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1\section{Measurement}2\label{sec:measurement}34This section defines every object used in the empirical analysis, in the order5in which the pipeline constructs them: returns, residuals, standardized6innovations, source shocks, isolation filters, propagation outcomes, networks,7and systemic co-jump minutes. Two principles govern all definitions. First,8\emph{strict lag discipline}: any parameter entering the transformation of9month-$m$ data---factor loadings, scale estimates, seasonal profiles, universe10membership, network weights---is estimated exclusively on data dated before11month $m$. Second, \emph{measurement before interpretation}: each construct is12defined so that its mechanical properties (denominators, censoring, base13rates) are known before any economic claim rests on it.1415\subsection{From bars to returns}1617The unit of observation is a one-minute bar during the regular session18(09:30--16:00 ET, truncated at the true exchange close on early-closing days).19Let $P_{i,t}$ be the last trade price of security $i$ in minute $t$.20One-minute log returns,21\begin{equation}22r_{i,t} \;=\; \log P_{i,t} - \log P_{i,t-1},23\end{equation}24are computed within ticker and trading day only: the first observed minute of25each stock-day has no return, and overnight intervals are never bridged, so no26intraday statistic mixes overnight information. Because the source records a27bar only when at least one trade occurs, a missing minute means ``no trade,''28not ``zero return''; returns are accordingly defined only across consecutively29observed minutes and missingness is carried, never imputed. Bars flagged for30OHLC inconsistency and all bars inside detected corporate-action windows31(Section~\ref{sec:data}) are excluded before any return is computed.3233All shock-level analysis operates on the lagged top-500 dollar-volume34universe. This choice trades breadth for interpretability: top-500 names trade35essentially every minute (median 382--390 bars per day from the mid-1990s36onward), so the cross-sectional response measures defined below have stable,37nearly complete denominators, and the asynchronicity that depresses38high-frequency comovement among thinner names \citep{epps1979comovements}39cannot masquerade as absent propagation. Results for top-1000 and top-300040universes appear in the robustness materials.4142\subsection{Residualization}43\label{sec:residualization}4445A raw extreme return is not evidence of an idiosyncratic event: it may load on46a market-wide movement. We therefore work with residuals from a single-factor47market model whose ingredients are constructed to avoid three mechanical48contaminations.4950\emph{Leave-one-out factor.} The market return at minute $t$, from the51perspective of security $i$, is the equal-weighted mean return of the universe52\emph{excluding} $i$:53\begin{equation}54r^{(-i)}_{m,t} \;=\; \frac{1}{N_t-1}\sum_{j\ne i} r_{j,t},55\end{equation}56where $N_t$ counts securities with an observable return at $t$. Without the57leave-one-out correction, a large own return mechanically moves the factor and58attenuates the measured shock; with 500 names the correction is small but we59apply it exactly.6061\emph{Lagged loadings.} The loading $\beta_i$ is the OLS slope of $r_{i,t}$ on62the equal-weighted market return, estimated on all minutes of month $m{-}1$ in63which the security has at least 300 valid return observations and the market64aggregate at least 50 contributing names; securities with insufficient history65receive $\beta_i=1$, and estimates are winsorized to $[-3,5]$. Residuals in66month $m$ are then67\begin{equation}68e_{i,t} \;=\; r_{i,t} - \beta_i\, r^{(-i)}_{m,t}.69\label{eq:residual}70\end{equation}71A single lagged loading per month is deliberately coarse: it cannot overfit72the event minutes it will later be used to classify, and any misspecification73it induces is absorbed by the placebo differencing described in74Section~\ref{sec:design}.7576\emph{Robust, seasonally adjusted scale.} Minute returns are strongly77leptokurtic and their dispersion follows the familiar intraday U-shape78\citep{andersen1997intraday}; a sample standard deviation would be dominated79by the very tail events we wish to detect, and an unadjusted scale would80over-detect shocks at the open. We therefore standardize by a81median-absolute-deviation scale interacted with a minute-of-day factor, both82from month $m{-}1$:83\begin{equation}84z_{i,t} \;=\; \frac{e_{i,t}}{1.4826\,\mathrm{MAD}_i \times f_{\tau(t)}},85\qquad86f_{\tau} \;=\; \frac{\mathrm{med}_i\,|e_{i,\cdot}/(1.4826\,\mathrm{MAD}_i)|_{\tau}}87{\overline{\mathrm{med}}},88\label{eq:zscore}89\end{equation}90where $\tau(t)$ is the minute of day, the numerator of $f_\tau$ is the91cross-sectional median absolute standardized residual at minute-of-day $\tau$92in month $m{-}1$, and $\overline{\mathrm{med}}$ normalizes the profile to unit93mean. The factor $1.4826$ makes the MAD consistent for the standard deviation94under normality; under the actual fat-tailed distribution, $z$ is a ranking95device whose thresholds are calibrated empirically, not a Gaussian quantile.96The same construction applied to the market return, with the prior month's97market MAD, yields the standardized market factor $z_{m,t}$ used in the98isolation filters.99100\subsection{Source shocks and isolation}101\label{sec:shockdef}102103A \emph{source shock} is an event $(i,t)$ with104\begin{equation}105|z_{i,t}| \;\ge\; 6,106\end{equation}107with the sign recorded and analyzed separately. The threshold is deliberately108extreme: it selects on the order of one event per stock-week109(\NEventsReal{} events over 410 months), deep enough into the tail that110bid--ask bounce alone cannot generate qualifying events for top-500 names111after 2001, while leaving enough events for precise inference within every112year. Nested thresholds ($|z|\ge8$, $|z|\ge12$) are reported in the113robustness table.114115Because the question concerns \emph{idiosyncratic} origins, two isolation116filters apply at the event minute:117\begin{enumerate}118\item \textbf{Calm market:} $|z_{m,t}| < 1$. The event does not coincide with119a market-factor movement.120\item \textbf{No simultaneous wave:} the share of eligible securities with121$|z_{j,t}|>4$ at minute $t$ is at most $2\%$, and at least 50 securities are122observable. The event is not part of a market-wide burst.123\end{enumerate}124Both filters are measured, not assumed: minutes excluded by filter (2) form125the systemic co-jump sample of Section~\ref{sec:cojumpdef}, so the design126partitions all extreme activity into an idiosyncratic branch and a systemic127branch rather than discarding the inconvenient one. The filters are128intentionally conservative about common information; the cost, acknowledged129throughout, is that any propagation operating \emph{through} an immediate130market-wide reaction is assigned to the systemic branch by construction.131132\subsection{Propagation outcomes}133\label{sec:outcomes}134135Fix an event $(i,t)$ and a horizon136$h\in\{1,2,5,10,15,30,60,120\}$ minutes. Receivers are all $j\ne i$ in the137universe with an observable bar at the measurement minute; outcomes require at138least 50 eligible receivers and are right-censored---recorded as missing, not139zero---whenever $t+h$ crosses the session close, so end-of-day events never140contribute mechanically attenuated responses. The outcomes are:141\begin{align}142\mathrm{Breadth}_{i,t}(h) &= \frac{1}{N_{t,h}-1}\sum_{j\ne i}143\mathbf{1}\{|z_{j,t+h}|>2\},144\label{eq:breadth}\\145\mathrm{Intensity}_{i,t}(h) &= \frac{1}{N_{t,h}-1}\sum_{j\ne i}|z_{j,t+h}|,146\label{eq:intensity}\\147\mathrm{Signed}_{i,t}(h) &= \frac{\operatorname{sign}(e_{i,t})}{N_{t,h}-1}148\sum_{j\ne i} z_{j,t+h}.149\label{eq:signed}150\end{align}151The receiver threshold $|z|>2$ is calibrated to the fat-tailed base rate: it152fires for roughly 7\% of stock-minutes unconditionally, high enough that153breadth is a well-populated share whose \emph{changes} are measured with154precision, low enough to respond to genuine tail activity. Cumulative breadth155replaces the point-in-time indicator with156$\mathbf{1}\{\max_{s\in(t,t+h]}|z_{j,s}|>2\}$, so a receiver that jumps at157$t+3$ and is quiet at $t+15$ still counts; its base rate is mechanically158higher (about 53\% at $h=15$) and its pre-event benchmark uses the mirrored159window $[t-5-h,\,t-5)$.160161Every outcome is also evaluated at $t-5$ (point-in-time) or on the mirrored162pre-window (cumulative). The pre-event value serves two distinct purposes: it163is the \emph{differencing baseline}---shocks arrive disproportionately in164minutes of elevated ambient activity, and the post-minus-pre difference165removes that level---and it is a \emph{falsification statistic}, since genuine166propagation cannot precede its cause.167168\subsection{Lagged correlation networks}169\label{sec:networkdef}170171For each month $m$ and security $i$, the neighbor set172$\mathcal{N}_i$ collects the twenty securities with the highest absolute173pairwise correlation with $i$'s residuals over month $m{-}1$ (pairwise174complete observations; neighbor sets require at least ten members present in175month $m$ to be used). Splitting equation~\eqref{eq:breadth} into176\begin{equation}177\mathrm{Breadth}^{\mathcal{N}}_{i,t}(h) = \frac{1}{|\mathcal{N}_i|}178\sum_{j\in\mathcal{N}_i}\mathbf{1}\{|z_{j,t+h}|>2\},179\qquad180\mathrm{Breadth}^{\overline{\mathcal{N}}}_{i,t}(h) = \text{same over }181j\notin\mathcal{N}_i\cup\{i\},182\end{equation}183yields the \emph{network differential}184$\mathrm{Breadth}^{\mathcal{N}}-\mathrm{Breadth}^{\overline{\mathcal{N}}}$,185the paper's sharpest instrument: transmission concentrated among twenty186connected names moves it one-for-one, while market-wide breadth dilutes the187same response by a factor of twenty-five. Because neighbor sets are lagged,188the differential cannot reflect the event's own contemporaneous comovement;189and because the identical differential is computed for placebo events, any190unconditional tendency of correlated stocks to co-move nets out.191192\subsection{Systemic co-jump minutes}193\label{sec:cojumpdef}194195A minute is a \emph{systemic co-jump} if the share of eligible securities with196$|z_{j,t}|>4$ exceeds 2\% (with at least 50 eligible names)---exactly the197complement of isolation filter (2). For each such minute we record the198affected shares at the $4\sigma$ and $6\sigma$ thresholds and the standardized199market return. Two summary series are used: the frequency of \emph{severe}200synchronization (affected share above 10\%) per 1,000 eligible minutes, and201the mean affected share within co-jump minutes. We emphasize severity and the202severe-minute count rather than the raw count at the 2\% threshold because203the latter is not comparable across regimes: idiosyncratic tail activity was204two orders of magnitude more frequent in the wide-tick 1990s, so a fixed205low threshold is breached almost continuously early in the sample for reasons206unrelated to synchronization.207208\subsection{What these measures can and cannot see}209\label{sec:measurement_limits}210211Three mechanical properties discipline interpretation. First, breadth212measures respond to \emph{tail} activity only; diffuse sub-threshold213responses would appear in intensity, which is reported alongside breadth214throughout. Second, all receiver measures condition on trading: a receiver215without a bar at $t+h$ leaves the denominator rather than entering as a216zero, so illiquidity cannot fabricate absorption. Third, the pre-2001 tick217regime (eighths and sixteenths) inflates both the level of $|z|$ activity and218its base rates; every cross-era comparison is therefore anchored on the219within-era placebo contrast, and the post-decimalization subsample reproduces220all headline results.221