Earth

Continents and their topography are shaped by plate tectonics and the resulting volcanic eruptions, both of which also affect climate and have played a key role in past great extinctions. Earth is being used here as a place carrier for land, both as available living space and as providing soil for forests, other “wild” areas, and agriculture for food supply, which accounts for roughly 40% of the planet’s land area, or 50 m km². Vegetation and soil together lock in about three times as much CO₂ as the atmosphere currently holds, as well as absorbing around one third of emissions. However, higher temperatures and greater rainfall will release some of that carbon (see Eglinton et al., 2021). Inner-city vertical and underground farms seem viable and may become essential providers of food as the climate warms (see Asseng et al., 2020).

Crops are frequently grown using artificial fertilizers (often made from methane and leaching nitrous oxide in runoff) and based on farmland created by deforestation, wetland draining, and mangrove removal, all adding to GHG emissions. Together with the methane given off by animal husbandry, especially by cattle, sheep, and goats, these all lead to substantial GHG emissions—humanity’s climate change “food print.” Reducing such emissions will not be easy, but steps can be taken.⁵ Land around volcanoes is fertile, so basalt dust could be added as a fertilizer and also absorbs atmospheric CO₂ (see Beerling et al., 2020; Nunes et al., 2014), and Biochar produced by pyrolysis of biomass and added to soil could also increase crop yields while reducing GHG emissions (see Woolf et al., 2010).

Climate change is increasing extreme land flooding from “rivers in the sky,” which can hold 15 times more water than the Mississippi River: Witness the massive floods in Australia during mid-March 2021 spanning a large area, with almost a meter of rain falling at Nambucca Heads over 6 days. There have also been massive floods recently in Europe and East Africa, bringing locust swarms and diseases like cholera.

Climate change is also increasing extreme drought, leading to a loss of crops, with the resultant stress on some plants like sorghums producing toxic hydrogen cyanide (see Shehab et al., 2020). Flooding and drought both lead to loss of soil, either from erosion or dust storms. Sea level rises cause coastal flooding, reducing usable land area, as well as forcing migration (see Figure 5).

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Figure 5. Rivers in the sky; dust storm; wild fire; coastal flooding.

Sources:(top left):https://psl.noaa.gov/arportal; (top right):https://public.wmo.int/en/our-mandate/focus-areas/environment/SDS; (bottom left):https://earthobservatory.nasa.gov/images/81919/rim-fire-california; (bottom right):https://en.wikipedia.org/wiki/Coastal_flooding

Air

Air is again a place carrier, here denoting the atmosphere, comprising mainly nitrogen (78%) and oxygen (21%), with greenhouse gases like water vapor (0.4%), carbon dioxide (CO₂), nitrous oxide (N₂O), and methane (CH₄), as well as ozone and some noble gases. Earth’s atmospheric blanket is essential to life, but seen from the space station, the atmosphere is a thin blue line round the planet, not much thicker than a sheet of paper around a soccer ball, so even small additional volumes of GHGs can have a large impact. Earth’s gravity and magnetic field are essential to retain its atmosphere against the solar wind and also to protect the ozone layer from damaging radiation.

Atmospheric gases have changed greatly over deep time, especially from volcanism and the exchange of CO₂ for oxygen through photosynthesis, so Earth’s range has included ice ages and tropical conditions, but Mars and Venus warn that atmospheric protection needs to be “just right.” Increased greenhouse gases generated by human activity, especially burning fossil fuels, are now the major cause of climate change. GHGs receive, then reradiate energy at different wavelengths between ultraviolet and longwave infrared: This reradiation is responsible for the atmospheric greenhouse effect. Eunice Foote (1856) showed that a flask of CO₂ heated greatly by the Sun, whereas those of water vapor and dry air did not, closely followed by the independent experimental evidence of John Tyndall (1859). Ortiz and Jackson (2020) analyzed Foote’s contribution and Levendis et al. (2020) suggested an alternative experiment.

Nitrous oxide emissions from nitrogen and phosphate fertilizers have doubled since 1970, so are about 7% of greenhouse gas emissions in 2020 (see Tian et al., 2020), but N₂O is nearly 300 times more potent per molecule than CO₂ as a greenhouse gas. Catalytic converters actually add to this growing problem while reducing toxic carbon monoxide (CO) vehicle exhaust emissions.

Atmospheric methane is now double the highest level over the past 800,000 years. CH₄ is about 20 times as powerful as CO₂ as a GHG, with a half-life in the upper atmosphere of around 15 years, gradually getting converted to CO₂. Estimates of methane in hydrates in 2020 are over 6 trillion tonnes, roughly twice the GHG equivalent in all fossil fuels: The release of even a small proportion of that could be disastrous.

Chlorofluorocarbons (CFCs) were destroying the ozone layer before the highly successful Montreal Protocol in 1987 greatly reduced their use. That agreement is a potential role model for a far more ambitious global commitment beyond the Paris Accord at the 21st Conference of the Parties (CoP21) to the 1992 United Nations Framework Convention on Climate Change. However, replacement refrigerants like halons and halocarbons, including hydrochlorofluorocarbons (HCFCs) and hydrofluorocarbons (HFCs), though less damaging to the ozone layer, are powerful greenhouse gases, meriting research for safer alternatives.

Fire

Fire is the place carrier for energy, currently obtained from burning vast volumes of fossil fuels. Humanity cannot continue to consume fossil fuels on the scale it does in the 21st century yet stay within the “carbon budget” required to achieve “net zero” GHG emissions, which is essential to prevent dangerous climate change. The resulting changing global temperatures are leading to increased frequency and severity of wild fires, from Australia, the Amazon, and California to Siberia, which is a potential tipping point from tundra melting and releasing methane. Wild fires create fire-induced thunderclouds, known as pyrocumulonimbus clouds, which increase aerosol pollutants trapped in the stratosphere and upper atmosphere, can generate fire tornados, and lead to flash floods while also igniting further spot fires and temporarily cooling the planet, like a moderate volcanic eruption. There is increased deforestation, especially in the Amazon and other tropical rainforests, with a loss of biodiversity, as well as increased GHG emissions from the burning of some of the forests.

The main hope is to replace fossil fuel-based energy by renewable sources from earth (using thermal energy such as ground-heat and air-source heat pumps), air (utilizing wind energy from onshore and offshore wind turbines), fire (from sunlight via solar cells, and nuclear, potentially including small modular reactors), and water (using hydroelectric energy from dams, diversion facilities, or pumped storage facilities). Table 2 records estimates of electricity-generating costs in $\pounds /\text{MWh}$ by different technologies. The costs of all renewable sources of electricity have been falling rapidly and seem likely to continue to do so. The share of U.K. electricity generated by renewables reached a peak of 60.5% in April 2020, according to National Grid data.

Zero GHG electricity generation from renewables is technically feasible but requires a huge increase in output and storage capacity (for windless cloudy periods), dependent on a very large investment. Sufficient supply could sustain electric transport, removing emissions from oil, and replace much household use of gas for heating and cooking. However, an electricity grid needs second by second balancing of electricity flows in an otherwise increasingly nonresilient system, dependent on highly variable renewable supplies, so both backup and instantaneously accessible storage are essential, suggesting an intelligent system that could exploit electric vehicle-to-grid storage (see Noel et al., 2019).

Water

Water may seem limitless and it surprises many people that in fact there is very little water, especially freshwater, on our planet. It is common to think that Earth is the “Blue Planet” where everyone is surrounded by an abundance of water, but widespread shallow oceans fool people: The Atlantic is only about 2.25 miles deep on average. The Pacific is wider and is deeper, at about 2.65 miles on average: At its deepest in the Challenger Deep of the Mariana Trench, it is roughly 6.8 miles down. In total, the Pacific holds 170 million cubic miles of water, just over half the 330 million cubic miles of water on Earth. If all sources of water on the planet were collected in a sphere, it would be just 860 miles in diameter. Consequently, it is easy to heat the oceans by emitting excessive volumes of CO₂ into the atmosphere (see Figure 1[c]), pollute them, fill them with plastic waste, and turn seawater into a weak carbonic acid. Ocean acidification impacts on many ocean species, especially calcifying organisms like oysters and corals, but also on fish and seaweeds, affecting the entire ocean food chain.

The worldwide ocean “conveyor belt” circulates heat and nutrients and carries oxygen to depths, maintaining the health of the oceans. A key driver is that warm water from the Gulf Stream moves north, evaporates, becomes saltier and denser, cools, and so sinks and flows south. Melting northern hemisphere ice could disrupt this circulation by diluting the denser salty water, as well as by increasing sea levels by about 18 cm by 2100 (see Hofer et al., 2020). Added to increasing volume from thermal expansion, rising sea levels have serious implications for coastal flooding, although the rises are not uniform, and recent coastal elevation measures have tripled estimates of global vulnerability to sea level rises (see Kulp & Strauss, 2019).

Conversely, southern ocean sea ice can dramatically lower ocean ventilation by reducing the atmospheric exposure time of surface waters and by decreasing the vertical mixing of deep ocean waters, leading to holding 40 ppm atmospheric CO₂ less at its maximum (see Stein et al., 2020). Currently, oceans absorb almost one third of anthropogenic CO₂ emissions, but as they warm, they will hold less.

Freshwater is a hugely important commodity and needs to be treated carefully. A total of about 19% of California’s electricity consumption goes toward water-related applications, such as treating, transporting, pumping, and heating. Additionally, about 15% of its in-state electricity generation comes from hydropower, yet the frequency of both high- and low-flow extreme streamflow events have increased significantly across the United States and Canada over the 20th and 21st centuries (see Dethier et al., 2020).

Kelp “forests” and seagrass “meadows” absorb and store CO₂, help offset increasing carbonic acidification as well as removing some pollutants, provide nurseries for young sea life, and help protect coasts against rising sea levels. Improving seaweed farming and raising aquaculture production by marine protection areas seem sensible, noting that offshore wind farms also act as marine reserves.

While the subsections have discussed each of earth, air, fire, and water separately, the interactions between them are obvious. Given their roles in climate change, they form the basis of many of the time series that are used to analyze climate change. The next section explores the problems of assuming such data are stationary when in fact they are highly nonstationary, in part due to the many changes already discussed.

Simulating SIS for an Intercept Shift

The results in Equation 19 are for a single draw of the time series, but Autometrics (see Doornik, 2009) can simulate SIS. Using the same settings as for Figure 10 and selecting step indicators at $0.5\%$ with no backtesting until the final stage, yields the estimates reported in Figure 11, with ${\tilde{\beta }}_i$ denoting coefficient estimates using indicator saturation estimation (ISE) where ${\hat{\beta }}_i$ is used to indicate OLS estimation with no ISE.

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Figure 11. (a) One time series from Equation 15; (b), (c) simulated recursive parameter estimates with estimated standard errors (ESEs) and Monte Carlo standard deviations (MCSDs) when applying SIS; (d) resulting forecast Chow tests.

The outcome is dramatically different from that in Figure 10. The estimate of ${\beta }_1$ is now relatively constant at between $0.4$ and $0.5$; the estimate of ${\beta }_0$ switches quickly from around unity to zero after $t=80$, and the forecast tests reject at about their nominal significance levels. The step indicator at $t=80$ was selected with a probability of $0.84$, rising to $0.97$ within $\pm 1$ of $t=80$. Irrelevant indicators were selected with probability $0.008$ (the empirical gauge), close to the nominal significance level of $0.005$, so there is almost no overfitting. The Monte Carlo standard deviations (MCSDs) are somewhat wider than the estimated standard errors (ESEs), so the latter slightly underestimate the uncertainty.

Simulating SIS for a Shift in Dynamics

A legitimate question is what if, instead of ${\beta }_0$ shifting, the intercept remained constant and the dynamic parameter ${\beta }_1$ changed? For example:

Setting the same parameter values for ${\beta }_0$, ${\beta }_1$, ${\sigma }_{ϵ}$, and $T_1=80$ as before, with ${\beta }_1^{*}=0.75$, the previous Monte Carlo is repeated, leading to Figure 12 for recursive estimation with no ISE.

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Figure 12. (a) A time series from Equation 20; (b), (c) recursive parameter estimates with estimated standard errors (ESEs) and Monte Carlo standard deviations (MCSDs); (d) forecast Chow tests.

Apart from Panel (a), the other panels are similar to those in Figure 10. The average simulation full-sample estimates from $M=1000$ replications are now

so are also closely similar to Equation 16. Indeed, for the single data draw, the subsample fits are identical to Equation 17 for the first period, but for the second subperiod,

so the estimate of ${\beta }_1$ is barely altered, although it had changed, but ${\hat{\beta }}_0$ is greatly shifted, reflecting the major break obvious in the data plot in Figure 12(a). Consequently, it may not surprise that SIS can again capture much of the shift in the intercept:

The simulation outcomes when applying SIS reinforce these conclusions. Figure 13 records the results when SIS is applied at a significance level of $\alpha =0.005$. Panel (b) shows that the estimates of ${\beta }_1$ increase after $t=80$, although they remain close to $0.5$ rather than $0.75$. As before, the estimates of ${\beta }_0$ change after $t=80$, gradually rising to 2 (Panel c), and the forecast Chow tests reject close to their nominal significance levels (Panel d). The step indicator at $t=80$ was selected with a probability of $0.46$, rising to $0.73$ within $\pm 1$ of $t=80$, $0.89$ within $\pm 2$, and $1.0$ within $\pm 3$ of $t=80$, so detection is more spread out. Irrelevant indicators were selected with probability $0.010$, somewhat above the nominal significance level of $0.005$, reflecting the issue being a shift in dynamics, of which only the induced location shift is detected. The MCSDs are wider than the ESEs, so the latter again underestimate the uncertainty.

Why Is SIS Effective for a Shift in Dynamics?

It may seem puzzling that SIS can help facing a shift in the dynamics, but the crucial modeling problems are those that arise from shifts in the long-run, or equilibrium, mean rather than in other parameters, as emphasized in the forecasting context by Clements and Hendry (1998, 1999). Indeed, notice that the dominant visual feature of both Panel (a) data graphs in Figures 12 and 13, respectively, are the sudden departures from their previous average locations. Let $\mu ={\beta }_0/\left(1-{\beta }_1\right)$, which shifts to ${\mu }_{{\beta }_0^{*}}^{*}={\beta }_0^{*}/\left(1-{\beta }_1\right)$ in Equation 15, so it goes from 2 to 0, as seen in Figure 10(a). Then ${\mu }_{{\beta }_1^{*}}^{*}={\beta }_0/\left(1-{\beta }_1^{*}\right)$ in Equation 20 changes from 2 to 4, as in Figure 12(a). Defining $\nabla {\mu }^{*}={\mu }^{*}-\mu$ and $\nabla {\beta }_1^{*}={\beta }_1^{*}-{\beta }_1$, the two cases can be written as:

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Figure 13. (a) A time series from Equation 20; (b), (c) simulated recursive parameter estimates with estimated standard errors (ESEs) and Monte Carlo standard deviations (MCSDs) when applying SIS; (d) resulting forecast Chow tests.

Expressed as mean zero deviations about their equilibrium means, when either DGP shifts, the problem is the sudden appearance of a nonzero intercept where previously it was zero. That is what SIS can correct. For the ${\beta }_0$ shift, removing the new intercept recreates the previous DGP, whereas the ${\beta }_1$ shift also induces a third term that has a mean of zero, so SIS cannot correct that. From Equations 24 and 25, predicted values for the step-indicator magnitudes can be calculated, leading to -1 and 0.5, respectively. For example, from Equation 19, $\mu =2$, so $S_{80}$ should have a coefficient of unity, not significantly different from the 0.8 found, whereas in Equation 23, it should be –0.5, close to the outcome.

While this illustration is for a simple setting, the principles generalize to more general models, including selecting across variables as well as indicators.

Tackling a Shift in Dynamics by Multiplicative Indicator Saturation

Although SIS offsets much of the effect of the changing dynamics, the more appropriate approach is to model the change in ${\beta }_1$, and the tool for doing so is multiplicative indicator saturation, denoted MIS (see, e.g., Castle et al., 2020; Kitov & Tabor, 2015). In MIS, $y_{\text{t}+1}$ is multiplied by almost every step indicator, creating the $T-2$ additional candidate variables $S_j\times y_{t-1}$ for $j=2,\dots ,T-1$. Thus, SIS is in fact MIS for the constant term, but is a special case of importance given the pernicious effect unmodeled location shifts have on estimated parameters and forecasts.

For one draw, applying selection at $0.1\%$ over the MIS regressors, the resulting model is

which almost exactly replicates the DGP. The outcome shows a shift in ${\widehat{\beta }}_1$ from $0.49$ before $t=80$ to $0.74$ after, with a constant intercept. The simulation outcomes are reported in Figure 14. The interaction indicator at $t=80$ was selected with a probability of $0.42$, rising to $0.62$ within $\pm 1$ of $t=80,0.81$ within $\pm 2$, and $0.89$ within $\pm 3$ of $t=80$. Irrelevant indicators were selected with probability $0.007$, which falls to $0.003$ excluding the indicators within $\pm 3$ of $t=80$. The MCSDs are again wider than the ESEs, so the latter underestimate the uncertainty.

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Figure 14. (a) A time series from Equation 20; (b), (c) simulated recursive parameter estimates with estimated standard errors (ESEs) and Monte Carlo standard deviations (MCSDs) when applying MIS; (d) resulting forecast Chow tests.

Tackling Changing Trends by Trend-Indicator Saturation

As noted in the section “Why Change Matters,” if humanity is to avoid catastrophic climate change, the present upward trends in GHG emissions must be reversed to become rapid downward trends. Thus, graphs of fossil fuel use and GHG emissions shaped like the $\cap$ in Figure 1(b) may become common, as may less extreme but still marked trend changes like those in Figure 1(a), (c), and (d). Economics, demography, and epidemiology have all experienced sudden large changes in trends, as Figure 15 illustrates.

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Figure 15. United Kingdom time series of (a) output per worker per year (productivity); (b) births per thousand of the population; (c) employment; (d) cumulative confirmed COVID-19 cases.

Productivity has increased at varying rates since 1860 but has stagnated since 2008, as highlighted by the ellipse, badly overforecasted by the U.K. Office of Budget Responsibility not adjusting to the “flatlining,” and much improved by the robust predictor proposed by Martinez et al. (2021). The birth rate was steadily increasing until the introduction of oral contraception in the mid-1960s, then fell sharply till the late 1970s and has fluctuated since. Employment has expanded greatly since 1860, with major fluctuations during world wars and severe depressions, but at a much more rapid rate since 2000 (which helps explain the productivity slowdown). Finally, Panel (d) shows dramatic changes in the trend of confirmed COVID-19 cases. To empirically model trends that change by unknown magnitudes at unknown points in time an unknown number of times requires a general tool like trend indicator saturation, denoted TIS (see Castle et al., 2019 and Walker et al., 2019 for an application). TIS is equivalent to applying multiplicative indicator saturation to a deterministic trend by interacting step indicators with the trend, $S_j\times t$, where $t$ is a deterministic trend. However, it merits a separate analysis to that of MIS because it is deterministic (like the constant) and also that ${\sum }_{t=1}^T{t}^2=\frac{1}{6}T(T+1)(2T+1)$, so grows at $O(T^3)$ as against $O(T^2)$ for sums of squares of stationary variables, and therefore has different gauge and potency properties to that of MIS.

Much macroeconomic modeling is in differences of variables or their logs to eliminate trends and possibly unit roots. However, the potency of detecting breaks is much higher when working in levels than in changes. To illustrate, an artificial trend break DGP is created to which TIS is applied to the levels of the time series, and the ability of TIS to detect the trend breaks is compared to the ability of SIS to detect location shifts in the changes of the time series.

Figure 16 records a single representative draw of the resulting time series, denoted $y_t$, $z_t$, $\mathrm{\Delta }{y}_t=y_t-y_{t-1}$, and $\mathrm{\Delta }{z}_t$, where

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Figure 16. Time series from Equations 27 and 28 of (a) $y_t$; (b) $z_t$; (c) $\mathrm{\Delta }{y}_t$; (d) $\mathrm{\Delta }{z}_t$.

and

so $z_t$ acts as the trend. Let ${\beta }_{0,t}=10$ and ${\beta }_{1,t}=1$ for $t=1,\dots ,60$, then ${\beta }_{0,t}=-70$ and ${\beta }_{1,t}=2$ for $t=61,\dots ,100$, with ${\gamma }_0=0$, ${\gamma }_1=1$, ${\sigma }_{ϵ}^2=1$, and ${\sigma }_{\nu }^2=0.001$ deliberately set to a tiny value to mimic a trend, as Figure 16(b) confirms.

“Ocular econometrics” on Figure 16 shows an obvious trend break in Panel (a), but not an obvious shift in the changes $\mathrm{\Delta }{y}_t$ in Panel (c). Applying TIS to the levels with a target nominal significance value of 0.001 picks up the shift, but SIS applied to the changes needs to be at 0.005 to do so. Doubling of the growth rate is a very large change, but smaller changes, such as a 50% increase, barely register in $\mathrm{\Delta }{y}_t$. Four cases are considered: (i) estimating the relation between $y_t$ and $z_t$ without allowing for the trend shift; (ii) estimating that relation with TIS at 0.001; (iii) regressing $\mathrm{\Delta }{y}_t={\psi }_0+{\psi }_1\mathrm{\Delta }{z}_t$; and (iv) estimating that last equation with SIS. Figure 17 shows the outcomes for (i) and (ii), and Figure 18 for the differenced cases (iii) and (iv).

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Figure 17. Actual and fitted values and residuals from (a), (b) regressing $y_t$ on $z_t$ without TIS, and (c), (d) regressing $y_t$ on $z_t$ with TIS.

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Figure 18. Actual and fitted values and residuals from selecting (a), (b) $\mathrm{\Delta }{y}_t$ from $\mathrm{\Delta }{z}_t$ with SIS; (c), (d) $y_t$ with TIS when $z_t$ is forced; (e), (f) $\mathrm{\Delta }{y}_t$ on SIS with $\mathrm{\Delta }{z}_t$ forced.

Not handling the break in the trend, as in Figure 17(a) and (b), is disastrous. For example, the deviations from trend would be unrelated to “excess demand” if $y_t$ was GDP. However, in Figure 17(c) and (d), the shift is detected by TIS.⁹ When there are shifts, ISEs aim to reveal what the processes were like in earlier periods. For TIS here, that is the trend that would have been found by an investigator having data up to $T=60$.

Figure 18 records two cases with SIS and one with TIS. In the top row, Panels (a) and (b) refer to the case where SIS is applied and $\mathrm{\Delta }{z}_t$ is not forced, so it is also selected jointly with the step indicators. Panels (e) and (f) in the bottom row refer to the case where SIS is applied but $\mathrm{\Delta }{z}_t$ is forced, which means that it is always retained in every selection, and only the step indicators are selected over. Panels (c) and (d) in the middle row are for the case with TIS when $z_t$ is forced, so only the trend indicators are selected over. SIS at 0.005 detects the shift, dropping $\mathrm{\Delta }{z}_t$ (when only the constant is forced), whereas $\mathrm{\Delta }{z}_t$ is retained with an unrestricted constant. An intercept is equally good here at representing the mean change, but would not work if $z_t$ was exactly the trend, so $\mathrm{\Delta }{z}_t$ was constant, whereas $\mathrm{\Delta }{z}_t$ would matter more than the intercept if ${\sigma }_{\nu }$ was larger. There is a small loss of fit if SIS is not used, and as the residual autocorrelation test is significant in all three cases, there is little sign of the step shift. Differencing doubles the error variance, so detecting the step shift rather than the trend break is harder, reflected in the need to use a less stringent nominal significance level.

Although the fit is similar between Figure 18(c) and (d) as measured by ${\hat{\sigma }}_{ϵ}$, there is nevertheless a substantial impact on forecasts of $y_t$ derived from $\mathrm{\Delta }{y}_t$, as Figure 19(b) and (d) shows for multistep forecasts. There is little difference between Figure 19(a) and (c) in the root mean square forecast errors (RMSFEs) for forecasts of the changes $\mathrm{\Delta }{y}_t$, but cumulating these forecasts to derive the levels’ forecasts leads to the RMSFE for the model without SIS being more than four times larger than that with SIS and those forecasts from case (iii) being systematically too low.

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Figure 19. Actual and forecast values for changes and derived levels from models of $\mathrm{\Delta }{y}_t$ on $\mathrm{\Delta }{z}_t$ without SIS (a), (b) denoted $\left(\hat{}\right)$, and with (c), (d) $\left(\stackrel{\sim }{}\right)$.

Figure 20 records multistep forecast values for the levels from models of $y_t$ on a forced constant and forced $z_t$, where Panel (a) is based on the model with TIS, Panel (b) is from the model without TIS, and Panel (c) are forecasts from a model with a step intercept correction from $t=84$ on. All three graphs also show robustified forecasts (see Hendry, 2006). The forecasts with TIS have dramatically smaller RMSFEs than those without. While the robust device is a great improvement for Panel (b) when TIS was not used, being based on differencing the data, its RMSFEs are similar to those in Figure 19 for the cumulated differenced forecasts without SIS (Panel b).

Models in differenced data face this risk of appearing to have few or no step shifts when estimated and produce respectable forecasts of future changes but suffer systematic forecast failure for the entailed levels of the data.

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Figure 20. Actual and forecast values for levels from models of (a) $y_t$ on a forced constant and forced $z_t$ with TIS $\left(\hat{}\right)$, and (b) without TIS $\left(\stackrel{\sim }{}\right)$, and (c) with a step intercept correction from $t=84(-)$.

Selecting Models

The framework begins by embedding the relevant subset of climate theory within a much more general model specification that allows for influences that are not explicitly included in the theory models. This could be due to the theory model itself being incorrect or incomplete, or it could be due to external effects that lie outside of the climate theory (e.g., volcanic eruptions or economic responses to a pandemic). The theory is the object of study, but the target for model selection is the data-generating process for the set of variables under analysis. In order to validly evaluate the objective or the postulated theory, the model must be as robust as feasible to outliers, shifts, potential omitted variables, nonlinearity, mis-specified dynamics, incorrect distributions, nonstationarity, and invalid conditioning.

The approach can orthogonalize all additional variables with respect to the theory variables, so the distributions of the estimators of parameters of the object of interest are unaffected by selection (see Hendry & Johansen, 2015). The search algorithm can retain without selection all the variables in theory model when selecting other features. This enables tighter than conventional selection significance levels, reducing the retention of irrelevant candidates without jeopardizing the retention of theory-relevant variables. This is a win–win situation. When the theory model is complete and correct, the resulting model will deliver precisely the same estimates as directly fitting it to data, even if selecting from a vastly larger specification, but if the theory is not complete and correct, the modeler would discover a better formulation.

The implication of commencing with much larger models than the theory would suggest is that the starting point for model selection would include more candidate variables, $N$, than the number of observations, $T$. A search algorithm is needed that can handle such a situation. Autometrics implements a block search algorithm to handle more variables than observations (see Doornik, 2009; Hendry & Doornik, 2014). The regressors are divided into subblocks but the theory variables are retained at every stage, selecting only over the putative irrelevant variables at a stringent significance level. The subblocks include expanding and contracting searches to handle correlated regressors that may need to enter jointly to be significant. It is almost costless to check large numbers of rival variables, so there are huge benefits if the initial specification is incorrect but the enlarged general unrestricted model nests the DGP.

Search is unavoidable as climate and economic variables are all interrelated with high correlations. There are many selection algorithms in the literature that assume a lack of correlation and therefore pursue single path searches. These include forward search and stepwise regression, 1-cut selection and backward elimination, and Lasso (see Tibshirani, 1996) and some of its variants. The benefits of Autometrics include formal tests for congruence to ensure all tests of reduction are valid, multipath search to avoid path dependence and ensure that the initial ordering of regressors does not matter, increased efficiency relative to estimating all $2^{\text{N}}$ possible models, which is infeasible in most cases as N becomes even moderately large, and a well-defined stopping point at terminal models using encompassing tests against the general unrestricted model to ensure there isn’t a substantial loss of information. On the principle of encompassing (see Bontemps & Mizon, 2008; Mizon & Richard, 1986) Doornik (2008) analyzes its role in automatic model selection. Although a multipath search is not as fast and simple as single path procedures and there is some dependence on how the blocking is implemented, increased computing power means that these costs are not large.

Three important automatic generalizations for specifying the initial general unrestricted model include (a) adding in many lags of the regressors to allow for a sequential factorization and the selection algorithm will then apply a lag length reduction stage, ensuring there are no unmodeled dynamics; (b) including nonlinear transformations of the regressors to allow for general unspecified forms of nonlinearity using polynomial and exponential expansions; and (c) a variety of indicator saturation estimators (ISEs) to model many different aspects of wide-sense nonstationarity. As illustrated earlier, each ISE is designed to match a specific problem: IIS to tackle outliers, SIS for location shifts, MIS for parameter changes, TIS for trend breaks, as well as designed indicator saturation for modeling phenomena with a regular pattern, such as detecting the impacts on temperature of volcanic eruptions (DIS; see Pretis et al., 2016). Importantly, saturation estimators can be used in combination, as demonstrated later, where IIS and SIS combined is called super-saturation (see Ericsson & Reisman, 2012; Kurle, 2019). All saturation estimators can be applied when retaining without selection a theory model that is the objective of a study, while selecting from other potentially substantive variables. Saturation estimators have seen applications across a range of disciplines, including dendrochronology, volcanology, geophysics, climatology, and health management, as well as economics, other social sciences, and forecasting. Although theory models are much better in many of these areas than in economics and other social sciences, modeling observational data faces most of the same problems, which is why an econometric tool kit can help.

The Retention Properties of Model Selection

Having established that commencing with large models ensures that many aspects of wide-sense nonstationarity in the data can be captured, it is important to assess the properties of selection from such a starting point. An effective model selection procedure should aim to select a congruent and encompassing model specification that retains the relevant variables, excludes irrelevant variables, and results in small mean square errors (MSE) for the estimated parameters of interest after model selection, regardless of whether additional correlated variables are included initially. This section considers how well such objectives can be achieved in practice.

Let $y_t$ be the variable of interest, with $z_t={\left(z_{1,t},\dots ,z_{N,t}\right)}^{\prime }$ including all possible explanatory variables, nonlinear functions, saturation estimators, and deterministic terms such as intercept or seasonals, and (for ease of notation) all lags of explanatory variables and nonlinear functions, such that

where $N>T$, and for convenience, the first $n$ regressors are defined as relevant in that they enter the DGP, and the subsequent $N-n$ regressors as irrelevant since they do not enter the DGP. It is assumed that ${ϵ}_t\sim \text{IN}\left[0,{\sigma }_{ϵ}^2\right]$, which is untestable given $N>T$ but should be satisfied by the inclusion of such a broad range of regressors in $z_t$. Apply a selection algorithm at a significance level $\alpha$ to Equation 29 for $M$ replications, and the OLS estimated coefficients retained in the selected model are denoted ${\tilde{\beta }}_i$ (where ${\tilde{\beta }}_i=0$ when $z_{i,t}$ is not retained in the final model). Letting $1_{({\tilde{\beta }}_{i,j}\ne 0)}$ denote an indicator function equal to unity when ${\tilde{\beta }}_{i,j}\ne 0$ and zero otherwise, the retention rate is given by

The gauge is defined as the average retention frequency under the null hypothesis (i.e., how many irrelevant variables are retained), given by

and the potency is defined as the average retention frequency under the alternative hypothesis, or how often relevant variables are retained:

The unconditional MSE is given by

and the conditional MSE, computed only over retained regressors, is given by

The theoretical gauge for IIS under the null hypothesis that there are no outliers is $\alpha T$. This extends to regressors; under the null that none of the $z_{\text{t}}$ are relevant so their noncentralities defined as $\text{E}\left[t_{{\beta }_i}\right]={\psi }_i$ are zero, then $\alpha N$ regressors should be retained on average. The empirical gauge, g, closely matches the theoretical gauge for tight $\alpha$, as shown in Hendry and Doornik (2014). However, if $\alpha$ is too loose, it can result in a higher g than the theoretical gauge due to a downward biased equation standard error by retaining too many irrelevant indicators. This can be corrected using the bias correction of Johansen and Nielsen (2009), but does not arise if $\alpha$ is appropriately controlled at $\alpha \le min\left[0.001;1/T\right]$. g will also be higher if SIS is applied, as two steps are needed to capture an outlier, and if IIS and SIS (super-saturation) are applied jointly, the gauge is roughly doubled. This can also be controlled by using a tighter $\alpha$. For non-indicator regressors, the choice of $\alpha$ depends on their number ($r$, say), often set as $\alpha \le min\left[0.01;1/r\right]$, and type, where a more stringent value may be set for selecting nonlinear functions of other included variables.

Finally, there is an additional cost to the gauge when applying mis-specification and encompassing tests; irrelevant regressors with estimated t statistics less than $c_{\alpha }$ may be retained to ensure these tests do not reject, adding to g (on average this can be an additional 1% of gauge). While the diagnostic and encompassing tests could be switched off, they are fundamental to ensuring a congruent and parsimonious specification so that the additional gauge costs are worthwhile.

As just argued, overfitting is not a concern despite commencing from very large $N$ as long as $\alpha$ is controlled. The difficulty is in retaining relevant variables which have low signal-to-noise ratios, denoted by their noncentralities, ${\psi }_i$. Tight $\alpha$ will mean that $c_{\alpha }$ is larger than conventional critical values, so variables at the margin of significance will be missed. However, there are a number of comments to put this into context. First, by applying saturation, the residual distributions will be approximately Normal, even if they were fat-tailed initially. As the Normal distribution has thin tails, critical values increase slowly as $\alpha$ decreases, so even using $\alpha =0.0001$ only leads to $c_a=4$. Second, theory variables can be retained without selection, so even if they have low ${\psi }_i$, they will always be included in the final selected model. Finally, the retention of low-significance relevant variables is not just a difficulty for model selection, as the same problem of retaining marginally significant regressors would be faced even if the DGP was the starting point. The costs of selection over and above the costs of inference (as measured by conventional t-testing on the DGP) are small and can even be negative (see Castle et al., 2011). The empirical potency p is close to theory powers for 1-off t-tests.

Unconditional estimates of DGP parameters will be downward biased for a variable with ${\psi }_i$ near $c_{{}_{\alpha }}$, comprising draws in which the variable is retained, so $\left|{\hat{t}}_{{\beta }_i}\right|>c_{\alpha }$ and draws in which the variable is not retained and therefore the estimated parameter is set to 0. However, plotting unconditional distributions, as in Hendry and Krolzig (2005), illustrates the quality of model selection in correctly classifying variables into DGP and non-DGP variables, where the latter have tiny UMSEs. The nonzero-mass distributions of the estimated parameters of DGP variables will be truncated Normals, decreasingly so as $\left|{\hat{t}}_{{\beta }_i}\right|$ increase.

Conditional on retaining the variables, estimates of DGP parameters will be upward biased, consisting only of draws where $\left|{\hat{t}}_{{\beta }_i}\right|>c_{\alpha }$. This bias can be corrected in orthogonal settings as the truncation value $c_{{}_{\alpha }}$is known, and bias correction also reduces CMSEs (see Hendry & Krolzig, 2003). Even without bias correction, the CMSEs are close to those for model selection commencing from the DGP, so again the issue is one of inference rather than selection. As long as $\alpha$ is controlled, the postselection distributions of parameter estimators are relatively unaffected by the search algorithm, being very similar to the corresponding postinference distributions from the DGP.

Forecasting Wide-Sense Nonstationary Time Series

“Conventional” economic forecasting uses a theory-based system that models the main variables of interest. Examples include a dynamic-stochastic general equilibrium (DSGE) model, a variant of a vector autoregression (VAR), or a simultaneous equations model. Some systems are closed in that all variables are modeled, but most are open with “offline” assumptions made about future values of unmodeled “exogenous” variables. Almost all economic systems are equilibrium-correction models (EqCMs) or differenced variants thereof. Clements and Hendry (1998, 1999) develop taxonomies of all forecast errors in closed systems to show that the key determinant of forecast failure is an unmodeled shift in the equilibrium mean. Hendry and Mizon (2012) extend the taxonomy to open systems and show that result still holds, but there are additional potential sources of forecast failure deriving from changes in the “exogenous” variables. Introductions to forecasting facing breaks are provided by Castle et al. (2016, 2019).

Nothing can solve the forecast failure resulting from an unanticipated shift in the equilibrium mean over the forecast horizon, of which the 21st century has already witnessed several, including the dot.com crash, the financial crisis, and the COVID-19 pandemic. Forecasting such shifts before they occur has not yet proved feasible, even if ex post claims that they were predicted abound. However, a number of approaches have been proposed to counter the problem of forecast failure due to in-sample shifts in equilibrium correction models, including differencing, and developing predictors that are “robust” after breaks (see, e.g., Castle et al., 2015; Hendry, 2006; Martinez et al., 2021). Pesaran et al. (2013) propose weighting observations when there are breaks near the forecast origin to minimize RMSFEs, allowing for both continuous and discrete parameter change (also see Pesaran & Timmermann, 2007). Nevertheless, avoiding systematic mis-forecasting has a cost in larger RMSFEs when there is no shift to offset. Given the dominance of breaks as the major source of forecast failure, the implications of model selection from a large set of explanatory variables have very little effect on forecast uncertainty, particularly given the similarity between postselection parameter distributions from the selected model and from the DGP. Indeed, model selection with saturation ensures unbiased deterministic terms at the forecast origin, thereby reducing forecast errors.

Exogeneity in Ice Ages Forecasts

As Pretis (2020) remarks:

Econometric studies beyond IAMs (integrated assessment models) are split into two strands: One side empirically models the impact of climate on the economy, taking climate variation as given . . . the other side models the impact of anthropogenic (e.g., economic) activity onto the climate by taking radiative forcing—the incoming energy from emitted radiatively active gases such as CO₂—as given. . . . This split in the literature is a concern as each strand considers conditional models, while feedback between the economy and climate likely runs in both directions. (p. 257)

Pretis (2021) addresses the exogeneity issue in more detail. Examples of approaches conditioning on climate variables such as temperature include Burke et al. (2015), Pretis et al. (2018), Burke et al. (2018), and Davis (2019). Hsiang (2016) reviews such approaches to climate econometrics. Examples from many studies modeling climate time series include Estrada et al. (2013), Kaufmann et al. (2011, 2013), and Pretis and Hendry (2013).¹⁰

The dynamic simultaneous system in Castle and Hendry (2020) for modeling and forecasting Antarctic ice volume and temperature as well as atmospheric CO₂ over the past 800,000 years of ice ages (see Figure 3) conditioned on the contemporaneous and lagged values of the Earth’s orbital variables of eccentricity, obliquity, and precession, shown in Figure 4. The open model taxonomy in Hendry and Mizon (2012) added nine potential sources of forecast error to the 10 that occur in closed models. They show that despite the in-sample forecasting model being correctly specified and all unmodeled variables (denoted by the vector $z_t$) are strongly exogenous with known future values, changes in dynamics can lead to forecast failure when the $z_t$ have nonzero means. Their possible impacts on the forecast accuracy of the system 100,000 years into the future are addressed.

To illustrate, let the DGP of a vector $y_t$ conditional on known $z_t$ be

which has a zero intercept to simplify the algebra. If $z_t$ has a nonzero mean, $\text{E}\left[z_t\right]=\mu$, then when ${\Psi }_1$ has all its eigenvalues inside the unit circle so that the process is stationary, the equilibrium mean is

In terms of deviations from equilibrium means:

If any of the parameters determining $\psi$ in Equation 36 change, then despite the future $z_t$ being super-strongly exogenous and known, shifts in that equilibrium mean will lead to forecast failure until the model is appropriately updated.

Forecasting after the dynamic parameter matrix shifts at $T+1$, so that the DGP becomes

from a forecast origin at $T$, where $y_T$ is known using

leads to the forecast error $e_{T+1|T}=y_{T+1}-{\hat{y}}_{T+1|T}$:

When the in-sample parameter estimates are unbiased so $\text{E}\left[{\hat{\Psi }}_1\right]={\Psi }_1$ and $\text{E}\left[{\hat{\Psi }}_2\right]={\Psi }_2$:

Thus, the equilibrium mean shifts when $\mu \ne 0$ and would also do so then if ${\Psi }_2$ shifted or if there was an intercept $\varphi \ne 0$ in the DGP. If neither $\mu$ nor $\varphi$ shift, then the key problem appears to be shifts in ${\Psi }_1$ and ${\Psi }_2$, although if ${\hat{y}}_T$ has to be estimated, then $y_T-{\hat{y}}_T$ could also cause a systematic forecast error, especially for multihorizon forecasts.

The Earth’s orbital drivers of eccentricity, obliquity, and precession are super-strongly exogenous and known into the distant future, albeit that a sufficiently large rogue object intruding into the solar system might perturb Earth’s orbit in unanticipated ways. Excluding that last possibility for the next 100,000 years so that neither $\mu$ nor ${\Psi }_2$ change, then the forecasting system being open does not by itself create additional problems. Instead, the change to the system from CO₂ being endogenously determined to being created anthropogenically is a fundamental shift in the DGP, seen in Figure 2. Castle and Hendry (2020) tackled this by computing two scenarios where the model for ice volume and temperature remains constant, but CO₂ is determined exogenously from ad 1000. The impact on the forecasts is dramatic, as Figure 21 shows, where Panel (a) is for ice under the three scenarios and Panel (b) for temperature.

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Figure 21. Three scenarios of different atmospheric CO₂ levels: Endogenous (long-dashed with error bars), and fixed at 400 ppm (dotted with error bands) and 560 ppm (thick dashed with error fans) for (a) ice volume; (b) temperature.

Forecasting 110,000 years ahead under ceteris paribus, so there is no human intervention, shown as the long-dashed line with $\pm 2{\overline{\sigma }}_f$ bars, mimics the ice age data. Forecasting conditional on atmospheric CO₂ remaining at 400 ppm (roughly the level in 2020), shown as the dotted line with solid $\pm 2{\tilde{\sigma }}_f$ error bands, leads to less ice than the ice age minimum over the past 800,000 years and higher temperatures than the peak of the ice ages. Finally, at 560 ppm (roughly representative concentration pathway, RCP8.5), shown as the thick dashed line with $\pm 2{\hat{\sigma }}_f$ fans, temperatures are far higher than during the ice ages and Antarctica is almost ice free.

Cointegration

The cointegrating or long-run relation was derived from that equation after transforming the indicators as noted. When mapping to a non-integrated specification that reparametrizes levels variables to first differences, step indicators should be included in the equilibrium correction mechanism (EqCM) to avoid cumulating to trends. However, they need to be led one period, as the EqCM will be lagged in the I(0) formulation. Impulse indicators and differenced step indicators are left unrestricted (see the survey articles by Hendry & Juselius, 2000, 2001; and Hendry & Pretis, 2016). Applications of cointegration analysis in climate econometrics include Kaufmann and Juselius (2010), Kaufmann et al. (2013), and Pretis (2020). This yields

All variables are significant at 1% other than $g$, which enters negatively, so given the other variables, the long-run effect of higher GDP is slightly lower emissions, perhaps reflecting the role of services in the United Kingdom. The coefficients of coal and oil are close to their values in Table 1. With four observations on $1-S_{2010}$, $S_{2010}$ is now precisely estimated and so can be included in ${\tilde{E}}_{LR}$. Figure 24 records the long-run estimated emissions from Equation 44, with actual emissions in Panel (a) and the mean zero long-run equilibrium error, ${\tilde{Q}}_t=E_t-{\tilde{E}}_{LR,t}$, in Panel (b).

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Figure 24. (a) $E_t$ and ${\tilde{E}}_{LR,t}$; (b) ${\tilde{Q}}_t=E_t-{\tilde{E}}_{LR,t}$ without impulse indicators, centered on a mean of zero.

Transforming to a model in first differences and the lagged EqCM from Equation 44, then reestimating, revealed a significant nonnormality test, so IIS was reapplied at 0.5%, which yielded (for 1861–2013, testing constancy over 2014–2017):

Changes in coal, oil, $k$, and $g$ all lead to changes in the same direction in emissions, which then equilibrates back to the long-run relation in Equation 44. Figure 25 provides a graphical description of the selected model. “Forecasts” are in inverted commas, since they are for the past, although the data points were outside the sample period used for selection and estimation and were not available when the previous study was undertaken. As Panel (a) is dominated by the fluctuations in the 1920s, Figure 26 plots the levels outcome from modeling, with the impulse and step indicator dates shown.

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Figure 25. (a) Actual, fitted, and forecast values for $\mathrm{\Delta }{E}_t$ from Equation 45; (b) residuals and forecast errors scaled by the equation standard error; (c) residual density and histogram with a Normal density; (d) residual autocorrelation.

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Figure 26. Actual, fitted, and forecast values for U.K. CO₂ emissions.

Forecast Evaluation

Figure 27(a) records the changes in CO₂ emissions and the fitted values from Equation 45, along with the 1-step conditional forecasts, denoted ${\widehat{\mathrm{\Delta }E}}_{T+h|T+h-1}$, with $\pm 2{\hat{\sigma }}_f$ shown as bars, estimating the model up to 2014. In the same panel, robust forecasts from Hendry (2006) (differencing the EqCM but without smoothing) are also reported. The forecasts are similar. Panel (b) integrates the forecasts to obtain 1-step ahead forecasts of the level of CO₂ emissions from both the model in Equation 45 (with $\pm 2{\hat{\sigma }}_f$ bands) and the robust forecasts. Again, the forecasts are similar, with no clear advantage to using either the econometric model or its robustified form. Panel (c) shifts the forecast origin back to 2009, so only data up to 2008 is available to estimate the model, although it was selected from the longer period. Again, the 1-step conditional forecasts from the econometric model, denoted ${\widehat{\mathrm{\Delta }E}}_{T+h|T+h-1}$ with $\pm 2{\hat{\sigma }}_f$ bars, are recorded along with the robust forecasts ${\widetilde{\mathrm{\Delta }E}}_{T+h|T+h-1}$. There is a clear benefit to using the robust forecasts over the longer forecast period. The conditional forecasts come before the implementation of the CCA2008 could have an effect, leading to forecasts that are too high from 2012 onward. The robust device remains accurate even when the CCA2008 is not explicitly modeled, as it differences out the previous in-sample mean for the change in CO₂ emissions, which was higher prior to the CCA2008, leading to biased 1-step ahead forecasts if not handled.

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Figure 27. Conditional 1-step forecasts for $\mathrm{\Delta }{E}_t$ (a) from Equation 45 over 2014–2017 (denoted $\hat{}$) with $\pm 2$ forecast standard error bars and the robust forecasts ${\widetilde{\mathrm{\Delta }E}}_{T+h|T+h-1}$; (b) the derived forecasts in levels with $\pm 2$ forecast standard error bands; (c) same as (a), but commencing in 2009.

The 1-step ahead forecasts highlight the impact of the CCA2008, which needs to be either modeled or accounted for (via robustification) for accurate forecasts. Modeling CO₂ emissions within a system allows for unconditional forecasts to be produced, so does not rely on data over the forecast period and hence is feasible ex ante. A vector autoregression (VAR) with two lags is sufficient to model the dynamics. Figure 28 plots the unconditional system 1-step and dynamic forecasts from a VAR in all five variables, either including the indicators found in Equation 45 or without the indicators. In the former case, all outliers and shifts will be captured in the VAR, whereas in the latter they will be ignored, which is a common approach in the economics literature that uses VARs as forecasting benchmarks. Panel (a) plots the 1-step ahead forecasts from the VAR without and with indicators, and Panel (b) records the dynamic forecasts. The importance of the step indicators is readily apparent, yielding huge reductions in RMSFEs and correcting the bias evident in the VAR without the steps. The role of the CCA2008 in producing a level shift reduction in CO₂ emissions is clear, so it needs to be modeled for accurate forecasting.

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Figure 28. (a) Outcomes and 1-step forecasts $\pm 2$ forecast standard errors as bars and fans, without and with indicators; (b) outcomes and h-step forecasts and a comparison with Cardt forecasts.

Figure 28 also records the forecasts for U.K. CO₂ emissions, comparing Cardt forecasts to the VAR forecasts, where the Cardt forecasting device is described in Doornik et al. (2020) and Castle et al. (2021). The sample estimation period for the Cardt forecasts is kept short (2005–2011) to avoid contaminating the persistence parameter estimates with earlier in-sample breaks. As a consequence, the uncertainty bands around the Cardt forecasts are very wide and thus are not shown. Extending the in-sample estimation period reduces the uncertainty bands but results in a larger bias due to unmodeled shifts in-sample, especially 2010. The Cardt forecasts are close to the VAR forecasts using SIS, although the VAR forecasts with SIS are preferred. The VAR with SIS has a mean forecast error of −3 compared to a mean error for the Cardt forecasts of −20. In contrast, the VAR model without SIS has a mean error of −83. The key characteristics of the Cardt forecasting device is that it dampens trends and growth rates, it averages across predictors, and it robustifies to breaks by over-differencing. As such, it has similar characteristics to the forecasts from the VAR with SIS, which models step shifts explicitly with step indicators and is superior to the VAR forecasts that do not model location shifts explicitly.

Super Exogeneity Tests

Parameter invariance is essential in policy models because without it, a model will mispredict under regime shifts. The concept of super exogeneity combines parameter invariance with valid conditioning, so it is crucial for policy (see Engle & Hendry, 1993; Hendry & Massmann, 2007; Hendry & Santos, 2010; Krolzig & Toro, 2002). To test the joint super exogeneity of the regressors, a natural procedure here is to check if any of the indicators in the conditional model enter the equations for the marginal variables: If so, the impacts of large changes in marginal variables are different from small changes. This test requires no ex ante knowledge of the timings or magnitudes of breaks or the data-generating processes (DGPs) of the marginal variables. The test has the correct size under the null of super exogeneity for a range of sizes of marginal-model saturation tests, and it has power to detect failures of super exogeneity when location shifts occur in the marginal models and the conditioning on them is invalid.

Having created a VAR, super exogeneity can be tested by a likelihood ratio test of the VAR, with the indicators only entering the equation for $E_t$ against entering every equation. This yields ${\chi }^2(37)=161**$, which strongly rejects. However, the rejection is due to the indicators for the 1920s also occurring in the equations for GDP and coal, which is not surprising as the postwar crash and the 1926 general strike affected both. Retaining $1_{1921}$, $1_{1926}$, $S_{1925}$, and $S_{1927}$ in the coal equation and $1_{1921}$ and $1_{1926}$ in GDP delivers ${\chi }^2(31)=25$, which is insignificant, implying the regressors in the CO₂ emissions model are super exogenous, so they are weakly exogenous and the parameters of the conditional model (Equation 45) are invariant to structural breaks in the marginal models.

Evaluating the U.K.’s 2008 Climate Change Act

The most important implication of the previously stated evidence is that substantial CO₂ reductions have been feasible, so far with little impact on GDP. The U.K.’s CCA2008 established the world’s first legally binding climate change target to reduce the U.K.’s GHG emissions (six gases, including CO₂, which is approximately 80% of the total) by at least 80% by 2050 from a 1990 baseline. The policy produced a series of 5-year carbon budgets, which are mapped to annual targets by starting 20 Mt above and ending 20 Mt below them in each period. Allowing 20% for other GHG emissions, these are called the Targets for CO₂. Figure 29(a) plots the Targets and CO₂. TargDiff denotes the difference between these targets and CO₂ emissions, and Equation 46 records the result from selecting step indicators by SIS to describe it over 2008–2020:

Thus, emissions were approximately 52 Mt below target after 2013 and fell another 50 Mt further below after 2019, part of which is undoubtedly due to the impacts from pandemic lockdowns. Nevertheless, these are large reductions. Farmer et al. (2019) suggest exploiting “sensitive intervention points” (SIPs) to accelerate the postcarbon transition and include the U.K.’s CCA2008 as a timely SIP with a large effect, corroborated here.

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Figure 29. (a) U.K. CO₂ emissions and CCA2008 CO₂ targets; (b) deviations from the target values with step indicators.

A similar approach could be used to evaluate the extent to which countries met their Paris Accord Nationally Determined Contributions (NDCs), given the appropriate data. The NDCs agreed at COP21 in Paris are insufficient to keep temperatures below 2 ºC, so they must be enhanced, and common time frames must be adopted to avoid a lack of transparency in existing NDCs (see Rowan, 2019). Since the baseline dates from which NDCs are calculated is crucial, 5-year NDC reviews and evaluation intervals are needed.

In 2019, the U.K. government revised its target to one of net zero GHG emissions by 2050, entailing no use of coal, oil, and natural gas, with no emissions coming from agriculture, construction, and waste (about 100 Mt p.a. in 2019) beyond what can be captured or extracted from the atmosphere. Increases in the capital stock could make the target harder to achieve unless they were carbon neutral. As capital embodies the technology at the time of its construction and is long-lived, transition to zero carbon has to be gradual and necessitates that new capital, and indeed new infrastructure in general, must be zero carbon producing. “Stranded assets” in carbon-producing industries face a problematic future as legislation imposes ever lower CO₂ emissions targets to achieve zero net emissions (see Pfeiffer et al., 2016), threatening “stranded workers,” as has happened historically for coal miners in the United Kingdom.

Implications of the U.K.’s CO₂ Emissions Model

Despite more candidate variables than observations, the econometric approach presented in this article developed a model to explain the United Kingdom’s extremely nonstationary CO₂ emissions time series data over 1860–2017 in terms of coal and oil usage, capital stock, and GDP. It was essential to take account of both stochastic trends and distributional shifts. Detection of major policy interventions by indicator saturation estimators yielded a congruent model of CO₂ emissions and accurate forecasts since the CCA2008 came into force. The policy implications highlight that CO₂ emissions are reducing rapidly, but far greater reductions are needed if the United Kingdom is to achieve its net-zero emissions target requiring all coal, oil, and natural gas to be eliminated or their GHG emissions sequestrated. The United Kingdom was a net CO₂ exporter through embodied CO₂ in the 19th century, but is now a net importer, although this component will decrease with falls in the GHG emissions of exporting countries.¹¹

The aggregate data provided little evidence of high costs to the large domestic reductions in CO₂ emissions—dropping by 202 Mt from 554 Mt in 2000 to 352 Mt (39%) by the end of 2019, before the pandemic—whereas real GDP rose by 39% over that period despite the “Great Recession.” As new “green” technologies are implemented, careful attention must be paid to local costs of lost jobs. Mitigating the inequality impacts of climate-induced changes has to be achieved to retain public support.