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\section{Methodology}
\label{sec:methodology}
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We bring six complementary research designs to bear, ranging from simple portfolio sorts to a
market-model event study. Each carries its own identifying assumptions and its own vulnerabilities: a
sort is transparent but model-free and silent on risk; a factor regression adjusts for risk but
imposes a linear factor structure; a panel with fixed effects controls for unobserved heterogeneity
but discards the cross-sectional variation that may matter most; an event study buys clean timing at
the cost of a single window. By design these weaknesses do not overlap, so a conclusion that survives
all six is far more credible than one resting on any one. Using the battery together lets us
triangulate the economic interpretation rather than lean on a single estimate.

\subsection{Portfolio sorts}
Each formation year we sort firms into quintiles on a given (within-year standardized) AI signal and
compute the equal-weighted (EW) and value-weighted (VW) forward return of each quintile, along with the
high-minus-low (Q5$-$Q1) spread that a self-financing long--short investor would have earned. We report
the time-series average of the annual spreads and test its significance with Newey--West
\citep{neweywest1987} standard errors (three lags) to account for any serial correlation induced by the
overlapping return windows. The appeal of the sort is that it imposes no model and makes no functional-form
assumption---it simply asks what high-AI firms earned relative to low-AI firms---and the EW--VW comparison
is itself informative: when the equal-weighted spread exceeds the value-weighted one, the effect is
concentrated in smaller index members rather than driven by the largest names.

\subsection{Factor-model spanning of a long--short portfolio}
A raw spread could in principle be no more than compensation for bearing known risks. To ask whether
the AI spread survives risk adjustment, we build an annually rebalanced, equal-weighted, self-financing
long--short portfolio (top minus bottom quintile), record its monthly return, and regress it on
(i)~the CAPM market factor and (ii)~the Fama--French five factors plus momentum (FF6). The intercept
(alpha) measures the average return the portfolio earned beyond what its loadings on these established
factors can explain, and the slopes reveal which risks the strategy is implicitly taking; we use
heteroskedasticity- and autocorrelation-consistent (HAC) standard errors throughout. A significant
alpha would indicate that AI disclosure is priced over and above standard exposures, while an
insignificant one would say the spread is spanned by known factors.

\subsection{Two-way fixed-effects panel regressions}
We estimate
\begin{equation}
  r_{i,t\to t+12} = \beta\, \text{AI}_{i,t} + \gamma' X_{i,t} + \alpha_i + \delta_t + \varepsilon_{i,t},
  \label{eq:panel}
\end{equation}
where $r$ is the winsorized forward return, $\text{AI}_{i,t}$ the standardized signal, $X$ the
controls, and $\alpha_i,\delta_t$ firm and year fixed effects. Standard errors are double-clustered by
firm and year \citep{petersen2009,cameron2011}. Firm fixed effects absorb all time-invariant
cross-firm differences, so $\beta$ identifies whether \emph{within-firm} changes in AI language
predict returns; year fixed effects absorb common shocks. Contrasting this estimate with the
cross-sectional ones isolates whether the signal lives across or within firms (H4).

\subsection{Fama--MacBeth regressions}
We also run annual cross-sectional regressions of the forward return on the AI signal and controls and
average the slopes following \citet{famamacbeth1973}, with Newey--West errors on the time series of
coefficients. This estimator weights each year equally and yields an interpretable
per-standard-deviation premium; plotting the annual slopes reveals the time profile of the premium
directly.

\subsection{Structural-break and difference-in-differences designs}
To test regime dependence (H2) we split the sample at the 30~November 2022 ChatGPT launch and compare
sub-period spreads, and we estimate pooled interaction models of the form
$r = \beta_1 \text{AI} + \beta_2 (\text{AI}\times \text{Post}) + \gamma' X + \varepsilon$, where
$\text{Post}=\mathbb{1}[\text{formation year}\ge 2022]$ and $\beta_2$ captures the change in the
premium. We complement this with a difference-in-differences design in which treatment is the firm's
\emph{pre-determined} AI exposure---fixed at its 2019--2021 average so it cannot respond to the
event---interacted with Post, within firm and year fixed effects. An event-time version interacts
pre-determined exposure with each year dummy (relative to 2021) to trace the dynamic response.

\subsection{Market-model event study}
Finally, we run a classic event study \citep{mackinlay1997} around the ChatGPT launch, which converts
the annual-frequency portfolio evidence into a high-frequency, cleanly dated test. The launch is close
to an ideal natural experiment for our purposes: it was a consumer-product event, unanticipated by
equity markets, that sharply raised the salience of AI without itself changing any firm's
fundamentals overnight. For each stock we estimate a market model ($r_{i,d}=a_i+b_i r^{m}_{d}+e_{i,d}$)
over the estimation window January--October 2022 and compute daily abnormal returns
$\text{AR}_{i,d}=r_{i,d}-(\hat a_i+\hat b_i r^m_d)$ over the event window November 2022--June 2024.
Cumulative abnormal returns (CAR) are compared across terciles of \emph{pre-event} AI disclosure---taken
from the FY2021 10-K, the last report filed before the launch and therefore immune to any reaction to
it---and we regress firm-level CAR on pre-event exposure with size, beta, and sector controls. Because
abnormal returns net out each firm's market-beta exposure, the event study isolates the
cross-sectional response to the AI-narrative shock from the contemporaneous (and initially negative)
market drift of 2022--2023, addressing the natural objection that high-AI firms simply rode a
beta-driven recovery.