The Network of Central Banks and Supervisors For Greening the Financial System (NGFS) is a supranational consortium of central banks and financial regulators; the United States is represented by the Federal Reserve, the OCC, the FDIC and the FHFA. The NGFS develops climate scenarios that are then used by its members to measure the resiliency of the banking system to climate risks. Thus, the Federal Reserve employed NGFS climate scenarios in its 2023 climate scenario exercise for large U.S. banks, and the ECB employed NGFS scenarios in its 2022 climate risk stress test of the Eurosystem balance sheet.
In early November this year, the NGFS updated the “damage function” component of its climate scenario, which projects the effect of rising temperatures and precipitation on global real income. The new damage function was based entirely on an academic paper published this year in the journal Nature.
The updated damage function predicts that global real income will be 19 percent lower by 2050 than it would have been with no climate change, regardless of whether current CO2 emission trends improve or worsen. The damage function projects 60 percent lower global real income by 2100 if current CO2 emission trends worsen. Going forward, these extreme economic assumptions in the scenarios would be used in climate stress tests of banks, practically ensuring that those tests will conclude that all global banks face material climate risks that would require higher capital levels.
This paper reviews the new damage function in the Nature paper and finds no basis for its projections. The statistical procedure used to justify the new damage model is arbitrary and could easily have produced a damage function that would have predicted much smaller losses of global real income. Those much smaller loss projections should not be taken seriously either. In our assessment of the damage function, we found very limited statistical evidence for any causation between the climate variables and material economic damage and no statistical evidence for the purported influence of the temperature variables that drive almost all the economic damage.
Our findings are consistent with the current state of academic damage function research, which is highly uncertain at this point. One recent academic paper tested 800 plausible specifications of climate damage functions, finding that damages ranged from large economic benefits to large economic losses.[1] With no academic consensus on climate damage functions, it is unclear why the NGFS would adopt the results of a single academic paper, especially given the massive ramifications for global economic growth.
Why Did the NGFS Update Its Damage Function?
The new damage function is based on academic research by Kotz, Levermann and Wenz[2], published in Nature in 2024. The adoption of the new model, which is discussed in a companion technical document[3], is intended to ensure that the NGFS climate scenarios remain state-of-the-art, incorporating the latest data and methodology. Damage functions are essential components of climate scenarios and models that perform cost-benefit analysis of climate policies, since they suggest how to trade off the costs of climate policy changes against economic losses caused by climate change.
The new model includes 10 distinct types of temperature and precipitation variables and incorporates the influence of climate change in past years on economic damage today, measured at a highly granular geographic level. These new features have given the NGFS confidence that the replacement damage function is much more accurate. In the NGFS release[4], Sabine Mauderer, Chair of the NGFS and First Deputy Governor of the Deutsche Bundesbank said, “we have significantly improved the accuracy of physical risk measurements.” Livio Stracca, Chair of the NGFS workstream “Scenario and Design Analysis” and Deputy Director, General Financial Stability at the European Central Bank, suggested the new damage function implies that transition risks are greater than previously supposed and that “the scenarios show that climate change is becoming a first order factor for our economies.”
Background on Damage Function Models
A damage function is a statistical relationship between climate variables and some measure of real output or real income. To estimate such a relationship, researchers gather data on output or income and climate variables such as temperature. One of the original and perhaps the most famous damage functions was formulated by Yale economist William Nordhaus in his DICE model[5], work for which he shared the Nobel Prize in Economics in 2018. The most recent vintage of the model, completed last year, continues to have a very simple damage function:
%Loss in global real GDP = 0.3467times ∆T2mean
Thus, if the global mean temperature rises by 3 degrees centigrade (perhaps by 2100), then the percent loss in global GDP would be
0.3467 times 32 = 0.3467 times 9 = 3.1%
Similarly, if global mean temperature went up 4.5C, then the loss in global real GDP would be 7%.
Nordhaus notes in the model documentation that estimating a damage function is challenging, since it is necessary to infer the relationship between temperature and GDP over the recent few decades and then extrapolate the relationship far into the future. Rather than using a strict econometric procedure to estimate the damage function, Nordhaus specifies the model by employing a combination of a survey of the academic literature on damage functions, a survey of literature on “tipping points,” which are potential non-linear impacts from climate change, and some judgmental adjustments. Nordhaus calibrates the damage function according to his view of the preponderance of the evidence as of 2023.
A potential limitation of Nordhaus’s approach is that it assumes that global mean temperature is sufficient to reliably infer the effect on GDP. Since climate variables such as temperature are available at highly granular levels, such as by city, county or province, should we not use this additional information? If measures of real GDP could also be obtained locally, then there would be far more data available to estimate the relationship between real GDP and temperature.
The New NGFS Damage Function
Recent damage function papers in the academic literature, including the NGFS damage function, have adopted this strategy. The NGFS damage function was estimated using the DOSE[6] database, which measures real GDP per capita-real income-for 83 countries and 1661 subregions over the years 1960 to 2019. Examples of subregions would be states in the U.S. or provinces in China.
Because the DOSE data is highly granular, it is possible to use many more climate variables than just global average temperature to measure the effect of climate change on real income. The NGFS damage function considers a variety of regional temperature and precipitation statistics observed in the regions in which real income is measured: 1) annual mean temperature in the region; 2) annual total precipitation in the region; 3) daily temperature variability over the year; 4) the number of wet days per year and 5) the total precipitation per year from extremely wet days.
The NGFS damage function does not include explicit squared terms as in the Nordhaus model, but it does include interaction terms such as the product of the change of annual mean temperature and the level of annual mean temperature. Multiplying these variables together ensures that the effect of the change in temperature becomes greater as the level of temperature rises. In all there are five interaction terms: 1) the change in temperature times the level of temperature; 2) the change in temperature variability times the level of temperature; 3) the change in precipitation times the level of precipitation; 4) the change in the number of wet days times the level of wet days; and 5) the change in extreme precipitation times the level of precipitation. Overall, there are 10 climate variables in all in the model, listed in Table 1 below.
Table 1
The NGFS damage function uses these 10 variables[7] to explain the annual real income growth for each of the 1,661 sub-regions. Besides allowing any of these 10 variables to affect real income growth in the year they occur, the model allows the terms to have a lagged influence: the temperature-based variables that occurred as much as 10 years ago can affect real income this year. For example, the temperature in Iowa in 1967 has an independent effect on GDP growth in Iowa in 1977 even after including the influence of the temperature-based variables in 1977, 1976, …1968.The precipitation variables as many as four years ago are allowed to affect real income growth today. The damage function uses three regression models which are combined by switching randomly between the regressions when economic damage is simulated into the future. Temperature terms can affect real income growth as much as eight, nine or 10 years in the future in the three versions of the regression model, with precipitation affecting real income growth as much as four years in the future in each model version.
Overall, as many as 74 climate variables can affect real income growth this year – the four temperature-based variables that occur this year, from last year, and from as many as 10 years ago, (4 x 11) and the six precipitation-based variables whose influence is felt this year, from last year and from as many as four years ago (6 x 5).
Relationship of New Damage Function to Old Damage Function
The new damage function is a generalization of the old NGFS damage function, which was based on research by Kalkuhl and Wenz (2020).[8] In Kalkuhl and Wenz (2020), loss to real income depended on ∆T and ∆T ∙ T. The new damage function includes lags of these variables up to 10 years as well as new temperature variables, ∆Tvar and ∆Tvar ∙ T̅, along with lags up to 10 years. Besides these additional temperature variables, the new model also includes the precipitation variables ∆P, ∆P ∙ P, ∆WD, ∆WD ∙ WD, ∆Pext, and ∆Pext ∙ T, along with four years of lags. Thus, the new damage model contains dramatically more variables than the old model.
Calibration of the New Damage Function
To calibrate the new damage function, coefficients on each climate variable and its lags are estimated from historical data on real income growth and the climate variables at a high spatial granularity. If we estimate a regression model for one of the three regression models, with eight lags on the temperature variables, we get the regression coefficients in Table 2.[9]
Table 2
To calculate an estimate of the loss to real income this year for a particular region, we collect the values of the temperature variables in the region of interest listed in Table 2 for this year and for each year in the past up to eight years ago. We would also collect the values of the precipitation variables this year and up to four years ago. We then take each climate variable, multiply it by its corresponding coefficient in Table 2 and then add. For example, we would take the change in temperature this year and multiply it by 0.00051. To this product, we would add the change in temperature last year multiplied by -0.00524. We continue this process until we have added up all 66 terms. That gives us the damage this year. Then we move to next year and repeat the process.
As we move forward in time, we will have to rely on climate models to give us projections of temperature and precipitation variables to use in calculations. We would repeat the process for each year from 2020 to 2080, giving a cumulative damage function for real income in the region we are focusing on. Then we repeat the process for all regions to get the effect on specific countries and globally. Because each year’s damage accumulates the effects of climate change in previous years, the damage compounds to large numbers. Oddly, not only does climate change up to 10 years in the past have independent effects on GDP today, as can be seen in Table 2, losses from climate change further in the past can have a bigger effect on real income today than climate change in more recent years.
Although the procedure outlined above is conceptually how damages are calculated, the model is more complicated. The data set is shuffled randomly 1000 times and then the model is re-estimated 1000 times, analogous to drawing a different hand from the shuffled deck. In the end, we have one thousand versions of Table 2.[10] After drawing a set of regression coefficients randomly, the Kotz et al (2024) procedure is to randomly select a future path of climate variables from a set of climate models that project climate variables into the future from 2020 to 2100. The future paths of the climate are selected for two scenarios: RCP 2.6 and RCP 8.5. RCP 2.5 is a scenario in which temperature rises until about 2050 and then levels off. RCP 8.5 is a scenario in which GHG emissions get much worse over time, implying a very high rise in temperature well over 3C. Repeating these simulations[11], we get the following results[12].
Figure 1
Figure 1 shows that global economic damage is fairly similar up to about 2050 under the two scenarios, with global damage leveling off at about 20 percent under RCP 2.6 but continuing to about 60 percent under RCP 8.5. The model simulations show that -19 percent is the “committed” economic damage under the model, i.e., considering the damage under both RCP 2.6 and RCP 8.5, -19 percent is the point at which the damage implied by the two scenarios can be statistically distinguished.A damage of -19 percent is inevitable regardless of the climate policy the world follows, according to the model. Figure 2 shows the difference between the old and new NGFS damage functions for various NGFS scenarios.[13]
Figure 2
Is the New Damage Function Really Better?
In its technical document, the NGFS offers some purported key advantages of the new damage function relative to the old damage function. The NGFS argues that since the new damage function is based on a variety of temperature and precipitation variables measured at a high geographic granularity, it should be more accurate. Moreover, the NGFS maintains that the inclusion of lags up to 10 years can capture the potential effects of climate change on real income that can persist for many years, an effect ignored by the old damage function.
Of course, whether including new climate variables in the new damage function is important depends on whether they practically matter in the damage projections. In Annex 4 of its technical document, the NGFS claims the additional variables contribute materially to the damage projections.[14] The NGFS analysis attempts to allocate the relative contributions of the climate variables to the global loss of 61.6 percent of real income produced by the new damage function under RCP 8.5 by 2100. The results of the NGFS analysis are shown in Table 3. According to the NGFS analysis, all of the variables in the model make material contributions to the final damage estimate, justifying the use of the new climate variables in the model.
Table 3
To check the NGFS estimates, we follow a different strategy. We can verify how much each set of variables matters by zeroing out coefficients of different categories of climate variables in the underlying regression models and then re-simulating the NGFS damage model. If we look at Table 4, in which we have circled the estimates for the second row, we can see that the ∆T ∙ T variables in the regression are good candidates for being highly influential. All the estimates are negative, the coefficient values are relatively large compared to the other coefficients and they are multiplied by the absolute level of temperature that is scaled by the change in temperature.
Table 4
To test the effect of that variable in the model, we take the 1000 simulated regressions without re-estimating them and zero out all coefficients other than the nine coefficients in the second row of Table 4. We then re-simulate the NGFS damage model along the RCP 2.6 and 8.5 scenarios, drawing randomly from the same set of climate models used to determine the NGFS damage function.[15] In this way, only the coefficients in the second row of Table 4 contribute to the projected damage. Figure 3 shows the result.
Figure 3
Figure 3 demonstrates that a damage model that only includes the scaled level of temperature terms from the regressions closely follows the full NGFS model that includes all the climate variables. In particular, the model simulations using all the climate variables estimated a global real income loss under RCP 8.5 by 2100 of 61.7 percent while the model simulations with only the scaled level of temperature variables produced a loss of 61 percent. This test contradicts the NGFS analysis that the newly added climate variables contribute significantly to the results.
As another check, we can pull out a temperature increase scenario under RCP 8.5 under one of the climate projection models[16] used by Katz et al (2024) to estimate the damage to 2100. In this experiment, for each year we predict the economic damage by using only the scaled level of temperature coefficients from Supplementary Table 4 in the Supplementary information in Katz et al (2024), i.e., we only include the current and lagged values of ∆T ∙T. Figure 4 demonstrates that the simplified regression model with only the temperature terms predicts the full simulation outcome well.
Figure 4
Although the NGFS touts the inclusion of a wide variety of temperature and precipitation variables as a key advantage of the new damage model, a full model simulation as well as a simplified check suggests that the newly included variables are irrelevant in practice. Thedamage function primarily depends on the level and change of temperature. Adding the additional climate variables does not matter practically.
How Much Does Each Lag of the Scaled Level of Temperature Matter?
Although the large number of lags of the one variable, the scaled level of temperature ∆T ∙T, explains the large damage estimates, it is not clear how much each lag contributes to the total. To answer this question, we zero out all coefficients reported in Table 4 other than the second row.[17] We then fully re-simulate the damage model, using the assumption of no lags, one lag, two lags, three lags, five lags and seven lags of ∆T ∙T. Figure 5 shows the results when either seven or five lags of ∆T ∙T only are kept in the model.
Figure 5
Figure 5 shows that seven lags in the temperature variables are sufficient to reproduce the full damage function. However, we start to see significantly lower damage with five lags.
Figure 6 shows the effect of including three, two, one or no lags in the model.
Figure 6
Starting with no lags of the ∆T ∙T temperature variables, Figures 5 and 6 show that the addition of each lag contributes significantly to the economic damage predicted by the model, with the effect of the lags saturating at about seven lags. The coefficients of the lags do not decline with time, but sometimes are greater than the contemporaneous effect. Adding up many lags, all of which are of the same order of magnitude, produces large economic losses. Thus, the statistical justification for including each lag of the scaled temperature in the model is vitally important if we are to believe the large economic damage results.
Is the Damage Model Statistically Justified?
Model Selection Criteria are Arbitrary
Kotz et al (2024) justify the included variables and their lags by using a combination of the Akaike Information Criterion (AIC) and related Bayesian Information Criterion (BIC).[18] The AIC and BIC criteria are goodness-of-fit measures that attempt to trade off goodness of fit against overfitting, i.e., including too many variables and lags in the regression model. The BIC penalizes the inclusion of additional variables more severely and is a more conservative measure.[19] Lower AIC and BIC measures are better. Kotz et al (2024) determine the number of lags to keep in the regressions by selecting models that minimize the AIC and BIC.
Kotz et al (2024) start with a model with 10 lags of temperature and 10 lags of precipitation variables and then remove one lag of one variable at a time. For example, Figure 7 plots AIC and BIC starting with 10 lags of all the variables and removing one lag of the change in temperature variables at a time, holding the lags of all other variables constant. Figure 7 shows that keeping all 10 lags of the change in temperature variable results in the minimum AIC and BIC, conditional on keeping all other variables at 10 lags. Figure 8 repeats the same procedure for the change in precipitation variable, suggesting that four lags of precipitation minimize the AIC and BIC.
Figure 7
Figure 8
Continuing with this lag selection methodology, Kotz et al (2024) find that models with four lags of precipitation variables and either eight, nine or 10 lags of temperature variables minimize the AIC and BIC[20]. They end up with three preferred specifications of regression model. Table 5 shows the AIC and BIC for each model.
Table 5
The model selection procedure employed by Kotz et al (2024) is arbitrary. Why start with ten lags of all variables and then remove one variable at a time, holding the lags of the others constant? There are many other ways to proceed. For example, if limiting overfitting is the concern, why not start with the simplest possible model and then add lags? Suppose we start with a candidate model with no lags and then we add them one by one, choosing the model that minimizes the AIC and BIC. Table 6 shows the result of applying this alternative procedure.
Table 6
The damage model with one lag of all temperature and precipitation variables has the lowest AIC and BIC statistics relative to the alternative models with no lags and two lags. More importantly, the model with one lag would be preferable to any of the NGFS damage model specifications in the first three columns of Table 6, since the AIC and BIC statistics are lower.
To benchmark the difference in economic damage implied by the one-lag damage model and the NGFS damage model, we run the temperature terms of the one-lag model on the representative RCP 8.5 scenario used in Figure 4. Figure 9 shows, not surprisingly, that the one-lag model produces 80 percent lower economic damage by 2100 compared to the NGFS damage model.
Figure 9
The Model Selection Process Did Not and Cannot Establish Statistical Significance of the Underlying Parameters
Kotz et al (2024) as well as the NGFS technical document emphasize that the goal of the damage model is to identify “robustly inferred causal relationships” that can be used “to project the exogenous impact of future climate conditions on the economy.”[21] However, the model selection procedure used to specify the details of the NGFS damage model is not relevant for establishing causation.
Kotz et al (2024) and the NGFS technical document imply that the use of the AIC and BIC helps to establish causation, although they do not spell out the argument as to how this would work. To infer causal relationships, it is necessary to estimate the magnitudes of the effects of climate variables on real income growth, ensuring that they are statistically significant. Kotz et al (2024) and the NGFS technical document do not make the explicit claim that the AIC and BIC analysis establishes causation between climate variables and economic damage by showing that the underlying estimates are statistically significant. But if they did make such an explicit claim, it would be invalid.
The AIC and BIC can be used to establish statistical significance of the parameters of the damage model, but only under the highly unrealistic assumption that the unobserved influences in the model are uncorrelated.[22] The methodology of restricting variables and preferring a model with a lower AIC or BIC is equivalent to running a hypotheses test, specifically a likelihood ratio test.[23] For the likelihood ratio test to be a valid statistical test, the unobserved influences in the panel regression model must be uncorrelated.
As emphasized by Cameron and Miller[24], panel data regression models, of which the NGFS damage model is an example, have a pronounced tendency to overstate statistical significance of variables if the correlation across the cross-sectional data—the subregions—and correlation across time—years—is not accounted for, leading to spurious statistical significance. In general, statistical hypothesis tests should be corrected for correlation, but the AIC and BIC are uncorrected. Since the underlying unobserved influences in the damage model are almost certainly correlated across subregions and across time, the AIC and BIC cannot be interpreted as statistical hypothesis tests that help to establish statistical significance, and by implication, causation. The AIC and BIC can only be used to make relative comparisons in sample between different model specifications, balancing goodness of fit against overfitting.
Important Coefficients Are Not Statistically Significant
Strangely, despite the emphasis on causation, Kotz et al (2024) do not focus on the statistical significance of the estimated parameters. Table 2, which corresponds to Supplementary Table 2 in Kotz et al (2024), does report statistical significance of the parameters. All but one of the coefficients in the second row, the variable that produces almost all the economic damage, are apparently statistically significant. However, these statistics are misleading. The reported standard errors[25] in Kotz et al (2024) are clustered by region, which assumes there are correlations of unobserved factors that affect regions[26]. However, the unobserved influences in the model are almost certainly also correlated across time, which is why it is common practice to cluster standard errors in the time dimension as well, i.e., double cluster the standard errors.
Recent research papers on similar panel regression-based climate damage functions have recognized that unobserved influences are correlated cross-sectionally and across time and have reported double clustered standard errors.Examples include Newell et al (2021), Burke et al (2015)[27] and Dell et al (2012).[28] In other climate-related econometric research that features panel regressions, Bolton and Kacperczyk (2023) [29] report their estimates of the existence and magnitude of a carbon premium in the equity markets after double clustering standard errors along the cross section and time dimensions. Similarly, Hopper (2024)[30] estimates a panel regression model that measures the effect of GHG emissions on probability of default and double clusters the standard errors across cross-sections and time. Kotz et al (2024) did not follow this best practice.
Table 7 reports the statistical significance levels that would have been obtained by double clustering the standard errors of the NGFS damage function regression model, compared to the singly clustered standard errors by region reported in Kotz et al (2024). When we single cluster or double cluster the standard errors, we do not change the estimates themselves. We only change the standard errors, which changes the statistical significance levels.
Table 7
As Table 7 shows, when accounting for correlation of unobserved influences across regions and time, a good bit of the statistical significance disappears. More concerning, the statistical significance of the terms that drive the most important coefficients on has vanished,[31] raising serious questions about the legitimacy of the damage function. Without robust statistical evidence on the causative effect of the coefficients, the large economic damage from the model disappears.
Conclusions
The academic literature on damage functions is much too uncertain to choose a new damage function that is obviously superior to the many alternatives. Instead, any new damage model should be specified as a hypothetical scenario assumption rather than as an established empirical fact.The NGFS could use a range of damage function models to incorporate different macroeconomic assumptions into its climate scenarios, as long as they are well-tested and plausible. To help ensure that the range of models selected is credible, the NGFS should employ an open process with extensive NGFS member review that is disclosed and subject to public comment before any new damage models are officially adopted.
The NGFS’s New Climate Damage Function- A Flawed Analysis With Massive Economic Consequences_Download
[1] Newell, R, Prest, B, and Sexton, S, “The GDP-Temperature relationship: Implications for climate change damages,” Journal of Environmental Economics and Management, (2021)
[2] Kotz, M, Levermann, A, and Wenz, L. “The economic commitment of climate change,” Nature 601, 2024, available at https://www.nature.com/articles/s41586-024-07219-0
[3] NGFS, “Damage functions, NGFS scenarios, and the economic commitment of climate change: An explanatory note,” November 2024, available at https://www.ngfs.net/sites/default/files/media/2024/11/05/ngfs_scenarios_explanatory_note_on_damage_functions.pdf
[4] https://www.ngfs.net/en/communique-de-presse/ngfs-publishes-latest-long-term-climate-macro-financial-scenarios-climate-risks-assessment-2024
[5] The most current version of the DICE model and documentation is available at https://yale.app.box.com/s/whlqcr7gtzdm4nxnrfhvap2hlzebuvvm/folder/196571686525
[6] Wenz, L, Carr, R, Kogel, N, Kotz, M, and Kalkuhl M, “DOSE-global data set of reported sub-national economic output,” Nature, 2023, available at https://www.nature.com/articles/s41597-023-02323-8
[7] The model is a panel regression that includes fixed effects, i.e., constants that differentiate the effect of the climate variables on the sub-regions and over time.
[8] Kalkuhl, M and Wenz, L, “The impact of climate conditions on economic production: evidence from a global panel of regions,” Journal of Environmental Economics and Management, (2020), available at https://www.sciencedirect.com/science/article/pii/S0095069620300838
[9] Table 2, which we have re-estimated, corresponds to Supplementary Table 2 in Kotz, M., Levermann A, and Wenz, L, “Supplementary information for: The economic commitment of climate change,” 2024, available at https://static-content.springer.com/esm/art%3A10.1038%2Fs41586-024-07219-0/MediaObjects/41586_2024_7219_MOESM1_ESM.pdf
[10] The same procedure is followed to estimate one thousand regressions with nine lags on the temperature variables and one thousand regressions with ten lags on the temperature variables. In the model simulations, regression coefficients are drawn randomly from one of the three models.
[11] We use the software and data provided by Kotz, Levermann, and Wenz (2024), available at https://zenodo.org/records/13957801
[12] This chart is a re-simulated version of Figure 1 in Kotz et al (2024).
[13] The Kotz et al damage function cannot be directly applied to the NGFS scenarios since it contains extra climate variables. It is ported to a function that depends on temperature and locations to be used within the NGFS scenarios.
[14] See NGFS technical document, page 49
[15] We also do the same for the thousand regressions with nine and 10 temperature variables and include these regressions in the simulation.
[16] We use the global circulation model ACCESS-ESM1-5 for this test
[17] We follow the same procedure for the other two regression models used in the damage function.
[18] If θ̂ is the estimated parameter vector of k variables, and lnL (θ̂)is the log-likelihood function, then AIC = – 2lnL(θ̂) + 2k. Similarly, BIC = – 2logL(θ̂) + kln(n), where n is the number of data points.
[19] The ln(n) term makes the BIC larger, especially for regressions that have a lot of data points n.
[20] The results of the AIC and BIC tests explains why Kotz et al use three regressions with eight, nine or 10 temperature variable lags and four lags of the precipitation variables, randomly switching between the three regressions when they simulate economic damage from the model.
[21] Kotz et al, Supplementary Information, pg 5. Also see NGFS Technical Document, pg 19
[22] In econometric terms, the error terms in the model must be homoscedastic and serially uncorrelated.
[23] If we have two nested models M1 and M2 with M1 ⊂ M2 with log-likelihoods lnL1 (θ̂1) and lnL2 (θ̂2) and k1 < k2 parameters, then we would select M1 if AIC(M1) < AIC(M2) or – 2lnL1(θ̂1) + 2k1 < – 2lnL2(θ̂2) + 2k2, which is a standard likelihood ratio test since LR = 2(lnL2(θ̂2) – lnL1(θ̂1)) < k2 – k1, distributed x2. Thus, when we select models that have lower AICs, we are implicitly performing likelihood ratio tests with a critical value of k2 – k1. Similarly, when we select models with lower BICs, we are implicitly performing likelihood ratio tests with a critical value of (k2 – k1)ln (n). Likelihood ratio tests are alternatives to the standard F and t tests.
[24] Cameron, A and Miller, D, “A Practitioner’s Guide to Cluster-Robust Inference,” Journal of Human Resources, 2015, available at https://jhr.uwpress.org/content/50/2/317
[25] We did not report the standard errors explicitly in Table 2 but rather color-coded what the standard errors imply for statistical significance.
[26] Supplementary tables 2,3 and 4 in Supplementary Information, Kotz et al (2024), report standard errors clustered at the region level only.
[27] Burke, M and Hsiang, S, “Global non-linear effects of temperature on economic production,” Nature, (2015)
[28] Dell, M, Jones, B, and Olken, B, “Temperature shocks and economic growth: evidence from the last century,” American Economic Journal of Macroeconomics, (2012)
[29] Bolton, P and Kacperczyk, M, “Global Pricing of Carbon Transition Risk,” Journal of Finance, August 2023, available at https://onlinelibrary.wiley.com/doi/10.1111/jofi.13272
[30] Hopper, G, “How Should Banks Manage Climate Transition Risk?” (2024), available at https://bpi.com/how-should-banks-manage-climate-transition-risk/
[31] Double clustering the standard errors in the nine and 10 lag regression models reported in Supplementary Tables 3 and 4 in Kotz et al (2024) produces similar results.
