Lead-Lag Networks From Return Series

An ordinary correlation graph treats every edge as symmetric: "A and B move together" carries no sense of which one moves first. But large, heavily-traded stocks often do genuinely move before smaller, less-traded ones react to the same news — large-cap price moves get incorporated quickly, small-caps lag because they trade less and get less attention. A lead-lag network captures exactly this asymmetry: instead of asking "do these two move together right now," it asks "does one's past return help predict the other's future return," and draws a directed edge from leader to follower when it does.

The analogy: the front and back of a marching band

Watch a marching band turn a corner from above: the row at the front of the turn changes direction first, and the rows further back visibly follow a beat later, each copying what the row ahead of it just did. A snapshot at any single instant might show the whole band roughly correlated in direction, but it's the sequence — front row turns, then the next row, then the next — that tells you who is leading and who is following. A lead-lag network is built by looking for exactly this kind of sequential following in return data: if stock A's return today reliably shows up echoed in stock B's return tomorrow, an edge is drawn from A to B, not the other way around.

Building the edge test: lagged regression / Granger-style causality

The standard way to test "does A lead B" is a simple lagged regression: regress B's return today on both B's own past return and A's past return, and check whether A's past return adds real predictive power:

$$r_{B,t} = \alpha + \beta_1 r_{B,t-1} + \beta_2 r_{A,t-1} + \varepsilon_t$$

In words: B's return today is explained partly by its own momentum ($\beta_1$ on its own lagged return) and partly by what A did yesterday ($\beta_2$ on A's lagged return). If $\beta_2$ is statistically distinguishable from zero — A's past return genuinely helps predict B's future return, beyond what B's own history already explains — the test concludes A Granger-causes B, and a directed edge $A \to B$ is added to the network. Running the same regression in the other direction (B's lagged return predicting A) tests the reverse edge independently; both, one, or neither direction can turn out significant.

Worked example 1: fitting the lagged regression by hand

Suppose over 5 days, stock A's returns (in %) were $[1.0, -0.5, 2.0, 0.5, -1.0]$ and stock B's returns the following day were $[0.6, -0.3, 1.1, 0.3, -0.5]$ — B consistently moves about 55-60% of A's prior move. Regressing $r_{B,t}$ on $r_{A,t-1}$ alone (ignoring B's own lag for simplicity): the slope estimate comes out close to $\beta_2 \approx 0.57$, and given how tightly the points track this ratio, the fit is strong — a high $R^2$, and a large t-statistic on $\beta_2$ relative to its standard error. This clears a significance threshold easily, so the test concludes $A \to B$: an edge is drawn from A to B in the lead-lag network, with edge weight often set to $\beta_2 = 0.57$ or to the t-statistic itself.

Worked example 2: testing the reverse direction finds nothing

Now regress $r_{A,t}$ on $r_{B,t-1}$ — does B's past return predict A's future return? Using the same five days, shift the pairing: does B's return on day $t{-}1$ predict A's return on day $t$? Given the data was constructed so that A moves first and B echoes it a day later, B's lagged values carry no independent information about A's next move beyond noise — the estimated slope comes out small and statistically indistinguishable from zero (say $\beta \approx 0.05$ with a t-statistic under 1). No edge is drawn $B \to A$. The result: a single directed edge $A \to B$, not a symmetric, undirected one — exactly the asymmetry a plain same-day correlation graph would have missed, since same-day correlation between A and B here would still show up as strongly positive in both directions.

The residuals visible in the regression fit above are exactly what a lead-lag test is scrutinizing: how much of tomorrow's move in the follower is explained by today's move in the candidate leader, versus how much is left as unexplained noise.

What this means in practice

Lead-lag networks are used to find genuine statistical arbitrage opportunities (trade the follower based on the leader's signal, since a real, persistent lead relationship implies short-term predictability) and to map how information or shocks propagate through a market — large caps leading smaller caps in the same sector, or an index leading its slower-to-update constituents. The central risk is overfitting a directional relationship in-sample and finding it vanishes out-of-sample: lead-lag effects driven by genuine trading-cost or liquidity asymmetries tend to be more stable than ones driven by noise, and any detected edge needs out-of-sample validation, not just an in-sample significant coefficient, before it is trusted for trading.

Practice

1. Using the same A and B return series, add a lag of B's own return to the regression from worked example 1 and discuss qualitatively how it might change the estimate of $\beta_2$.
2. Explain in one sentence why an ordinary same-day correlation graph cannot distinguish "A leads B" from "B leads A."
3. A backtest shows a strong lead-lag edge in-sample that disappears entirely out-of-sample. Name one likely cause.