Panel.Estimand­Characterization.Flexible­DIDMundlak

Saturated untreated regression recovers the untreated potential outcome on treated cells. If no anticipation holds: the treated and untreated potential-outcome means agree on every cell in the untreated-outcome regression's design and conditional parallel trends holds — the mean untreated potential outcome admits an additive cohort/time fixed-effects representation given covariates, then on any treated cohort-time cell (g,t) covered by the saturated untreated regression S, the fitted value S.m0 g t c equals the mean untreated potential outcome Y0Mean g t c, for every covariate cell c.

POLS cell coefficients equal imputation. On any treated cohort-time cell (g,t), the flexible POLS cell coefficient equals the imputation residual mean at that cell.

ETWFE cell coefficients equal imputation. On any treated cohort-time cell (g,t), the extended two-way-fixed-effects (ETWFE) cell coefficient equals the imputation residual mean at that cell.

Cell-level characterization: imputation, POLS, and ETWFE agree with the ATT cell. If no anticipation holds, conditional parallel trends holds — the mean untreated potential outcome admits an additive cohort/time fixed-effects representation given covariates, and (g,t) is a treated cohort-time cell covered by the untreated-regression witness S and the imputation/POLS/ETWFE estimands E, then the imputation, POLS, and ETWFE cell coefficients at (g,t) all equal the ATT cell τ_gt.

Aggregate characterization: every weighting of imputation, POLS, and ETWFE equals the weighted ATT aggregate. If no anticipation holds and conditional parallel trends holds — the mean untreated potential outcome admits an additive cohort/time fixed-effects representation given covariates, then for any treated-cell weighting function a, the a-weighted aggregates of the imputation, POLS, and ETWFE cell estimands all equal the a-weighted ATT aggregate.

Headline finite-cell characterization (Wooldridge, Theorem B). If no anticipation holds: the treated and untreated potential-outcome means agree on every cell in the untreated-outcome regression's design and conditional parallel trends holds — the mean untreated potential outcome admits an additive cohort/time fixed-effects representation given covariates, then, given the saturated untreated regression S and the POLS/ETWFE finite-cell residual normal equations carried by E, on every treated cohort-time cell the flexible imputation, POLS, and ETWFE estimands all equal the ATT cell, and consequently every treated-cell weighted aggregate of the three estimands equals the correspondingly weighted ATT aggregate.

Wooldridge finite-panel scalar TWFE-two-way-Mundlak equivalence. For a scalar two-way-fixed-effects panel regression problem P with optional time-constant controls Zvar and time-only controls Mvar, given any pooled two-way Mundlak regression fit stated by its normal equations, that fit's coefficient on the regressor equals the two-way-fixed-effects coefficient.

Existence of a TWFE solution. For a vector two-way-fixed-effects problem, whose residualized Gram matrix is nonsingular by assumption, the closed-form coefficient P.betaTWFE solves the matrix normal equation defining the TWFE coefficient.

Full-rank uniqueness of the vector TWFE coefficient. For a vector TWFE problem P with nonsingular residualized Gram matrix, if a coefficient vector β satisfies the coordinate-wise TWFE normal equation — in every coordinate the double-demeaned regressor is orthogonal to the double-demeaned residual, then β equals the closed-form vector TWFE coefficient P.betaTWFE.

The imputation estimand equals a population conditional expectation. On a treated cohort-time cell (g,t), suppose the cohort event is measurable, each covariate cell is measurable, the covariate cells are pairwise disjoint, the covariate cells cover the whole sample space, the treatment-effect integrand Δ is integrable, each finite covariate weight equals the conditional probability of that covariate cell given the cohort event, and each finite cell residual — the observed mean minus the fitted untreated mean — equals the within-cell conditional mean of Δ given the cohort event and that covariate cell. Then the imputation cell estimand thetaImp g t equals the population conditional expectation E[Δ | cohortEvent].

On a treated cell in a population-built system, the fitted untreated mean equals the population conditional mean of the untreated potential outcome. Fix a population model on a sample space Ω, with population outcomes Y0pop, Ygpop, Yobspop; suppose every cell event cellEvent g t c is measurable, every cell has strictly positive probability mass, and the three population outcomes are each integrable on every cell. Suppose also the cohort share is strictly positive on every treated cell, the covariate weights are nonnegative and sum to one within each cohort, and the observed outcome agrees pointwise with the cohort-g outcome on treated cells and with the untreated outcome on untreated cells (pointwise consistency). If the finite cell system P is exactly the one built from this population data by StaggeredATTCells.ofPopulation and, for a saturated untreated regression S on P, no anticipation holds and conditional parallel trends holds, then on any treated cell (g,t), the saturated regression's fitted value S.m0 g t c equals the population conditional mean E[Y0pop | cellEvent g t c], for every covariate cell c.

On an untreated cell in a population-built system, the fitted untreated mean equals the population conditional mean of the untreated potential outcome. Fix a population model on a sample space Ω, with population outcomes Y0pop, Ygpop, Yobspop; suppose every cell event cellEvent g t c is measurable, every cell has strictly positive probability mass, and the three population outcomes are each integrable on every cell. Suppose also the cohort share is strictly positive on every treated cell, the covariate weights are nonnegative and sum to one within each cohort, and the observed outcome agrees pointwise with the cohort-g outcome on treated cells and with the untreated outcome on untreated cells (pointwise consistency). If the finite cell system P is exactly the one built from this population data by StaggeredATTCells.ofPopulation and, for a saturated untreated regression S on P, no anticipation holds and conditional parallel trends holds, then on any untreated cell (g,t), the saturated regression's fitted value S.m0 g t c equals the population conditional mean E[Y0pop | cellEvent g t c], for every covariate cell c.

Wooldridge finite-panel K-vector TWFE-two-way-Mundlak equivalence (Theorem A). For a K-vector two-way-fixed-effects panel regression problem P with optional time-constant controls Zvar and time-only controls Mvar, given any pooled two-way Mundlak regression fit stated by its normal equations, that fit's coefficient vector on the regressors equals the K-vector two-way-fixed-effects coefficient vector.