Minimum-Distance Estimation on a Cohort-by-Event-Time Lattice
Author
Apoorva Lal
Published
July 16, 2026
Abstract
Staggered-adoption event studies present an unresolved bias-variance tradeoff. A pooled event-study regression is precise and can fail to estimate an interpretable average treatment-effect curve when effects differ across adoption cohorts. A saturated cohort-by-event-time estimator accommodates that heterogeneity and can be noisy when there are many small cohorts. This paper proposes an adaptive estimator between those endpoints. Starting from a saturated heterogeneity-robust estimate, we construct minimum-distance projections onto a nested lattice of adjacent cohort and event-time partitions. Every candidate estimates the same support-aware aggregate event-study curve. A Lepskii rule selects the lowest-dimensional candidate whose curve agrees statistically with every finer comparable model. Cross-model covariance and a simultaneous critical value are obtained from the saturated regression’s cluster influence functions. Hyperparameter grids are tuned by honest nested cluster splitting against an independent saturated target estimate. Simulations with nine uneven adoption cohorts show that the selector recovers cohort-only, time-only, and tensor-product structure while approaching saturation when pooling distorts the target. The method is implemented as a pyfixest-compatible Python class.
1. Introduction
The recent difference-in-differences literature has clarified why a conventional two-way fixed-effects (TWFE) event study can fail to estimate a meaningful average of heterogeneous treatment effects under staggered adoption (Goodman-Bacon 2021; Chaisemartin and D’Haultfœuille 2020; Goldsmith-Pinkham, Hull, and Kolesár 2024). A direct response is to estimate treatment effects separately for every adoption cohort and event time, then aggregate the resulting cells using transparent nonnegative weights (Sun and Abraham 2021; Callaway and Sant’Anna 2021; Wooldridge 2021). That response protects the estimand and introduces a potentially large collection of noisy cohort-time coefficients.
This paper treats the choice between pooled and saturated event studies as an adaptive estimation problem. The motivating joint-test approach in Lal (2025) asks whether cohort-specific deviations from a common event-study curve are jointly zero. The test supplies a useful binary diagnostic. Rejection leaves the appropriate degree of partial pooling unresolved, and nonrejection leaves the magnitude of pooling bias unresolved. The resulting estimation problem is to determine how much structured heterogeneity the data can support.
The selection logic follows Lepskii’s comparison-to-refinement principle (Lepski 1990; Lepski and Spokoiny 1997; Goldenshluger and Lepski 2011). It is also related to recent work on adaptation to misspecification, where efficiency gains from a restricted model are weighed against detectable departures from that model (Armstrong, Kline, and Sun 2025). Our setting differs because the candidate class is a two-dimensional partial order and every model must be mapped to the same dynamic treatment-effect target.
We propose a two-axis partial-pooling estimator. The first axis groups adjacent adoption cohorts. The second groups adjacent event times, while preserving the pre/post boundary. Crossing the two partitions produces rectangular restrictions on the cohort-by-event-time treatment-effect surface. The coarsest candidates have a few interpretable blocks; the finest candidate is saturated. All candidates are linear minimum-distance projections of one saturated estimate, and all are mapped to the same support-aware aggregate event-study target.
The estimator has three useful features. First, one expensive fixed-effects regression produces the entire candidate path. Second, because every candidate is a linear transformation of the same saturated estimator, cross-model covariance is available exactly from a shared influence-function stack. Third, a cluster multiplier bootstrap can calibrate one critical value simultaneously over candidate models, refinements, and event times.
The paper makes four contributions.
It formulates partial pooling over a nested cohort-time lattice while holding the reported event-study estimand fixed.
It derives the minimum-distance candidate estimators and their joint covariance from a saturated heterogeneity-robust fit.
It gives a Lepskii selection rule and an honest nested-splitting procedure for tuning the candidate grid without using final-sample performance.
It implements the procedure in Python and evaluates risk, oracle regret, coverage, selection, and computation in a repeated many-cohort experiment.
The selected complexity pertains to a particular aggregate target. Distinct cohort effects may cancel under that target, making pooling useful for the aggregate curve while leaving the latent surface heterogeneous. Claims about full-surface homogeneity require a surface-level selection criterion. This distinction is central to interpreting the results.
2. Setup and Target
The introduction motivates a bias-variance tradeoff whose solution requires all candidate estimators to answer the same causal question. This section fixes that question, records which cohort-event cells are supported, and defines the saturated heterogeneity-robust estimator from which every later candidate is constructed. Its outputs are the base coefficient vector \hat\theta, covariance \hat\Sigma, influence contributions \hat\psi_c, and target map L used by the adaptive method in Section 3.
2.1 Staggered adoption
Units i=1,\ldots,n are observed in periods t=1,\ldots,T. Treatment adoption is absorbing:
D_{it}=1\{t\geq G_i\},
where G_i\in\mathcal G\cup\{\infty\} is the adoption date and G_i=\infty denotes a never-treated unit. Event time is s=t-G_i. Let \mathcal G_T=\{g_1<\cdots<g_J\} denote treated cohorts and let \mathcal S denote the reported event-time window, excluding the normalized reference period s=-1.
We maintain no anticipation, parallel trends for untreated potential outcomes, and overlap for every reported cell. The current implementation uses never-treated observations as the comparison group in a saturated interaction regression. The minimum-distance construction applies without change when the base vector and influence functions come from another identified heterogeneity-robust estimator.
2.2 Supported cells
Let \mathcal I\subseteq\mathcal G_T\times\mathcal S be the set of estimable cells. A cell enters \mathcal I only when it reaches a prespecified minimum support and the saturated design retains its coefficient. Unsupported cells receive zero target weight, and the remaining weights are renormalized over supported cohorts at each event time. This exposes the changing target population and prevents extrapolation of late event times for late adopters.
Stack supported effects in \theta\in\mathbb R^p. The reported dynamic ATT is
Post-treatment observations beyond the reporting window are trimmed, isolating the fixed effects from unmodeled tail effects. Earlier leads remain in the estimation sample under no anticipation. Let \hat\theta be the retained saturated coefficient vector and \hat\Sigma its cluster-robust covariance.
We assume the finite-dimensional asymptotic linear representation
where c indexes independent clusters. The scaling is absorbed into \hat\psi_c. In pyfixest, these contributions are constructed from the cluster sums of residualized regressor scores and the regression bread.
2.4 The estimator in miniature
Consider three adoption cohorts and three post-treatment event times. Suppress one lead period, which remains in its own block throughout the example and is included in each E count below. Figure 1 places the saturated cell estimates beside the weights used to form the aggregate target.
Figure 1: A miniature saturated effect surface and its event-time-specific target weights. Clustered standard errors are 0.35 for the early and middle cohorts and 0.45 for the late cohort. Interactive tooltips report the estimate, standard error, and target weight in every cell.
The left panel has an evident low-dimensional pattern: the early and middle cohorts are similar, their first two effects are nearly flat, and the late cohort follows a different path. The right panel shows why cohort structure can matter for the aggregate curve. The late cohort receives 50 percent of the weight at s=0 and 10 percent at s=2. Applying the displayed weights gives
\hat\tau_{\mathrm{sat}}=(1.73,\ 1.48,\ 1.92).
Figure 2 puts four restrictions on a refinement ladder. C1-E2 uses one post-treatment rectangle. C1-E3 splits short and long event times. C2-E3 also splits the late cohort from the early-middle block. C3-E4 is saturated. Every move right splits an existing rectangle, and the lower panel maps each restricted surface back to the same support-weighted curve.
Figure 2: The refinement ladder and its common-target curves. Black outlines delimit shared post-treatment coefficients. Dashed aggregate curves correspond to rejected candidates; green marks the selected C2-E3 model and black marks saturation. Interactive tooltips report cell estimates, restrictions, target weights, aggregate effects, and differences from saturation.
In this illustration, the first two restrictions visibly move the aggregate curve. The C2-E3 curve remains compatible with its saturated refinement, so the rule stops there. Its four post-treatment coefficients are 1.05 and 2.45 for the early-middle and late short-run blocks, and 2.05 and 0.70 for their long-run blocks. Applying the original target weights gives
\hat\tau_{C2-E3}=(1.75,\ 1.47,\ 1.92),
which nearly overlays the saturated aggregate using four post-treatment coefficients in place of nine. The full procedure is visible in the two figures: estimate transparent cohort-event cells, project them onto nested rectangles, map every projection to the same target, and retain the least complex curve that agrees with all refinements. Section 3 develops the projection, comparison covariance, and simultaneous critical value used for that decision.
3. Method: Adaptive Partial Pooling on a Lattice
This section develops the paper’s estimator from beginning to end. Section 2 delivered a saturated vector \hat\theta, its covariance \hat\Sigma, and the common target map L. The task now is to use restrictions on \theta to reduce variance without silently changing L\theta. We first build a nested collection of interpretable restrictions, then project every restricted estimate back to the same target curve, use Lepskii comparisons to decide how much refinement the data require, separate that within-lattice decision from hyperparameter tuning and inference, and finally state the formal limits of the procedure. Together, these consecutive stages form one adaptive estimator.
3.1 Constructing candidate restrictions
The goal of this first stage is to turn the unrestricted cohort-by-event-time surface from Section 2 into an ordered set of plausible lower-dimensional surfaces. This supplies the approximation class for the paper’s bias-variance problem: coarse members pool many cells and can be precise, while fine members retain more heterogeneity and approach the saturated estimator. The nesting structure introduced here is also what makes the later Lepskii comparisons directional and interpretable.
Nested adjacent partitions
Start with a toy event study containing three adoption cohorts, g\in\{3,5,7\}, and five reported event times, s\in\{-2,0,1,2,3\}. The reference period s=-1 is normalized away. The cohort partitions in this example are
Figure 3: A three-cohort example of nested adjacent partitions. A colored run is one block. Moving downward only splits existing blocks: \mathcal P refines adoption cohorts and \mathcal Q refines event times.
The distinction is concrete. \mathcal P_a says which neighboring adoption cohorts must share coefficients. \mathcal Q_b says which neighboring event times must share coefficients. For example, \mathcal P_2 allows the late cohort to differ from the early and middle cohorts, while \mathcal Q_3 allows the lead, short-run response, and long-run response to differ.
A finer partition splits existing blocks, and every finer block remains inside one coarser block. Thus \mathcal Q_3 refines \mathcal Q_2 because each block of \mathcal Q_3 lies entirely inside a block of \mathcal Q_2. The implementation constructs each path recursively to preserve this containment. Independent equal-width binning at every dimension can violate nesting. The protected break at zero keeps leads and lags in separate blocks.
Crossing partitions into coefficient rectangles
Crossing a cohort partition with an event-time partition turns the two one-dimensional groupings into a restricted treatment-effect surface. In the toy truth below, the early and middle cohorts share a path, the late cohort has a different path, and the post-treatment response has short- and long-run phases. The middle candidate, \mathcal P_2\times\mathcal Q_3, therefore matches the six true coefficient rectangles exactly.
Figure 4: Crossing cohort and event-time partitions. Each black outline is one shared coefficient. The coarse model has two rectangles, the structure-matching model has six, and the saturated model has fifteen. Cell colors and labels show the restricted coefficient value.
The left panel pools all post-treatment cells into one coefficient. The middle panel splits both axes just enough to recover the toy structure. The right panel assigns a separate coefficient to every displayed cohort-event cell. A refinement may split rectangles along the cohort axis, the event-time axis, or both. It weakens restrictions and raises dimension.
Formally, candidate m=(a,b) imposes equality within rectangles of \mathcal P_a\times\mathcal Q_b. Let H_m be the p\times d_m expansion matrix satisfying
\theta=H_m\eta_m
when the restriction is exact. With incomplete support, H_m contains rows for cells in \mathcal I, and parameters correspond to nonempty rectangles.
The partial order is
m\preceq f
\quad\Longleftrightarrow\quad
\mathcal P_f\text{ refines }\mathcal P_m
\text{ and }
\mathcal Q_f\text{ refines }\mathcal Q_m.
This containment relation organizes the candidates as a lattice. We label a candidate with K cohort blocks and R event-time blocks as C_K\text{-}E_R; for example, the middle toy model is C2-E3.
3.2 Mapping every candidate to the common target
The partitions above describe restrictions on the cohort-event surface. The reported object is an aggregate event-study curve. This stage turns each candidate restriction into an estimator of the same \tau=L\theta defined in Section 2. Holding the target fixed ensures that differences between coarse and fine models measure the consequences of pooling under one common estimand.
Minimum-distance projection
It helps to keep the three spaces separate. The vector \eta_m contains one coefficient per nonempty rectangle, \theta contains one coefficient per supported cohort-event cell, and \tau contains one coefficient per reported event time. The matrices below move between those spaces.
Object
Dimension
Role
H_m
p\times d_m
expands d_m rectangle coefficients into p supported cells
P_m
p\times p
replaces a saturated cell surface by its restricted fitted surface
L
q\times p
aggregates supported cells into the fixed q-period target curve
D_m=LP_m
q\times p
maps the saturated estimate directly to candidate m’s target curve
For positive semidefinite metric W, estimate the restricted parameters by
\hat\eta_m
=
(H_m'WH_m)^+H_m'W\hat\theta,
where + denotes a Moore-Penrose inverse. Define
P_m
=
H_m(H_m'WH_m)^+H_m'W.
The first equation chooses the rectangle coefficients whose expanded surface H_m\eta_m is closest to \hat\theta in the geometry defined by W. The second collects that calculation into the projection map P_m. The same base fixed-effects regression supplies every candidate.
Thus \hat\theta_m is useful for inspecting the restricted cohort-time surface, whereas \hat\tau_m is the object compared and ultimately reported. Every \hat\tau_m has the same coordinates and target weights; only the amount of pooling used to estimate it changes.
The production estimator uses feasible GLS, W=\hat\Sigma^+. If model m is correct, this is the efficient minimum-distance estimator under the saturated Gaussian approximation. Under misspecification, it estimates a projection:
A support-weighted projection is also available as a diagnostic. When the same support weights define both the within-event-time projection and L, cohort pooling leaves the aggregate target numerically invariant and offers no meaningful variance gain for that target. GLS weighting converts the equality restrictions into an efficiency gain under the restricted model, with an explicit bias cost away from it.
Joint covariance across candidates
The next selection stage compares candidate curves to one another and therefore requires the covariance of their difference in addition to each curve’s marginal standard errors. All candidates are transformations of the same saturated estimate and are highly correlated; preserving that dependence is one of the main practical benefits of constructing the entire lattice from one base fit.
Adding the two marginal variances would ignore their common saturated fit and substantially overstate comparison noise.
Endpoints and the external comparator
The lattice contains several familiar restrictions:
one cohort block with saturated event time is a common-cohort dynamic curve;
one lead block and one post block with one cohort block is a static pooled restriction, subject to the lead normalization;
saturated cohorts and saturated event times reproduce \hat\theta exactly;
intermediate rectangles represent adoption regimes and dynamic phases.
The common-cohort minimum-distance endpoint differs algebraically from a naive TWFE event study. The simulations estimate naive TWFE in a separate regression and use it as an external comparator outside the lattice.
3.3 Choosing complexity with Lepskii comparisons
The preceding stages produce many estimates of one target curve: coarse candidates have lower sampling variance when their restrictions are adequate, whereas fine candidates are less exposed to pooling bias. This stage decides which precision gains the data support. Its role in the overall paper is to turn the candidate lattice from a menu that a researcher must choose manually into an estimator with a reproducible data-dependent complexity choice.
The selector asks whether relaxing a coarse candidate’s restrictions changes the reported target curve by more than can be explained by the shared sampling noise in \hat\theta. This target-specific comparison allows pooling when latent heterogeneity aggregates away and triggers refinement when heterogeneity materially changes the aggregate target.
Lepskii’s principle answers that question by starting from the precision-favoring end of the lattice. A candidate remains viable only if its curve agrees with all of its finer, less restricted comparators after accounting for uncertainty. The selected model is the least complex viable candidate. This implements a one-sided burden of proof: additional parameters are retained only when a refinement reveals a target-relevant discrepancy.
Pairwise target-curve discrepancies
For a coarse candidate m and a comparable refinement f\succeq m, define
\hat\Delta_{m,f}=\hat\tau_m-\hat\tau_f.
At the population level the corresponding difference is
\Delta_{m,f}
=
L(P_m-P_f)\theta.
If the restriction imposed by m is adequate for the target, both candidates have the same target limit and \Delta_{m,f}=0. If pooling in m distorts the target, at least one refinement can produce a nonzero difference. We compare only nested pairs because f\succeq m has a precise meaning: f relaxes some of the equalities imposed by m and imposes none that conflict with them. Each incomparable candidate receives its own refinement checks. Direct specification comparisons use nested pairs.
Section 3.2 supplied the covariance \hat S_{m,f} of the difference. For each event time, the standardized discrepancy is therefore
The implementation summarizes the entire curve comparison by its largest absolute standardized discrepancy on a prespecified selection set \mathcal S_0:
Large T_{m,f} means that some target coordinate changes too much to attribute to estimation noise; small T_{m,f} supports retaining the coarser candidate. The default \mathcal S_0 contains supported post-treatment event times and excludes the normalized reference period and the weakest tail. This set is part of the scientific decision: it identifies the portion of the curve whose approximation quality governs pooling. The reported curve may include additional event times.
Every finer model contributes information to the comparison. A noisy saturated curve can mask a departure that is visible in a well-chosen intermediate refinement. For example, in the toy surface, a cohort split can reveal the late cohort’s different response before every cohort-event cell is estimated separately. The rule therefore compares each candidate with every finer member of the lattice.
Simultaneous cluster multiplier calibration
The statistic above is searched over event times, over refinements of a given candidate, and ultimately over all candidates considered for selection. A pointwise normal cutoff yields excessive rejection under this search. One cluster multiplier process approximates the joint null distribution of the entire family.
For draw r, generate independent cluster multipliers \xi_c^{(r)} with mean zero and variance one. The perturbation of the saturated vector is
Reusing the draw is important: it preserves the dependence across models and event times induced by their common saturated fit. After standardization by \hat S_{m,f,ss}^{1/2}, let Z^{*(r)} be the maximum absolute perturbation over all comparable (m,f) pairs and all s\in\mathcal S_0. The critical value is the empirical 1-\alpha quantile,
This one threshold accounts jointly for the event-time maximum, every refinement comparison, and the fact that acceptance is checked for multiple candidate models. The bootstrap is used here to calibrate the selection rule; Section 3.4 develops uncertainty statements for the selected treatment-effect curve.
Acceptance and model choice
Candidate m is accepted when
T_{m,f}\leq\hat c_{1-\alpha}
\quad\text{for every }f\succeq m.
Acceptance records statistical agreement with every permitted relaxation at the target-curve level. Surface-level equality remains unassessed. The saturated model is accepted vacuously because its refinement set is empty, which guarantees at least one accepted candidate.
Among accepted candidates, the procedure selects the model with the fewest estimable rectangles. Ties are resolved by fewer cohort blocks, then fewer event-time blocks, then lower estimated target variance. In the toy example, the coarse \mathcal P_1\times\mathcal Q_2 candidate should fail once its pooled curve is compared with refinements that isolate the late cohort and the dynamic phases. If \mathcal P_2\times\mathcal Q_3 then agrees with every finer curve, the rule stops there and avoids the variance cost of fifteen saturated cells.
The full within-lattice procedure can be read as six operations:
Fit the saturated surface once and recover \hat\theta, \hat\Sigma, and cluster influence contributions \hat\psi_c.
Build each candidate map D_m=LP_m and its common-target curve \hat\tau_m=D_m\hat\theta.
For every nested pair, compute \hat\Delta_{m,f} and its shared-fit covariance \hat S_{m,f}.
Use common multiplier draws to obtain one simultaneous critical value.
Retain candidates that agree with every finer comparator.
Report the least complex retained candidate as \hat m and its curve as \hat\tau_{\hat m}.
3.4 Tuning the procedure and reporting uncertainty
Section 3.3 selects a model inside a specified lattice, conditional on choices such as the partition path, projection metric, and selection level. This stage addresses a different problem: how those procedural choices are set and what uncertainty statement is attached to the resulting curve. Keeping the two layers separate is necessary for the paper’s adaptive goal. A simultaneous cutoff accounts for searching within one lattice. Searching across alternative lattices requires independent validation.
selection level \alpha, multiplier distribution, bootstrap draws
fixed for the primary specification; varied in sensitivity analysis
The minimum cell support that defines the target is structural. The minimum support allowed for an intermediate pooled rectangle can be tuned because it changes the candidate grid without changing L\theta.
The implemented basis-family comparison remains within nested piecewise-constant models. The balanced family bisects the largest available block; the front-loaded family uses narrower early post-treatment blocks before allocating resolution to the tail. Splines or other nested smooth bases can enter through the same H_m interface. The reported experiment uses the two piecewise-constant families.
This classification prevents tuning from redefining the scientific question. Structural choices determine the estimand or identifying design and are fixed. Candidate-grid choices determine which bias-variance compromises the selector may consider and can be ranked by independent predictive evidence. Calibration choices govern the conservativeness and numerical accuracy of the search and are prespecified for the primary analysis.
Honest nested cluster splitting
The tuning objective is the risk of the whole adaptive procedure under a hyperparameter configuration h. Each training fold therefore runs the complete procedure from Section 3.3, including its own multiplier calibration and model choice. An independent saturated estimate from the validation fold supplies a restriction-free target benchmark with sampling noise.
Clusters are stratified by adoption cohort, including the never-treated group. An outer split reserves one set of clusters exclusively for inference. The remaining clusters are divided into inner folds. For hyperparameter configuration h and validation fold v, fit the adaptive estimator on the training folds and estimate the saturated target independently in fold v.
For fixed event-time weights a_s, use the noise-corrected loss
Because training and validation clusters are independent, subtracting the validation variance removes this noise floor and leaves the target risk of the training procedure in expectation. Stratification keeps the fold target populations as similar as possible; remaining differences in support are a source of approximation error. The implementation uses inverse-validation-variance weights by default to limit the influence of weakly estimated validation coordinates, and reports equal-weight sensitivity.
Each training fit calibrates its own pathwise multiplier critical value. After inner validation chooses a configuration, that configuration is fit on the full tuning sample to select one lattice model. The resulting model key is then held fixed and estimated on the untouched outer inference sample. Hyperparameters and the model key depend exclusively on the tuning clusters and validation loss.
The outer split adds an independent inference layer to cross-validation. It creates a final sample on which both the hyperparameters and the discrete model choice are predetermined. This clean separation of model construction and uncertainty calculation costs precision.
Three levels of inference
The preceding selection and tuning steps create several possible meanings of a standard error. The software separates precision gains from pooling, protection against model selection, and protection against projection bias into three inferential objects.
Model conditional. Treat the selected model as fixed and use D_{\hat m}\hat\Sigma D_{\hat m}'. These intervals provide model-conditional diagnostics. Unconditional post-selection coverage requires an additional adjustment.
Honest split sample. Use the tuning clusters to select hyperparameters and a model, then estimate on the outer inference sample. Conditional intervals are honest with respect to the tuning and model-selection stage, at a direct efficiency cost. They cover the original target L\theta only when the fixed selected model’s projection bias is negligible. Projection bias persists after sample splitting.
Bootstrap after selection. On the full sample, perturb the saturated estimate with cluster multipliers, rerun selection in every draw, and form percentile intervals from selected curves. This captures discrete model selection conditional on the observed projection matrices. The implementation holds those matrices fixed in each draw, yielding an empirical selection adjustment. Uniformly valid confidence bands require stronger theory.
3.5 What the method guarantees
The previous stages define the complete estimator and its inferential variants. This final part of the method section states the scope of the formal claims. The results explain why candidate curves have tractable joint limits and when the selector removes overly coarse models. The claims cover a fixed finite lattice. Adaptive minimaxity over a growing nonparametric class and uniformly valid full-sample post-selection inference remain open.
Assumptions
The assumptions deliberately attach to the saturated estimator and the finite collection of linear maps built above. They allow the partial-pooling machinery to inherit identification and asymptotic normality without imposing a separate data-generating model for every candidate.
Assumption A (identified saturated limit). Under no anticipation, parallel trends, and overlap, \hat\theta\overset p\to\theta for the supported cells.
Assumption B (joint linearity). The saturated estimator has a cluster-level asymptotic linear representation, \hat\Sigma is consistent, and relevant contrast variances are bounded away from zero after removing deterministic normalizations.
Assumption C (finite nested lattice). The cohort and event-time paths are finite, deterministic, and nested. The target map L and selection set \mathcal S_0 are fixed.
Assumption D (bootstrap validity). The cluster multiplier process consistently approximates the joint Gaussian limit of all standardized (m,f,s) contrasts.
Proposition 1: candidate limits
The first result identifies the estimand of each restricted candidate. It makes the source of any pooling bias explicit and links the projection in Section 3.2 to the common target used by selection.
If \theta\in\operatorname{col}(H_m), then P_m(W)\theta=\theta and \hat\tau_m is consistent for L\theta. Otherwise its probability limit has target bias B_m=L[P_m(W)-I]\theta.
Proposition 2: joint covariance
The second result establishes a joint law for the whole candidate path. It is the formal basis for the difference covariance \hat S_{m,f} and the common multiplier process.
For any finite set of candidates,
\sqrt n
\begin{pmatrix}
\hat\tau_{m_1}-LP_{m_1}\theta\\
\vdots\\
\hat\tau_{m_K}-LP_{m_K}\theta
\end{pmatrix}
\rightsquigarrow
N(0,\Omega),
where block (k,l) of \Omega is D_{m_k}\Sigma D_{m_l}'. This follows by applying a finite collection of linear maps to the saturated influence function. Feasible GLS contributes only higher-order error under consistency and regularity of \hat W.
Proposition 3: exact-structure selection
The third result describes the clean benchmark in which one candidate is the lowest-dimensional adequate representation of the target. It formalizes the selector’s intended behavior in the exact block designs used later as diagnostic finger exercises.
Let m_0 be the lowest-complexity candidate whose target limit equals the target and whose finer candidates share that target limit. Suppose every lower- complexity candidate has at least one comparable refinement and event time for which the standardized population contrast diverges, while Assumptions A-D hold. Then the simultaneous rule rejects every lower-complexity candidate with probability approaching one and accepts m_0 with probability at least 1-\alpha+o(1). With a vanishing selection level satisfying the usual power condition, selection of m_0 is consistent.
Approximate structure is empirically central. When every candidate is an approximation, bias and variance jointly determine the desirable model. The natural oracle is
for reporting loss A. A sharp oracle inequality on the two-dimensional partial order requires additional monotonicity linking target bias and refinement. Without that monotonicity, the theoretical claim stops at exact structure. The numerical study reports regret relative to the best feasible lattice candidate under repeated-sample risk.
4. Implementation
Section 3 defines the estimator as a sequence of partitions, linear projections, joint comparisons, tuning decisions, and uncertainty calculations. This section translates those mathematical objects into the software used in the paper. Its goal is to present the proposed procedure as an operational pyfixest extension for applications and simulations, and to identify the diagnostics exposed to an applied user before the numerical evaluation begins.
LepskiiEventStudy subclasses pyfixest’s DID interface. One saturated pyfixest fit supplies coefficients, CRV1 covariance, residualized scores, and the bread. The implementation verifies numerically that the cluster influence stack reproduces the pyfixest covariance before using it for calibration.
The result object exposes all candidate curves, the acceptance lattice, pairwise statistics, selected partitions, unsupported cells, and conditional or adjusted intervals. LepskiiTuner implements the stratified outer split and inner noise-corrected risk comparison. Tests cover partition nesting, protected breaks, target mappings, unsupported cells, covariance reconstruction, both oracle endpoints, calibration reproducibility, honest cluster separation, and Altair schema validation.
5. Numerical Experiment
The preceding section establishes how the estimator is computed. This section constructs a controlled many-small-cohort experiment to determine whether those computations solve the paper’s substantive problem: retaining precision when pooling restrictions are useful and moving toward saturation as those restrictions fail. It first defines the treatment-effect surfaces and candidate lattice, then fixes the competing estimators and repeated-sample criteria used in the results.
5.1 Design
The experiment uses T=36 periods and nine treated cohorts adopting in periods 8,10,\ldots,24. Cohort shares are uneven and sum to 0.82; the remaining units are never treated. The main run uses n=1{,}200 units and 50 replications per DGP. Unit, time, and serially correlated idiosyncratic shocks are generated by the repository’s existing panel DGP utility.
The cohort path is
K\in\{1,2,3,5,9\},
and the event-time path is
R\in\{2,3,5,9,17\}.
The 17-bin endpoint separates every reported nonreference event time. Selection uses lags 0 through 11 and 399 Rademacher cluster multiplier draws. The same number of draws is used for bootstrap-after-selection intervals.
The suite includes exact finger exercises and less favorable patterns:
complete homogeneity, with oracle candidate C1-E2;
three cohort-only blocks, aligned with C3-E2;
a common short-versus-long response, aligned with C1-E3;
an early/late long-run interaction, aligned with C2-E5;
smooth gradients, delayed responses, local shocks, and F-test-style log, shark-fin, and sinusoidal curves;
three designs in which every cohort has a distinct path.
