Conceptual Winsorizing for SCC

• Conceptual winsorizing is an approach that replaces outlier social cost of carbon estimates with economically feasible upper bounds.
• It employs two bounds—the ability-to-pay and Leviathan Tax limits—tying SCC values to GDP, emissions, and tax revenue.
• The method adapts over time as economies decarbonize, trimming only conceptually implausible estimates for improved policy robustness.

Conceptual winsorizing is an approach to handling outliers in published estimates of the social cost of carbon (SCC) by replacing arbitrary statistical cutoffs with economic or conceptual upper bounds. In Richard S.J. Tol’s formulation, “the social cost of carbon is either a willingness to pay, which cannot exceed the ability to pay, or a proposed carbon tax, which cannot raise more revenue than all other taxes combined.” Estimates above these bounds are reset to the relevant bound because they are conceptually or economically implausible rather than merely statistically extreme (Tol, 10 Aug 2025).

1. Definition and scope

Traditional winsorizing is the statistical practice of limiting extreme data points by replacing values above or below a chosen threshold with the nearest value within that threshold. In the SCC setting, Tol proposes conceptual winsorizing as an alternative criterion: the question is not whether an estimate lies in the top $1\%$, but whether it can exist in principle given the interpretation of SCC and the economic constraints implied by that interpretation (Tol, 10 Aug 2025).

The proposal is motivated by the observation that there are many published estimates of the social cost of carbon and that some are clear outliers, the result of poorly constrained models. Conceptual winsorizing is designed to remove high outliers using bounds tied to feasibility and interpretation. The paper’s abstract states that conceptual winsorizing successfully removes high outliers and that it slackens as economies decarbonize, slowly without climate policy, faster with (Tol, 10 Aug 2025).

This use of winsorization differs from the standard robust-statistical literature, where winsorization usually caps observations at percentile or threshold values to reduce sensitivity to tail behavior. In that broader literature, winsorization is a data transformation technique in which extreme values are “shrunken” toward the center, and it is used in settings such as robust covariance estimation, credibility theory, and importance sampling (Lafit et al., 2022, Zhao et al., 2023, Orenstein, 2018). Conceptual winsorizing retains the capping logic but changes the basis of the cap.

2. Contrast with percentile winsorizing

The central distinction is between a threshold defined by statistical rank and a threshold defined by economic meaning. Percentile winsorizing trims a fixed tail fraction whether or not those observations are substantively implausible. Conceptual winsorizing may trim none or many observations, depending on whether estimates violate feasibility conditions (Tol, 10 Aug 2025).

Aspect	Percentile winsorizing	Conceptual winsorizing
Threshold basis	Statistical percentiles	Economic concepts or real-world feasibility
Justification	Statistical or mathematical	Economic or conceptual
Tail treatment	Always trims a fixed proportion	May trim none or all outliers depending on context
Geographic variation	Blind to country differences	Can be applied at the country level
SCC interpretation	Not interpretation-specific	Tied to willingness to pay or feasible tax policy

In this formulation, conceptual winsorizing is not simply a robust summary-statistic device. It is interpretation-dependent. If SCC is read as willingness to pay, one upper bound follows; if it is read as a proposed carbon tax, another upper bound follows. A common misconception is therefore to treat conceptual winsorizing as percentile winsorizing with a different cutoff. The paper instead frames it as an economically grounded feasibility restriction (Tol, 10 Aug 2025).

3. Conceptual upper bounds for the social cost of carbon

Tol develops two main upper bounds. The first is the Ability to Pay Upper Bound, also labeled “Weitzman-Winsor.” It interprets SCC as willingness to pay for mitigation. The key claim is that willingness to pay cannot exceed total income per unit of carbon emitted. For a country $c$, the upper bound is

$W_c^{WTP} = \frac{Y_c}{E_c},$

where $Y_c$ is GDP and $E_c$ is total carbon emissions of country $c$ (Tol, 10 Aug 2025).

The second is the Leviathan Tax Upper Bound, also labeled “Hobbes-Winsor.” It interprets SCC as the rate for a carbon tax. The underlying criterion is that the revenue from a carbon tax cannot exceed the total current tax revenue without fundamentally transforming the size of government. If $\tau_c$ is the country’s average tax rate as a fraction of GDP, then

$W_c^{tax} = \tau_c \frac{Y_c}{E_c}.$

Any SCC estimate above this value is trimmed to this maximum (Tol, 10 Aug 2025).

The paper also gives numerical illustrations. For 2019, world average carbon intensity is 11,571 tC/$and the maximum willingness to pay for SCC is $11,571/tC. Governments collected 13.8% of income in taxes globally, so a carbon tax of $1,594/tC would suffice to replace all other taxes. The paper states that most published estimates of SCC are below these bounds, but that a significant minority are not, warranting conceptual winsorizing (Tol, 10 Aug 2025).

4. Formal implementation

Let $S_i$ denote a published SCC estimate and let $c$0 denote the relevant conceptual upper bound for country $c$1, chosen either from the willingness-to-pay or tax-based construction. Tol’s generalized implementation formula is

$c$2

The operation is therefore a bounded replacement rule: for each estimate, use the lower of the reported value or the conceptual upper bound, and then compute emission-weighted averages across countries (Tol, 10 Aug 2025).

This formulation makes explicit that conceptual winsorizing is not a pure one-sample transformation in the usual robust-statistics sense. The upper bound is external to the empirical SCC distribution and is instead a function of GDP, emissions, and tax rates. This suggests that the method is best understood as a constraint derived from the meaning assigned to SCC, rather than as a generic tail-robust estimator.

The paper also emphasizes that the procedure can be applied at the country level using local GDP and tax rates. That feature distinguishes it from percentile winsorizing, which is blind to country differences (Tol, 10 Aug 2025).

5. Decarbonization and the slackening of the bound

A defining temporal feature of conceptual winsorizing is that its severity is not fixed. The paper states that conceptual winsorizing slackens as economies decarbonize, slowly without climate policy, faster with (Tol, 10 Aug 2025). The mechanism is straightforward in the paper’s own terms: with economic growth and decarbonization, carbon intensity declines, so the maximum willingness to pay per tonne and the maximum feasible carbon tax per tonne both rise.

Under the ability-to-pay bound, a decline in $c$3 raises $c$4. Under the tax-based bound, the same decline raises $c$5, holding tax rates fixed. The conceptual cutoff therefore becomes less restrictive over time, and fewer published SCC estimates are classified as outliers (Tol, 10 Aug 2025).

This temporal adaptivity contrasts with percentile winsorizing, where the rule is fixed unless the percentile threshold is changed. In conceptual winsorizing, the bound evolves with the economy. The paper presents this as a substantive advantage because the restriction tightens or relaxes in line with changing ability to pay and changing carbon intensity rather than in line with an arbitrary tail fraction.

6. Implications, limitations, and methodological context

The principal implication is that conceptual winsorizing removes only those outliers that are economically, conceptually, or politically infeasible, rather than trimming high estimates merely for statistical reasons. The paper argues that this keeps attention on which SCC estimates are plausible or implementable and avoids over-reducing the SCC mean because of uncritically high but unrealistic values (Tol, 10 Aug 2025).

The method is also presented as improving policy robustness. Since the bounds are rooted in economic reality and political feasibility, SCC estimates used for policy become less influenced by long tails that may reflect modeling errors or unrealistic assumptions. At the same time, the paper notes an equity issue: using global averages can mask within-country disparities, richer countries can afford higher SCC, and this alignment with ability to pay comes with caveats regarding distributional effects (Tol, 10 Aug 2025).

Within the broader winsorization literature, conceptual winsorizing occupies a distinct position. Standard winsorizing replaces extremes with boundary values, retaining sample size and controlling the influence of outliers (Zhao et al., 2023). In robust covariance estimation, multivariate winsorization shrinks extremes toward the center while respecting data geometry (Lafit et al., 2022). In importance sampling, winsorization clips large weights to trade bias for variance (Orenstein, 2018). Conceptual winsorizing inherits the basic replacement operation but relocates the decision rule from empirical tail behavior to external conceptual feasibility.

That shift is the method’s defining feature. It is neither a denial of robust statistics nor a variant of percentile trimming under another name. It is a proposal to winsorize by interpretation: if SCC is a willingness to pay, it cannot exceed the ability to pay; if SCC is a carbon tax, it cannot raise more revenue than all other taxes combined (Tol, 10 Aug 2025).