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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. Using multiple designs with different identifying assumptions guards
against any single method's weaknesses and lets us triangulate the economic interpretation.

\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, plus the
high-minus-low (Q5$-$Q1) spread. 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 in the overlapping return windows. Sorts are model-free and transparent, and the
EW--VW comparison reveals whether an effect is concentrated in larger or smaller firms.

\subsection{Factor-model spanning of a long--short portfolio}
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 unexplained by known risk factors; we use
heteroskedasticity- and autocorrelation-consistent (HAC) standard errors. A significant alpha would
indicate pricing beyond standard exposures.

\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. 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 (from the FY2021 10-K), and we regress firm-level
CAR on pre-event exposure with size, beta, and sector controls. Because abnormal returns net out
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.