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\section{Methodology}
\label{sec:methodology}
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We bring six research designs to bear, ranging from portfolio sorts to a market-model event study.
Each carries its own identifying assumptions and vulnerabilities: a sort is transparent but 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. A conclusion that
survives all six is more credible than one resting on any one of them, but only if the inference
within each design is sound, and this section is explicit about where it is fragile.

\subsection{Portfolio sorts}
Each formation year $Y$ we sort firms into quintiles of a within-year standardized signal and
compute the equal-weighted (EW) and value-weighted (VW) forward return of each quintile. The annual
high-minus-low spread and its time-series average are
\begin{equation}
  \text{LS}_{Y} \;=\; \bar r_{Q5,Y}-\bar r_{Q1,Y}, \qquad
  \overline{\text{LS}} \;=\; \frac{1}{N_Y}\sum_{Y}\text{LS}_{Y},
  \label{eq:spread}
\end{equation}
and we test $\overline{\text{LS}}=0$ with Newey--West standard errors with three lags
\citep{neweywest1987} on the series of annual spreads. The appeal of the sort is that it imposes no
functional form; the EW--VW comparison indicates whether the effect is concentrated in smaller index
members. The limitation is sample size: the full sample has $N_Y=15$ annual spreads, the
pre-period twelve, and the post-period three. A Newey--West $t$-statistic computed on three
observations has no useful sampling distribution, so for the post-period we report it only for
completeness and rely instead on the cohort-by-cohort values, on leave-one-out means, on pooled
firm-level regressions with standard errors clustered by firm, and on the monthly long--short
series described next.

\subsection{Factor-model spanning of a long--short portfolio}
A raw spread could be compensation for bearing known risks. We build an annually rebalanced,
equal-weighted, self-financing long--short portfolio (top minus bottom quintile), record its
monthly return $\text{LS}_m$, and estimate
\begin{equation}
\begin{aligned}
  \text{LS}_m \;=\;& \alpha + \beta_{M}\,\text{MKT}_m + \beta_{S}\,\text{SMB}_m + \beta_{H}\,\text{HML}_m
  + \beta_{R}\,\text{RMW}_m \\
  &+ \beta_{C}\,\text{CMA}_m + \beta_{U}\,\text{MOM}_m + \beta_{T}\,\text{TECH}_m + u_m ,
\end{aligned}
  \label{eq:alpha}
\end{equation}
with heteroskedasticity- and autocorrelation-consistent standard errors (three lags). We report the
CAPM version, the Fama--French five factors plus momentum (FF6; \citealp{famafrench2015,carhart1997}),
and FF6 augmented with a technology-sector factor $\text{TECH}_m$, the equal-weighted monthly return
of the Information Technology constituents of the panel minus the equal-weighted return of all
constituents. The sector factor is not a priced risk factor in the usual sense; it is a diagnostic
that asks whether the AI long--short is distinguishable from a bet on the technology sector during
the sample. The intercept $\alpha$ is the average monthly return beyond what the loadings explain.
Because the monthly series has 180 observations (36 in the post-period), it supports inference that
the three annual post-period spreads cannot; we also report the Sharpe ratio of the series in each
sub-period.

\subsection{Two-way fixed-effects panel regressions}
We estimate
\begin{equation}
  r_{i,Y+1} = \beta\, \text{AI}_{i,Y} + \gamma' X_{i,Y} + \alpha_i + \delta_Y + \varepsilon_{i,Y},
  \label{eq:panel}
\end{equation}
where $r_{i,Y+1}$ is the winsorized forward return of equation~\eqref{eq:fwd}, $\text{AI}_{i,Y}$ the
standardized signal, $X$ the standardized controls, and $\alpha_i,\delta_Y$ firm and year fixed
effects. Standard errors are double-clustered by firm and year
\citep{petersen2009,thompson2011,cameron2011}; with fifteen year clusters the time dimension of
the clustering is thin, and we report firm-clustered versions where the distinction matters. Firm
fixed effects absorb all time-invariant cross-firm differences, so $\beta$ identifies whether
within-firm changes in AI language predict returns; year 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 and average the slopes \citep{famamacbeth1973},
\begin{equation}
  r_{i,Y+1} = a_Y + b_Y\, \text{AI}_{i,Y} + c_Y' X_{i,Y} + e_{i,Y}, \qquad
  \hat b = \frac{1}{N_Y}\sum_Y \hat b_Y ,
  \label{eq:fm}
\end{equation}
with Newey--West errors on the time series of $\hat b_Y$. 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, and adding GICS-sector dummies to the cross-sectional
regression gives the within-sector version of the slope.

\subsection{Structural-break and difference-in-differences designs}
To test regime dependence (H2) we split the sample at the ChatGPT launch and compare sub-period
spreads, and we estimate the pooled interaction model
\begin{equation}
  r_{i,Y+1} = \beta_1\, \text{AI}_{i,Y} + \beta_2\, (\text{AI}_{i,Y}\times \text{Post}_Y)
  + \gamma' X_{i,Y} + \delta\,\text{Post}_Y + \varepsilon_{i,Y},
  \qquad \text{Post}_Y=\mathbb{1}[Y\ge 2022],
  \label{eq:break}
\end{equation}
where $\beta_2$ is the change in the premium per standard deviation of the signal, with standard
errors clustered by formation year. We complement this with a difference-in-differences design in
which treatment is the firm's pre-determined exposure $\overline{\text{AI}}_i^{\,19\text{--}21}$, the
average of its composite over the filings of 2019--2021, standardized across firms, so that it
cannot respond to the event:
\begin{equation}
  r_{i,Y+1} = \theta\,\bigl(\overline{\text{AI}}_i^{\,19\text{--}21}\times\text{Post}_Y\bigr)
  + \gamma' X_{i,Y} + \alpha_i + \delta_Y + \varepsilon_{i,Y},
  \label{eq:did}
\end{equation}
estimated with firm and year fixed effects and double-clustered errors, and with a binary version
in which treatment is above-median pre-determined exposure. An event-time version replaces the
single interaction with interactions of the fixed exposure with each formation-year dummy relative
to 2021,
\begin{equation}
  r_{i,Y+1} = \sum_{\tau\neq 2021}\theta_\tau\,\bigl(\overline{\text{AI}}_i^{\,19\text{--}21}\times\mathbb{1}[Y=\tau]\bigr)
  + \gamma' X_{i,Y} + \delta_Y + \varepsilon_{i,Y},
  \label{eq:evt}
\end{equation}
which traces the dynamic response and, in the years before 2022, tests the parallel-trends
assumption. Because each $\theta_\tau$ is identified within a single year, standard errors for
equation~\eqref{eq:evt} must be clustered by firm, not by year; the first version of this paper
clustered by year, which made the cluster-robust variance of each $\theta_\tau$ degenerate and the
confidence bands spuriously narrow.

\subsection{Market-model event study}
Finally, we run an event study \citep{mackinlay1997} around the ChatGPT launch, which converts the
annual-frequency portfolio evidence into a daily, cleanly dated test. The launch was a
consumer-product event, unanticipated by equity markets, that raised the salience of AI without
itself changing any firm's fundamentals overnight. For each stock we estimate the market model
\begin{equation}
  r_{i,d}=a_i+b_i\, r^{m}_{d}+e_{i,d}
  \label{eq:mm}
\end{equation}
over the estimation window 3~January--31~October 2022 (209 trading days, SPY as the market), and
compute abnormal and cumulative abnormal returns over the event window 1~November 2022--30~June
2024,
\begin{equation}
  \text{AR}_{i,d}=r_{i,d}-(\hat a_i+\hat b_i r^m_d), \qquad
  \text{CAR}_i(T)=\sum_{d=1}^{T}\text{AR}_{i,d}.
  \label{eq:car}
\end{equation}
CARs are compared across terciles of pre-event AI exposure measured from the filings of calendar
year 2021, all dated before the launch, and firm-level CARs are regressed on exposure with size,
beta, and sector controls. The firm-level Welch $t$-test used in the first version treats the
firms as independent, which they are not when they share an event date
\citep{brownwarner1985,kolari2010}. We therefore also treat the equal-weighted High-minus-Low
abnormal return as a single daily series $\text{AR}^{HL}_d$ and compute
\begin{equation}
  t_{\text{CDA}}(T)=\frac{\sum_{d=1}^{T}\text{AR}^{HL}_d}{\hat\sigma_{HL}\sqrt{T}},
  \label{eq:cda}
\end{equation}
where $\hat\sigma_{HL}$ is the daily standard deviation of the same portfolio series in the
estimation window (the crude dependence adjustment of \citealp{brownwarner1985}) or, to allow for
event-induced variance \citep{boehmer1991}, in the event window. We also report the average
pairwise correlation of estimation-window residuals, which governs the overstatement of the
independence-based test \citep{kolari2010}. Horizons of 250 and 395 trading days are long for an
event study; at such horizons the market model's intercept, estimated in the 2022 bear market,
makes all CARs drift negative, and only differences between terciles are interpretable
\citep{fama1998,kothariwarner2007}.