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1% ============================================================================2\section{Methodology}3\label{sec:methodology}4% ============================================================================56We bring six complementary research designs to bear, ranging from simple portfolio sorts to a7market-model event study. Each carries its own identifying assumptions and its own vulnerabilities: a8sort is transparent but model-free and silent on risk; a factor regression adjusts for risk but9imposes a linear factor structure; a panel with fixed effects controls for unobserved heterogeneity10but discards the cross-sectional variation that may matter most; an event study buys clean timing at11the cost of a single window. By design these weaknesses do not overlap, so a conclusion that survives12all six is far more credible than one resting on any one. Using the battery together lets us13triangulate the economic interpretation rather than lean on a single estimate.1415\subsection{Portfolio sorts}16Each formation year we sort firms into quintiles on a given (within-year standardized) AI signal and17compute the equal-weighted (EW) and value-weighted (VW) forward return of each quintile, along with the18high-minus-low (Q5$-$Q1) spread that a self-financing long--short investor would have earned. We report19the time-series average of the annual spreads and test its significance with Newey--West20\citep{neweywest1987} standard errors (three lags) to account for any serial correlation induced by the21overlapping return windows. The appeal of the sort is that it imposes no model and makes no functional-form22assumption---it simply asks what high-AI firms earned relative to low-AI firms---and the EW--VW comparison23is itself informative: when the equal-weighted spread exceeds the value-weighted one, the effect is24concentrated in smaller index members rather than driven by the largest names.2526\subsection{Factor-model spanning of a long--short portfolio}27A raw spread could in principle be no more than compensation for bearing known risks. To ask whether28the AI spread survives risk adjustment, we build an annually rebalanced, equal-weighted, self-financing29long--short portfolio (top minus bottom quintile), record its monthly return, and regress it on30(i)~the CAPM market factor and (ii)~the Fama--French five factors plus momentum (FF6). The intercept31(alpha) measures the average return the portfolio earned beyond what its loadings on these established32factors can explain, and the slopes reveal which risks the strategy is implicitly taking; we use33heteroskedasticity- and autocorrelation-consistent (HAC) standard errors throughout. A significant34alpha would indicate that AI disclosure is priced over and above standard exposures, while an35insignificant one would say the spread is spanned by known factors.3637\subsection{Two-way fixed-effects panel regressions}38We estimate39\begin{equation}40 r_{i,t\to t+12} = \beta\, \text{AI}_{i,t} + \gamma' X_{i,t} + \alpha_i + \delta_t + \varepsilon_{i,t},41 \label{eq:panel}42\end{equation}43where $r$ is the winsorized forward return, $\text{AI}_{i,t}$ the standardized signal, $X$ the44controls, and $\alpha_i,\delta_t$ firm and year fixed effects. Standard errors are double-clustered by45firm and year \citep{petersen2009,cameron2011}. Firm fixed effects absorb all time-invariant46cross-firm differences, so $\beta$ identifies whether \emph{within-firm} changes in AI language47predict returns; year fixed effects absorb common shocks. Contrasting this estimate with the48cross-sectional ones isolates whether the signal lives across or within firms (H4).4950\subsection{Fama--MacBeth regressions}51We also run annual cross-sectional regressions of the forward return on the AI signal and controls and52average the slopes following \citet{famamacbeth1973}, with Newey--West errors on the time series of53coefficients. This estimator weights each year equally and yields an interpretable54per-standard-deviation premium; plotting the annual slopes reveals the time profile of the premium55directly.5657\subsection{Structural-break and difference-in-differences designs}58To test regime dependence (H2) we split the sample at the 30~November 2022 ChatGPT launch and compare59sub-period spreads, and we estimate pooled interaction models of the form60$r = \beta_1 \text{AI} + \beta_2 (\text{AI}\times \text{Post}) + \gamma' X + \varepsilon$, where61$\text{Post}=\mathbb{1}[\text{formation year}\ge 2022]$ and $\beta_2$ captures the change in the62premium. We complement this with a difference-in-differences design in which treatment is the firm's63\emph{pre-determined} AI exposure---fixed at its 2019--2021 average so it cannot respond to the64event---interacted with Post, within firm and year fixed effects. An event-time version interacts65pre-determined exposure with each year dummy (relative to 2021) to trace the dynamic response.6667\subsection{Market-model event study}68Finally, we run a classic event study \citep{mackinlay1997} around the ChatGPT launch, which converts69the annual-frequency portfolio evidence into a high-frequency, cleanly dated test. The launch is close70to an ideal natural experiment for our purposes: it was a consumer-product event, unanticipated by71equity markets, that sharply raised the salience of AI without itself changing any firm's72fundamentals overnight. For each stock we estimate a market model ($r_{i,d}=a_i+b_i r^{m}_{d}+e_{i,d}$)73over the estimation window January--October 2022 and compute daily abnormal returns74$\text{AR}_{i,d}=r_{i,d}-(\hat a_i+\hat b_i r^m_d)$ over the event window November 2022--June 2024.75Cumulative abnormal returns (CAR) are compared across terciles of \emph{pre-event} AI disclosure---taken76from the FY2021 10-K, the last report filed before the launch and therefore immune to any reaction to77it---and we regress firm-level CAR on pre-event exposure with size, beta, and sector controls. Because78abnormal returns net out each firm's market-beta exposure, the event study isolates the79cross-sectional response to the AI-narrative shock from the contemporaneous (and initially negative)80market drift of 2022--2023, addressing the natural objection that high-AI firms simply rode a81beta-driven recovery.82