Figure 8 · Bayesian posterior on the Day-0 effect

Large negative effects are inconsistent with the data

The chart shows the range of announcement-day effects consistent with how ExxonMobil normally trades. The most likely effect is near zero. A governance discount worse than −2% is essentially inconsistent with the observed data.

Observed Day-0: +0.021% 95% of the time, the true effect was in this range Worse than −2%: 0.4% −4% −3% −2% −1% 0 +1% +2% +3% +4% −1.48% +1.53% Possible Day-0 announcement effect (percentage points)
+0.02%
Best guess at the true effect
±1.5%
Anything bigger is inconsistent with the data at the 95% credibility level
0.4%
Chance the real loss was worse than −2%
Sources & methodology notes
Bayesian posterior on the Day-0 effect (normal-normal conjugate)Prior: θ ∼ N(0, σpre2),   σpre = 0.77%
Likelihood: AR0 | θ ∼ N(θ, σobs2)
Posterior: θ | AR0 = +0.021% ∼ N(+0.02%, 0.54%2)
P(θ < −2pp) = 0.004  ·  P(θ < −3pp) < 0.001

The posterior visualized above rests on three methodological pillars: the Bayesian conjugate-prior framework for inference on a single mean parameter, the empirical-event-study tradition within which the Day-0 abnormal return is taken, and the synthetic-control variance estimator that calibrates the posterior’s scale. Canonical references are listed below.

  1. George E. P. Box & George C. Tiao, Bayesian Inference in Statistical Analysis (Wiley 1973). Canonical reference for the normal-normal conjugate-prior framework used here. Establishes that a flat prior on a normal-mean parameter combined with a normal likelihood yields a normal posterior with mean equal to the observed value and standard deviation equal to the data-generating process’s; precisely the construction visualized.
  2. Andrew Gelman et al., Bayesian Data Analysis ch. 2 (3d ed. 2013). Reference for the normal-normal conjugate posterior used here. With a flat (improper) prior centered at zero and a likelihood whose variance is estimated from the pre-period gap distribution, the posterior on the Day-0 effect is normal with mean equal to the observed Day-0 gap (+0.021%) and standard deviation equal to σpre (0.7676%).
  3. Alberto Abadie, Alexis Diamond & Jens Hainmueller, Synthetic Control Methods for Comparative Case Studies, 105 J. Am. Stat. Ass’n 493 (2010). Underlying synthetic-control specification that produces the Day-0 gap (+0.021%) feeding the posterior mean. doi.org/10.1198/jasa.2009.ap08746.
  4. Stephen J. Brown & Jerold B. Warner, Using Daily Stock Returns: The Case of Event Studies, 14 J. Fin. Econ. 3 (1985). Foundational empirical-event-study paper establishing the daily-returns research design within which the Day-0 abnormal return feeding this posterior is computed. Their small-sample variance results inform the σpre estimator used in the posterior’s scale.
  5. James M. Patell, Corporate Forecasts of Earnings Per Share and Stock Price Behavior: Empirical Tests, 14 J. Acct. Res. 246 (1976). Provides the frequentist parallel to this posterior — the Patell-z test on standardized abnormal returns, reported in the article’s multi-window battery (Figure 1 grid, fn.~27). Bayesian and frequentist bounds agree on the null finding.
  6. Donald J. Schuirmann, A Comparison of the Two One-Sided Tests Procedure and the Power Approach, 15 J. Pharmacokinetics & Biopharm. 657 (1987). Source of the TOST equivalence tests reported alongside this posterior in the article’s matched-pair footnote (fn.~26). The TOST equivalence bound at ±2 pp is mutually consistent with this posterior’s P(effect < −2%) = 0.004.
  7. Shane Goodwin, Read the Fine Print: What ExxonMobil’s Proxy Actually Says About Texas Redomiciliation, Columbia Law School Blue Sky Blog (May 2026); replication kit on file with the SMU Corporate Governance Initiative. Companion paper; fn.~29 derives the posterior from the 220-day pre-period synthetic-control gap distribution (N = 220, σpre = 0.77%).

Data attribution. Underlying daily adjusted closing prices sourced from S&P Capital IQ (IQ_CLOSEPRICE_ADJ feed). Pre-period covers 220 trading days ending T−1 = March 9, 2026.

Source: Author’s calculations from S&P Capital IQ daily adjusted closing prices; posterior derivation in Goodwin (2026), footnote on Bayesian credible intervals.