\section{Measurement}
\label{sec:measurement}

This section defines every object used in the empirical analysis, in the order
in which the pipeline constructs them: returns, residuals, standardized
innovations, source shocks, isolation filters, propagation outcomes, networks,
and systemic co-jump minutes. Two principles govern all definitions. First,
\emph{strict lag discipline}: any parameter entering the transformation of
month-$m$ data---factor loadings, scale estimates, seasonal profiles, universe
membership, network weights---is estimated exclusively on data dated before
month $m$. Second, \emph{measurement before interpretation}: each construct is
defined so that its mechanical properties (denominators, censoring, base
rates) are known before any economic claim rests on it.

\subsection{From bars to returns}

The unit of observation is a one-minute bar during the regular session
(09:30--16:00 ET, truncated at the true exchange close on early-closing days).
Let $P_{i,t}$ be the last trade price of security $i$ in minute $t$.
One-minute log returns,
\begin{equation}
r_{i,t} \;=\; \log P_{i,t} - \log P_{i,t-1},
\end{equation}
are computed within ticker and trading day only: the first observed minute of
each stock-day has no return, and overnight intervals are never bridged, so no
intraday statistic mixes overnight information. Because the source records a
bar only when at least one trade occurs, a missing minute means ``no trade,''
not ``zero return''; returns are accordingly defined only across consecutively
observed minutes and missingness is carried, never imputed. Bars flagged for
OHLC inconsistency and all bars inside detected corporate-action windows
(Section~\ref{sec:data}) are excluded before any return is computed.

All shock-level analysis operates on the lagged top-500 dollar-volume
universe. This choice trades breadth for interpretability: top-500 names trade
essentially every minute (median 382--390 bars per day from the mid-1990s
onward), so the cross-sectional response measures defined below have stable,
nearly complete denominators, and the asynchronicity that depresses
high-frequency comovement among thinner names \citep{epps1979comovements}
cannot masquerade as absent propagation. Results for top-1000 and top-3000
universes appear in the robustness materials.

\subsection{Residualization}
\label{sec:residualization}

A raw extreme return is not evidence of an idiosyncratic event: it may load on
a market-wide movement. We therefore work with residuals from a single-factor
market model whose ingredients are constructed to avoid three mechanical
contaminations.

\emph{Leave-one-out factor.} The market return at minute $t$, from the
perspective of security $i$, is the equal-weighted mean return of the universe
\emph{excluding} $i$:
\begin{equation}
r^{(-i)}_{m,t} \;=\; \frac{1}{N_t-1}\sum_{j\ne i} r_{j,t},
\end{equation}
where $N_t$ counts securities with an observable return at $t$. Without the
leave-one-out correction, a large own return mechanically moves the factor and
attenuates the measured shock; with 500 names the correction is small but we
apply it exactly.

\emph{Lagged loadings.} The loading $\beta_i$ is the OLS slope of $r_{i,t}$ on
the equal-weighted market return, estimated on all minutes of month $m{-}1$ in
which the security has at least 300 valid return observations and the market
aggregate at least 50 contributing names; securities with insufficient history
receive $\beta_i=1$, and estimates are winsorized to $[-3,5]$. Residuals in
month $m$ are then
\begin{equation}
e_{i,t} \;=\; r_{i,t} - \beta_i\, r^{(-i)}_{m,t}.
\label{eq:residual}
\end{equation}
A single lagged loading per month is deliberately coarse: it cannot overfit
the event minutes it will later be used to classify, and any misspecification
it induces is absorbed by the placebo differencing described in
Section~\ref{sec:design}.

\emph{Robust, seasonally adjusted scale.} Minute returns are strongly
leptokurtic and their dispersion follows the familiar intraday U-shape
\citep{andersen1997intraday}; a sample standard deviation would be dominated
by the very tail events we wish to detect, and an unadjusted scale would
over-detect shocks at the open. We therefore standardize by a
median-absolute-deviation scale interacted with a minute-of-day factor, both
from month $m{-}1$:
\begin{equation}
z_{i,t} \;=\; \frac{e_{i,t}}{1.4826\,\mathrm{MAD}_i \times f_{\tau(t)}},
\qquad
f_{\tau} \;=\; \frac{\mathrm{med}_i\,|e_{i,\cdot}/(1.4826\,\mathrm{MAD}_i)|_{\tau}}
{\overline{\mathrm{med}}},
\label{eq:zscore}
\end{equation}
where $\tau(t)$ is the minute of day, the numerator of $f_\tau$ is the
cross-sectional median absolute standardized residual at minute-of-day $\tau$
in month $m{-}1$, and $\overline{\mathrm{med}}$ normalizes the profile to unit
mean. The factor $1.4826$ makes the MAD consistent for the standard deviation
under normality; under the actual fat-tailed distribution, $z$ is a ranking
device whose thresholds are calibrated empirically, not a Gaussian quantile.
The same construction applied to the market return, with the prior month's
market MAD, yields the standardized market factor $z_{m,t}$ used in the
isolation filters.

\subsection{Source shocks and isolation}
\label{sec:shockdef}

A \emph{source shock} is an event $(i,t)$ with
\begin{equation}
|z_{i,t}| \;\ge\; 6,
\end{equation}
with the sign recorded and analyzed separately. The threshold is deliberately
extreme: it selects on the order of one event per stock-week
(\NEventsReal{} events over 410 months), deep enough into the tail that
bid--ask bounce alone cannot generate qualifying events for top-500 names
after 2001, while leaving enough events for precise inference within every
year. Nested thresholds ($|z|\ge8$, $|z|\ge12$) are reported in the
robustness table.

Because the question concerns \emph{idiosyncratic} origins, two isolation
filters apply at the event minute:
\begin{enumerate}
\item \textbf{Calm market:} $|z_{m,t}| < 1$. The event does not coincide with
a market-factor movement.
\item \textbf{No simultaneous wave:} the share of eligible securities with
$|z_{j,t}|>4$ at minute $t$ is at most $2\%$, and at least 50 securities are
observable. The event is not part of a market-wide burst.
\end{enumerate}
Both filters are measured, not assumed: minutes excluded by filter (2) form
the systemic co-jump sample of Section~\ref{sec:cojumpdef}, so the design
partitions all extreme activity into an idiosyncratic branch and a systemic
branch rather than discarding the inconvenient one. The filters are
intentionally conservative about common information; the cost, acknowledged
throughout, is that any propagation operating \emph{through} an immediate
market-wide reaction is assigned to the systemic branch by construction.

\subsection{Propagation outcomes}
\label{sec:outcomes}

Fix an event $(i,t)$ and a horizon
$h\in\{1,2,5,10,15,30,60,120\}$ minutes. Receivers are all $j\ne i$ in the
universe with an observable bar at the measurement minute; outcomes require at
least 50 eligible receivers and are right-censored---recorded as missing, not
zero---whenever $t+h$ crosses the session close, so end-of-day events never
contribute mechanically attenuated responses. The outcomes are:
\begin{align}
\mathrm{Breadth}_{i,t}(h) &= \frac{1}{N_{t,h}-1}\sum_{j\ne i}
\mathbf{1}\{|z_{j,t+h}|>2\},
\label{eq:breadth}\\
\mathrm{Intensity}_{i,t}(h) &= \frac{1}{N_{t,h}-1}\sum_{j\ne i}|z_{j,t+h}|,
\label{eq:intensity}\\
\mathrm{Signed}_{i,t}(h) &= \frac{\operatorname{sign}(e_{i,t})}{N_{t,h}-1}
\sum_{j\ne i} z_{j,t+h}.
\label{eq:signed}
\end{align}
The receiver threshold $|z|>2$ is calibrated to the fat-tailed base rate: it
fires for roughly 7\% of stock-minutes unconditionally, high enough that
breadth is a well-populated share whose \emph{changes} are measured with
precision, low enough to respond to genuine tail activity. Cumulative breadth
replaces the point-in-time indicator with
$\mathbf{1}\{\max_{s\in(t,t+h]}|z_{j,s}|>2\}$, so a receiver that jumps at
$t+3$ and is quiet at $t+15$ still counts; its base rate is mechanically
higher (about 53\% at $h=15$) and its pre-event benchmark uses the mirrored
window $[t-5-h,\,t-5)$.

Every outcome is also evaluated at $t-5$ (point-in-time) or on the mirrored
pre-window (cumulative). The pre-event value serves two distinct purposes: it
is the \emph{differencing baseline}---shocks arrive disproportionately in
minutes of elevated ambient activity, and the post-minus-pre difference
removes that level---and it is a \emph{falsification statistic}, since genuine
propagation cannot precede its cause.

\subsection{Lagged correlation networks}
\label{sec:networkdef}

For each month $m$ and security $i$, the neighbor set
$\mathcal{N}_i$ collects the twenty securities with the highest absolute
pairwise correlation with $i$'s residuals over month $m{-}1$ (pairwise
complete observations; neighbor sets require at least ten members present in
month $m$ to be used). Splitting equation~\eqref{eq:breadth} into
\begin{equation}
\mathrm{Breadth}^{\mathcal{N}}_{i,t}(h) = \frac{1}{|\mathcal{N}_i|}
\sum_{j\in\mathcal{N}_i}\mathbf{1}\{|z_{j,t+h}|>2\},
\qquad
\mathrm{Breadth}^{\overline{\mathcal{N}}}_{i,t}(h) = \text{same over }
j\notin\mathcal{N}_i\cup\{i\},
\end{equation}
yields the \emph{network differential}
$\mathrm{Breadth}^{\mathcal{N}}-\mathrm{Breadth}^{\overline{\mathcal{N}}}$,
the paper's sharpest instrument: transmission concentrated among twenty
connected names moves it one-for-one, while market-wide breadth dilutes the
same response by a factor of twenty-five. Because neighbor sets are lagged,
the differential cannot reflect the event's own contemporaneous comovement;
and because the identical differential is computed for placebo events, any
unconditional tendency of correlated stocks to co-move nets out.

\subsection{Systemic co-jump minutes}
\label{sec:cojumpdef}

A minute is a \emph{systemic co-jump} if the share of eligible securities with
$|z_{j,t}|>4$ exceeds 2\% (with at least 50 eligible names)---exactly the
complement of isolation filter (2). For each such minute we record the
affected shares at the $4\sigma$ and $6\sigma$ thresholds and the standardized
market return. Two summary series are used: the frequency of \emph{severe}
synchronization (affected share above 10\%) per 1,000 eligible minutes, and
the mean affected share within co-jump minutes. We emphasize severity and the
severe-minute count rather than the raw count at the 2\% threshold because
the latter is not comparable across regimes: idiosyncratic tail activity was
two orders of magnitude more frequent in the wide-tick 1990s, so a fixed
low threshold is breached almost continuously early in the sample for reasons
unrelated to synchronization.

\subsection{What these measures can and cannot see}
\label{sec:measurement_limits}

Three mechanical properties discipline interpretation. First, breadth
measures respond to \emph{tail} activity only; diffuse sub-threshold
responses would appear in intensity, which is reported alongside breadth
throughout. Second, all receiver measures condition on trading: a receiver
without a bar at $t+h$ leaves the denominator rather than entering as a
zero, so illiquidity cannot fabricate absorption. Third, the pre-2001 tick
regime (eighths and sixteenths) inflates both the level of $|z|$ activity and
its base rates; every cross-era comparison is therefore anchored on the
within-era placebo contrast, and the post-decimalization subsample reproduces
all headline results.