spb/wp15_uqo

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 $t$. One-minute20log returns,21\begin{equation}22r_{i,t} \;=\; \log P_{i,t} - \log P_{i,t-1},23\label{eq:return}24\end{equation}25are computed within ticker and trading day only: the first observed minute of26each stock-day has no return, and overnight intervals are never bridged, so no27intraday statistic mixes overnight information. Because the source records a28bar only when at least one trade occurs, a missing minute means ``no trade,''29not ``zero return''; returns are defined only across consecutively observed30minutes and missingness is carried, never imputed. Bars flagged for OHLC31inconsistency and all bars inside detected corporate-action windows32(Section~\ref{sec:data}) are excluded before any return is computed.3334All shock-level analysis operates on the lagged top-500 dollar-volume35universe. This choice trades breadth for interpretability: top-500 names trade36in nearly every minute from the mid-1990s onward (median 382--390 bars per37day), so the cross-sectional response measures defined below have stable,38nearly complete denominators, and the asynchronicity that depresses39high-frequency comovement among thinner names40\citep{epps1979comovements,lo1990nonsynchronous} cannot masquerade as absent41propagation. Even so, the number of receivers with an observable return at a42given minute averages \NEligNineties{} in the 1990s and about43\NEligMean{} over the full sample, and every share below is computed over44observed receivers only.4546\subsection{Residualization}47\label{sec:residualization}4849A raw extreme return is not evidence of an idiosyncratic event: it may load on50a market-wide movement. We therefore work with residuals from a single-factor51market model whose ingredients are constructed to avoid three mechanical52contaminations.5354\emph{Leave-one-out factor.} The market return at minute $t$, from the55perspective of security $i$, is the equal-weighted mean return of the universe56excluding $i$,57\begin{equation}58r^{(-i)}_{m,t} \;=\; \frac{1}{N_t-1}\sum_{j\ne i} r_{j,t},59\label{eq:loo}60\end{equation}61where $N_t$ counts securities with an observable return at $t$ and $r_{m,t}$62without the superscript denotes the full equal-weighted mean. Without the63leave-one-out correction a large own return mechanically moves the factor and64attenuates the measured shock; with several hundred names the correction is65small but we apply it exactly.6667\emph{Lagged loadings.} The loading $\beta_i$ is the OLS slope of $r_{i,t}$ on68$r_{m,t}$ over all minutes of month $m{-}1$ in which the market aggregate has69at least 50 contributing names, provided security $i$ has at least 300 valid70return observations in that month; securities with insufficient history71receive $\beta_i=1$, and estimates are winsorized to $[-3,5]$. Residuals in72month $m$ are then73\begin{equation}74e_{i,t} \;=\; r_{i,t} - \beta_i\, r^{(-i)}_{m,t}.75\label{eq:residual}76\end{equation}77A single lagged loading per stock-month is deliberately coarse: it cannot78overfit the event minutes it will later be used to classify, and any79misspecification it induces---including the downward bias in betas estimated80from nonsynchronous minute returns \citep{scholes1977estimating}---is common81to real and placebo minutes of the same stock-month and is absorbed by the82placebo contrast of Section~\ref{sec:design}.8384\emph{Robust, seasonally adjusted scale.} Minute returns are strongly85leptokurtic and their dispersion follows the familiar intraday U-shape86\citep{andersen1997intraday}; a sample standard deviation would be dominated87by the very tail events we wish to detect, and an unadjusted scale would88over-detect shocks at the open. Let $\mathrm{MAD}_i$ be the median absolute89deviation of $e_{i,\cdot}$ from its median over month $m{-}1$, and let90$u_{i,t}=e_{i,t}/(1.4826\,\mathrm{MAD}_i)$. The minute-of-day factor is the91cross-sectional median of $|u|$ at each minute of day $\tau$ in month $m{-}1$,92normalized to unit mean over the session,93\begin{equation}94f_{\tau} \;=\; \frac{\operatorname{med}_{(i,t):\,\tau(t)=\tau}\,|u_{i,t}|}95{\frac{1}{390}\sum_{\tau'}\operatorname{med}_{(i,t):\,\tau(t)=\tau'}\,|u_{i,t}|},96\qquad97z_{i,t} \;=\; \frac{e_{i,t}}{1.4826\,\mathrm{MAD}_i \, f_{\tau(t)}}.98\label{eq:zscore}99\end{equation}100The cross-sectional median makes the periodicity estimate robust to the jumps101it is later used to detect, in the spirit of \citet{boudt2011robust}. The102constant $1.4826$ makes the MAD consistent for the standard deviation under103normality; under the actual fat-tailed distribution, $z$ is a ranking device104whose thresholds are calibrated empirically, not a Gaussian quantile. A105security whose prior-month residuals are exactly zero in more than half of106the minutes has $\mathrm{MAD}_i=0$ and is excluded from the month, which acts107as a staleness screen that binds mainly in the pre-decimalization years. The108same construction applied to $r_{m,t}$, with the prior month's market MAD,109yields the standardized market factor $z_{m,t}$ used in the isolation110filters.111112\subsection{Source shocks and isolation}113\label{sec:shockdef}114115A \emph{source shock} is an event $(i,t)$ with $|z_{i,t}|\ge 6$, with the sign116recorded and analyzed separately. The threshold is deliberately extreme: it117selects \NEventsReal{} events over 410 months, about \EventsPerStockWeek{}118per stock-week in the top-500 universe, deep enough into the tail that119bid--ask bounce alone cannot generate qualifying events for top-500 names120after 2001, while leaving enough events for precise inference within every121year. Nested thresholds ($|z|\ge8$, $|z|\ge12$) are reported in the122robustness table.123124Because the question concerns idiosyncratic origins, two isolation filters125apply at the event minute. The first requires a calm market factor,126$|z_{m,t}|<1$, so that the event does not coincide with a market-wide127movement. The second requires that no simultaneous wave be under way: the128share of observable securities with $|z_{j,t}|>4$ at minute $t$ is at most1292\%, and at least 50 securities are observable. Both filters are measured, not130assumed: minutes excluded by the second filter form the systemic co-jump131sample of Section~\ref{sec:cojumpdef}, so the design partitions all extreme132activity into an idiosyncratic branch and a systemic branch rather than133discarding the inconvenient one. The filters are intentionally conservative134about common information; the cost, acknowledged throughout, is that any135propagation operating through an immediate market-wide reaction is assigned136to the systemic branch by construction. The filters do not, however, require137that the source be the only $|z|\ge6$ event in its minute: with a 2\% cap on138simultaneous $|z|>4$ names, up to about nine other securities may move139sharply at $t$. In practice \ShareSingleton\% of events are alone in their140minute and the mean number of source events per event minute is141\MeanCoArrival{}; Section~\ref{sec:results} reports every estimate142separately for singleton minutes.143144\subsection{Propagation outcomes}145\label{sec:outcomes}146147Fix an event $(i,t)$ and a horizon148$h\in\{1,2,5,10,15,30,60,120\}$ minutes. Receivers are all $j\ne i$ in the149universe with an observable bar at the measurement minute; outcomes require at150least 50 eligible receivers and are right-censored---recorded as missing, not151zero---whenever $t+h$ crosses the session close, so end-of-day events never152contribute mechanically attenuated responses. With $N_{t+h}$ the number of153observable receivers, the outcomes are154\begin{align}155\mathrm{Breadth}_{i,t}(h) &= \frac{1}{N_{t+h}-1}\sum_{j\ne i}156\mathbf{1}\{|z_{j,t+h}|>2\},157\label{eq:breadth}\\158\mathrm{Intensity}_{i,t}(h) &= \frac{1}{N_{t+h}-1}\sum_{j\ne i}|z_{j,t+h}|,159\label{eq:intensity}\\160\mathrm{Signed}_{i,t}(h) &= \frac{\operatorname{sign}(e_{i,t})}{N_{t+h}-1}161\sum_{j\ne i} z_{j,t+h}.162\label{eq:signed}163\end{align}164Breadth counts receivers with a \emph{tail exceedance}, a standardized165residual beyond two in absolute value; we avoid the word ``jump'' for this166object, since the threshold is far below any formal jump-test cutoff. The167threshold is calibrated to the fat-tailed base rate: in placebo windows about168\BaseBreadthShare{} of receivers exceed it, high enough that breadth is a169well-populated share whose changes are measured with precision, low enough170to respond to genuine tail activity. Intensity captures diffuse171sub-threshold responses, and the signed outcome captures directional172comovement with the source. Cumulative breadth replaces the point-in-time173indicator with $\mathbf{1}\{\max_{s\in(t,t+h]}|z_{j,s}|>2\}$, so a receiver174that moves at $t+3$ and is quiet at $t+15$ still counts; its base rate is175mechanically higher (about \CumBaseFifteen\% at $h=15$ in placebo windows) and its pre-event benchmark uses176the mirrored window $[t-5-h,\,t-5)$.177178Every outcome is also evaluated at $t-5$ (point-in-time) or on the mirrored179pre-window (cumulative). The pre-event value is the differencing baseline:180shocks arrive disproportionately in minutes of elevated ambient activity, and181the post-minus-pre difference removes that level. It is also informative in182its own right, and Section~\ref{sec:results} reports the pre-event level of183each arm rather than asserting their equality.184185\subsection{Lagged correlation networks}186\label{sec:networkdef}187188For each month $m$ and security $i$, the neighbor set $\mathcal{N}_i$ collects189the twenty securities with the highest absolute pairwise correlation with $i$'s190residuals $e$ over month $m{-}1$ (pairwise complete observations; a neighbor191set is used only if at least ten of its members are present in month $m$).192Splitting equation~\eqref{eq:breadth} into193\begin{align}194\mathrm{Breadth}^{\mathcal{N}}_{i,t}(h) &= \frac{1}{|\mathcal{N}_i|}195\sum_{j\in\mathcal{N}_i}\mathbf{1}\{|z_{j,t+h}|>2\},196\label{eq:nbreadth}\\197\mathrm{Breadth}^{\overline{\mathcal{N}}}_{i,t}(h) &= \frac{1}{N_{t+h}-|\mathcal{N}_i|-1}198\sum_{j\notin\mathcal{N}_i\cup\{i\}}\mathbf{1}\{|z_{j,t+h}|>2\},199\label{eq:nbreadth_other}200\end{align}201yields the \emph{network contrast}202$\mathrm{Breadth}^{\mathcal{N}}-\mathrm{Breadth}^{\overline{\mathcal{N}}}$,203the paper's sharpest instrument: transmission concentrated among twenty204connected names moves it one-for-one, while market-wide breadth dilutes the205same response by a factor of about twenty. Because neighbor sets are lagged,206the contrast cannot reflect correlations estimated on the event itself; and207because the identical contrast is computed for placebo events, any208unconditional tendency of correlated stocks to be more or less active than209the rest of the universe nets out in the real-minus-placebo210\emph{network differential} of Section~\ref{sec:design}.211212Two properties of this construction must be kept in view. First, neighbors213are identified from one-minute residual correlations, and the precision of214that identification is not constant over time: when the Epps effect is215strong, as it was in the 1990s (Section~\ref{sec:results}), one-minute216correlations are dominated by asynchronous trading and discreteness, and the217twenty ``closest'' names are a noisier selection of true economic peers than218they are in the 2010s. Secular changes in the network differential therefore219combine changes in transmission with changes in how well the network is220measured. Second, the network contrast is computed at $t+h$ only; its221pre-event counterpart at $t-5$ was not stored by the pipeline and is listed222among the analyses left for future work.223224\subsection{Systemic co-jump minutes}225\label{sec:cojumpdef}226227A minute is a \emph{systemic co-jump minute} if the share of observable228securities with $|z_{j,t}|>4$ exceeds 2\% (with at least 50 observable229names)---exactly the complement of the second isolation filter. For each such230minute we record the affected shares at the $4\sigma$ and $6\sigma$231thresholds and the standardized market return. Two summary series are used:232the frequency of \emph{severe} synchronization (affected share above 10\%)233per 1,000 observable minutes, and the mean affected share within co-jump234minutes. We emphasize severity and the severe-minute count rather than the235raw count at the 2\% threshold because the latter is not comparable across236regimes: idiosyncratic tail activity was two orders of magnitude more237frequent in the wide-tick 1990s, so a fixed low threshold is breached almost238continuously early in the sample for reasons unrelated to synchronization.239The same discreteness also inflates the severe-minute count before 2001, and240we do not compare the 1990s with later decades on this series.241242\subsection{What these measures can and cannot see}243\label{sec:measurement_limits}244245Three mechanical properties discipline interpretation. First, breadth246responds to tail activity only; diffuse sub-threshold responses appear in247intensity, which is reported alongside breadth. Second, all receiver measures248condition on trading: a receiver without a bar at $t+h$ leaves the249denominator rather than entering as a zero, so illiquidity cannot fabricate250absorption. Third, the pre-2001 tick regime inflates both the level of $|z|$251activity and its base rates; every cross-era comparison is therefore anchored252on the within-era placebo contrast, and the post-decimalization subsample is253reported for every headline estimate.254