1 Introduction
The Arctic is warming about twice as fast as the global average (Osborne2018). This phenomenon of Arctic amplification in surface air temperature is closely connected to a dramatic multidecade reduction in Northern sea ice. Indeed, since accurate satellite measurements began in 1978, the extent of Arctic summer sea ice has shrunk by about 40 percent, a loss in area comparable to the western continental United States. This drop in sea ice is one of the most conspicuous warning signs of ongoing climate change. In addition, the reduction of sea ice plays a critical role in the pace of future global climate change and has important implications for the polar region and the rest of the world.
At a regional level, diminishing sea ice alters polar ecosystems and habitats and introduces major economic opportunities and risks. For example, new deposits of natural gas, petroleum, and other natural resources will become accessible for extraction, emission, and possible spillage (Petrick2017). Also, reduced ice coverage facilitates tourism and the use of Arctic shipping routes, which are shorter than traditional passages via the Suez or Panama Canals. These new routes reduce sailing times but increase Arctic environmental risks from, for example, discharges, spills, and soot deposits (Bekkers2016). Finally, melting sea ice will have geopolitical consequences for Arctic sealane control (ebinger2009).
Although these proximal effects are important, the farreaching implications of diminished Arctic sea ice for regulating global climate and weather are even more consequential. Higher Arctic temperatures promote thawing and erosion of the polar permafrost, which can result in the release of large amounts of carbon dioxide and methane and provide a significant impetus to further global warming (Tanskietal2019). Increasing Arctic temperatures also hasten the melting of the Greenland ice sheet, further pushing up sea levels (Trusel2018). In addition, less sea ice and more open water diminishes the reflectivity (or albedo) of the Arctic region, so that, over time, a greater share of solar heat is absorbed by the earth, which leads to increased temperatures worldwide and further Arctic amplification (Hudson2011). Finally, a warming Arctic and loss of ice cover can alter the global dynamics of ocean and air streams, and this effect already appears to be changing weather patterns at subpolar latitudes (PetoukhovandSemenov2010) and weakening thermohaline ocean circulation including the current that warms Europe (LiuFedorov2019).
In brief, the loss of Arctic sea ice is not just a stark indicator of a changing climate, but it also plays an integral role in the timing and intensity of further global climate change. Not surprisingly then, the downward trend in Arctic sea ice has been the subject of hundreds of research studies. The forecasting literature alone is voluminous and impressive in both methodology and substance.^{1}^{1}1Recent research includes petty2017, Onoetal2018, SerrezeAndMeier2019, and Ionita2019. Ongoing prediction research forums include the Sea Ice Prediction Network at https://www.arcus.org/sipn and the Polar Prediction Project at https://www.polarprediction.net. Although the importance of accurate polar prediction is hard to overstate, substantial uncertainty still remains about the future evolution of sea ice. Indeed, obtaining a deeper understanding of Arctic sea ice loss has been called a “grand challenge of climate science” (Katssovetal2010).
Much forwardlooking sea ice analysis has been based on largescale climate models, which represent of the fundamental physical, chemical, and biological drivers of the earth’s climate. These models attempt to capture the dynamics of the oceans, atmosphere, cryosphere, and land surface at a high frequency and a granular level of geographic and spatial detail. Such structural physical models are invaluable for understanding climate variation, determining event and trend attribution, and assessing alternative scenarios. However, from a forecasting perspective, climate models have generally underestimated the amount of lost sea ice in recent decades (Stroeve2007; Stroeve2012; Jahn2016; and Rosenblum2017). In addition, longrange sea ice projections can differ widely across climate models (StroeveNotz2015).
Given the global significance of Arctic conditions and the progress yet to be made on structural global climate models, we provide a statistical analysis of the longrun future evolution of Arctic sea ice. There is already some evidence that smallscale statistical models with no explicitly embedded physical science can have some success in forecasting Arctic sea ice (Guemasetal2016; Wangetal2016). Our work is distinguished by its use of intrinsically stochastic “unobserved components” models, with detailed attention to trend, seasonality, and serial correlation. Based on several decades of satellite data, we provide statistical forecasts of the future loss of Arctic sea ice.^{2}^{2}2Our analysis does not examine prediction skill in realtime repeated forecasting, as in Ionita2019, but considers very longrange projections at a point in time, as in Guemasetal2016 . Importantly, these forecasts provide probability assessments of a range of longrun outcomes and quantify both model parameter uncertainty and intrinsic uncertainty. Of particular interest are probability assessments of the timing of an icefree Arctic, an outcome with vital economic and climate consequences (Massonnet2012; Snape2014; and Jahn2016). For this analysis, we also introduce a novel statistical modeling mechanism – a shadow ice interpretation – that allows us to readily account for the zero lower bound on the extent of Arctic sea ice in our model.
Our resulting distributional forecasts suggest an icefree Arctic summer is more likely than not within two decades – much sooner than the projections from many largescale climate models. In particular, we contrast our statistical forecasts with projections from the ensemble of model simulations conducted for the fifth Coupled Model Intercomparison Project (CMIP5) – a highlyregarded central source for international global climate model projections. On average, these climate models envisage icefree Arctic conditions close to the end of the century (assuming a range of businessasusual carbon emissions paths). Thus, besides their relevance for environmental and economic planning, our probability assessments may also provide a useful benchmark for assessing or calibrating global climate models going forward.
We proceed as follows. In section 2, we introduce a linear statistical model and use it to produce longrange sea ice point forecasts. In section 3, we introduce the “shadow ice” concept to account for the zeroice lower bound. In section 4, we generalize to a nonlinear (quadratic) statistical model and to interval forecasts that incorporate several forms of uncertainty. In section 5, we compare our statistical model forecasts to global climate model forecasts with particular attention to hard versus soft landings at zero ice. In section 6, we make probabilistic assessments of several sea ice scenarios. We conclude in section 7.
2 A Linear Statistical Model and Point Forecasts
Arctic sea ice has been continuously monitored since 1978 using satellitebased passive microwave sensing, which is unaffected by cloud cover or a lack of sunlight. For a polar region divided into a grid of individual cells, the satellite data provide a brightness reading for each cell, which can be converted into fractional ice surface coverage estimates for each cell. Sea ice extent,
– a common measure of total ice area – is the total area of all cells with at least 15 percent ice surface coverage. That is, rounds down cells with measured coverage of less than 15 percent to zero and rounds up cells that pass the 15 percent threshold to full coverage.^{3}^{3}3Another measure of Arctic ice is sea ice area, which adds together the measured fractions of icecovered areas of all cells that pass the 15 percent threshold. For details, see StroeveNotz2015. The uprounding in is effectively a bias correction, as melting pools on summer ice surfaces can be mistaken for icefree open water. Our analysis uses monthly average data from November 1978 through October 2019 from the National Snow and Ice Data Center (NSIDC). The NSIDC data use the NASA team algorithm to convert the satellite microwave brightness readings into measured ice coverage (Fettereretal2017).^{4}^{4}4We interpolate the missing December 1987 and January 1988 observations with fitted values from a regression on trend and monthly dummies estimated using the full data sample.
Figure 1 plots the time series of Arctic – each monthly average observation is a dot – with an estimated linear trend superimposed. The clear downward trend is accompanied by obvious seasonality. A more subtle feature is the possible time variation in the seasonal effects, which may be trending at different rates and possibly nonlinearly. These effects turn out to be of interest in a complete statistical representation of the dynamics of sea ice.
A simple initial representation to capture this variation is a linear statistical model with twelve intercepts, one for each month, each of which may be differently trending, and potentially serially correlated stochastic shocks:
(1) 
where the
’s are monthly dummy variables (
in month and 0 otherwise, ) and is a time dummy (). Model (1) – and other versions below – are estimated by maximizing the Gaussian likelihood. Detailed regression results for model (1) are in column (6) of Table A1 in Appendix A.Figure 2 shows the resulting linear trends for all twelve months, highlighting March in blue and September in red. All of the monthly trends slope downward – an indication of a warming climate – and are highly significant. The slopes of the linear trends also differ across months (SerrezeAndMeier2019; Cavalierietal2012). In particular, the summer months of July through October have notably steeper downward sloping trends than the winter months of December through May. The estimated September trend, for example, is twice as steep as the March trend, and the difference is highly statistically significant. These linear trends are also extrapolated out of sample (shaded gray) through the end of the century. For example, September sea ice extent is projected to reach zero just after 2072.
Such linear point forecasts are a useful first step, but they can be improved by allowing for nonlinearity in the trends and by quantifying forecast uncertainty – as described in section 4. First, however, we elucidate a “shadow ice” modeling approach that takes into account the fact that the measured amount of sea ice is bounded below by zero.
3 A Shadow Ice Interpretation
One consideration for downward trending statistical models for Arctic sea ice is that the measured amount of sea ice will always be nonnegative. In contrast, extrapolations of simple trending models will eventually push into negative territory. There are various functional forms that can be used to model such bounded time series, and the appropriate representation depends very much on the details of the realworld phenomenon under examination.^{5}^{5}5One modeling approach is to rescale the bounded time series data to the real line using, say, a logratio transformation (Wallis1987)
. Alternatively, a time series can be modeled in the original bounded sample space using, for example, the beta autoregressive model of
Rocha2008. Some bounds act like reflecting barriers, so the variable of interest spends very little time at the constraint. Other bounds are absorbing states, and once reached, they may be sustained for some time.With positive amounts of sea ice, fluctuations in SIE can serve as a rough approximation for changes in the amount of thermal energy in the Arctic; that is, hotter and colder surface temperatures are reflected in less or more ice, respectively. However, this connection breaks down when the ice disappears: While SIE is fixed at zero, the surface temperature of the Arctic ocean can continue to warm. Furthermore, the warmer the ocean becomes, the less likely there will be a quick return of sea ice, which is indicative of a partiallyabsorbing state.
To account for this effect, the negative values of sea ice produced by a statistical model can be viewed as a rough expression of ocean temperature. Thus, we redefine the lefthand side variable of the unconstrained model as a shadow surface ice extent, . We view as a notional variable that equals measured surface ice when positive, but that may also go negative to represent ocean thermal energy more broadly. Formally, to translate negative modelbased sea ice values into nonnegative sea ice observations, our shadow ice model modifies the unconstrained model (1) to respect the zero lower bound for ice:
(2) 
That is, we now interpret our earlier unconstrained linear model of surface ice as a model of shadow ice, , so that the observed extent of sea ice, SIE, is the maximum of and zero.^{6}^{6}6A similar framework has been successfully applied in finance to model nominal interest rates near their zero lower bound (CR2014; BR2016).
The shadow ice model respects the nonlinearity of observed ice at the zero lower bound but retains tractability. It also allows us to translate the longrange forecasts from downward trending models like model (1) – including distributional projections – into observed data that are always nonnegative. As a matter of physical interpretation, very negative values of shadow ice extent, , represent environments in which the thermal content of the Arctic ocean is high enough that an immediate return to a positive SIE is unlikely. This shadow ice structure provides an intuitive and simple approximation of the thermodynamics of the Arctic ocean transition between sea ice and open water and serves as a useful modeling tool for observed SIE dynamics.^{7}^{7}7Wangetal2016 take a different approach by simply constraining the model by the lower bound (for sea ice concentration in their case), so negative predicted values are is simply set to zero. In a dynamic model with lagged sea ice, this procedure will result in a representation of a physical bound that is much closer to a reflecting barrier.
4 A Quadratic Statistical Model and Interval Forecasts
A downward linear trend is a common representation of the secular decline in Arctic sea ice, but linearity is not assured by the physical science. There are a variety of climate feedback mechanisms that could hasten or retard the pace of sea ice loss. The wellknown ice albedo effect occurs as sea ice cover is reduced, and the resulting darker ocean surface absorbs more energy, which in turn further reduces sea ice (Stroeve2012). This feedback effect amplifies sea ice seasonality and may progressively steepen the downward trend in over time (Schroder2014). The geography of the Arctic Ocean, which is constrained by land masses that can partially block expanding winter ice becomes less relevant as sea ice shrinks, and relaxing this constraint may allow greater seasonal variation and a steeper downward sea ice trend (SerrezeAndMeier2019). However, there are offsetting negative feedback mechanisms – associated, for example, with increased cloud cover – that could slow the rate of sea ice loss over time (IPCC2019). Indeed, StroeveNotz2015 argue against trend amplification in favor of trend constancy (linearity) or trend attenuation. Moreover, as described in the next section, longrange projections from largescale global climate models appear dominated by feedback mechanisms that slow the rate of September sea ice loss over time.^{8}^{8}8More extreme forms of nonlinearity – such as discontinuous breaks, tipping points, and thresholds – are possible but viewed as less likely (StroeveNotz2015). GoldsteinEtAl18 argue that Arctic sea ice is best modeled by steplike shifting means at fitted breakpoints. However, modified statistical information criteria that properly account for the implicit flexibility of such breakpoints – following Hall2013 – do not favor such a shifting mean models relative to a linear trend.
The lack of a complete understanding of the drivers of Arctic sea ice recommends consideration of a flexible empirical specification, so we generalize from linear to quadratic trends:
(3) 
We label model (3) as the “general” quadratic model as no constraints are imposed on the twelve quadratic () parameters. The linear model (2) of course emerges as a special (constrained) case, when .
Figure 3 shows the estimation fits and forecasts from the general quadratic model, by month, with March and September highlighted in blue and red, respectively. The trend curvatures for all months are strikingly similar: the trends for all months decrease at an increasing rate. That is, the estimated coefficients are negative for every month, indicating that is diminishing at an increasing rate. (Detailed estimation results of model (3) appear in column (1) of Table A1 in Appendix A.) The size of the estimated negative coefficients on the quadratic trend terms and their statistical significance vary by month. The most negative and significant coefficients are in the summer months of August, September, and October, and these months show the greatest trend rates of decline. An test of joint hypothesis that produces a value of 0.00.^{9}^{9}9Because and are correlated, an insignificant coefficient would not necessarily imply that nonlinearity is unimportant. Relative to a linear trend model, the nonlinear trend model forecasts lower sea ice at long horizons. FebruaryApril point forecasts nevertheless remain well above zero through the century, but AugustOctober point forecasts approach zero much more quickly. Indeed the quadratic September point forecast hits zero in 2045.
Table 1 summarizes the results from an indepth statistical investigation of the summer and nonsummer differences in quadratic trend curvature using two standard model selection criteria: the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). The AIC and BIC
are estimates of outofsample forecasting performance (meansquared error), formed by penalizing estimates of insample forecasting performance for degrees of freedom used in model fitting and differing only in the precise penalty applied
(Diebold2007). The table reports these model selection criteria for six versions of the quadratic trend model (3) with various equality constraints imposed on the quadratic coefficients (with corresponding estimation results in Table A1 in Appendix A).The models favored by AIC and BIC – that is, those with smaller values – are very similar and involve summer and nonsummer restrictions. The AIC selects a model with equal ’s for the nine nonsummer months (NovemberJuly) and unconstrained ’s for the three summer months (AugustOctober). The BIC, which penalizes degrees of freedom more harshly, selects a slightly more constrained model, with the nonsummer ’s again constrained to be equal and the three summer ’s also constrained to be equal. Compared to the unconstrained version of equation (3) – the general model – both the AIC and BIC prefer specifications with some summer and nonsummer equality constraints imposed. Still, the constraints are not very binding, so their imposition saves degrees of freedom without substantially degrading fit. All told, the results of Table 1 suggest a “simplified” quadratic model, namely model (3) with both summer and nonsummer quadratic coefficients constrained separately to equality: and .
(1)  (2)  (3)  (4)  (5)  (6)  
NONE  Seq  NSeq  Seq+NSeq  ALLeq  ALL0  
AIC  0.0673 [3]  0.0651 [4]  0.0913 [1]  0.0877 [2]  0.0639 [5]  0.0569 [6] 
BIC  0.2569 [6]  0.2421 [5]  0.1647 [2]  0.1513 [1]  0.1665 [4]  0.1649 [3] 
Notes: We show AIC and BIC vaules for the quadratic model with various equality constraints imposed on the quadratic coefficients . “NONE” denotes no constraints, which corresponds to the general quadratic model (3). “Seq” (“Summer equal”) denotes AugustOctober equal (). “NSeq” (“NonSummer equal”) denotes NovemberJuly equal (). “Seq+NSeq” denotes summer months equal and nonsummer months (separately) equal, which corresponds to the simplified quadratic model. “ALLeq ” denotes all months equal (). “ALL0” denotes all months 0 (), which corresponds to the linear model (1). Model ranks appear in brackets, where [1] denotes the best (smallest) criterion value. We show in boldface the best two models according to each criterion.
Figure 4 shows the simplified quadratic model fits and forecasts. The simplified model point forecasts are very similar to those of the general quadratic model in Figure 3, with a zeroice September also reached in 2045, but the rank ordering of the months in terms of is better preserved going forward. Going beyond these point forecasts, an important advantage of a formal statistical approach is that it can quantify the amount of future uncertainty. Figure 4 supplements the simplified quadratic trend point forecasts with interval or probability density forecasts. When making longhorizon interval forecasts, it is crucial to account for parameter estimation error, because its deleterious effects grow with the forecast horizon. For example, although parameter estimation error may have small effects on 6monthahead intervals, it will be greatly compounded for 600monthahead intervals. An estimate of the timestandard deviation of the forecast error, which accounts for parameter estimation error, is , where
is the standard error of the regression,
is a 361 column vector of time
righthandside variables, is a 36 matrix whose columns contain the regression’s righthandside variables over time, and is sample size.^{10}^{10}10See for example Johnston1972, pp. 153155, for derivation of this canonical result. We use to produce the pointwise prediction intervals of Figure 4. Under normality of the shocks underlying the simplified quadratic model, theintervals are approximate 95 percent confidence intervals.
^{11}^{11}11Normality of theshocks does not appear unreasonable, as the simplified quadratic model residuals have skewness and kurtosis of 0.17 and 3.73, respectively. We also obtained similar results without assuming normality via bootstrap simulation, which we discuss in section
6. The intervals widen rapidly; indeed the September interval starts to include zero before 2040.^{12}^{12}12In contrast, interval forecasts that fail to account for parameter estimation uncertainty quickly approach (by about 12 months ahead) the fixedwidth interval , where is the estimated unconditional standard deviation of the disturbance in equation (3), and fail to widen with forecast horizon.Finally, in Figure 5 we zoom in on the 20602062 (36month) segment of the simplified quadratic model shadow ice forecast. The point forecast trends down and is seasonally below zero by then. (Indeed, as already discussed, the point forecast is seasonally below zero well before then.) Moreover, the intervals widen over time, and entire intervals are seasonally below zero by then. Hence it appears that, with near certainty, summer will vanish by 2060. This result is also clear from our earlier Figure 4, but Figure 5 highlights it in a different and complementary way.
5 Statistical versus Climate Model Projections
Of the many analyses of the longterm future evolution of Arctic ice, most have focused on projections from largescale climate models. Such models are based on the underlying physical, chemical, and biological processes that govern the dynamics of weather and climate across ocean, air, ice, and land. The models fit an immense number of variables at a high temporal frequency and a granular spatial scale (e.g., a 30minute time interval and a 100km worldwide grid). Dozens of scientific groups around the world have constructed and currently maintain such models. Occasionally, these groups conduct concurrent simulations as part of the Coupled Model Intercomparison Project (CMIP) that involve common sets of inputs including carbon emissions scenarios. The most recently completed iteration or phase of this project is the fifth one, denoted CMIP5 (Taylor2012). The CMIP5 model comparison study was the main source of climate projections included by the International Panel on Climate Change (IPCC) in its landmark Fifth Assessment Report.
Arctic is a key variable projected by climate models, and the models included in CMIP5 are generally judged to provide a better fit to Arctic sea ice than earlier CMIP iterations.^{13}^{13}13Stroeve2007 describes the poor sea ice fit of the CMIP3 models, while the somewhat improved fit of the CMIP5 models is noted by in Stroeve2012. Figure 6 shows three projections for September constructed as averages across sets of CMIP5 global climate models. These multimodel mean projections are constructed under three different scenarios, or Representative Concentration Pathways (RCPs), for future greenhouse gas concentrations. The brown, yellow, and blue lines are averages of the models, respectively, under a high level of carbon emissions (RCP8.5), a medium level (RCP6.0, yellow), and a low level (RCP4.5, blue).^{14}^{14}14The climate model data are described in SerrezeAndMeier2019 and Stroeve2012 and were kindly provided by Andrew Barrett at the National Snow and Ice Data Center. The model sets averaged for each scenario are not identical, with the RCP4.5, RCP6.0, and RCP8.5 scenarios based on 26, 8, and 25 climate models, respectively. The first two higher emissions scenarios are viewed as more likely businessasusual outcomes and are the most relevant to compare to statistical projections that extend the historical sample of past data and assume a continuation of the world economy’s current population and development trajectories.^{15}^{15}15In the RCP8.5 scenario, continuing increases in greenhouse gas emissions through the end of the century raise the 2100 global average temperature by about 4.06.0C above preindustrial levels (USGCRP2018). In the RCP4.5 scenario, greenhouse gas emissions level off before midcentury, and the 2100 global average temperature is approximately 2.03.0C above preindustrial levels.
The solid red line in Figure 6 shows the September trend from the simplified quadratic model estimated on the full sample from November 1978 to October 2019. The dotted lines bracketing this forecast provide an approximate 95 percent confidence interval. However, the CMIP5 climate model projections are only based on data through 2005 and do not include the past dozen or so years of sea ice observations. For comparability to these climate model projections, we reestimated the simplified quadratic model using data from 1978 through 2005, and the September trend from this pre2006 model is shown as the dashed red line. An interesting first result is that from the close conjunction of the two simplified model statistical projections (the solid and dashed red lines), the addition of data from 2006 to 2019 does not lead to a significant revision in the statistical trend model. The very modest differences between the pre2006 estimated quadratic trend and the fullsample version is an indication of the stability and suitability of the simplified statistical model.
Comparing the statistical and climate model projections in Figure 6 reveals two salient features. First, throughout the forecast period, the fullsample statistical model projection is significantly lower than any of the climate multimodel mean projections. Indeed, the climate model mean projections are well outside the 95 percent confidence intervals. Relative to the statistical model estimated on data before 2006 or the fullsample version, the climate model means are overestimating the 2019 trend by about 1.0 million km^{2}. This wedge is projected to increase dramatically over the next two decades. Indeed, the pre2006 and fullsample statistical trend models project zero ice in 2042 and 2044, respectively, but none of the climate model mean projections reach a completely icefree Arctic in this century. Notably, even assuming a high level of emissions (RCP8.5) – which is a scenario with continuing increases in average global surface temperatures throughout this century – the multimodel mean projection never reaches zero Arctic summer sea ice.
Second, in contrast to the statistical projection, the climate model mean projections show a decreasing rate of ice loss over time – that is, a concave rather than convex structure. Specifically, the climate model projections all display a roughly linear decline for the first couple of forecast decades (through around 2040) and then start to level out. For the lower emissions RCP4.5 scenario, the leveling out and deceleration of sea ice partly reflects a slowdown in the pace of global temperature increases. This effect is not at work in the RCP8.5 scenario as temperatures steadily climb through 2100. However, very close to zero sea ice extent, the leveling out of in RCP8.5 appears to reflect the hypothesized difficulty of melting the thick sea ice clinging near northern coastlines – notably in Greenland and Canada. Climate models generally assume that these coastal regions will retain sea ice for a time even after the open Arctic Sea is free of ice. Therefore, a common definition of “icefree” or “nearly icefree” in the literature is a threshold of 1.0 million km^{2} rather than zero SIE (WangOverland2009). Still, even with this higher threshold, the three climate model mean projections only reach a nearly icefree Arctic in 2068, 2089, and 2100 for successively lower emissions scenarios, respectively. In contrast, the pre2006 statistical projection reaches the higher 1.0 million km^{2} threshold in 2037, and the fullsample statistical model reaches that level in 2039.
Some have argued that climate models generally do well in representing the largescale evolution of Arctic sea ice (StroeveNotz2015), but a number of studies have noted that the CMIP5 global climate models overpredicted the amount of Arctic sea ice (Massonnet2012; StroeveEtAl2012b; SerrezeAndMeier2019). That overprediction continues, and its source is not well understood. One proposed correction to this overprediction has been to focus on the models that fit the historical data better according to certain metrics (WangOverland2012). However, there is no agreed upon model selection criterion. Also, Rosenblum2017 discount such model selection because models with more accurate sea ice readings also tend to overpredict global temperatures, so the selected climate models may be getting sea ice loss right for the wrong reason. Finally, it should be noted that from a statistical viewpoint, focusing on a simple average forecast from many models has been shown to be a robust prediction strategy (DieboldAndShin2019).
In some respects, the wide differences between the statistical and climate model projections are not too surprising. In climate models, the monthly observations on total Arctic SIE are a highlevel output from complex, nonlinear, granular representations of the relevant underlying science. Obtaining good SIE predictions from these models requires correctly specifying a host of detailed subsidiary processes. Such a bottomup modeling procedure has important advantages in structural interpretation and counterfactual scenario analysis. However, in a variety of disciplines, a bottomup procedure, which carries the possibility that small misspecifications can accumulate and affect highlevel aggregates, has not been found to improve prediction relative to a topline procedure that directly models the object of interest (Diebold2007). Thus, based on broad previous experience, we believe that direct statistical projections of Arctic SIE are likely to be relatively accurate.
6 Probability Assessments of an IceFree Arctic
An advantage of a formal statistical model is its ability to make probability density forecasts for a range of possible reduced Arctic sea ice scenarios. Of particular interest are probability assessments of an icefree or nearly icefree Arctic, that is, the probability that equals zero or is less than or equal to some threshold , respectively. Formally, such event probabilities can be denoted as , which represents the probability that sea ice extent is less than or equal to in month
. We estimate these scenario probability distributions using the simplified quadratic model and a stochastic simulation procedure that accounts for parameter estimation uncertainty and allows for potentially nonGaussian serially correlated stochastic shocks. From a given set of simulated paths, we estimate the event probabilities of interest as the proportion of simulated paths in which the event occurs out of the total number of paths.
^{16}^{16}16We provide details on this simulation procedure in Appendix B. For further discussion of the econometrics of threshold event probabilities, see BR2016.An event that has attracted much attention in the literature is the initial occurrence of an icefree or nearly icefree September. We calculate the probability for each September with date so that and for all . Specifically, for a given and simulation , we determine the year in which September first reaches , and then we cumulate across across all simulations to build a distributional estimate. The red lines in Figure 7 provide the resulting probability distributions for an initial “icefree” Arctic September for , , and , that is, for progressively more lenient definitions of “icefree.” As noted above, the middle value of , which represents Arctic sea ice of less than million , is a popular benchmark in the literature (WangOverland2012). For that particular threshold, the statistical model produces a distribution centered at 2039.^{17}^{17}17Both the mean and median round to 2039 despite a slightly skewed distribution toward longer times. Of particular interest is the probability distribution of dates taking into account model parameter uncertainty and stochastic shock uncertainty, and this distribution shows about a 60 percent probability of an effectively icefree September Arctic occurring in the 2030s.
The distribution with is bracketed on either side by distributions that use the higher and lower thresholds. The distribution has a median date of 2044 and a 95 percent range from 2039 to 2053. The distribution has a median date of 2033 and an earlier and slightly narrower 95 percent range from 20230 to 2039. The climate modeling literature has pointed to several factors that underpin the uncertainty in the timing of an initial September icefree Arctic including natural climate variability, emissions path uncertainty, and model uncertainties related to sea ice dynamics among other elements (SerrezeAndMeier2019). These factors are at least partially accounted for in our analysis.
The multimodel mean CMIP5 climate projections described above are well outside the abovedescribed distributional ranges. For a threshold of 1 million km^{2}, the mean projections reach a nearly icefree Arctic in 2068 and 2089 under the RCP8.5 and RCP6.0 scenarios, respectively. The former date is denoted by a vertical brown bar in Figure 7. The range of dates across individual models is also extremely wide, stretching well over a century (Jahn2016). To narrow this range, researchers have omitted models with poor performance using a variety of sea ice metrics. With such model selection procedures, the range of dates for a first nearly icefree September is narrowed greatly to a 20year span that runs from the 2040s to 2060s (Massonnet2012; ThackHall2019).^{18}^{18}18Again, the caveats of Rosenblum2017 regarding such model selection are relevant. Even such a carefully circumscribed span is roughly a decade later than the simplified quadratic trend model produces. Moreover, the climate model simulation exercises are not designed to yield formal error estimates or measures of uncertainty as the spread of climate model forecasts in an ensemble is insufficient to completely characterize forecast uncertainty.
Finally, we note the availability of density forecasts for a variety of richer joint scenarios of interest. As one example, Jahn2016 describe other definitions of “icefree” that involve, for example, 5year running means. Alternatively, “icefree” may require no ice for several consecutive months, for example, to accommodate meaningful freight shipping, tourism, mining, and commercial fishing (Aksenov2017). For example, the strong Autumn demand for international freight shipping to satisfy yearend Western holiday consumer demand could make a multimonth icefree Arctic shipping lane of interest. In this case, the probability distribution of the initial occurrence of an “icefree” summer – a joint icefree August, September, and October – could be relevant. Figure 7 shows this distribution in black assuming an icefree threshold of . This density is notably shifted right – and more rightskewed – compared to the September scenarios.
7 Concluding Remarks
A rapidly warming Arctic is an ominous sign of the broader climate change caused by human activity, but declining Arctic sea ice also has an important influence on the pace and intensity of future climate change. Using statistical models, we have provided probabilistic projections of 21stcentury Arctic sea ice that account for both intrinsic uncertainty and parameter estimation uncertainty. These projections indicate that summertime Arctic sea ice will quickly diminish and disappear – with about a 60 percent probability of an effectively icefree September Arctic occurring within two decades.
By contrast, the average projection from leading climate models implies an initial seasonal icefree Arctic several decades later – even assuming a businessasusual emissions path. The slow and decreasing pace at which largescale climate models reach an icefree Arctic may be a serious shortcoming, and such conservative projections of sea ice loss could be a misleading guide for global climate policy.^{19}^{19}19Our statistical results are very much in line with the concerns of Stroeve2007 regarding the possibility that the slow projected decline in Arctic sea ice by climate models suggests they are underestimating the effects of greenhouse gases. There are numerous examples, across many disciplines, showing that parsimonious statistical representations can provide forecasts that are at least as accurate as the ones from detailed structural models.^{20}^{20}20A classic example from economics is Nelson1972, who showed that simple statistical models forecast the economy as well as largescale structural models based on economic theory. However, rather than treat statistical models as just forecast competitors to climate models, there is very likely to be scope to use them as complementary representations going forward. The mechanisms governing Arctic sea ice loss and connecting that loss to atmospheric, oceanic, and permafrost responses are not fully captured in climate models. Statistical models may be able to help assist in bridging such gaps until a more complete understanding is available. In addition, statistical models may also be able to provide a benchmark for model performance that can be used for model calibration or tuning. Finally, the statistical models may also play a useful role in helping select among various climate models. All this suggests that further statistical model research extended to a multivariate setting – say, jointly treating Arctic sea ice and other variables – may be of particular interest.
Appendix A Detailed Regression Estimation Results
empty
(1)  (2)  (3)  (4)  (5)  (6)  

NONE  Seq  NSeq  Seq+NSeq  ALLeq  ALL0  
15.1121*  15.1372*  15.0865*  15.0922*  15.0373*  15.2274*  
15.9436*  15.9570*  15.9404*  15.9457*  15.8894*  16.0804*  
16.0792*  16.0820*  16.0049*  16.0100*  15.9524*  16.1446*  
15.3521*  15.3443*  15.2278*  15.2326*  15.1741*  15.3674*  
13.8027*  13.7832*  13.7604*  13.7650*  13.7056*  13.9001*  
12.4776*  12.4443*  12.4352*  12.4397*  12.3796*  12.5749*  
10.4243*  10.3733*  10.4804*  10.4849*  10.4243*  10.6202*  
8.0945*  8.0202*  8.1441*  8.0754*  8.2547*  8.4508*  
7.4698*  7.3645*  7.5179*  7.4201*  7.6022*  7.7976*  
9.1633*  9.2520*  9.2146*  9.3080*  9.4932*  9.6870*  
11.4307*  11.4872*  11.4928*  11.4997*  11.4488*  11.6374*  
13.5642*  13.6042*  13.5673*  13.5735*  13.5205*  13.7097* 
(Continued on next page.)
(1)  (2)  (3)  (4)  (5)  (6)  
NONE  Seq  NSeq  Seq+NSeq  ALLeq  ALL0  
0.0026  0.0029*  0.0023  0.0024*  0.0017  0.0040*  
0.0023  0.0024  0.0022*  0.0023*  0.0016  0.0039*  
0.0027  0.0028  0.0018*  0.0019  0.0012  0.0035*  
0.0031  0.0030  0.0016  0.0016  0.0009  0.0033*  
0.0019  0.0016  0.0014  0.0014  0.0007  0.0031*  
0.0028  0.0024  0.0023*  0.0024*  0.0016  0.0040*  
0.0034*  0.0028*  0.0041*  0.0042*  0.0034*  0.0058*  
0.0021  0.0012  0.0027  0.0019  0.0040*  0.0064*  
0.0030*  0.0018  0.0036*  0.0024*  0.0045*  0.0069*  
0.0006  0.0016  0.0012  0.0023*  0.0044*  0.0068*  
0.0021  0.0028*  0.0029*  0.0030*  0.0023*  0.0046*  
0.0023  0.0028  0.0024*  0.0024*  0.0017  0.0041*  
2.72E06  2.11E06  3.40E06  3.29E06  4.78E06*  0  
3.31E06  2.99E06  3.40E06  3.29E06  4.78E06*  0  
1.53E06  1.47E06  3.40E06  3.29E06  4.78E06*  0  
3.24E07  5.17E06  3.40E06  3.29E06  4.78E06*  0  
2.38E06  2.84E06  3.40E06  3.29E06  4.78E06*  0  
2.39E06  3.18E06  3.40E06  3.29E06  4.78E06*  0  
4.78E06  5.96E06*  3.40E06  3.29E06  4.78E06*  0  
8.57E06*  1.03E05*  7.35E06*  8.96E06*  4.78E06*  0  
7.89E06*  1.03E05*  6.69E06*  8.96E06*  4.78E06*  0  
1.24E05*  1.03E05*  1.11E05*  8.96E06*  4.78E06*  0  
4.99E06*  3.57E06  3.40E06  3.29E06  4.78E06*  0  
3.45E06  2.48E06  3.40E06  3.29E06  4.78E06*  0  
0.7302*  0.7284*  0.7298*  0.7270*  0.7288*  0.7461*  
0.0468*  0.0473*  0.0472*  0.0478*  0.0491*  0.0497* 
Notes: We show maximumlikelihood estimates of the parameters of the general quadratic model (3), with various equality constraints imposed on the quadratic coefficients . “NONE” denotes no constraints, which corresponds to the general quadratic model (3). “Seq” (“Summer equal”) denotes AugustOctober equal (). “NSeq” (“NonSummer equal”) denotes NovemberJuly equal (). “Seq+NSeq” denotes summer months equal and nonsummer months (separately) equal, which corresponds to the simplified quadratic model. “ALLeq” denotes all months equal (). “ALL0” denotes all months 0 (), which corresponds to the linear model (1). In each column, we show constrained parameter estimates in boldface. “*” denotes significance at the ten percent level.
Appendix B Simulation Methods
We estimate scenario probability distributions using the simplified quadratic model (that is, the quadratic model (3) subject to the constraints and ) and a simulation procedure that accounts for parameter estimation uncertainty and allows for potentially nonGaussian serially correlated stochastic shocks. To describe this simulation procedure, it is useful to rewrite the quadratic model (3) in a more concise notation:
where and
. We estimate the model by maximum likelihood using the historical data sample , as discussed earlier, and then we simulate future paths based on the estimated model. Simulation proceeds as follows:

Draw from the sampling distribution of the estimated parameter vector , and form , .

Draw by sampling with replacement from the observed ’s (with equal weights ), .

Draw from the sampling distribution of , and form , , using the last observed historical residual as the initial condition.

Form , .
Finally, we estimate event probabilities of interest as the relative frequency of occurrence across the simulated paths.
Comments
There are no comments yet.