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
 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).
 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. .
Table 4. Long-run Equilibrium Relations Implied by the Cointegrating Relations
| ||Cointegrating Relations Normalized for Equilibrium Adjusting Variable||Rate of Equilibrium Correction|
|CR#1||Tempt = 0.83CO2t +μ1t||−0.495|
|CO2t = 1.027Tempt +μ1t||−0.137|
|CR#2||CH4t = –0.89Icet +0.18Ecct +μ2t||−0.495|
|CR#3||Icet = −0.97CO2t − 1 −0.10Ecct − 1 +μ3t||−0.105|
|CO2t = −1.03Icet − 1 −0.10Ecct − 1 +μ3t||−0.204|
|CR#4||Fet = 0.16Cat −0.85Levelt −2.02*Oblt +0.08*Prect +μ4t||−0.447|
|CR#5||SSTt = −0.25Fet −1.00Nat −0.24SunSumt +μ5t||−0.189|
|CR#6||SO4t = 0.72Fet +μ6t||−0.156|
|Fet = 1.04SO4t +μ6t||−0.090|
|CR#7||Cat = −0.71*Levelt −0.25Ecct +μ7t||−0.166|
|Levelt = −1.41Cat −0.0.35Ecct +μ7t||−0.028|
|CR#8||Levelt = −2.5*CO2t −2.01Icet +1.29SSTt +μ8t||−0.300|
|CR#9||CO2t = −2.02Nat −0.43SO4t −0.77SSTt +μ9t||−0.229|
|Nat = −0.49CO2t −0.21SO4t −0.38SSTt +μ9t||−0.330|
|CR#10||Icet = −0.76*Ecct +4.46Oblt +2.88SunSumt +μ10t||−0.037|
 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
 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].
 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.
 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].
 Statistical results are consistent with the general outline of the iron hypothesis. CR #6, which can be interpreted as a cointegrating relation for , shows a positive relation between 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.  who conclude that there is only a weak bivariate correlation between sulfate and dust flux over the last 150 Kyr.
 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 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].
 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 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. .
 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 , 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].
 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  but larger than estimates generated by Morales Maqueda and Rahmstorf  and Kurahashi-Nakamura et al. .
 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
 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
 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].
 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.
 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.
 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  explains the transition from isotopic stage 6 to 5 using summer insolation gradients. Similarly, Masson-Delmotte et al.  relates rapid changes in Greenland to large-scale changes in atmospheric circulation.
4.5 Sea Level
 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.  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).
 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].