Lagged-Response Regression¶
Time-series test — instantiates Information Set Specification and Completeness Verification
Tests whether old information still predicts later price or signal movement after the supposed incorporation window.
A Lagged-Response Regression puts a single, sharp question to a whole time series: once a piece of information is old enough that it should already be fully priced, does it still predict where the signal goes next? Its defining move is to regress future signal movement on a lagged information variable and read the coefficient. If incorporation were complete by the declared window, that coefficient should be statistically zero — the past information carries no residual predictive power. A coefficient that stays significantly nonzero past the window is a direct measurement of incomplete or slow incorporation. Unlike an event study or drift analysis, it is not anchored to a discrete dated release; it works across the continuous series, treating the lag structure of the predictor as the thing being estimated. Its verdict is a statement about latency and completeness — the response wasn't finished on time — not a claim that a tradeable profit exists.
Example¶
An analyst studies pricing on a large used-car marketplace and suspects that public reliability information diffuses too slowly into listing prices. Her lagged information variable is the change in a model-year's published reliability rating; her outcome is the later week-over-week change in average listing price for that model. She regresses price changes on the reliability change lagged one week, two weeks, three, four. Efficient incorporation would predict that by, say, two weeks — her declared update window, reflecting how long relisting and negotiation take — the lagged coefficient is indistinguishable from zero. Instead it stays clearly positive through week four: a reliability upgrade published a month ago still predicts listing prices ticking up. The estimated lag profile even maps the pathway — prices adjust in steps as sellers relist and buyers renegotiate, not all at once. The scoped conclusion is a latency statement: reliability information is not fully incorporated into this marketplace's prices within the two-week window; real incorporation takes closer to five.
How it works¶
- Pick the lagged predictor. Choose the information variable whose residual predictive power you want to test, and the horizon after which it should carry none.
- Regress forward movement on the lag. Fit future signal changes on the lagged information variable across the whole series, at several lag lengths.
- Read the coefficient against the window. A coefficient still significant beyond the declared incorporation window is evidence the response was incomplete on schedule.
- Trace the lag profile as pathway. How the coefficient decays over successive lags describes how the information diffuses — the incorporation channel's real speed.
Tuning parameters¶
- Lag length and horizon — how far back the predictor is lagged and how far forward the outcome runs. Too short and slow incorporation is missed; too long and the regression chases spurious long-range correlations.
- Predictor definition — the exact form of the lagged information variable (level, change, surprise). This is the boundary: it declares which information class is on trial for residual predictability.
- Overlap and autocorrelation handling — whether overlapping windows and serial correlation are corrected, which governs whether a "significant" coefficient is real.
- Control set — what other predictors are held constant so the lagged coefficient isn't absorbing an omitted common factor.
When it helps, and when it misleads¶
Its strength is that it needs no discrete event: it can detect slow, diffuse under-incorporation spread across a whole series that an event-window test would never catch, and its lag profile doubles as a map of how fast information really travels. It is the natural test of weak-form-style predictability[n1] extended to any public information variable.
Its failure mode is spurious predictability. Overlapping windows, persistent regressors, data-snooping across many candidate lags, and the well-known small-sample bias in predictive regressions can all manufacture a "significant" lagged coefficient from thin air. It is also easy to over-read: a nonzero coefficient is evidence of incomplete latency, not automatically a profit — transaction costs may swallow it whole. The guarding discipline is to pre-declare the predictor and window, correct for overlap and autocorrelation, and report the finding as a completeness statement rather than a trading signal.
How it implements the components¶
information_set_boundary_statement— the choice of lagged predictor variable is the boundary: it declares exactly which information class is being tested for residual predictability.incorporation_pathway_model— the estimated lag profile reveals the real channel and speed by which the information diffuses into the signal (relisting, search, renegotiation).latency_and_update_window— the declared horizon after which the lagged coefficient should be zero; a coefficient surviving past it measures incomplete incorporation.
It does NOT quantify the residual return you could earn or rewrite the efficiency claim — that opportunity-and-logging work is Post-Announcement Drift Analysis, via residual_opportunity_probe and efficiency_claim_revision_log; and it does not score a reaction at a fixed dated event — that is Event-Study Information-Response Test. Its nearest twin is the drift analysis: this regression is a continuous coefficient test on a lagged predictor across the whole sample, tied to no discrete event, whereas drift is an event-anchored cumulative abnormal return continuing in a surprise's direction.
Related¶
- Instantiates: Information Set Specification and Completeness Verification — supplies a continuous, event-free measure of residual predictability and true incorporation latency.
- Sibling mechanisms: Event-Study Information-Response Test · Post-Announcement Drift Analysis · Abnormal-Return / Residual Model · Arbitrage Opportunity Scan · Information-Source Inventory Matrix · Market-Microstructure Order-Book Probe · Cross-Market Information-Leakage Check
Editorial Notes¶
Form Classification¶
Form family: Analysis, Modeling & Optimization
Rationale: Lagged-Response Regression operates as a computation, comparison, model, or analytic representation used to infer, estimate, or choose because it tests whether old information still predicts later price or signal movement after the supposed incorporation window
Independent corroboration: The frozen evidence defines Lagged-Response Regression as 'Tests whether old information still predicts later price or signal movement after the supposed incorporation window', so its operative form is Analysis, Modeling & Optimization.
Nearest alternative: Experiment, Test & Rehearsal — It estimates residual prediction from an existing historical series rather than deliberately perturbing a live target.
Review outcome: Independent reviewer agreement; medium confidence.
Origin Attribution¶
Primary origin: Economics & Finance
Origin pattern: Cross-disciplinary synthesis
Present-day reach: Multi-domain
Rationale: Financial econometrics developed regressions testing whether stale information retains predictive power after an incorporation window.
Related originating lineages:
- Statistics & Experimental Design — Time-series inference supplied lag testing, null hypotheses, and uncertainty estimates.
Review resolution: Both independent reviews place the primary lineage in economics_finance. The queued differences (domain_reach_disagreement) concern secondary metadata rather than primary provenance. The final retains statistics_experimental_design only where a reviewer supplied a formative-lineage rationale; downstream application by itself is not treated as origin. origin_mode=cross_disciplinary_synthesis records the relationship among origin traditions, while domain_reach=multi_domain records application breadth separately. encyclopedia_synthesis=false reflects whether either reviewer identified a corpus-specific synthesis, and confidence=high preserves the more cautious evidence assessment.
Review outcome: Reconciled after independent review; high confidence.
Notes¶
[n1] Weak-form efficiency holds that future returns cannot be predicted from past prices or already-public information; a significant lagged-predictor coefficient is direct evidence against it — subject to the caveat that predictive regressions are prone to small-sample and overlapping-window bias that can fake such a coefficient. ↩