spb/wp17_uqo

1% ============================================================================2\section{Methodology}3\label{sec:methodology}4% ============================================================================56We bring six research designs to bear, ranging from portfolio sorts to a market-model event study.7Each carries its own identifying assumptions and vulnerabilities: a sort is transparent but silent8on risk; a factor regression adjusts for risk but imposes a linear factor structure; a panel with9fixed effects controls for unobserved heterogeneity but discards the cross-sectional variation that10may matter most; an event study buys clean timing at the cost of a single window. A conclusion that11survives all six is more credible than one resting on any one of them, but only if the inference12within each design is sound, and this section is explicit about where it is fragile.1314\subsection{Portfolio sorts}15Each formation year $Y$ we sort firms into quintiles of a within-year standardized signal and16compute the equal-weighted (EW) and value-weighted (VW) forward return of each quintile. The annual17high-minus-low spread and its time-series average are18\begin{equation}19  \text{LS}_{Y} \;=\; \bar r_{Q5,Y}-\bar r_{Q1,Y}, \qquad20  \overline{\text{LS}} \;=\; \frac{1}{N_Y}\sum_{Y}\text{LS}_{Y},21  \label{eq:spread}22\end{equation}23and we test $\overline{\text{LS}}=0$ with Newey--West standard errors with three lags24\citep{neweywest1987} on the series of annual spreads. The appeal of the sort is that it imposes no25functional form; the EW--VW comparison indicates whether the effect is concentrated in smaller index26members. The limitation is sample size: the full sample has $N_Y=15$ annual spreads, the27pre-period twelve, and the post-period three. A Newey--West $t$-statistic computed on three28observations has no useful sampling distribution, so for the post-period we report it only for29completeness and rely instead on the cohort-by-cohort values, on leave-one-out means, on pooled30firm-level regressions with standard errors clustered by firm, and on the monthly long--short31series described next.3233\subsection{Factor-model spanning of a long--short portfolio}34A raw spread could be compensation for bearing known risks. We build an annually rebalanced,35equal-weighted, self-financing long--short portfolio (top minus bottom quintile), record its36monthly return $\text{LS}_m$, and estimate37\begin{equation}38\begin{aligned}39  \text{LS}_m \;=\;& \alpha + \beta_{M}\,\text{MKT}_m + \beta_{S}\,\text{SMB}_m + \beta_{H}\,\text{HML}_m40  + \beta_{R}\,\text{RMW}_m \\41  &+ \beta_{C}\,\text{CMA}_m + \beta_{U}\,\text{MOM}_m + \beta_{T}\,\text{TECH}_m + u_m ,42\end{aligned}43  \label{eq:alpha}44\end{equation}45with heteroskedasticity- and autocorrelation-consistent standard errors (three lags). We report the46CAPM version, the Fama--French five factors plus momentum (FF6; \citealp{famafrench2015,carhart1997}),47and FF6 augmented with a technology-sector factor $\text{TECH}_m$, the equal-weighted monthly return48of the Information Technology constituents of the panel minus the equal-weighted return of all49constituents. The sector factor is not a priced risk factor in the usual sense; it is a diagnostic50that asks whether the AI long--short is distinguishable from a bet on the technology sector during51the sample. The intercept $\alpha$ is the average monthly return beyond what the loadings explain.52Because the monthly series has 180 observations (36 in the post-period), it supports inference that53the three annual post-period spreads cannot; we also report the Sharpe ratio of the series in each54sub-period.5556\subsection{Two-way fixed-effects panel regressions}57We estimate58\begin{equation}59  r_{i,Y+1} = \beta\, \text{AI}_{i,Y} + \gamma' X_{i,Y} + \alpha_i + \delta_Y + \varepsilon_{i,Y},60  \label{eq:panel}61\end{equation}62where $r_{i,Y+1}$ is the winsorized forward return of equation~\eqref{eq:fwd}, $\text{AI}_{i,Y}$ the63standardized signal, $X$ the standardized controls, and $\alpha_i,\delta_Y$ firm and year fixed64effects. Standard errors are double-clustered by firm and year65\citep{petersen2009,thompson2011,cameron2011}; with fifteen year clusters the time dimension of66the clustering is thin, and we report firm-clustered versions where the distinction matters. Firm67fixed effects absorb all time-invariant cross-firm differences, so $\beta$ identifies whether68within-firm changes in AI language predict returns; year effects absorb common shocks. Contrasting69this estimate with the cross-sectional ones isolates whether the signal lives across or within70firms (H4).7172\subsection{Fama--MacBeth regressions}73We also run annual cross-sectional regressions and average the slopes \citep{famamacbeth1973},74\begin{equation}75  r_{i,Y+1} = a_Y + b_Y\, \text{AI}_{i,Y} + c_Y' X_{i,Y} + e_{i,Y}, \qquad76  \hat b = \frac{1}{N_Y}\sum_Y \hat b_Y ,77  \label{eq:fm}78\end{equation}79with Newey--West errors on the time series of $\hat b_Y$. This estimator weights each year equally80and yields an interpretable per-standard-deviation premium; plotting the annual slopes reveals the81time profile of the premium directly, and adding GICS-sector dummies to the cross-sectional82regression gives the within-sector version of the slope.8384\subsection{Structural-break and difference-in-differences designs}85To test regime dependence (H2) we split the sample at the ChatGPT launch and compare sub-period86spreads, and we estimate the pooled interaction model87\begin{equation}88  r_{i,Y+1} = \beta_1\, \text{AI}_{i,Y} + \beta_2\, (\text{AI}_{i,Y}\times \text{Post}_Y)89  + \gamma' X_{i,Y} + \delta\,\text{Post}_Y + \varepsilon_{i,Y},90  \qquad \text{Post}_Y=\mathbb{1}[Y\ge 2022],91  \label{eq:break}92\end{equation}93where $\beta_2$ is the change in the premium per standard deviation of the signal, with standard94errors clustered by formation year. We complement this with a difference-in-differences design in95which treatment is the firm's pre-determined exposure $\overline{\text{AI}}_i^{\,19\text{--}21}$, the96average of its composite over the filings of 2019--2021, standardized across firms, so that it97cannot respond to the event:98\begin{equation}99  r_{i,Y+1} = \theta\,\bigl(\overline{\text{AI}}_i^{\,19\text{--}21}\times\text{Post}_Y\bigr)100  + \gamma' X_{i,Y} + \alpha_i + \delta_Y + \varepsilon_{i,Y},101  \label{eq:did}102\end{equation}103estimated with firm and year fixed effects and double-clustered errors, and with a binary version104in which treatment is above-median pre-determined exposure. An event-time version replaces the105single interaction with interactions of the fixed exposure with each formation-year dummy relative106to 2021,107\begin{equation}108  r_{i,Y+1} = \sum_{\tau\neq 2021}\theta_\tau\,\bigl(\overline{\text{AI}}_i^{\,19\text{--}21}\times\mathbb{1}[Y=\tau]\bigr)109  + \gamma' X_{i,Y} + \delta_Y + \varepsilon_{i,Y},110  \label{eq:evt}111\end{equation}112which traces the dynamic response and, in the years before 2022, tests the parallel-trends113assumption. Because each $\theta_\tau$ is identified within a single year, standard errors for114equation~\eqref{eq:evt} must be clustered by firm, not by year; the first version of this paper115clustered by year, which made the cluster-robust variance of each $\theta_\tau$ degenerate and the116confidence bands spuriously narrow.117118\subsection{Market-model event study}119Finally, we run an event study \citep{mackinlay1997} around the ChatGPT launch, which converts the120annual-frequency portfolio evidence into a daily, cleanly dated test. The launch was a121consumer-product event, unanticipated by equity markets, that raised the salience of AI without122itself changing any firm's fundamentals overnight. For each stock we estimate the market model123\begin{equation}124  r_{i,d}=a_i+b_i\, r^{m}_{d}+e_{i,d}125  \label{eq:mm}126\end{equation}127over the estimation window 3~January--31~October 2022 (209 trading days, SPY as the market), and128compute abnormal and cumulative abnormal returns over the event window 1~November 2022--30~June1292024,130\begin{equation}131  \text{AR}_{i,d}=r_{i,d}-(\hat a_i+\hat b_i r^m_d), \qquad132  \text{CAR}_i(T)=\sum_{d=1}^{T}\text{AR}_{i,d}.133  \label{eq:car}134\end{equation}135CARs are compared across terciles of pre-event AI exposure measured from the filings of calendar136year 2021, all dated before the launch, and firm-level CARs are regressed on exposure with size,137beta, and sector controls. The firm-level Welch $t$-test used in the first version treats the138firms as independent, which they are not when they share an event date139\citep{brownwarner1985,kolari2010}. We therefore also treat the equal-weighted High-minus-Low140abnormal return as a single daily series $\text{AR}^{HL}_d$ and compute141\begin{equation}142  t_{\text{CDA}}(T)=\frac{\sum_{d=1}^{T}\text{AR}^{HL}_d}{\hat\sigma_{HL}\sqrt{T}},143  \label{eq:cda}144\end{equation}145where $\hat\sigma_{HL}$ is the daily standard deviation of the same portfolio series in the146estimation window (the crude dependence adjustment of \citealp{brownwarner1985}) or, to allow for147event-induced variance \citep{boehmer1991}, in the event window. We also report the average148pairwise correlation of estimation-window residuals, which governs the overstatement of the149independence-based test \citep{kolari2010}. Horizons of 250 and 395 trading days are long for an150event study; at such horizons the market model's intercept, estimated in the 2022 bear market,151makes all CARs drift negative, and only differences between terciles are interpretable152\citep{fama1998,kothariwarner2007}.153