\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},
\label{eq:return}
\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 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
in nearly every minute from the mid-1990s onward (median 382--390 bars per
day), 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,lo1990nonsynchronous} cannot masquerade as absent
propagation. Even so, the number of receivers with an observable return at a
given minute averages \NEligNineties{} in the 1990s and about
\NEligMean{} over the full sample, and every share below is computed over
observed receivers only.

\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
excluding $i$,
\begin{equation}
r^{(-i)}_{m,t} \;=\; \frac{1}{N_t-1}\sum_{j\ne i} r_{j,t},
\label{eq:loo}
\end{equation}
where $N_t$ counts securities with an observable return at $t$ and $r_{m,t}$
without the superscript denotes the full equal-weighted mean. Without the
leave-one-out correction a large own return mechanically moves the factor and
attenuates the measured shock; with several hundred 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
$r_{m,t}$ over all minutes of month $m{-}1$ in which the market aggregate has
at least 50 contributing names, provided security $i$ has at least 300 valid
return observations in that month; 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 stock-month is deliberately coarse: it cannot
overfit the event minutes it will later be used to classify, and any
misspecification it induces---including the downward bias in betas estimated
from nonsynchronous minute returns \citep{scholes1977estimating}---is common
to real and placebo minutes of the same stock-month and is absorbed by the
placebo contrast of 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. Let $\mathrm{MAD}_i$ be the median absolute
deviation of $e_{i,\cdot}$ from its median over month $m{-}1$, and let
$u_{i,t}=e_{i,t}/(1.4826\,\mathrm{MAD}_i)$. The minute-of-day factor is the
cross-sectional median of $|u|$ at each minute of day $\tau$ in month $m{-}1$,
normalized to unit mean over the session,
\begin{equation}
f_{\tau} \;=\; \frac{\operatorname{med}_{(i,t):\,\tau(t)=\tau}\,|u_{i,t}|}
{\frac{1}{390}\sum_{\tau'}\operatorname{med}_{(i,t):\,\tau(t)=\tau'}\,|u_{i,t}|},
\qquad
z_{i,t} \;=\; \frac{e_{i,t}}{1.4826\,\mathrm{MAD}_i \, f_{\tau(t)}}.
\label{eq:zscore}
\end{equation}
The cross-sectional median makes the periodicity estimate robust to the jumps
it is later used to detect, in the spirit of \citet{boudt2011robust}. The
constant $1.4826$ 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. A
security whose prior-month residuals are exactly zero in more than half of
the minutes has $\mathrm{MAD}_i=0$ and is excluded from the month, which acts
as a staleness screen that binds mainly in the pre-decimalization years. The
same construction applied to $r_{m,t}$, 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 $|z_{i,t}|\ge 6$, with the sign
recorded and analyzed separately. The threshold is deliberately extreme: it
selects \NEventsReal{} events over 410 months, about \EventsPerStockWeek{}
per stock-week in the top-500 universe, 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 idiosyncratic origins, two isolation filters
apply at the event minute. The first requires a calm market factor,
$|z_{m,t}|<1$, so that the event does not coincide with a market-wide
movement. The second requires that no simultaneous wave be under way: the
share of observable securities with $|z_{j,t}|>4$ at minute $t$ is at most
2\%, and at least 50 securities are observable. Both filters are measured, not
assumed: minutes excluded by the second filter 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 through an immediate market-wide reaction is assigned
to the systemic branch by construction. The filters do not, however, require
that the source be the only $|z|\ge6$ event in its minute: with a 2\% cap on
simultaneous $|z|>4$ names, up to about nine other securities may move
sharply at $t$. In practice \ShareSingleton\% of events are alone in their
minute and the mean number of source events per event minute is
\MeanCoArrival{}; Section~\ref{sec:results} reports every estimate
separately for singleton minutes.

\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. With $N_{t+h}$ the number of
observable receivers, 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}
Breadth counts receivers with a \emph{tail exceedance}, a standardized
residual beyond two in absolute value; we avoid the word ``jump'' for this
object, since the threshold is far below any formal jump-test cutoff. The
threshold is calibrated to the fat-tailed base rate: in placebo windows about
\BaseBreadthShare{} of receivers exceed it, high enough that breadth is a
well-populated share whose changes are measured with precision, low enough
to respond to genuine tail activity. Intensity captures diffuse
sub-threshold responses, and the signed outcome captures directional
comovement with the source. 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 moves at $t+3$ and is quiet at $t+15$ still counts; its base rate is
mechanically higher (about \CumBaseFifteen\% at $h=15$ in placebo windows) 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 is the differencing baseline:
shocks arrive disproportionately in minutes of elevated ambient activity, and
the post-minus-pre difference removes that level. It is also informative in
its own right, and Section~\ref{sec:results} reports the pre-event level of
each arm rather than asserting their equality.

\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 $e$ over month $m{-}1$ (pairwise complete observations; a neighbor
set is used only if at least ten of its members are present in month $m$).
Splitting equation~\eqref{eq:breadth} into
\begin{align}
\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\},
\label{eq:nbreadth}\\
\mathrm{Breadth}^{\overline{\mathcal{N}}}_{i,t}(h) &= \frac{1}{N_{t+h}-|\mathcal{N}_i|-1}
\sum_{j\notin\mathcal{N}_i\cup\{i\}}\mathbf{1}\{|z_{j,t+h}|>2\},
\label{eq:nbreadth_other}
\end{align}
yields the \emph{network contrast}
$\mathrm{Breadth}^{\mathcal{N}}-\mathrm{Breadth}^{\overline{\mathcal{N}}}$,
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 about twenty. Because neighbor sets are lagged,
the contrast cannot reflect correlations estimated on the event itself; and
because the identical contrast is computed for placebo events, any
unconditional tendency of correlated stocks to be more or less active than
the rest of the universe nets out in the real-minus-placebo
\emph{network differential} of Section~\ref{sec:design}.

Two properties of this construction must be kept in view. First, neighbors
are identified from one-minute residual correlations, and the precision of
that identification is not constant over time: when the Epps effect is
strong, as it was in the 1990s (Section~\ref{sec:results}), one-minute
correlations are dominated by asynchronous trading and discreteness, and the
twenty ``closest'' names are a noisier selection of true economic peers than
they are in the 2010s. Secular changes in the network differential therefore
combine changes in transmission with changes in how well the network is
measured. Second, the network contrast is computed at $t+h$ only; its
pre-event counterpart at $t-5$ was not stored by the pipeline and is listed
among the analyses left for future work.

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

A minute is a \emph{systemic co-jump minute} if the share of observable
securities with $|z_{j,t}|>4$ exceeds 2\% (with at least 50 observable
names)---exactly the complement of the second isolation filter. 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 observable 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.
The same discreteness also inflates the severe-minute count before 2001, and
we do not compare the 1990s with later decades on this series.

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

Three mechanical properties discipline interpretation. First, breadth
responds to tail activity only; diffuse sub-threshold responses appear in
intensity, which is reported alongside breadth. 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 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 is
reported for every headline estimate.