0 · Front Matter

0.1 Abstract of the proposed study

A large empirical literature reports a positive risk-adjusted return to portfolios formed on environmental, social and governance (ESG) scores. That literature is almost uniformly conditioned on a single rating vendor. Because vendor scores for the same issuer correlate at levels far below the 0.99 achieved by credit rating agencies: Berg, Kölbel and Rigobon (2022) report pairwise correlations of 0.38 to 0.71 across six major providers , the "ESG portfolio" is not a well-defined object. It is a vendor-specific object.

This study asks whether the reported ESG alpha survives a change of vendor, holding the mandate, the universe, the optimiser, the rebalancing calendar and the constraint level fixed. The identifying variation is the choice of rating provider and nothing else. If a mandate written as "top-quartile ESG" produces materially different portfolios, materially different factor loadings, and a non-zero tracking error but a statistically zero return differential across two reputable providers, then the ESG alpha documented in the literature is not a property of corporate sustainability. It is a property of a proprietary algorithm, and the resulting sector and style tilts are an uncompensated active risk borne by the beneficiary.

0.2 Primary research questions

0.3 Hypotheses

The study is designed to be capable of disproving its own motivating conjecture. Each hypothesis is stated with the null that would falsify the divergence thesis.

• H1 (Systematic bias). . Rejection establishes that the two vendors load differently on observable characteristics, i.e. that divergence is not white noise.
• H2 (Portfolio friction). active share between Portfolio A and Portfolio B . The theoretical prior under bivariate normality and the observed correlation range is an active share of 39%–58% (§A.1), so H2 is close to a manipulation check.
• H3 (No compensation). in the FF5 model. Failure to reject H3 while H2 is rejected is the paper's central result: divergence generates risk without return.
• H4 (Sector-bet dominance). the systematic (factor + sector) block explains of . Rejection establishes that the divergence is a bet, not diversifiable churn.
• H5 (Friction dominance). rating-migration-induced turnover drag bp p.a. Rejection quantifies the price of the divergence in the currency fiduciaries actually use.
• H6 (Alpha attenuation). The single-vendor ESG–return coefficient is attenuated by , where is the inter-rater reliability ratio; under the observed correlation range this implies 29%–62% attenuation (§A.2). Instrumenting one vendor's score with another's isolates the common component . If is materially smaller in magnitude than the rescaling would predict, the single-vendor coefficient was being carried by the vendor-specific bias term rather than by common ESG content, which is the divergence thesis restated at the firm level.

0.4 Contribution relative to the existing literature

1. The divergence literature (Phase 1) documents disagreement at the score level. This protocol carries it through a constrained optimiser to the portfolio level, which is where fiduciary consequences are realised. The mapping from score dispersion to portfolio dispersion is nonlinear (a threshold operator) and is derived in closed form in §1.4.
2. It separates divergence into a priced-risk channel (systematic tilt) and a transaction-cost channel (idiosyncratic churn), which the existing literature conflates.
3. It supplies a falsification test, synthetic rater pairs with matched cross-sectional correlation, that distinguishes a genuine divergence effect from the mechanical consequence of any two imperfectly correlated screens.
4. It is designed against a point-in-time rating archive, addressing the retroactive-restatement problem documented by Berg, Fabisik and Sautner (ECGI WP; SSRN 3722087), which invalidates most published ESG backtests built on current-vintage vendor files.

0.5 Workflow decomposition

The four phases are specified as separable work packages with explicit hand-off artefacts. Where a multi-agent implementation is used, the boundaries below are the natural assignment; notation is fixed centrally in §A.9 so that outputs remain composable.

Phase 1: Theoretical Framework and Literature Synthesis

1.1 The anchor: Berg, Kölbel and Rigobon (2022)

Citation. Berg, F., Kölbel, J. F., and Rigobon, R. (2022). "Aggregate Confusion: The Divergence of ESG Ratings." Review of Finance, 26(6), 1315–1344. DOI: 10.1093/rof/rfac033.

What the paper establishes. Working with six providers (KLD (discontinued 2017), Sustainalytics, Vigeo Eiris (now Moody's ESG), RobecoSAM (now S&P Global), Asset4 (now Refinitiv/LSEG) and MSCI) the authors map every provider indicator into a common taxonomy of 64 categories and decompose observed rating disagreement into three orthogonalised channels:

Pairwise correlations across providers range from 0.38 to 0.71. The paper further documents a rater effect: a provider's overall assessment of a firm propagates into its category-level assessments, so measurement error is structured within rater rather than independent across categories.

Why the 56/38/6 split is the load-bearing fact for this study. The three channels have entirely different portfolio consequences, and the existing literature does not distinguish them:

• Weights divergence (6%) is economically benign. It is a monotone re-mixing of a common information set. Rank orderings are largely preserved and it produces very little set-membership churn.
• Scope divergence (38%) is a systematic distortion. A provider that scores a category the other omits has, in effect, added a characteristic to the objective function. Because ESG categories map non-uniformly onto sectors (e.g. carbon intensity onto Energy and Utilities; data privacy onto Information Technology; supply-chain labour onto Consumer Discretionary), scope divergence necessarily loads onto GICS sectors. Scope divergence is the sector-bet channel.
• Measurement divergence (56%) is the dominant channel and is mixed: the rater effect implies a firm-level systematic component (a provider's house view, correlated with disclosure volume and firm size), while the residual is classical noise. Measurement divergence is therefore the source of both the size/quality tilt and the turnover churn, and separating its two parts is the econometric heart of Phase 2.

Methodological extension proposed here. Berg et al. obtain their shares from a sequential substitution of rater-specific elements with common elements. Sequential decompositions are order-dependent: the share attributed to scope depends on whether measurement is harmonised first. This study re-estimates the decomposition using a Shapley value over the three channels, averaging the marginal contribution of each channel across all substitution orders:

where is the reduction in obtained by harmonising the channels in . The Shapley shares satisfy efficiency, , and are order-invariant. Reporting alongside the published 56/38/6 is a direct, falsifiable replication contribution.

1.2 A structural measurement model of the rating

Let denote the latent, mandate-relevant ESG quality of firm at time , the object every provider claims to measure. Let be a vector of observable firm characteristics (log market capitalisation, disclosure intensity, sector indicators, R&D intensity, leverage, geographic revenue mix, analyst coverage). The rating produced by provider is modelled as:

Two elements of this specification carry the analysis.

The bias vector . This is the algorithmic fingerprint of the provider, the projection of its scope choices, weighting scheme and rater effect onto observable characteristics. The industry-standard characterisations are directly testable restrictions on :

• MSCI rewards disclosure capacity and compliance infrastructure. Restriction: and .
• Sustainalytics gap-fills with sub-industry averages where firm-specific data are absent. Restriction: conditional on sub-industry, the within-industry variance of is compressed relative to , i.e. , with the compression increasing in the firm's missing-data density.

These are hypotheses about vendor behaviour, not assumptions. §2.5 specifies their estimation and §2.6 their falsification.

The divergence decomposition. Define observed divergence in normalised units:

The two components have qualitatively different portfolio consequences, and the entire economic argument of the paper rests on this separation:

Because is a linear function of persistent characteristics, it does not mean-revert and cannot be diversified away by holding more names; it is removed only by neutralising the portfolio to . Because is transient, it is diversified in the cross-section but is not free, it is paid for at every rebalance, in the currency of Phase 4.

1.3 The errors-in-variables problem and its consequence for reported ESG alpha

Suppose the true cross-sectional relation between latent ESG quality and expected excess return is . A researcher who runs the regression on a single vendor's score recovers not but an attenuated coefficient:

This identity is the reason inter-rater correlation is not merely a curiosity. Under the symmetric-noise benchmark, the observed 0.38–0.71 correlation range is the reliability ratio, implying that published single-vendor ESG coefficients are attenuated toward zero by 29% to 62% (§A.2). Two implications follow, and they cut in opposite directions, which is precisely why the study must test both:

1. A genuine ESG effect is understated. Correcting for measurement error should magnify a real . This is the logic of the instrumental-variables correction pursued by Berg, Kölbel, Pavlova and Rigobon in their noise-correction work (NBER WP 30562; SSRN 3941514), who instrument one provider's rating with another's.
2. A spurious ESG effect is manufactured. If what a screen actually selects is rather than , then the "ESG alpha" is the return to the characteristics in (size, profitability, low leverage, high disclosure) repackaged. This is testable: the FF5 loadings of the screened portfolio will absorb it, and will vanish.

The instrument and its exclusion restriction. Using as an instrument for requires . This is not innocuous: both providers ingest the same corporate sustainability reports, CDP submissions and regulatory filings, so a common input error violates exclusion. The protocol therefore requires:

• Over-identification with a third provider ( from LSEG/Refinitiv, ISS ESG or S&P Global CSA), enabling a Hansen test of the over-identifying restrictions;
• A Hausman test of OLS against IV;
• Reporting of the first-stage -statistic against the Stock–Yogo weak-instrument criterion. Note that the first-stage equals , so even the weakest observed reliability implies a strong first stage in a firm-level panel; weak identification binds only in short, small cross-sections such as a single month's regression;
• An explicit statement, in the limitations section, that a rejected test is evidence of correlated vendor error, itself a finding of interest, since correlated error means the market cannot diversify vendor choice.

1.4 From score dispersion to portfolio dispersion: the threshold operator

Portfolio mandates do not apply the score linearly. They apply a threshold: "hold only issuers in the top quartile." This is the single most consequential and least analysed step in the chain, because a threshold operator amplifies dispersion in the region where it bites.

Let be jointly standard normal with correlation after normalisation (§2.4), and let the eligibility rule be with . The probability that a firm survives both screens is:

where is the bivariate standard normal CDF. The conditional retention rate, the fraction of Portfolio A's names that Portfolio B also holds, and the Jaccard similarity of the two eligible sets are:

For two equal-weighted portfolios of equal cardinality, the active share of Cremers and Petajisto (2009) reduces exactly to .

Evaluating this at the correlation range reported by Berg et al. yields the study's ex ante motivation, computed numerically in §A.1:

Two portfolios written to the identical mandate, over the identical universe, at the identical constraint level, differ in roughly two-fifths to three-fifths of their holdings purely as a function of which vendor's file was loaded. For calibration, an active share of 39%–58% is in the range that Cremers and Petajisto associate with genuinely active management, here obtained with no active view whatsoever. This is the quantity H2 formalises, and it is the reason the study is worth running.

The amplification result. Note that across the relevant range: at the scores agree strongly, yet 39% of holdings differ. The threshold operator converts moderate score disagreement into severe portfolio disagreement, because disagreement is concentrated exactly at the cut-point where the mass of the distribution is greatest. This nonlinearity is the mechanism by which "modest" rating divergence becomes a first-order portfolio problem, and it is the primary theoretical contribution of Phase 1.

1.5 Foundational literature: verified sources

The following are verified against publisher records. Each entry states the specific role it plays in this protocol.

Divergence and its measurement

1. Berg, Kölbel and Rigobon (2022), Review of Finance 26(6), 1315–1344. DOI: 10.1093/rof/rfac033.: Anchor. Supplies the scope/measurement/weights decomposition and the correlation range that calibrates §1.4.

2. Christensen, D. M., Serafeim, G., and Sikochi, A. (2022). "Why Is Corporate Virtue in the Eye of the Beholder? The Case of ESG Ratings." The Accounting Review, 97(1), 147–175. DOI: 10.2308/TAR-2019-0506.: Establishes that greater ESG disclosure is associated with greater rating disagreement, not less. This is the direct antecedent for the disclosure term in and a warning against the intuition that mandatory disclosure regimes will resolve divergence mechanically.

3. Chatterji, A. K., Durand, R., Levine, D. I., and Touboul, S. (2016). "Do Ratings of Firms Converge? Implications for Managers, Investors and Strategy Researchers." Strategic Management Journal, 37(8), 1597–1614. DOI: 10.1002/smj.2407.: The pre-Berg benchmark. Distinguishes theorisation (what a rater means by social performance) from commensurability (how it converts that construct to a number). Maps onto scope and measurement divergence respectively, and provides an independent, earlier-vintage estimate of disagreement, useful for testing whether divergence has narrowed over time.

Asset pricing consequences of disagreement

4. Gibson Brandon, R., Krueger, P., and Schmidt, P. S. (2021). "ESG Rating Disagreement and Stock Returns." Financial Analysts Journal, 77(4), 104–127. DOI: 10.1080/0015198X.2021.1963186.: The closest existing work to RQ3. Treats disagreement as a firm-level characteristic and relates it to returns. This protocol departs from it in the object of study: Gibson Brandon et al. price disagreement as a characteristic; this study prices the portfolio-construction consequence of disagreement, which is a distinct and, for a fiduciary, more actionable quantity.

5. Avramov, D., Cheng, S., Lioui, A., and Tarelli, A. (2022). "Sustainable Investing with ESG Rating Uncertainty." Journal of Financial Economics, 145(2), 642–664. DOI: 10.1016/j.jfineco.2021.09.009.: Supplies the equilibrium theory. ESG uncertainty raises the market premium, depresses demand for ESG assets, and raises both CAPM alpha and effective beta. Provides the theoretical prior that divergence is not return-neutral in equilibrium, and thus the alternative hypothesis against which H3's null is tested.

6. Serafeim, G., and Yoon, A. (2023). "Stock Price Reactions to ESG News: The Role of ESG Ratings and Disagreement." Review of Accounting Studies, 28(3), 1500–1530. DOI: 10.1007/s11142-022-09675-3.: Establishes the information-processing channel: consensus ratings predict future ESG news, but predictive ability degrades in the presence of rater disagreement, and price reactions to ESG news weaken. Motivates the interpretation of a null as evidence that neither vendor holds an informational edge.

Equilibrium pricing of sustainability preferences (for interpreting the sign of any surviving alpha)

7. Pedersen, L. H., Fitzgibbons, S., and Pomorski, L. (2021). "Responsible Investing: The ESG-Efficient Frontier." Journal of Financial Economics, 142(2), 572–597. DOI: 10.1016/j.jfineco.2020.11.001.: Supplies the constrained-optimisation framework this protocol operationalises in Phase 3, and the language of the ESG-adjusted efficient frontier along which the shadow price is measured.

8. Pástor, Ľ., Stambaugh, R. F., and Taylor, L. A. (2022). "Dissecting Green Returns." Journal of Financial Economics, 146(2), 403–424. DOI: 10.1016/j.jfineco.2022.07.007.: Critical for interpretation. Realised green outperformance in the sample window is driven by unanticipated shifts in climate preferences, not by higher expected returns; theory implies green stocks should earn lower expected returns. Any positive in-sample must therefore be tested against a sentiment-shock control before it is called alpha.

9. Bolton, P., and Kacperczyk, M. (2021). "Do Investors Care About Carbon Risk?" Journal of Financial Economics, 142(2), 517–549. DOI: 10.1016/j.jfineco.2021.05.008.: Documents a carbon premium in unscaled emissions, providing a candidate omitted risk factor for the augmented specification in §3.6.

Data integrity, the papers that discipline Phase 2

10. Aswani, J., Raghunandan, A., and Rajgopal, S. (2024). "Are Carbon Emissions Associated with Stock Returns?" Review of Finance, 28(1), 75–106. DOI: 10.1093/rof/rfad013.: Shows that returns correlate with vendor-estimated emissions but not firm-disclosed emissions, and that vendor estimates track financial fundamentals. This is the sharpest published demonstration that ESG-data "alpha" can be an artefact of vendor modelling, and it is the direct methodological precedent for this study's central conjecture. (Note for completeness: the same issue carries a Comment and a Reply; the exchange should be cited in full, as the disagreement is itself informative about the fragility of vendor-data inference.)

11. Berg, F., Fabisik, K., and Sautner, Z. "Is History Repeating Itself? The (Un)Predictable Past of ESG Ratings." ECGI Finance Working Paper; SSRN 3722087. Working paper, verify publication status at time of submission.: Documents retroactive restatement of historical ESG ratings by a major provider, such that a backtest run today on current-vintage files does not reproduce what an investor could have known in real time. This is the single most important methodological constraint on Phase 2 and the reason point-in-time archives are non-negotiable.

12. Berg, F., Kölbel, J. F., Pavlova, A., and Rigobon, R. "ESG Confusion and Stock Returns: Tackling the Problem of Noise." NBER Working Paper 30562; SSRN 3941514. Working paper, verify publication status at time of submission.: The noise-correction methodology underlying §1.3.

Econometric and portfolio-construction methods

13. Fama, E. F., and French, K. R. (2015). "A Five-Factor Asset Pricing Model." Journal of Financial Economics, 116(1), 1–22. DOI: 10.1016/j.jfineco.2014.10.010.: The benchmark model of Phase 3. Note the authors' own finding that HML becomes redundant in the presence of RMW and CMA; §3.7 addresses the resulting multicollinearity.
14. Cremers, K. J. M., and Petajisto, A. (2009). "How Active Is Your Fund Manager? A New Measure That Predicts Performance." The Review of Financial Studies, 22(9), 3329–3365. DOI: 10.1093/rfs/hhp057.: Active share, used in §1.4 and §3.8.
15. Novy-Marx, R., and Velikov, M. (2016). "A Taxonomy of Anomalies and Their Trading Costs." The Review of Financial Studies, 29(1), 104–147. DOI: 10.1093/rfs/hhv063.: The transaction-cost framework of Phase 4, including the buy/hold-spread (banding) result that motivates §4.4.
16. Harvey, C. R., Liu, Y., and Zhu, H. (2016). "…and the Cross-Section of Expected Returns." The Review of Financial Studies, 29(1), 5–68. DOI: 10.1093/rfs/hhv059.: Multiple-testing discipline; the hurdle applied in §5.3.
17. Ledoit, O., and Wolf, M. (2008). "Robust Performance Hypothesis Testing with the Sharpe Ratio." Journal of Empirical Finance, 15(5), 850–859.: The studentised bootstrap test for the difference of two Sharpe ratios, which is the correct inference for comparing Portfolio A with Portfolio B. (DOI not independently confirmed in this pass; verify before submission.)

Literature required but not cited here, by type. Consistent with the no-hallucination guardrail, the following are categories of source the protocol requires; specific works must be located and verified by the research team rather than assumed:

• The canonical GRS joint-alpha test (Gibbons, Ross and Shanken, Econometrica, 1989) and the Fama–MacBeth (1973) cross-sectional procedure, both standard, both to be cited from the primary source.
• HAC covariance estimation (Newey and West, 1987, and the 1994 automatic-bandwidth extension) and the stationary bootstrap (Politis and Romano, 1994).
• Covariance shrinkage (Ledoit and Wolf, on linear shrinkage toward structured targets) and weight-constraint regularisation (Jagannathan and Ma, on the equivalence between no-short-sale constraints and shrinkage).
• Delisting-return bias correction (Shumway, 1997, Journal of Finance), required for CRSP handling in §2.7.
• Low-frequency spread estimators (Corwin and Schultz high–low estimator; Abdi and Ranaldo close–high–low estimator) as alternatives to TAQ-based effective spreads.
• Square-root market-impact estimation (Almgren et al.; Frazzini, Israel and Moskowitz on live institutional trading costs) for the calibration of in §4.3.
• Practitioner divergence studies from index providers and asset managers (e.g. work published in the Journal of Portfolio Management and Financial Analysts Journal on ESG rating dispersion), useful for out-of-academic-sample corroboration but to be treated as non-peer-reviewed evidence.

Phase 2: Data Architecture and Algorithmic Bias Normalisation

2.1 Design principle

Every design choice in this phase is subordinated to one rule: the panel must contain only information an investor could have acted on at the close of month . Divergence research is unusually exposed to violations of this rule, because ESG vendor files are delivered as current-vintage databases in which historical scores have been restated, coverage has been backfilled, and methodologies have been retro-applied. A study that fails here does not produce a biased estimate; it produces an estimate of a strategy that was never investable.

2.2 Asset universe

Primary universe: Russell 1000, point-in-time constituents.

Robustness universe: S&P 500. The S&P 500 is run as a secondary specification, not the primary, for three reasons: committee-based inclusion introduces selection endogeneity correlated with governance quality; the smaller cross-section reduces the number of names per quartile from ~250 to ~125, roughly -inflating the idiosyncratic tracking error; and its large-cap skew mechanically compresses the disclosure-intensity variation that identifies . Where the S&P 500 result differs from the Russell 1000 result, the difference is itself reportable evidence on the size-dependence of algorithmic bias.

Coverage-conditioned estimation universe. All primary tests are run on the dual-covered intersection: firms with a valid, in-force rating from both providers at . The coverage-attrition table (§2.8) is a required exhibit, because conditioning on dual coverage is itself a selection and its magnitude must be visible to the reader.

2.3 The methodology-break problem: and a mandatory sample split

Two structural breaks in vendor methodology fall inside a 2016–2026 window. Neither can be ignored, and neither is adequately handled by a linear time control.

Required design response.

1. Primary sample: 2018-10 through 2026-06 (93 months), the period over which the Sustainalytics ESG Risk Rating is methodologically stable in construct. This is the head-to-head window.
2. Legacy sample: 2016-01 through 2018-08, run separately with the legacy Sustainalytics ESG Rating, reported as a robustness exhibit only. Results are never pooled across the September 2018 boundary.
3. May-2024 break: an event study, not a nuisance. Because the vendor announced the change and quantified its impact, the induced turnover is a clean natural experiment on methodology-driven, fundamentals-orthogonal rebalancing. Estimate abnormal turnover and abnormal return in the month window around 30 May 2024, and report the friction attributable to a single unilateral vendor decision as a standalone exhibit in Phase 4. This converts a data problem into the protocol's most direct evidence on H5.
4. MSCI methodology actions must be handled symmetrically. MSCI ESG Ratings are published as a AAA–CCC letter grade underpinned by a continuous industry-adjusted score; both the letter thresholds and the underlying key-issue weights are subject to periodic annual methodology review. The research team must obtain the vendor's dated methodology-change log and construct a corresponding break calendar before estimation. Every break date enters the specification as an indicator interacted with the eligible-set-change dummy.

Consequence for the primary window: 93 months is a short sample for alpha inference. §3.9 reports the resulting minimum detectable effect and prescribes the remedies. This is disclosed at the design stage rather than discovered at the referee stage.

2.4 Normalisation: making incommensurable scales comparable

The two vendors produce scores that differ in direction (higher-is-better vs lower-is-better), support (bounded 0–10 industry-adjusted vs an unbounded-in-principle risk score), construct (peer-relative vs absolute) and granularity (heavily discretised vs quasi-continuous). Any comparison must therefore be invariant to monotone transformation. Three steps, applied in order, at every month , using only the cross-section available at .

Step 1: Sign harmonisation. Sustainalytics' ESG Risk Rating is a risk measure in which lower is better. Define the sign-harmonised score so that higher is better for both vendors:

This step is the most common silent error in the applied literature. A sign inversion converts the study's central result into its exact negation and will not announce itself in any diagnostic. The protocol requires a hard assertion in code: in every month, with the job failing if the assertion is violated.

Step 2: Rank / normal-score transformation (primary specification). Because the underlying scales are ordinal-incommensurable, the primary normalisation is the van der Waerden normal-score transform using the Blom plotting position, applied cross-sectionally within month:

where is taken over the dual-covered cross-section at with mid-ranks for ties. This transform is invariant to any strictly increasing reparameterisation of either vendor's scale, which is exactly the invariance required when one construct is peer-relative and the other absolute. It also renders the bivariate-normal geometry of §1.4 an assumption about the copula only, which is testable rather than imposed.

Step 3: Winsorised cross-sectional Z-score (robustness specification). For comparability with the applied literature and to preserve cardinal information about the magnitude of divergence:

All headline results are reported under both transforms. Divergence between them is diagnostic of tail-driven rather than central-tendency-driven disagreement, and is reported as such.

2.4.1 The industry-neutralisation decision: a design fork that must not be collapsed

Standard practice is to z-score within GICS industry group. For this study, that would destroy the object of interest. The sector bet induced by scope divergence is precisely what RQ2 measures; industry-neutralising the score removes it before it can be estimated. The protocol therefore constructs and carries both layers throughout, and treats their difference as an estimand rather than a nuisance:

The contrast between the portfolio built on and the portfolio built on identifies the sector-allocation channel by construction: any difference in tracking error, factor loading or alpha between the two is attributable to the sector content of the rating. This is the cleanest available identification of the "uncompensated sector bet" in the study's title, and it requires no additional data.

2.5 Estimating the algorithmic bias vector

Estimate, by month and then pooled, a seemingly-unrelated system in which the same characteristic vector explains each vendor's normalised score:

The characteristic vector is specified ex ante and frozen before estimation:

• market capitalisation (the size/attention channel)
• Disclosure intensity: count of non-missing ESG data points reported, or a vendor-independent disclosure score; plus an indicator for publication of a standalone sustainability report and for third-party assurance
• GICS sector indicators (10 or 11 sectors, dated per §2.6)
• R&D / assets, capex / assets, leverage, gross profitability, asset growth (the FF5 characteristic space, so that any style tilt is visible)
• Foreign sales share, employee count, unionisation-free operational scale proxies
• Analyst coverage, institutional ownership, index membership indicators
• Firm age since first CRSP appearance

Test of H1. Stack the system and test equality of the bias vectors by a Wald statistic on the difference, with standard errors clustered two-way by firm and month:

Individual coefficient contrasts deliver the vendor-behaviour restrictions of §1.2 as directional -tests: and under the "MSCI rewards compliance capacity" characterisation.

Decomposing observed divergence. The fitted system yields the operational split of §1.2:

This share is the Phase 2 headline statistic and the bridge to Phase 3.

2.6 Detecting industry-average gap-filling

The hypothesis that Sustainalytics substitutes sub-industry averages where firm-specific evidence is absent generates three distinct, independently testable signatures. All three should be run; agreement across them constitutes evidence, any one alone does not.

(a) Variance-ratio test. Under gap-filling, within-sub-industry dispersion is compressed:

Scale-dependence is removed by applying the test to the normalised rather than raw scores, and by reporting a rank-based Fligner–Killeen statistic alongside the parametric ratio.

(b) Shrinkage-intensity regression. Regress each firm's deviation from its sub-industry mean on its missing-data density (the fraction of the vendor's own indicator set that is unreported by the firm):

A negative , firms that disclose less sit closer to their peer average, is the signature of gap-filling. The identical regression on provides the vendor contrast; is the directional prediction.

(c) Point-mass / clustering diagnostic. Test for excess probability mass at sub-industry medians using a discontinuity-style density test on (median-centred), reporting excess kurtosis and the mass within of zero against the empirical null generated by permuting industry labels.

2.7 Point-in-time integrity, look-ahead and survivorship

Identifier crosswalk. The linkage must be historical, not current.

Rating availability lag. Vendor files carry both a rating as-of date and a publication/dissemination date; these differ, sometimes by weeks. The eligible set at month is formed only from ratings whose dissemination date precedes the formation date:

A month variant is run as a conservatism check; a material difference between and results indicates the strategy depends on rapid vendor-file ingestion, which is itself a reportable operational finding.

Archival vintages. Where the vendor supplies an as-was archive (monthly snapshot files), that archive is the primary source and the current-vintage file is used only to measure restatement. Define the restatement diagnostic:

Reporting the distribution of , and re-running the headline test on current-vintage data to show how much the measured alpha changes, is a required exhibit. It converts the Berg–Fabisik–Sautner critique from a caveat into a quantified sensitivity. If an as-was archive cannot be obtained for a vendor, that vendor's results must be labelled as subject to unquantified restatement bias, and the paper must say so in the abstract.

Survivorship and delisting. Portfolio membership is determined by the PIT universe at formation and is not conditioned on survival to the end of the sample. Delisted names are held to the delisting date and the CRSP delisting return DLRET is applied. Where DLRET is missing and the delisting code indicates a performance-related delisting (CRSP codes 500 and 520–584), the standard correction of the delisting-bias literature (Shumway, 1997) is applied; a variant and a "missing-set-to-zero" variant are reported as bounds. Proceeds are reinvested pro rata across the surviving eligible set at the delisting date.

2.8 Missing data

Coverage is missing not at random: the probability of being rated rises with size, index membership, disclosure and analyst attention, the same variables that constitute . Naive imputation is therefore not a neutral repair; it manufactures the bias under study.

Prohibited. Imputing a missing vendor score from that vendor's own industry mean. This is mechanically identical to the gap-filling behaviour being tested in §2.6, and would guarantee a false positive.

Required, in this order:

1. Primary, complete-case on the dual-covered intersection. Report the attrition table: universe size, MSCI coverage, Sustainalytics coverage, intersection, by year, in both counts and market-cap share. Coverage differentials across vendors are a first-order finding in their own right.
2. Selection quantification: Heckman-type model. Estimate a first-stage probit for dual coverage and include the inverse Mills ratio in the score regressions of §2.5. The exclusion restriction (a variable affecting coverage but not the score) is the weak link; index-membership timing and analyst-coverage shocks are candidates, and the identifying assumption must be argued explicitly rather than asserted.
3. Robustness, multiple imputation. MICE with imputations using only PIT-available covariates, combined by Rubin's rules:

4. Bounds, worst-case sensitivity. Report Manski-style worst-case bounds on obtained by assigning uncovered firms the most and least favourable feasible score. If the headline conclusion survives the bounds, coverage selection is not driving it; if it does not, the paper must report the result as bounded rather than point-identified.

2.9 Deliverable of Phase 2

A single reproducible panel, PANEL_PIT, keyed on (PERMNO, month-end), containing: sign-harmonised and both-normalisation vendor scores under universe-wide and industry-neutral variants; eligibility flags , ; the decomposed divergence ; dated GICS; the FF5 characteristic set; CRSP returns with delisting adjustment; liquidity fields (ADV, effective spread, quoted depth); coverage and restatement flags; and a per-cell provenance stamp recording vendor file vintage and dissemination date. No downstream analysis may read any file other than PANEL_PIT.

Phase 3: Portfolio Optimisation and Econometric Testing

3.1 Inputs to the optimiser

Covariance matrix. With and an estimation window of 60 months, the sample covariance is singular or near-singular and its extreme eigenvalues are severely biased. Two estimators are carried in parallel and results reported under both:

Linear shrinkage. Shrink the sample covariance toward a structured target (constant-correlation, or a single-index target):