cools water directly, and indirectly by increasing surface albedo. A similar effect may generate the negative relation with Fe.

5 Conclusion

[55] The high-dimensional CVAR suggests a caveat about the general strategy used to identify and quantify mechanisms thought to drive glacial cycles. To date, many of these efforts focus on bivariate relations. As indicated by the large differences between the single equation and total system effects of changes in CO2 and sea ice, feedbacks in the climate system can amplify linkages that have small direct effects. Under these conditions, a single equation approach may (1) miss a third variable that drives a correlation between two other variables, and (2) therefore understate their importance in the climate system. Because the cointegrating relations and the differenced variables in the CVAR generally are close to orthogonal, the results do not suffer from the usual collinearity problem. Thus, the multivariate approach will reduce bias due to omitted variables and collinearity.

[56] Despite these advantages, we recognize that statistical analyses are not definitive. Clearly, physical models are necessary to understand the linkages suggested by statistical models. Hopefully, the statistical results presented here can help modelers focus on the physical linkages that may be most important and test the importance of mechanisms suggested by their models against the observational record.