A lead-lag relationship is an ordinary correlation with one series shifted in time. If series A at day t correlates with series B at day t+1, A is said to lead B by a day.
corr(A[t], B[t + k]) for a range of offsets k
The appeal is obvious. A signal that moves first is a signal you can trade on. The difficulty is that a time offset is the weakest possible evidence of a mechanism, and several ordinary features of market structure produce offsets with no information in them.
Reading a lag profile
You compute the correlation across a range of offsets and look at the shape, not at the single largest value.
| Offset | Correlation | What the shape suggests |
|---|---|---|
| -3 to -1 | Near zero | No reverse relationship |
| 0 | 0.31 | Contemporaneous co-movement |
| +1 | 0.24 | A candidate lead |
| +2 | 0.19 | Decaying, which is consistent with a real effect |
| +3 onward | Near zero | Effect does not persist |
A smooth decay is more believable than a spike at one offset with nothing around it. An isolated spike at k = +4 and noise everywhere else is usually the best of many offsets you tested, which makes it a multiple-testing artefact.
Time zones are not information
The clearest Indian example is the relationship between US indices and the Nifty. The US session closes overnight in IST, so US moves appear to lead the Indian open with a high correlation. That is a consequence of non-overlapping sessions rather than a prediction. By the time the NSE opens, the information is already in the opening print, and the lag correlation describes a gap you cannot trade. The same applies to any pair whose exchanges keep different hours.
Ask what could be traded in the interval between the lead and the lag. If the answer is nothing, because the second market gapped to the new level before you could act, the relationship is an accounting identity rather than an edge.
Caveats
A lead-lag correlation is not causation. Both series may be reacting to a third driver, and the offset may reflect nothing more than which one reprices faster. Liquidity differences alone create apparent leads, since large caps absorb information before smallcaps do.
The split-sample test is the minimum bar. Compute the profile on the first half of the sample and the second half separately. A relationship present in one half and absent in the other is luck, and that is the common outcome when the lag was chosen after looking at the data.
Testing many offsets across many pairs is a large scan. Ten pairs at ten offsets is a hundred tests, and the best of a hundred needs a t-statistic near 2.8 before it says anything. Faster information flow also compresses lags toward zero over time, so a lead documented on older data may already have closed.