Randomization Tests in Switchback Experiments

Randomization T ests in Switc h bac k Exp erimen ts ∗ Jizhou Liu PHBS Business Sc ho ol P eking Universit y jizhou.liu@phbs.pku.edu.cn Liang Zhong F acult y of Business and Economics The Univ ersity of Hong Kong samzl@hku.hk F ebruary 27, 2026 Abstract Switc hbac k exp erimen ts—alternating treatmen t and con trol ov er time—are widely used when unit-lev el randomization is infeasible, outcomes are aggregated, or user in terference is unav oid- able. In practice, exp erimen tation must supp ort fast pro duct cycles, so teams often run studies for limited durations and mak e decisions with mo dest samples. At the same time, outcomes in these time-indexed settings exhibit serial dep endence, seasonalit y , and o ccasional heavy-tailed sho c ks, and temporal interference (carryo v er or an ticipation) can render standard asymptotics and naiv e randomization tests unreliable. In this paper, w e develop a randomization-test frame- w ork that deliv ers ﬁnite-sample v alid, distribution-free p -v alues for several n ull hypotheses of in terest using only the known assignment mechanism, without parametric assumptions on the outcome pro cess. F or causal eﬀects of interests, w e imp ose t wo primitive conditions—non- an ticipation and a ﬁnite carryo ver horizon m —and construct conditional randomization tests (CR Ts) based on an ex ante p o oling of design blo cks into “sections,” whic h yields a tractable conditional assignment law and ensures imputabilit y of fo cal outcomes. W e pro vide diagnostics for learning the carryo v er window and assessing non-anticipation, and we introduce studentized CR Ts for a session-wise weak null that accommodates within-session seasonality with asymp- totic v alidit y . P ow er appro ximations under distributed-lag eﬀects with AR(1) noise guide design and analysis c hoices, and simulations demonstrate fav orable size and p o w er relative to common alternativ es. Our framework extends naturally to other time-indexed designs. KEYW ORDS: Causal inference; Conditional Randomization T est; Switch bac k Experiments. ∗ W e wan t to thank Jinglong Zhao for his helpful commen ts. All errors are our own. 1 In tro duction Randomized exp erimen ts pla y a central role in guiding decision making in online and operational systems suc h as mark etplaces, transp ortation platforms, and digital adv ertising ( Blake and Co ey , 2014 ; Blake et al. , 2015 ; Gordon et al. , 2019 ; Kohavi et al. , 2020 ; W ager and Xu , 2021 ; Bo jinov and Gupta , 2022 ; Johari et al. , 2022 ; Li et al. , 2022 ; Christensen et al. , 2023 ; T ang et al. , 2024 ; Maso ero et al. , 2026 ). Y et in many of these environmen ts, unit-level randomization is infeasible, outcomes are observ ed only in aggregate, or interference across users is una voidable. These constraints motiv ate switchb ack exp eriments , in which a platform alternates betw e en treatmen t and control o v er time rather than across individuals ( Bo jino v et al. , 2023 ; Jiang and Ding , 2025 ; Lin and Ding , 2025 ). Reliable inference in these time-indexed exp erimen ts is b oth practically imp ortan t and metho d- ologically c hallenging. Platforms often need to make decisions quickly—sometimes within da ys or w eeks—which limits the av ailable sample size ( Koha vi et al. , 2020 ; Gupta et al. , 2019 ). A t the same time, outcomes form a time series with serial dep endence, seasonal patterns, and o ccasionally hea vy-tailed sho c ks, so normal appro ximations can b e po orly calibrated at realistic horizons; for example, Bo jino v et al. ( 2023 ) do cument settings in whic h calibration ma y require on the order of T = 1200 p eriods. Inference is further complicated by temp oral interference: outcomes ma y dep end on recen t treatmen ts (carry ov er) or ev en future sc heduled assignmen ts (an ticipation), making many n ull h yp otheses non-sharp and inv alidating naive Fisher randomization tests ( A they et al. , 2018 ; Basse et al. , 2019 ; Zhong , 2024 ). T o address these c hallenges, this pap er dev elops a randomization-test framew ork for several n ull hypotheses of in terest in switch bac k exp erimen ts. Our approac h yields ﬁnite-sample v alid, distribution-free p -v alues using only the known assignmen t mechanism, without imp osing paramet- ric assumptions on the outcome pro cess. The goal is to mak e randomization inference op erationally reliable in time-indexed exp erimen ts conducted o ver realistic, and often short, horizons. T o conduct inference for the total treatmen t eﬀect—deﬁned as the eﬀect of sustained expo- 1 sure to the in terven tion—we require only t wo standard assumptions on potential outcomes: non- an ticipation, whic h rules out dep endence on future assignments, and a ﬁnite carryo ver horizon, whic h allo ws outcomes to dep end on recen t treatmen t history but not on distant past assignmen ts. Within regular switc h back exp eriments in the sense of Bo jinov et al. ( 2023 ), w e construct con- ditional randomization tests that remain ﬁnite-sample v alid and can be implemen ted via simple Mon te Carlo resampling ( A they et al. , 2018 ; Basse et al. , 2019 ; Puelz et al. , 2021 ; Basse et al. , 2024 ; Liu et al. , 2025 ). Our approac h p o ols design blo c ks into candidate sections in adv ance of observing the assignmen t path and conditions on those that are constan t under the realized sc hedule, yielding a tractable randomization law and a v alid reference distribution for outcomes not con taminated b y carry ov er. T o mitigate the concern of p oten tial violation of the assumptions, we also prop ose tw o comple- men tary diagnostics for the temporal structure of in terference. First, w e dev elop a family of tests for the n ull hypothesis of at most m -perio d carry ov er eﬀects and exploit the nested structure of these n ulls to obtain a sequen tial pro cedure that controls the family-wise error rate while deliv er- ing an in terpretable estimate of treatmen t memory length. Second, to assess the non-anticipation restriction, we adapt the unconditional Pairwise Imputation–based Randomization T est (PIR T) of Zhong ( 2024 ) to switc hbac k schedules, using a notion of imputable times where p otential outcomes can b e paired across schedules under the an ticipation n ull. In ligh t of the practical interest for testing av erage eﬀect, w e also address w eak-n ull inference, where the target is a mean-zero restriction on treatmen t eﬀects in the p ost–burn-in regime. W e in tro duce a session-wise weak n ull that requires mean-zero eﬀects at each within-session p osition among the fo cal p erio ds (e.g., Monda y vs. T uesda y), accommo dating within-session seasonalit y in practice. W e construct studentized CR Ts for this n ull and establish asymptotic v alidity under mild momen t and stabilization conditions ( Li and Ding , 2017 ; W u and Ding , 2021 ; Zhao and Ding , 2021 ). Finally , we study p o wer under a sup erp opulation mo del with distributed-lag treatmen t eﬀects 2 and AR(1) noise. The resulting appro ximations clarify how blo c k length, burn-in, and prede- termined p ooling jointly determine the signal-to-noise ratio of the studentized CR Ts, providing practical guidance for design and analysis c hoices. Monte Carlo simulations corroborate these insigh ts: the proposed CR T main tains near-nominal size under b oth Gaussian and heavy-tailed sho c ks, while Fisher tests under a missp eciﬁed sharp null ov er-reject in hea vy tails and Horvitz– Thompson asymptotic tests are often conserv ativ e at realistic horizons; in terms of pow er, the CR T matches asymptotic tests under Gaussian noise and dominates size-correct alternatives under hea vy-tailed disturbances. F or diagnostics, the carryo v er CR T ac hieves accurate size with pow er increasing in eﬀect magnitude and sample size, and the PIR T-based non-an ticipation test con trols size and delivers meaningful p o wer; together with our theoretical approximations, these ﬁndings highligh t block length, burn-in, and section p o oling as ﬁrst-order design lev ers for sensitivit y in time-indexed exp erimen ts. This pap er contributes to the growing literature on switch back and other time-indexed ex- p erimen tal designs. Bo jinov et al. ( 2023 ) study regular switc h back designs and dev elop largely asymptotic inference pro cedures. Related design-based analyses for experiments with temp oral structure include Jiang and Ding ( 2025 ) and Lin and Ding ( 2025 ). By contrast, we adopt a Fish- erian randomization p erspective and construct conditioning-based tests that deliv er ﬁnite-sample exact, distribution-free p -v alues for standard switch back schedules. The remainder of the paper is organized as follows. Section 2 describ es the setup and nota- tion. Section 3 develops ﬁnite-sample v alid CR Ts for total treatment eﬀects, carry ov er length and non-an ticipation. Section 4 dev elops studentized CR Ts for session-wise weak n ull hypotheses and pro vides asymptotic v alidity results. Section 5 deriv es p o wer appro ximations under a superp opu- lation mo del. Section 6 illustrates the sim ulation results. Section 7 concludes. 3 2 Setup and Notation 2.1 Switc h back exp erimen ts and regular blo c k randomization W e observ e a single outcome time series o v er T measuremen t p eriods. Let [ T ] := { 1 , 2 , . . . , T } index time, and let W := ( W 1 , . . . , W T ) ∈ { 0 , 1 } T denote the treatmen t assignment path, where W t = 1 indicates treatmen t is “on” during p eriod t and W t = 0 indicates treatment is “oﬀ.” F or an y candidate assignmen t path w = ( w 1 , . . . , w T ) ∈ { 0 , 1 } T , let Y t ( w ) ∈ R be the p otential outcome at time t that would b e observed if the assignment path were w . The observ ed outcome satisﬁes the consistency relationship Y obs t = Y t ( W obs ) , t ∈ [ T ] , where W obs denotes the realized assignment. Throughout, for integers 1 ≤ a ≤ b ≤ T , we write w a : b := ( w a , . . . , w b ) for a sub vector, and w e use 1 d and 0 d to denote length- d all-ones and all-zeros vectors, resp ectiv ely . A switchb ack exp eriment is a time-based randomized exp erimen t in which the treatment is held constan t o ver contiguous blo cks of time and may switc h only at pre-sp eciﬁed times. In this pap er, w e w ork with the class of r e gular switchb ack exp eriments of Bo jino v et al. ( 2023 ). F ollowing Bo jinov et al. ( 2023 ), let T = { t 0 = 1 < t 1 < · · · < t K } ⊆ [ T ] , t K +1 := T + 1 , b e a deterministic set of switc h times. These induce K + 1 design blo c ks B k := { t : t k ≤ t ≤ t k +1 − 1 } , k = 0 , 1 , . . . , K. 4 Let Q = ( q 0 , . . . , q K ) ∈ (0 , 1) K +1 denote the block-lev el treatmen t probabilities. Deﬁnition 2.1 (Regular switc h back exp erimen t) . Fix ( T , Q ) as ab ove. A r e gular switchb ack ex- p eriment dr aws indep endent blo ck-level assignments { W ( k ) } K k =0 with W ( k ) ∼ Bernoulli( q k ) , k = 0 , 1 , . . . , K, and then sets W t := W ( k ) for al l t ∈ B k . Equivalently, W is blo ckwise c onstant on e ach B k , with indep endent blo ck lab els. Deﬁnition 2.1 co v ers common switch back implementations in whic h time is partitioned in to in terv als (e.g., hours, da ys, or w eeks), and each interv al is independently assigned treatment with a p ossibly time-v arying probabilit y . W e will use W 0 :=  w ∈ { 0 , 1 } T : w is blockwise constant on each B k  (1) to denote the set of assignmen t paths that respect the design blocks. 2.2 Assumptions on temp oral in terference Switc hbac k experiments are t ypically used when outcomes ma y dep end on recen t treatmen t history , so we allow for temp or al interfer enc e across p erio ds through t wo primitive restrictions on p oten tial outcomes. Assumption 2.1 (Non-an ticipating potential outcomes) . F or an y t ∈ [ T ], an y preﬁx w 1: t ∈ { 0 , 1 } t , 5 and any tw o con tinuation paths w ′ t +1: T , w ′′ t +1: T ∈ { 0 , 1 } T − t , Y t ( w 1: t , w ′ t +1: T ) = Y t ( w 1: t , w ′′ t +1: T ) . Assumption 2.1 rules out dep endence of Y t on future treatmen t assignments. It is a timing restriction: p oten tial outcomes at time t are allow ed to dep end arbitrarily on the history up to t , but not on { w t +1 , . . . , w T } . Assumption 2.2 ( m -carry ov er eﬀects) . There exists a ﬁxed and given m ∈ { 0 , 1 , . . . , T − 1 } suc h that for an y t ∈ { m + 1 , m + 2 , . . . , T } , any w t − m : t ∈ { 0 , 1 } m +1 , and any t wo histories w ′ 1: t − m − 1 , w ′′ 1: t − m − 1 ∈ { 0 , 1 } t − m − 1 , Y t ( w ′ 1: t − m − 1 , w t − m : T ) = Y t ( w ′′ 1: t − m − 1 , w t − m : T ) . Assumption 2.2 bounds the “memory” of the treatment: outcomes at time t ma y dep end on the most recen t m + 1 treatment indicators ( w t − m , . . . , w t ) but not on assignmen ts more than m p eriods in the past. T ogether, Assumptions 2.1 and 2.2 imply a lo cal dep endence structure: for all t ≥ m + 1, the potential outcome Y t ( w ) dep ends on w only through the length-( m + 1) window w t − m : t . In particular, if t w o assignmen t paths w and w ′ satisfy w t − m : t = w ′ t − m : t , then Y t ( w ) = Y t ( w ′ ) for all t ≥ m + 1. Remark 2.1 (In terpretation of m ) . The parameter m is the maximal carryo v er horizon: if treat- men t switches at time s , then its eﬀect on outcomes may p ersist through time s + m , but it do es not aﬀect Y t for t > s + m via channels other than the con temp oraneous and recen t treatmen t indicators. The case m = 0 corresp onds to no carry ov er b eyond the current p eriod. 6 3 Main Results 3.1 Ov erview of Conditional Randomization T ests The classical Fisher Randomization T est (FR T) sim ulates the randomization distribution of a test statistic under the known assignmen t mechanism and compares it to the observed statistic ( Im b ens and Rubin , 2015 , Chapter 5). This procedure is ﬁnite-sample v alid for sharp null hypotheses, i.e., n ulls that allo w the analyst to impute all missing potential outcomes under ev ery assignmen t in the design supp ort. In switc hbac k exp erimen ts with temporal in terference, man y h yp otheses of interest are not sharp o ver the full assignment space. F or example, a null that only links p oten tial outcomes under tw o particular assignmen t paths (e.g., all-treated v ersus all-control) do es not determine p oten tial outcomes under intermediate switching paths. As a result, a naiv e FR T that resamples the full assignmen t path generally cannot be implemen ted by imputation. Conditional randomization tests (CR Ts) address this problem by restricting the resampling pro cedure to a c onditioning event that renders the n ull h yp othesis sharp on a subset of units and assignmen ts. In the terminology of Basse et al. ( 2019 ), a conditioning ev ent takes the form C = ( U , W ) , where U is a subset of units (here, time p eriods) and W is a subset of assignments. In the literature, U and W are commonly referred to as the fo c al units and the fo c al assignments , respectively . The analyst sp eciﬁes a c onditioning me chanism p ( C | W ), whic h ma y be random and may depend on the realized assignmen t W . The CR T then samples assignments from the conditional law p ( W | C ) ∝ p ( C | W ) p ( W ) , (2) and c omputes a test statistic using only outcomes on U . Finite-sample v alidity follows when the 7 test statistic is imputable on C under the n ull, meaning that its v alue under an y w ∈ W can b e computed from the observed outcomes given the n ull restriction ( Basse et al. , 2019 , Theorem 1). Tw o general recip es are commonly used to construct conditional randomization tests. First, the framework of Basse et al. ( 2019 ) allows for non-de gener ate conditioning mec hanisms, in which p ( C | W ) is gen uinely randomized. A v alid CR T then draws a single conditioning even t C ∼ p ( · | W obs ) and samples assignments from the induced conditional distribution ( 2 ). The main drawbac k is computational: ev aluating and sampling from ( 2 ) can be exp ensiv e when the conditional ran- domization space is large or lacks exploitable structure. Second, Puelz et al. ( 2021 ) study CR Ts under de gener ate conditioning mechanisms of the form p ( C | W ) = 1 { W ∈ W } , for some (p ossibly data-dep enden t) restricted randomization space W . In this regime, v alidity hinges on an invarianc e requirement: the conditioning even t used for resampling m ust not c hange as we v ary assignmen ts within the restricted space. Equiv alen tly , if we view the conditioning even t as a mapping C ( w ) generated from an assignmen t w , then in v ariance requires C ( w ) = C ( W obs ) for all w ∈ W , (3) so that the conditioning ev en t is constant across all fo c al assignments considered by the test. Our switch bac k procedures fall into this second class. W e emplo y a degenerate conditioning mec hanism— C is a deterministic function of the realized assignment path—and our main tec hnical task is to construct C so that (i) the n ull b ecomes imputable on the selected fo cal p erio ds U and (ii) the inv ariance condition ( 3 ) holds for the resulting restricted assignment set. In particular, our conditioning even ts are built from pr e determine d time partitions (ﬁxed ex ante ), so that re- sampling treatmen t labels within focal assignments cannot c hange the ev ent itself. This yields a 8 tractable conditional assignmen t law and p ermits eﬃcient Monte Carlo CR T implemen tation while main taining ﬁnite-sample v alidity under the n ulls considered b elo w. 3.2 T esting total treatmen t eﬀects W e b egin with randomization tests for the partially sharp null hypothesis of no total treatment eﬀect in a switch back exp erimen t: H tot 0 : Y t ( 1 T ) = Y t ( 0 T ) for all t ∈ [ T ] , (4) where 1 T and 0 T denote the constan t all-ones and all-zeros assignmen t paths. Under Assump- tions 2.1 – 2.2 , ( 4 ) is equiv alent to Y t ( 1 m +1 ) = Y t ( 0 m +1 ) for all t ≥ m + 1 b ecause Y t ( · ) dep ends on w only through the last m + 1 en tries. The key c hallenge is that, under carry ov er, ( 4 ) is not directly imputable for all p eriods: changing the assignment at time t can aﬀect outcomes at times t + 1 , . . . , t + m . T o construct a conditioning ev ent that isolates imputable outcomes, we introduce predetermined time se ctions . F ormally , a se ction is a contiguous time interv al [ s, e ] whose endp oints are ﬁxed ex ante and that is formed b y merging consecutive design blo c ks. W e will restrict atten tion to p erio ds t whose relev an t treatment history ( W t − m , . . . , W t ) lies en tirely within a section that is constant under the realized assignment. Fix a predetermined family of disjoin t sections S pre = { [ s 1 , e 1 ] , . . . , [ s J , e J ] } , 1 ≤ s 1 ≤ e 1 < s 2 ≤ · · · < s J ≤ e J ≤ T , constructed ex ante (indep enden t of W ). Eac h section is obtained by p ooling consecutiv e design blo c ks, and m ust satisfy: 9 (i) ( Mer ge d fr om design blo cks ) F or eac h j there exist 0 ≤ a j ≤ b j ≤ K suc h that [ s j , e j ] = { t : t a j ≤ t ≤ t b j +1 − 1 } . (ii) ( L ength c onstr aint ) e j − s j ≥ m (equiv alently , | [ s j , e j ] | ≥ m + 1). In simulation, S pre is constructed deterministically from the randomization design and the pre- sp eciﬁed carryo v er horizon m . Sp eciﬁcally , w e p ool consecutiv e design blocks greedily un til the p ooled length is at least m + 1, then rep eat this pro cedure on the remaining blocks. This pro duces a disjoin t cov er of [ T ] and ﬁxes section boundaries indep enden tly of the realized assignmen t path and all outcomes. 1 Giv en a realized assignmen t W , let S ( W ) ⊆ S pre denote the sub collection of predetermined sections that are constant under W : S ( W ) = n [ s j , e j ] ∈ S pre : W t = W t ′ for all t, t ′ ∈ [ s j , e j ] o . Because S pre is ﬁxed ex ante , the map W 7→ S ( W ) is fully determined by the realized assignmen t and inv olv es no analyst discretion. Within each constant section, only the last e − s − m + 1 perio ds are free of carryo ver contam- ination from outside the section. Accordingly , deﬁne the set of fo cal units U ( W ) = [ [ s,e ] ∈S ( W ) { t ∈ [ T ] : s + m ≤ t ≤ e } . (5) If S ( W ) = ∅ , then U ( W ) = ∅ and no fo cal units are av ailable. In a CR T, fo cal units are the units on whic h the test statistic is computed. The av ailability of fo cal units dep ends on the design, the carry ov er horizon m , and the realized assignment path. 1 An y deterministic rule dep ending only on the design and m w ould be v alid. W e adopt greedy p ooling b ecause it guaran tees admissible section length while minimizing po oling. 10 Short sections relative to m may yield U ( W ) = ∅ with non-negligible probability . T o guide implemen tation, practitioners ma y ev aluate ex ante the exp ected num b er of fo cal units under the randomization design, see section 5 for a detailed discussion. Next, w e describ e how we construct the fo cal assignment space, o ver which w e p erform Monte Carlo resampling of treatment assignmen ts. Conditioning on the realized constant sections S ( W obs ), w e restrict resampling to assignments that (i) remain blockwise constan t, (ii) are constan t within eac h realized constant section, and (iii) match the observed assignmen t outside those sections: W  S ( W obs )  = n w ∈ { 0 , 1 } T : (a) w ∈ W 0 , (b) w t = w t ′ for all t, t ′ ∈ [ s, e ] , for all [ s, e ] ∈ S ( W obs ) , (c) w t = w obs t for all t / ∈ [ [ s,e ] ∈S ( W obs ) [ s, e ] o . (6) This construction is tailored to imputabilit y: under H 0 and Assumptions 2.1 – 2.2 , the outcomes { Y obs t : t ∈ U ( W obs ) } are unaﬀected by c hanges to w within W ( S ( W obs )) outside the fo cal histories. Let T ( { Y t } t ∈U , w ) be any test statistic computed from the fo cal outcomes and the assignment path. 2 Algorithm 1 describ es a Monte Carlo CR T that samples from the conditional assignmen t la w on W ( S ( W obs )) and computes a one-sided p -v alue. The next lemma characterizes the conditional law of a merged section under indep enden t blo ck Bernoulli assignment. Lemma 3.1 (Conditional law on a constan t merged section) . Supp ose blo cks W ( k ) ar e indep endent with W ( k ) ∼ Bern( q k ) . F or a mer ge d se ction c overing blo cks a, . . . , b , c onditional on the event { W ( a ) = · · · = W ( b ) } , the c ommon value is Bernoul li with pr ob ability p = Q b k = a q k Q b k = a q k + Q b k = a (1 − q k ) . 2 A common choice is a w eighted diﬀerence-in-means comparing focal outcomes in treated v ersus control perio ds, p ossibly with in verse-probabilit y weigh ts if { q k } v ary across blo c ks. 11 Algorithm 1 Conditional randomization test for treatmen t eﬀects (regular switch back) Input: Observ ed { Y obs t } T t =1 , W obs ; switch times T with blo c ks { B k } K k =0 ; blo c k probabilities Q ; realized sections S ( W obs ) = { [ s j , e j ] } j ∈J ( W obs ) . Output: ˆ p . Construct U ( W obs ) by ( 5 ) and compute T obs := T ( { Y obs t } t ∈U ( W obs ) , W obs ). F or eac h section [ s j , e j ], ﬁnd ( a j , b j ) with [ s j , e j ] = S b j k = a j B k , and set p j := Q b j k = a j q k Q b j k = a j q k + Q b j k = a j (1 − q k ) . Set count ← 0. for b = 1 to M do Initialize W ( b ) ← W obs . for j ∈ J ( W obs ) do Dra w Z ( b ) j ∼ Bernoulli( p j ) and set W ( b ) t ← Z ( b ) j for all t ∈ [ s j , e j ]. end Compute T ( b ) := T ( { Y obs t } t ∈U ( W obs ) , W ( b ) ). if T ( b ) ≥ T obs then coun t ← count + 1. end end Return ˆ p := (count + 1) / ( M + 1). Mor e over, acr oss disjoint se ctions these c onditional dr aws ar e indep endent. Theorem 3.1 (Finite-sample v alidity) . Supp ose the assignment fol lows Deﬁnition 2.1 and Assump- tions 2.1 and 2.2 hold. Consider testing the p artial ly sharp nul l ( 4 ) . L et C ( W ) := ( S ( W ) , U ( W )) denote the c onditioning event. L et ˆ p b e the Monte Carlo p -value r eturne d by Algorithm 1 . Then, under H 0 , P  ˆ p ≤ α    C ( W ) = C ( W obs )  ≤ α for al l α ∈ (0 , 1) . Theorem 3.1 is a ﬁnite-p opulation statement: p oten tial outcomes are treated as ﬁxed, and ran- domness enters only through the switc hbac k assignment mechanism and the Mon te Carlo resam- pling. The result follows from (i) imputability of fo cal outcomes under H tot 0 giv en the conditioning ev ent and (ii) correct simulation of the conditional assignmen t la w via Lemma 3.1 ( Basse et al. , 2019 ; Athey et al. , 2018 ). Remark 3.1 (Predetermined sections and inv ariance) . Theorem 3.1 relies on the inv ariance re- quiremen t for degenerate CR Ts ( Puelz et al. , 2021 ): the conditioning even t m ust b e the same for ev ery focal assignmen t considered b y the test. This is wh y w e ﬁx a predetermined collection of disjoint candidate sections S pre indep enden t of W , and then deﬁne S ( W ) only by selecting those 12 predetermined sections that are constant under W . Because section boundaries are ﬁxed ex an te, an y w ∈ W ( S ( W obs )) preserves the same selected sections (and hence the same fo cal set), so C ( w ) = C ( W obs ). If sections were deﬁned adaptively from W (e.g., via maximal runs of all-ones and all-zeros paths), resampling could create longer consecutive all-ones/all-zeros paths, changing C and breaking in v ariance. Remark 3.2 (Perm utation implementations under homogeneous section probabilities) . Algorithm 1 samples section lab els using their conditional treatment probabilities { p j } . If the design is homo- geneous in the sense that q k ≡ q and ev ery realized section in S ( W obs ) merges the same num b er of design blo cks (equiv alently , has the same length in block units), then p j ≡ p for all selected sections. In this case the selected section labels are exc hangeable, and one may implement the CR T via a simple p erm utation: condition additionally on the treated count P j ∈J Z j and p erm ute the observed lab els across the selected sections. When the section probabilities are not all equal, exc hangeability fails. A con venien t exact alternativ e is str atiﬁe d p ermutation : partition the selected sections into strata with common p j , condition on the treated coun t within eac h stratum, and p erm ute lab els only within strata. This retains the simplicit y of p erm utations while respecting heterogeneous assignment probabilities. 3.3 T esting the carry o ver horizon The CR T in Section 3.2 assumes a kno wn carry ov er horizon m . In applications, how ever, m is rarely kno wn a priori, and it is often important to assess whether the carryo ver windo w used to deﬁne fo cal outcomes is plausibly long enough. This section dev elops randomization tests for the n ull h yp othesis that carry ov er eﬀects v anish after m p erio ds. 3 3 Bo jinov et al. ( 2023 ) also prop ose a test for identifying the order of carryo v er eﬀects. Their approach, how ever, requires a sp ecialized design implemented across diﬀeren t units—thus necessitating changes to the exp erimen tal proto col—and its v alidity is primarily asymptotic. In contrast, our metho d applies directly to existing exp erimental designs and yields ﬁnite-sample v alidity , which is particularly imp ortan t when sample sizes are mo dest. 13 Deﬁnition 3.1 (Null of m -carryo ver eﬀects H m 0 ) . Fix m ∈ { 0 , 1 , . . . , T − 1 } and assume no- anticip ation as in Assumption 2.1 . The nul l hyp othesis of at most m -p erio d c arryover eﬀe cts, denote d H m 0 , states that for any t ∈ { m + 1 , . . . , T } , any c ontinuation p ath w t − m : t ∈ { 0 , 1 } m +1 , and any two histories w ′ 1: t − m − 1 , w ′′ 1: t − m − 1 ∈ { 0 , 1 } t − m − 1 , Y t ( w ′ 1: t − m − 1 , w t − m : t ) = Y t ( w ′′ 1: t − m − 1 , w t − m : t ) . Deﬁnition 3.1 makes precise the intuition that, under H m 0 , outcomes at time t are unaﬀected b y assignmen ts more than m perio ds in the past. W e reuse the predetermined section family S pre = { [ s j , e j ] } J j =1 from Section 3.2 and index sections in increasing time order. T o decouple the randomized lab els from the outcomes used for testing, we employ an “alternating holdout” device: we hold out ev ery other predetermined section and use the remaining sections as fo cal outcome sections. Deﬁne the fo cal sections as { [ s 2 i , e 2 i ] } ⌊ J / 2 ⌋ i =1 . Within each focal section, w e k eep exactly those times whose last m lags remain in the same section: U ( m ) = ⌊ J / 2 ⌋ [ i =1 { t ∈ [ T ] : s 2 i + m ≤ t ≤ e 2 i } . (7) Under H m 0 , outcomes { Y t ( w ) : t ∈ U ( m ) } dep end only on as signmen ts inside their own focal section, and hence they are in v arian t to changes in assignment outside that section. Lemma 3.2 (Lo cal dep endence of Y t under H m 0 ) . If t ∈ [ s 2 i + m, e 2 i ] , then under H m 0 the quantity Y t ( w ) dep ends only on the assignments w t − m : t , and in p articular w t − m : t ⊆ [ s 2 i , e 2 i ] . Ther efor e, changing assignments outside [ s 2 i , e 2 i ] c annot change Y t . 14 Let W 0 b e the blo c kwise-constan t assignmen t set in ( 1 ). Conditioning on S pre (whic h is pre- determined), w e deﬁne the randomization space by keeping the assignment ﬁxe d on fo cal sections while allowing the remaining blo c ks to v ary according to the design: W ( m ) = n w ∈ W 0 : w t = W obs t for all t ∈ [ s 2 i , e 2 i ] , i = 1 , . . . , ⌊ J / 2 ⌋ o . (8) A natural test statistic aggregates fo cal outcomes section-b y-section and treats the treatment lev els of the pr e c e ding (non-fo cal) sections as randomized lab els. F or each ev en-num b ered section 2 i , deﬁne the fo cal mean ¯ Y obs 2 i = 1 |U ( m ) 2 i | e 2 i X t = s 2 i + m Y obs t , U ( m ) 2 i = { t : s 2 i + m ≤ t ≤ e 2 i } , whic h av erages observed outcomes ov er times whose last m lags remain within the focal section. F or the preceding non-fo cal section 2 i − 1, deﬁne its sec tion lab el as the treatment assignment at its ﬁnal time p oin t, Z 2 i − 1 := W e 2 i − 1 . Because treatment is blo c kwise constan t within each section, Z 2 i − 1 indexes the common treatmen t lev el applied throughout the en tire preceding section [ s 2 i − 1 , e 2 i − 1 ]. When the assignment probabilities are known from the design, w e form an in verse-probabilit y- w eighted con trast across adjacent section pairs: T m ( W ) = 1 ⌊ J / 2 ⌋ ⌊ J / 2 ⌋ X i =1  Z 2 i − 1 ¯ Y obs 2 i Pr( Z 2 i − 1 = 1) − (1 − Z 2 i − 1 ) ¯ Y obs 2 i 1 − Pr( Z 2 i − 1 = 1)  . Under H m 0 , fo cal outcomes depe nd only on assignments within their own section, and hence are indep enden t of the treatment level in the preceding section under the randomization distribution. The statistic T m ( W ) therefore tests for residual dep endence of fo cal outcomes on treatment exp osure 15 Algorithm 2 Conditional randomization test for m -carry ov er (regular switch bac k) Input: Observ ed { Y obs t } T t =1 , W obs ; predetermined sections S pre = { [ s j , e j ] } J j =1 ; carry ov er horizon m . Output: b p m . Construct U ( m ) b y ( 7 ). Compute T obs m := T m ( { Y obs t } t ∈U ( m ) , W obs ). Set count ← 0. for b = 1 to M do Dra w an assignmen t path W ( b ) ∼ W | W ∈ W ( m ) (i.e., keep W ﬁxed on eac h fo cal sec- tion [ s 2 i , e 2 i ] and resample the remaining blo c ks according to the known switc hbac k design). Compute T ( b ) m := T m ( { Y obs t } t ∈U ( m ) , W ( b ) ). if T ( b ) m ≥ T obs m then coun t ← count + 1. end end Return b p m := (coun t + 1) / ( M + 1). in adjacen t prior sections. The corresp onding randomization distribution is obtained b y resampling the non-fo cal section lab els in W ( m ) while holding the fo cal outcomes { Y obs t : t ∈ U ( m ) } ﬁxed. 4 W e no w make explicit ho w to obtain the p -v alue for testing a giv en H m 0 . Fix m and treat the fo cal set U ( m ) in ( 7 ) and the conditional assignmen t space W ( m ) in ( 8 ) as the conditioning ev en t. Under H m 0 and Lemma 3.2 , the focal outcomes { Y obs t : t ∈ U ( m ) } are inv ariant to re-randomizing assignmen ts on the non-fo cal sections, so any statistic that dep ends on the observ ed outcomes only through U ( m ) and on assignments only through the randomized lab els is imputable. Algorithm 2 giv es a Monte Carlo CR T that samples from the conditional assignmen t la w on W ( m ) and returns a one-sided p -v alue. The v alidit y of Algorithm 2 follo ws directly from the same argumen t as in Theorem 3.1 . In particular, for each ﬁxed m , it yields a v alid level- α test of H m 0 . In man y applications, how ev er, m is unknown, and the ob jective is to iden tify a plausible minimal horizon ¯ m b ey ond whic h carry ov er eﬀects v anish. Because the h yp otheses { H m 0 } M m =0 are nested (Proposition 3.1 ), false n ulls m ust o ccur b efor e true nulls. This monotonic structure allows us to run the tests sequen tially using the p -v alues { b p m } from Algorithm 2 and stop at the ﬁrst non-rejection, as formalized in Algorithm 3 . Theorem 3.2 then guarantees FWER control (Deﬁnition 3.2 ) without multiplicit y corrections. 4 An y statistic measurable with resp ect to fo cal outcomes and the randomized lab els, and imputable under Lemma 3.2 , yields a v alid conditioning-based randomization test. 16 Prop osition 3.1 (Nestedness) . Supp ose ther e exists ¯ m ∈ { 0 , . . . , M } such that H ¯ m 0 is true. Then H m 0 is true for al l m ≥ ¯ m . Deﬁnition 3.2 (FWER under nested n ulls) . L et φ = ( φ 0 , . . . , φ M ) b e a multiple testing rule, wher e φ m ( W obs ) = 1 denotes r eje ction of H m 0 . L et ¯ m denote the smal lest index such that H ¯ m 0 is true. The family-wise err or r ate (FWER) is FWER = Pr  ∃ m ≥ ¯ m such that φ m ( W obs ) = 1  , the pr ob ability of r eje cting at le ast one true nul l hyp othesis. Algorithm 3 Sequential testing under nested carryo ver nulls Inputs : Carry ov er test statistics { T m } M m =0 , observed assignment W obs , observed outcomes Y obs , and the kno wn assignmen t mec hanism. Set : ˆ m ← 0. for m = 0 to M − 1 do Compute b p m via Algorithm 2 . if b p m ≤ α then set ˆ m ← m + 1 and reject H m 0 ; end else break; end end Output: Estimated memory length ˆ m . Theorem 3.2. A lgorithm 3 c ontr ols the family-wise err or r ate at level α . Algorithm 3 eﬀectively provides a low er b ound (or conserv ative estimate) for the carry ov er horizon m , rather than p oin t iden tiﬁcation. That being said, it remains useful in practice. F or example, Bo jinov et al. ( 2023 ) note that when m is unkno wn, it is generally preferable to c ho ose a v alue slightly larger than the true m rather than substantially smaller. This suggests that ev en a conserv ativ e estimate or low er b ound on m can b e informative for exp erimen tal design and inference, since underestimating the carry ov er horizon can lead to more serious distortions than mo dest o verestimation. 17 3.4 T esting non-an ticipation Sections 3.2 – 3.3 treat non-an ticipation as a maintained assumption. Here w e construct a ﬁnite- sample v alid test of this restriction. Because an ticipation concerns dep endence on futur e assign- men ts, constructing con venien t conditioning even ts for conditional randomization tests is diﬃcult without strong structural restrictions. W e therefore emplo y an unconditional randomization test based on the Pairwise Imputation–based Randomization T est (PIR T) of Zhong ( 2024 ), adapted to switc hbac k schedules. Imputable times. Under Assumption 2.1 , p oten tial outcomes at time t dep end only on the assignmen t preﬁx w 1: t . Th us if tw o schedules agree through time t , the corresp onding p oten tial outcomes must coincide. Deﬁnition 3.3 (Imputable times) . F or sche dules w , w ′ ∈ { 0 , 1 } T , deﬁne I ( w , w ′ ) =  t : w 1: t = w ′ 1: t  . Times in I ( w , w ′ ) are precisely those at whic h the non-an ticipation n ull implies equality of p oten tial outcomes across the t w o sc hedules and hence supp ort imputation-based inference. Preﬁx-preserving randomization. Indep enden tly drawn switch back schedules t ypically di- v erge quic kly , pro ducing few imputable times. T o ensure adequate ov erlap, w e restrict randomiza- tion to sc hedules that share an initial preﬁx with the observed assignment. Fix L ∈ { 0 , . . . , T } . Giv en W obs , alternativ e sc hedules W ∗ are drawn from the design distribu- tion sub ject to W ∗ 1: L = W obs 1: L . Assignmen ts after time L follo w the original design mec hanism. This guaran tees { 1 , . . . , L } ⊆ I ( W obs , W ∗ ) for ev ery dra w. 18 Cen tered signed-score statistic. Let s t ( w ) = 2 w t +1 − 1 for t < T , and deﬁne I ( w ′ , w ) = I ( w ′ , w ) ∩ { 1 , . . . , T − 1 } . Giv en observ ed outcomes Y obs = Y ( W obs ), deﬁne T NA  Y obs , w , w ′  = 1 | I ( w ′ , w ) | X t ∈ I ( w ′ , w ) s t ( w )  Y obs t − ¯ Y obs I ( w ′ , w )  , (9) where ¯ Y obs A = | A | − 1 P t ∈ A Y obs t , and T NA = + ∞ if I ( w ′ , w ) = ∅ . This statistic dep ends only on outcomes at imputable times and is therefore pairwise imputable under the non-an ticipation n ull. Centering remo ves sensitivit y to the o verall lev el of outcomes on the imputable set and ensures the statistic is driv en b y asso ciation betw een outcomes and the next-p eriod assignment sign. Under an ticipation alternatives where outcomes increase with future treatment, this asso ciation tends to b e p ositiv e, yielding p o wer while preserving ﬁnite-sample v alidity . PIR T pro cedure. Let W ∗ b e drawn from the preﬁx-preserving design. Deﬁne the pairwise statistic A = T NA ( Y obs , W ∗ , W obs ) , B = T NA ( Y obs , W obs , W ∗ ) . The PIR T p -v alue is p = P ( A ≥ B ) , with probabilit y tak en o ver the preﬁx-preserving randomization distribution. Two-sided v ariants use | A | ≥ | B | . Finite-sample v alidity . Under non-an ticipation, observed outcomes coincide with the corre- sp onding p oten tial outcomes at all imputable times for every pair ( W obs , W ∗ ). Because the statistic 19 dep ends only on imputable outcomes and the randomization distribution is known, the resulting test controls size exactly in ﬁnite samples under the conditional design. See Zhong ( 2024 ) for general v alidit y results. 4 T esting W eak-null Hyp otheses F rom this section on ward, w e sp ecialize to switch back designs in which the treatment is randomized o ver equal-length, naturally deﬁned design blo c ks (e.g., days or weeks); w e refer to these blo c ks as sessions 5 . This restriction is motiv ated b y common practice and b y in terpretabilit y: the w eak- n ull h yp otheses we consider are form ulated position-by-position within a session (e.g., Monda y vs. T uesday), whic h is meaningful only when the design blo c ks correspond to a stable time unit. The goal of this section is to clarify which w eak (mean-zero) null h yp otheses are testable using randomization-based inference in switc h back exp erimen ts with carry ov er. Because weak n ulls are not sharp, ﬁnite-sample exact CR Ts are generally una v ailable; instead, w e use studentized CR Ts to obtain asymptotic size con trol ( Ch ung and Romano , 2013 ; Li and Ding , 2017 ; DiCiccio and Romano , 2017 ; Zhao and Ding , 2021 ; W u and Ding , 2021 ). W e show that studen tized CR Ts can test: (i) a join t w eak n ull requiring mean-zero eﬀects at each within-session fo cal position, (ii) a w eak null requiring mean-zero av erage eﬀects ov er the fo cal perio ds at the end of eac h session, and (iii) p osition-wise nulls for an y individual fo cal p osition. W e also highlight an imp ortan t limitation: without further structure, we generally cannot test a global weak null that av erages treatmen t eﬀects ov er the entire time series. 5 W e use blo cks to denote the (p ossibly unequal-length) design interv als induced by the switc h times. In con trast, sessions are equal-length, naturally deﬁned blo c ks (used only in this section), while se ctions (or p ooled sections) are predetermined unions of consecutive blo c ks used for conditioning in our CR Ts. 20 4.1 W eak-null h yp otheses under ﬁxed-length sessions Fix a session length L and a carryo ver length m with L > m , and assume for simplicit y that T = J L for some in teger J ≥ 1. Sessions are deterministic in terv als [ s j , e j ] := { ( j − 1) L + 1 , . . . , j L } , j = 1 , . . . , J, and the switch back design assigns a constant treatmen t lab el Z j ∈ { 0 , 1 } to eac h session j with kno wn probabilit y p j := Pr( Z j = 1) ∈ (0 , 1). Under the main tained non-an ticipation and m - carry ov er restrictions, p eriods near the start of a session ma y dep end on the previous session’s assignmen t. Accordingly , w e focus on the within-session fo c al window at the end of eac h session, U j := { t ∈ [ T ] : s j + m ≤ t ≤ e j } , n := |U j | = L − m, and the ov erall focal set U := ∪ J j =1 U j . Index fo cal times in session j by ℓ = 1 , . . . , n via t j,ℓ := s j + m + ℓ − 1. F or z ∈ { 0 , 1 } , deﬁne the fo cal potential outcome at within-session p osition ℓ under the constant path z T b y Y j,ℓ ( z ) := Y t j,ℓ ( z T ) , z ∈ { 0 , 1 } . Under m -carry ov er and L > m , Y j,ℓ ( z ) can b e interpreted as the outcome at the ℓ th fo cal p osition in session j when the session is assigned z (the relev an t ( m + 1)-p erio d treatment history lies entirely within the session’s constant segmen t). Deﬁne the cross-session av erage eﬀect at within-session fo cal p osition ℓ as τ ℓ := 1 J J X j =1 { Y j,ℓ (1) − Y j,ℓ (0) } , ℓ = 1 , . . . , n. W e consider three empirically relev an t w eak-null h yp otheses: 21 (1) Joint session-wise w eak n ull (strong). The strongest fo cal w eak null requires mean-zero eﬀects at e ach within-session focal p osition: H sw 0 : τ ℓ = 0 for all ℓ = 1 , . . . , n. (10) This n ull is attractive in practice when within-session seasonalit y is imp ortan t (e.g., da y-of-w eek eﬀects), b ecause it rules out cancellation across positions. (2) F o cal-av erage w eak n ull (w eaker). Let ¯ Y j ( z ) := n − 1 P n ℓ =1 Y j,ℓ ( z ) denote the session-lev el mean p oten tial outcome ov er the fo cal window, and deﬁne the corresp onding focal-av erage eﬀect τ U := 1 J J X j =1 { ¯ Y j (1) − ¯ Y j (0) } . W e will also test the weak er n ull H U 0 : τ U = 0 . (11) Since τ U = n − 1 P n ℓ =1 τ ℓ , the strong joint n ull H sw 0 implies H U 0 , but not conv ersely . The null H U 0 is directly aligned with the focal set U used by our CR Ts and is therefore a natural target for studen tized randomization inference. (3) P osition-wise w eak n ulls. F or an y fo cal p osition ℓ ∈ { 1 , . . . , n } , w e can test H 0 ,ℓ : τ ℓ = 0 , and, more generally , test subsets of p ositions b y restricting atten tion to { τ ℓ : ℓ ∈ L} for any L ⊆ { 1 , . . . , n } . 22 Wh y the global w eak n ull is generally not testable. A common estimand under constan t paths is the global post–burn-in a v erage eﬀect ¯ Y m ( w ) := 1 T − m T X t = m +1 Y t ( w ) , τ glob m := ¯ Y m ( 1 T ) − ¯ Y m ( 0 T ) , and the corresp onding global null H glob 0 : τ glob m = 0. In switch back designs with carry ov er, how ever, v alid randomization-based inference is naturally tied to fo cal perio ds whose relev ant treatment histories are con tained within constan t segmen ts. A studen tized CR T built on the focal set U targets τ U ; under the global n ull τ glob m = 0, the fo cal-a verage eﬀect τ U need not b e zero (e.g., under within-session heterogeneity or sign-rev ersing patterns), so the randomization distribution of a fo cal statistic is not generally cen tered at zero and uniform v alidity cannot be guaranteed without additional structure linking fo cal and non-fo cal p erio ds. That said, when the outcome pro cess is appro ximately stationary across within-session p ositions and m is small relative to L (so n = L − m is close to L ), τ U can b e close to the global av erage eﬀect, making H U 0 a practically informativ e pro xy in many applications. 4.2 A studen tized CR T for the fo cal-a v erage null Let Z obs j ∈ { 0 , 1 } denote the realized session assignment and p j := Pr( Z j = 1) its known design probabilit y . Deﬁne the observed session-level fo cal mean ¯ Y obs j := 1 n X t ∈U j Y obs t . Under the ﬁxed-length session design and m -carry ov er, ¯ Y obs j = ¯ Y j ( Z obs j ) for eac h j , so { ¯ Y j (1) , ¯ Y j (0) } J j =1 are well-deﬁned session-level p oten tial outcomes for the fo cal window. W e estimate the fo cal-a v erage 23 eﬀect τ U using the Horvitz–Thompson (HT) estimator ˆ τ HT := 1 J J X j =1 Z obs j ¯ Y obs j p j − (1 − Z obs j ) ¯ Y obs j 1 − p j ! , (12) and studentize using the conserv ative upp er-bound form b V up := 1 J 2 J X j =1 ( ¯ Y obs j ) 2 Z obs j p 2 j + 1 − Z obs j (1 − p j ) 2 ! , T stud := ˆ τ HT q b V up . (13) The studentized CR T compares T obs stud to its randomization distribution obtained by resampling Z ∗ 1 , . . . , Z ∗ J indep enden tly with Z ∗ j ∼ Bernoulli( p j ) and recomputing T ∗ stud while holding { ¯ Y obs j } ﬁxed. The resulting one-sided Mon te Carlo p -v alue is denoted ˆ p w . (Here, the session partition and fo cal set are deterministic, so “conditioning” reduces to conditioning on a deterministic ev ent.) This pro cedure can b e view ed as the ﬁxed-length-session analogue of the CR T in Section 3.2 , with studen tization added to handle w eak n ulls. Assumption 4.1 (W eak-null asymptotics under ﬁxed-length sessions) . The carry ov er length m and session length L are ﬁxed with L > m , and T = J L → ∞ so that J → ∞ . There exists p ∈ (0 , 1 / 2) such that p ≤ p j ≤ 1 − p for all j and all T . Let ¯ Y j (1) and ¯ Y j (0) denote the session-level focal mean p oten tial outcomes ab ov e and deﬁne M j := max {| ¯ Y j (1) | , | ¯ Y j (0) |} . Assume: (i) ( Uniform fourth-moment b ound ) There exists C < ∞ suc h that sup J ≥ 1 1 J P J j =1 M 4 j ≤ C . (ii) ( Stabilization ) The empirical measures ν J := 1 J P J j =1 δ ( p j , ¯ Y j (1) , ¯ Y j (0)) con verge weakly to some probabilit y measure ν on [ p, 1 − p ] × R 2 . (iii) ( Nonde gener acy ) If ( P , Y 1 , Y 0 ) ∼ ν , then σ 2 S := E ν   r 1 − P P Y 1 + r P 1 − P Y 0 ! 2   > 0 . 24 Assumption 4.1 (ii) is a standard triangular-arra y stabilization condition in randomization CL Ts: it requires that as J → ∞ , the empirical distribution of session-lev el p oten tial outcomes (and as- signmen t probabilities) con verges to a stable limiting “p opulation.” This assumption is weak er than indep endence or stationarit y of the outcome process; it formalizes the idea that the experimental en vironment do es not drift arbitrarily as more sessions are observed. It rules out, for example, systematic time trends in treatment eﬀects or assignment probabilities, progressively heavier-tailed session means, or structural regime c hanges that make early and late sessions incomparable. Theorem 4.1 (Asymptotic v alidit y for the fo cal-av erage w eak n ull) . Supp ose Deﬁnition 2.1 holds with e qual-length sessions of length L > m , and Assumptions 2.1 and 2.2 hold. If Assumption 4.1 holds, then under H U 0 in ( 11 ) , lim sup T →∞ P ( ˆ p w ≤ α ) ≤ α for al l α ∈ (0 , 1) , wher e ˆ p w is the Monte Carlo p -value c ompute d fr om the r andomization distribution of T stud in ( 13 ) . In p articular, the same test is asymptotic al ly valid under the str onger joint nul l H sw 0 in ( 10 ) . 4.3 P osition-wise and join t tests across within-session p ositions The statistic T stud targets the focal-av erage eﬀect τ U and can therefore ha v e low p o wer against alternativ es where eﬀects v ary across within-session p ositions and cancel in the av erage (e.g., sign- rev ersing patterns). T o prob e heterogeneity and to test the strong join t n ull H sw 0 more directly , it is natural to use position-wise and joint tests. P osition-wise tests. Fix ℓ ∈ { 1 , . . . , n } and let Y obs j,ℓ := Y j,ℓ ( Z obs j ) denote the observ ed outcome at fo cal p osition ℓ in session j . Consider the HT estimator ˆ τ ℓ, HT := 1 J J X j =1 Z obs j Y obs j,ℓ p j − (1 − Z obs j ) Y obs j,ℓ 1 − p j ! , 25 and the upper-b ound v ariance estimator b V ℓ, up := 1 J 2 J X j =1 ( Y obs j,ℓ ) 2 Z obs j p 2 j + 1 − Z obs j (1 − p j ) 2 ! , T ℓ := ˆ τ ℓ, HT q b V ℓ, up . A studen tized CR T for the p osition-wise null H 0 ,ℓ : τ ℓ = 0 is obtained b y the same resampling sc heme: resample Z ∗ 1 , . . . , Z ∗ J indep enden tly with Z ∗ j ∼ Bernoulli( p j ), hold { Y obs j,ℓ } J j =1 ﬁxed, and recompute T ∗ ℓ . The resulting Monte Carlo p -v alue is asymptotically v alid under the same conditions as Theorem 4.1 . The same construction applies to any subset of fo cal p ositions b y restricting ℓ to a set L . Join t tests via quadratic-form ( F -type) statistics. T o test the strong joint n ull H sw 0 (or, more generally , τ ℓ = 0 for all ℓ ∈ L for a subset L ), one can com bine the v e ctor of position-wise eﬀect estimators using an F -type quadratic form. Let Y obs j := ( Y obs j, 1 , . . . , Y obs j,n ) ⊤ and deﬁne the v ector HT estimator b τ HT := ( ˆ τ 1 , HT , . . . , ˆ τ n, HT ) ⊤ . A natural matrix analogue of ( 13 ) is b Σ up := 1 J 2 J X j =1 Y obs j ( Y obs j ) ⊤ Z obs j p 2 j + 1 − Z obs j (1 − p j ) 2 ! , and an omnibus statistic is T F := b τ ⊤ HT b Σ − 1 up b τ HT , with the conv ention that a generalized inv erse ma y b e used when needed (e.g., when n is large relativ e to J ). A randomization p -v alue is obtained b y recomputing T ∗ F under each resampled assignmen t v ector Z ∗ = ( Z ∗ 1 , . . . , Z ∗ J ). Such quadratic-form tests are standard in randomization inference for m ultiv ariate outcomes; see, e.g., Ding and Dasgupta ( 2017 ). These joint tests are t ypically m uc h more sensitive than fo cal av erages to structured or sign-rev ersing alternativ es. F ormal asymptotic v alidity results for the p osition-wise and join t studentized CR Ts describ ed ab o v e are stated and pro ved in App endix D.7 . 26 5 P o w er analysis under a sup erp opulation mo del Sections 3 and 4 establish v alidity of our randomization tests under a ﬁnite-p opulation framework. This section studies pow er under a sup erpopulation model for the p oten tial-outcome time series. The goal is to obtain in terpretable approximations that clarify ho w blo c k length and predetermined p ooling en ter the signal-to-noise ratio of the studen tized CR Ts. Similarly to Section 4 , we consider a simple setup where the design blo cks hav e a ﬁxed length and are longer than the carry o ver memory length m . 6 F ormally , ﬁx an exp erimen t length T and a blo c k length L ≥ 1 such that T = M L for some integer M . Deﬁne design blo c ks B k := { ( k − 1) L + 1 , . . . , k L } , k = 1 , . . . , M . Assume a regular switch back design with constan t assignmen t probabilit y q ∈ (0 , 1): W ( k ) ind ∼ Bernoulli( q ) , W t = W ( k ) for all t ∈ B k . (14) Fix a po oling size r ≥ 1 and assume r | M . Set J := M /r and deﬁne predetermined po oled sections [ s j , e j ] := j r [ k =( j − 1) r +1 B k = { ( j − 1) r L + 1 , . . . , j r L } , j = 1 , . . . , J, eac h of length rL . Let E j denote the ev ent that p o oled section j is blockwise constant: E j = { W (( j − 1) r +1) = · · · = W ( j r ) } . On E j , let Z j denote the common blo c k v alue. Lemma 5.1 (Conditional p o oled-section la w under predetermined p o oling) . Under ( 14 ) , for e ach 6 In this section, we use the term “blo c k” for ﬁxed-length design interv als rather than “sessions,” since blocks need not correspond to naturally o ccurring time units (e.g., days or weeks) to supp ort meaningful w eak null hypotheses as in Section 4 . 27 j , P ( Z j = 1 | E j ) = p ( r ; q ) := q r q r + (1 − q ) r . Mor e over, acr oss disjoint se ctions the p airs { ( E j , Z j ) } ar e indep endent, and henc e c onditional on { E j : j ∈ J ( W ) } the lab els { Z j : j ∈ J ( W ) } ar e indep endent Bernoulli( p ( r ; q )) . Fix a burn-in parameter m ≥ 0 and assume rL > m . Each p ooled section con tributes n := r L − m (15) fo cal time p oints (the last n p erio ds in the po oled section). W e posit the distributed-lag mo del Y t ( w ) = µ + m 0 X ℓ =0 β ℓ w t − ℓ + ε t , t = 1 , . . . , T , (16) where m 0 ≥ 0 is the true carry ov er horizon, µ ∈ R is a baseline level, and { ε t } is a stationary mean-zero pro cess indep enden t of the assignmen t mec hanism. F or closed-form expressions, assume AR(1) errors: ε t = ρ ε t − 1 + u t , u t iid ∼ (0 , σ 2 u ) , | ρ | < 1 , (17) with σ 2 ε := σ 2 u / (1 − ρ 2 ). Under constant paths, the total long-run eﬀect is τ tot := E [ Y t ( 1 T ) − Y t ( 0 T )] = m 0 X ℓ =0 β ℓ . The carryo v er n ull H m 0 corresp onds to β ℓ = 0 for all ℓ > m . Remark 5.1 (Asymptotic regime) . Throughout, w e consider M → ∞ with ( L, r , m, m 0 , q ) ﬁxed, so T = M L → ∞ and J = M /r → ∞ . F or the total-eﬀect test, the n umber of usable (constan t) 28 p ooled sections J tot := |J ( W ) | is random, where J ( W ) := { j : E j } ; under ( 14 ), J tot /J → π r ( q ) with π r ( q ) = q r + (1 − q ) r , and hence J tot → ∞ with high probabilit y . In the pro of, we ﬁrst deriv e a ﬁnite-p opulation CL T conditional on the realized p otential outcomes, then use the DGP only to approximate the resulting random v ariance functionals and obtain an unconditional normal ap- pro ximation. This is not a superp opulation CL T: the estimand remains sample-dep endent, and the DGP is in vok ed solely to mak e v ariance and design/pow er trade-oﬀs transparen t—not to redeﬁne the target of inference. T otal-eﬀect statistic. Conditional on the realized set of usable p o oled sections, let J tot = |J ( W ) | denote their num b er and write Z j ∈ { 0 , 1 } for the (common) p ooled-section assignment lab el on eac h j ∈ J ( W ). Let ¯ Y obs j denote the mean outcome o v er the n fo cal perio ds in p o oled section j . With constant p = p ( r ; q ) and equal n , the p o oled-section HT estimator equals ˆ τ HT := 1 J tot X j ∈J ( W ) Z j ¯ Y obs j p − (1 − Z j ) ¯ Y obs j 1 − p ! , (18) and we studentize using b V up := 1 J 2 tot X j ∈J ( W ) ( ¯ Y obs j ) 2  Z j p 2 + 1 − Z j (1 − p ) 2  , T tot := ˆ τ HT q b V up . (19) m -carry o ver statistic (paired predetermined p o oled sections). F ollowing the carry ov er test in Section 3.3 , w e use an alternating holdout: even p o oled sections provide outcomes, and the immediately preceding o dd po oled sections provide randomized labels. Let J e := ⌊ J / 2 ⌋ and consider the J e adjacen t pairs  [ s 2 j − 1 , e 2 j − 1 ] , [ s 2 j , e 2 j ]  for j = 1 , . . . , J e . F or each ev en p o oled 29 section 2 j , deﬁne the fo cal mean ¯ Y obs 2 j := 1 n e 2 j X t = s 2 j + m Y obs t , n := r L − m. Deﬁne the randomized label from the preceding o dd p ooled section as the assignmen t on its last blo c k: Z 2 j − 1 := W ((2 j − 1) r ) ∈ { 0 , 1 } . Under ( 14 ), Pr( Z 2 j − 1 = 1) = q and { Z 2 j − 1 } J e j =1 are indep enden t. W e estimate the tail carry ov er signal using the HT contrast ˆ δ m := 1 J e J e X j =1 Z 2 j − 1 ¯ Y obs 2 j q − (1 − Z 2 j − 1 ) ¯ Y obs 2 j 1 − q ! , (20) and studentize with the corresp onding upp er-bound form b V ( m ) up := 1 J 2 e J e X j =1 ( ¯ Y obs 2 j ) 2  Z 2 j − 1 q 2 + 1 − Z 2 j − 1 (1 − q ) 2  , T m := ˆ δ m q b V ( m ) up . (21) W e ﬁrst record the v ariance of the av erage of n consecutive AR(1) errors: σ 2 ¯ ε ( n, ρ ) := V ar 1 n n X i =1 ε i ! = σ 2 ε n 2 " n + 2 n − 1 X h =1 ( n − h ) ρ h # . (22) Prop osition 5.1 (Po w er for the total-eﬀect test) . Assume ( 14 ) , the pr e determine d p o ole d-se ction c onstruction, and the DGP ( 16 ) – ( 17 ) . Assume the analyst burn-in satisﬁes m ≥ m 0 so that fo c al me ans ar e unc ontaminate d by c arryover. L et φ tot b e the one-side d level- α CR T that r eje cts for lar ge T tot in ( 19 ) . Then, c onditional on J tot , P ( φ tot = 1 | J tot ) ≈ 1 − Φ  z 1 − α − µ tot ( J tot ) σ tot  , (23) 30 wher e p = p ( r ; q ) , n = r L − m , and µ tot ( J tot ) := τ tot √ J tot q σ 2 ¯ ε ( n,ρ ) p (1 − p ) + ( µ + τ tot ) 2 p + µ 2 1 − p , σ 2 tot := σ 2 ¯ ε ( n, ρ ) + { µ + (1 − p ) τ tot } 2 σ 2 ¯ ε ( n, ρ ) + pµ 2 + (1 − p )( µ + τ tot ) 2 . (24) Mor e over, 0 < σ 2 tot ≤ 1 and σ 2 tot → 1 under lo c al alternatives τ tot = o (1) . Prop osition 5.2 (Po w er for the m -carry ov er test) . Assume ( 14 ) , the pr e determine d p o ole d-se ction c onstruction, and the DGP ( 16 ) – ( 17 ) . Assume m 0 > m and m 0 ≤ r L (25) so that, after the m -p erio d burn-in, any lag le aving an even p o ole d se ction c an r e ach only into the imme diately pr e c e ding o dd p o ole d se ction. F or the c arryover test statistic ( 20 ) – ( 21 ) , only the assignment on the last design blo ck of the pr e c e ding o dd p o ole d se ction is use d as the r andomize d lab el. Deﬁne the c orr esp onding tail-signal functional δ m ( n ) := 1 n m 0 X ℓ = m +1 min { L, ℓ − m } β ℓ , n = r L − m, (26) and let φ m b e the one-side d level- α CR T that r eje cts for lar ge T m in ( 21 ) . Then, as J → ∞ , P ( φ m = 1) ≈ 1 − Φ  z 1 − α − µ m ( J ) σ m  , (27) wher e J e = ⌊ J / 2 ⌋ and µ m ( J ) := δ m ( n ) √ J e q v ( m ) up , σ 2 m := σ 2 δ v ( m ) up , (28) with π 1 = q , π 0 = 1 − q , and v ( m ) up := E [ ¯ Y (1) 2 ] π 1 + E [ ¯ Y (0) 2 ] π 0 , (29) 31 σ 2 δ := E  1 π 1 − 1  ¯ Y (1) 2 +  1 π 0 − 1  ¯ Y (0) 2 + 2 ¯ Y (1) ¯ Y (0)  . (30) Her e ¯ Y (1) and ¯ Y (0) denote the even-se ction fo c al me an under the c ounterfactual intervention that the last design block of the pr e c e ding o dd p o ole d se ction is set to 1 versus 0 , r esp e ctively (with al l other blo ck assignments and the err or pr o c ess fol lowing their original laws). P o oling more blocks (larger r ) increases the within-section fo cal sample size n = r L − m and reduces σ 2 ¯ ε ( n, ρ ) via within-section a v eraging, but it also reduces the n umber of po oled sections J = M /r (and hence the num b er of o dd–ev en pairs J e = ⌊ J / 2 ⌋ used b y the carry o ver diagnostic). F or the total-eﬀect test, usable p o oled sections m ust be constant; the selection probability is π r ( q ) = q r + (1 − q ) r , so J tot ∼ Binomial( J, π r ( q )) and E [ J tot ] = J π r ( q ). F or the carryo ver test based on ( 20 )–( 21 ), p ooling aﬀects p o wer primarily through the tail-signal functional δ m ( n ) in ( 26 ) and through the v ariance of the even-section fo cal mean: eac h lag co eﬃcien t β ℓ with ℓ > m con tributes to δ m ( n ) only through those fo cal times whose lag- ℓ exp osure falls in the last blo c k of the preceding o dd p o oled section, which yields w eights capp ed at L and a 1 /n dilution from a v eraging o ver n fo cal p eriods. Propositions 5.1 and 5.2 quantify how these eﬀects (signal dilution, v ariance reduction from larger n , and fewer pairs J e when r grows) jointly determine the resulting noncen trality parameter. Giv en pilot data (or pre-exp erimen t time series), one can estimate the noise parameters in ( 17 ) (e.g., ρ and σ 2 ε , hence σ 2 ¯ ε ( n, ρ )) and sp ecify a plausible eﬀect scale (e.g., τ tot for the total-eﬀect test or a tail proﬁle { β ℓ : ℓ > m } for the carryo v er test). F or eac h candidate r , compute n = r L − m and π r ( q ), approximate the distribution of J tot ∼ Binomial( J, π r ( q )), and plug these in to ( 23 )–( 27 ) (with J tot replaced by E [ J tot ] or av eraged o v er simulated dra ws of J tot ). A practical constraint is to a void o verly aggressive p ooling that yields too few usable sections (or too few informative odd sections), e.g., b y requiring E [ J tot ] and J e to exc eed minimal thresholds for stable studentization and accurate Mon te Carlo calibration. In Prop ositions 5.1 – 5.2 , experimental design enters p ow er through the inv erse-probability factors 32 1 /p and 1 / (1 − p ) (and, under cen tered outcomes and lo cal alternativ es, essentially through p (1 − p ) in the noise term), so highly im balanced designs can substan tially inﬂate the studen tization term and reduce the resulting noncentralit y parameter. When outcomes are approximately centered (so µ ≈ 0) and eﬀect sizes are modest, a balanced assignmen t p ≈ 1 / 2 is therefore t ypically near- optimal. More generally , practitioners can plug pilot estimates in to the pow er appro ximations ( 23 )–( 27 ) and select p (or, in the p o oled-section setup, the underlying blo c k-level fraction q , which determines p = p ( r ; q ) and E [ J tot ] = J π r ( q )) b y numerically maximizing the resulting p ow er proxy sub ject to op erational constraints (e.g., minimum allo cation rules). 6 Sim ulation This section ev aluates the ﬁnite-sample size control and p o w er properties of the prop osed testing pro cedures. W e study three classes of hypotheses: (i) the null of no total treatment eﬀect, (ii) the m -carry ov er restriction, and (iii) the no-an ticipation restriction. The sim ulation design is intended to assess both v alidity and p o wer in time-series en vironmen ts c haracterized by temp oral dependence and heavy-tailed disturbances. In this sim ulation study , we adopt a p oten tial-outcomes framew ork similar to Bo jinov et al. ( 2023 ), but allow for a richer stochastic structure in the disturbance term in order to stress-test ﬁnite-sample v alidit y . P otential outcomes are generated according to Y t ( w 1: T ) = µ + α t + T X s =1 δ ( t +1 − s ) w s + ε t α t 1 { w t − m : t ∈ { 0 m +1 , 1 m +1 }} , t = 1 , . . . , T . (31) Here µ is a constan t baseline lev el, α t is a deterministic time eﬀect, δ ( t +1 − s ) are non-stochastic carry ov er co eﬃcien ts go verning the eﬀect of treatmen t assignments at diﬀeren t lags, w s ∈ { 0 , 1 } denotes treatment assignmen t at time s , and ε t is a sto chastic disturbance. The indicator term allo ws the scale of the disturbance to dep end on whether treatmen t has remained constan t ov er 33 the most recen t m + 1 perio ds, thereb y generating treatmen t-history–dep enden t heteroskedasticit y . Diﬀeren t conﬁgurations of the carryo ver co eﬃcients corresp ond to diﬀerent null hypotheses and alternativ e eﬀect structures. Throughout, we set µ = 0 and α t = log t . The n umber of p eriods is T ∈ { 60 , 120 , 180 , 240 , 300 } , represen ting exp erimen tally realistic horizons at whic h asymptotic appro ximations ma y b e unreli- able. W e consider t wo disturbance speciﬁcations: 1. ε t i.i.d. ∼ N (0 , 1), 2. ε t i.i.d. ∼ t (1), representing hea vy-tailed shocks. W e ﬁx the treatmen t assignment mec hanism to the optimal regular switch bac k design of Bo jinov et al. ( 2023 ). Switc h points o ccur at times { 1 , 2 m + 1 , 3 m + 1 , . . . , ( n − 2) m + 1 } , with switc hing probability 1 / 2 at eac h decision p oin t. Holding this assignmen t rule ﬁxed across all conﬁgurations allows us to fo cus on the ﬁnite-sample b eha vior of the inference pro cedures rather than design c hoice. In all sim ulation experiments, w e implemen t the pre-determined sectioning rule describ ed in Section 3.2 . Concretely , giv en the design blo c ks { B k } K k =1 and carry ov er horizon m , we p o ol con- secutiv e blo c ks greedily un til the p ooled length is at least m + 1, pro ducing a disjoint co ver of [ T ]. 6.1 Null of no total treatmen t eﬀect T o ev aluate size and pow er under H tot 0 , w e ﬁx the carryo ver horizon at m = 2 and assume it is correctly speciﬁed in the analysis as in Bo jinov et al. ( 2023 ). W e set δ ( p ) = 0 for p / ∈ { 1 , 2 , 3 } and 34 imp ose δ (1) = δ (2) = δ (3) = δ, δ ∈ { 0 , 1 , 2 , 3 } . When δ = 0, the n ull ( 4 ) holds and empirical rejection frequencies measure test size. When δ > 0, rejection frequencies measure p o wer. Unlik e the sharp-null conﬁguration considered in Bo jinov et al. ( 2023 ), our sp eciﬁcation do es not imply a sharp null when δ = 0, allo wing a direct comparison betw een Fisher randomization tests and conditional randomization tests under a non- sharp null. F or eac h parameter conﬁguration, w e generate one treatment assignment path and compute observ ed outcomes from ( 31 ). W e then apply four testing pro cedures: 1. the prop osed conditional randomization test (CR T), 2. a Fisher randomization test imposed under a misspeciﬁed sharp nul l, 3. the unconditional PIR T pro cedure of Zhong ( 2024 ) with rejection threshold α , 4. the Horvitz–Thompson asymptotic test of Bo jinov et al. ( 2023 ). F or asymptotic inference, w e use the conserv ative v ariance upper b ound proposed in Bo jino v et al. ( 2023 ). All tests are conducted at nominal lev el α = 0 . 05 using the Horvitz–Thompson estimator. Eac h conﬁguration is replicated 1,000 times, and rejection rates are reported as Mon te Carlo av erages. Figure 1 rep orts rejection frequencies across sample sizes and error distributions. The CR T main tains size con trol across all conﬁgurations. In con trast, the Fisher randomization test exhibits substan tial ov er-rejection under hea vy-tailed disturbances, reﬂecting misspeciﬁcation of the sharp n ull when treatmen t eﬀects are not fully imputable. This pattern is consisten t with the theoretical results of Athey et al. ( 2018 ). Both the unadjusted PIR T at level α and the Horvitz–Thompson asymptotic test displa y conserv ative b eha vior in sev eral conﬁgurations. 35 Figure 1: Rejection F requencies Under the Null of no total treatmen t eﬀect Notes: Each panel rep orts Monte Carlo rejection frequencies based on 1,000 replications at nominal level α = 0 . 05. The solid horizontal line marks the nominal size. “CR T (exact)” denotes the prop osed conditional randomization test. “Asymptotic” denotes the Horvitz–Thompson test of Bo jino v et al. ( 2023 ). “FR T (misspec.)” denotes the Fisher randomization test imp osed under a sharp null. “PIR T ( α )” denotes the unconditional PIR T pro cedure ev aluated at lev el α . In terms of p ow er, the CR T p erforms competitively across all scenarios. Under Gaussian distur- bances, the CR T, Fisher test, and asymptotic test exhibit similar p o w er, while PIR T is somewhat less pow erful. Under heavy-tailed disturbances, the CR T ac hiev es the highest p o wer among pro- cedures that main tain v alid size. Although the Fisher test ma y app ear more pow erful in some conﬁgurations, this reﬂects size distortion rather than gen uine pow er gains. Ov erall, the results indicate that the prop osed CR T ac hieves reliable size con trol while main- taining strong pow er, and remains robust to heavy-tailed sho c ks and ﬁnite-sample dep endence. 6.2 T ests for m -carry o ver and an ticipation This section ev aluates the ﬁnite-sample size and pow er of the t wo diagnostic pro cedures proposed ab o v e. W e study the m -carryo v er restriction using the conditional randomization test (CR T), and 36 the non-anticipation restriction using the PIR T pro cedure. Carry o ver diagnostic. T o assess the m -carry ov er test, w e ﬁx the true carry ov er horizon at m = 2 and ev aluate the n ull that treatmen t eﬀects v anish b ey ond this horizon. The data-generating pro cess in tro duces carryo ver eﬀects strictly beyond the tested horizon by setting δ ( p ) = 0 for p / ∈ { 4 , 5 , 6 } , δ (4) = δ (5) = δ (6) = δ, with δ ∈ { 0 , 1 , 2 , 3 } . Th us δ = 0 corresponds to the null of no carryo v er b ey ond m = 2, while δ > 0 generates violations of the n ull. Inference is conducted using the proposed CR T. Empirical rejection frequencies therefore measure size when δ = 0 and p o wer when δ > 0. 7 An ticipation diagnostic. T o ev aluate the non-an ticipation test, we in tro duce dep endence on future treatment assignments b y setting δ ( p ) = 0 for p / ∈ {− 2 , − 1 , 0 } , δ ( − 2) = δ ( − 1) = δ (0) = δ, again with δ ∈ { 0 , 1 , 2 , 3 } . The null of non-anticipation holds when δ = 0 and is violated when δ > 0. Inference is conducted using the prop osed PIR T procedure with preﬁx-preserving randomization and holdout length L = 2 m + 1 = 5. Figure 2 rep orts rejection frequencies across sample sizes and error distributions for the carry ov er diagnostic. The CR T maintains accurate size con trol across all conﬁgurations. P ow er increases with the magnitude of the violation and with the sample size, reac hing approximately 60% in the largest designs under Gaussian disturbances. Figure 3 rep orts rejection frequencies for the non-anticipation diagnostic using the PIR T pro- 7 F or both carryo ver and an ticipation diagnostics we use the same outcome mo del. W e do not compare directly with the metho d of Bo jinov et al. ( 2023 ) (Section 4.4), since their approach requires a diﬀerent exp erimen tal design tailored to each hypothesis. 37 Figure 2: Rejection F requencies Under the Carryo v er Null and Alternativ es Notes: Each panel rep orts Monte Carlo rejection frequencies based on 1,000 replications at nominal level α = 0 . 05. The solid horizontal line marks the nominal size. Figure 3: Rejection F requencies Under the Anticipation Null and Alternativ es Notes: Each panel rep orts Monte Carlo rejection frequencies based on 1,000 replications at nominal level α = 0 . 05. The solid horizontal line marks the nominal size. 38 cedure. Size control remains close to nominal across designs. Po wer increases with the magnitude of anticipation eﬀects but do es not v ary monotonically with the total horizon, b ecause the eﬀective sample size is driven by the num b er of imputable times—i.e., sto c hastic o verlaps of preﬁxes—rather than b y T p er se. Unlike carry ov er testing, where fo cal sets are deterministic given the design, an- ticipation testing inherits additional ﬁnite-sample v ariabilit y from this random o verlap structure, so the count of imputable times can remain limited ev en as T grows, leading to irregular p o wer patterns across realizations. Ov erall, the results indicate that the proposed pro cedures pro vide reliable ﬁnite-sample size con trol and non trivial p o w er in realistic time-series environmen ts. A more detailed theoretical analysis of pow er for PIR T-based diagnostics in dep endent time-series settings is left for future w ork. 7 Conclusion W e study inference for switc hbac k exp eriments run ov er realistic operational horizons, where se- rial dep endence, hea vy-tailed shocks, and temporal in terference can undermine classical appro x- imations. Our contribution is a randomization-test framew ork that yields ﬁnite-sample v alid, distribution-free p -v alues using only the kno wn assignmen t mechanism and t w o primitiv e condi- tions on p otential outcomes: non-an ticipation and a ﬁnite carryo ver horizon. W e develop random- ization tests for total eﬀects alongside diagnostics for carry ov er length and non-an ticipation. F or a verage-eﬀect targets, w e develop studentized CR Ts that accommo date within-session seasonalit y and establish asymptotic v alidity under mild conditions. W e complemen t the theory with p o wer approximations under distributed-lag eﬀects with AR(1) noise, whic h highligh t blo c k length, burn-in, and section po oling as ﬁrst-order design lev ers for sensitivit y . Monte Carlo evidence is consistent with these insights, indicating reliable ﬁnite-sample p erformance of the prop osed tests across realistic settings. 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Zhao, A. and Ding, P. (2021). Cov ariate-adjusted ﬁsher randomization tests for the a v erage treatmen t eﬀect. Journal of Ec onometrics , 225 278–294. 43 Zhong, L. (2024). Unconditional randomization tests for interference. arXiv pr eprint arXiv:2409.09243 . 44 A Additional Results A.1 A regression-based studentized statistic for the w eak n ull In practice, switch back analyses are often rep orted through a regression coeﬃcient and its robust standard error. Because assignmen t probabilities are kno wn b y design, w e can deﬁne a regression- based studen tized statistic that is n umerically identical to a weigh ted least squares coeﬃcient/SE pair and can be used inside the same CR T resampling scheme. This provides a conv enient imple- men tation of a studen tized CR T for the fo cal-a verage n ull H U 0 . Deﬁne the in verse-probabilit y w eight ω obs j := Z obs j p j + 1 − Z obs j 1 − p j , j = 1 , . . . , J . Consider the w eighted regression of ¯ Y obs j on an in tercept and Z obs j : ( ˆ α reg , ˆ τ reg ) := arg min α,τ ∈ R J X j =1 ω obs j  ¯ Y obs j − α − τ Z obs j  2 . (32) Deﬁne the IPW-normalized weigh ted means ˆ µ 1 := P J j =1 Z obs j ¯ Y obs j p j P J j =1 Z obs j p j , ˆ µ 0 := P J j =1 (1 − Z obs j ) ¯ Y obs j 1 − p j P J j =1 (1 − Z obs j ) 1 − p j . (33) Then ˆ τ reg = ˆ µ 1 − ˆ µ 0 . Let ˆ r j, 1 := ¯ Y obs j − ˆ µ 1 for Z obs j = 1 and ˆ r j, 0 := ¯ Y obs j − ˆ µ 0 for Z obs j = 0. The heteroskedasticit y- robust (HC0) v ariance simpliﬁes to b V reg := P j : Z obs j =1 ˆ r 2 j, 1 p 2 j  P J j =1 Z obs j p j  2 + P j : Z obs j =0 ˆ r 2 j, 0 (1 − p j ) 2  P J j =1 1 − Z obs j 1 − p j  2 . (34) 45 Deﬁne the regression-based studentized statistic T reg := ˆ τ reg q b V reg . (35) Using T reg as the test statistic inside the same Monte Carlo resampling scheme yields a practical alternativ e to T stud . In particular, the regression-based v ariance uses within-arm residual v ariation and can be less sensitive to large baseline outcome lev els. Corollary A.1 (Asymptotic v alidity of the regression-based w eak-null test) . Under the ﬁxe d-length session setup, Assumptions 2.1 and 2.2 , and Assumption 4.1 (with a mild additional nonde gener acy c ondition), the Monte Carlo p -value b ase d on T reg satisﬁes, under H U 0 in ( 11 ) , lim sup T →∞ P ( ˆ p reg ≤ α ) ≤ α, for al l α ∈ (0 , 1) . B Pro of of Theorem 3.1 W e prov e Theorem 3.1 b y verifying the imputabilit y condition for our test statistic under the con- ditioning even t C ( W ) = ( U ( W ) , W  S ( W )  ), and the inv ariance prop erty of the conditioning even t under fo cal resampling. Then, w e apply the general v alidity result for conditional randomization tests in Basse et al. ( 2019 ). Let W obs denote the realized assignment and let C obs := C ( W obs ) =  U ( W obs ) , W  S ( W obs )  . Let W ( C obs ) denote the conditional randomization space W ( C obs ) := { w ∈ { 0 , 1 } T : C ( w ) = C obs } . 46 Algorithm 1 samples W ( b ) from the conditional assignment distribution W | {C ( W ) = C obs } and computes the Mon te Carlo p -v alue based on the realized statistic T obs := T  { Y obs t } t ∈U ( W obs ) , W obs  . Throughout, under the sharp n ull H 0 , the full sc hedule of potential outcomes is ﬁxed (non- sto c hastic) and the only randomness is through W . W e use the notion of imputability from Basse et al. ( 2019 ). In our setting, it suﬃces to show that, under H 0 , the v alue of the statistic T ( { Y obs t } t ∈U ( W obs ) , w ) can be computed for an y w ∈ W ( C obs ) using the observ ed outcomes, i.e., without requiring unobserved p oten tial outcomes. Fix w ∈ W ( C obs ). By deﬁnition of W ( C obs ), U ( w ) = U ( W obs ) =: U obs . Therefore, the statistic under w only depends on outcomes at time points t ∈ U obs . No w take any t ∈ U obs . By construction of U ( W ) in ( 5 ), there exists a section [ s, e ] ∈ S ( W obs ) suc h that s + m ≤ t ≤ e . Hence the last m + 1 assignmen ts are constan t within that section: W obs t = W obs t − 1 = · · · = W obs t − m . Because w ∈ W ( C obs ) implies S ( w ) = S ( W obs ), the corresponding section [ s, e ] is also a section under w , and therefore w t = w t − 1 = · · · = w t − m . Under Assumptions 2.1 and 2.2 , for an y assignmen t path ˜ w , the potential outcome at time t dep ends only on the ( m + 1)-length history ( ˜ w t − m , . . . , ˜ w t ). Under H tot 0 , w e hav e equality of 47 p oten tial outcomes for the constan t histories, Y t ( 1 m +1 ) = Y t ( 0 m +1 ) for all t ∈ [ T ] . Since b oth W obs and w generate a constan t ( m + 1)-length history at time t , it follo ws that Y t ( w ) = Y t  W obs  . In other w ords, for every focal time t ∈ U obs , the p oten tial outcome under w equals the observ ed outcome Y obs t : Y t ( w ) = Y obs t for all t ∈ U obs , w ∈ W ( C obs ) . Consequen tly , for any w ∈ W ( C obs ), the statistic v alue T  { Y t ( w ) } t ∈U obs , w  is equal to T  { Y obs t } t ∈U obs , w  , whic h is computable from the observ ed outcomes on U obs and the h yp othesized assignmen t v ector w . This v eriﬁes imputability of the test statistic under the conditioning ev en t C ( W ) = C obs in the sense of Basse et al. ( 2019 ). The in v ariance prop ert y is also v eriﬁed by noting that U ( w ) = U ( W obs ) and S ( w ) = S ( W obs ) for all w ∈ W ( C obs ). Giv en imputabilit y under H 0 , Theorem 1 (and the asso ciated Monte Carlo implementation re- sult) of Basse et al. ( 2019 ) implies that the conditional randomization test that samples assignmen ts from W | {C ( W ) = C obs } yields a v alid p -v alue: for all α ∈ (0 , 1), P  ˆ p ≤ α    C ( W ) = C obs  ≤ α, 48 whic h is exactly the claim of Theorem 3.1 . B.1 Conditional assignmen t la w on realized constan t merged sections This subsection v eriﬁes that Algorithm 1 samples from the correct conditional assignment distri- bution under the regular switc hbac k design, conditional on the realized section structure used for the test. Pr o of of L emma 3.1 . Fix a merged section that cov ers block indices a, . . . , b . By assumption, the blo c k assignments are independent with W ( k ) ∼ Bernoulli( q k ). Deﬁne the ev ent that the section is constan t at the block level: E a : b := { W ( a ) = · · · = W ( b ) } . On E a : b there are exactly tw o p ossibilities: either all blocks equal 1 or all blocks equal 0. Let A a : b := { W ( a ) = · · · = W ( b ) = 1 } , B a : b := { W ( a ) = · · · = W ( b ) = 0 } . Then E a : b = A a : b ˙ ∪ B a : b is a disjoin t union. By indep endence, P ( A a : b ) = b Y k = a q k , P ( B a : b ) = b Y k = a (1 − q k ) , and therefore P ( E a : b ) = b Y k = a q k + b Y k = a (1 − q k ) . Let Z denote the common v alue on the section under E a : b , i.e. Z := W ( a ) on E a : b . Then P ( Z = 1 | E a : b ) = P ( A a : b ) P ( E a : b ) = Q b k = a q k Q b k = a q k + Q b k = a (1 − q k ) . 49 Hence, conditional on E a : b , the section’s common assignmen t is Bernoulli with probabilit y p = Q b k = a q k Q b k = a q k + Q b k = a (1 − q k ) , and the en tire block v ector ( W ( a ) , . . . , W ( b ) ) equals ( Z, . . . , Z ). W e now pro ve the claimed indep endence across disjoint sections. Let sections j = 1 , . . . , J b e disjoin t at the block lev el, where section j cov ers blo cks a j , . . . , b j , and deﬁne E j := { W ( a j ) = · · · = W ( b j ) } , Z j := W ( a j ) on E j . Because the underlying blo c ks are indep endent and the collections of indices { a j , . . . , b j } are disjoin t across j , the random v ectors { W ( a j ) , . . . , W ( b j ) } are mutually independent, and the ev en ts { E j } are also m utually independent. F or an y ( z 1 , . . . , z J ) ∈ { 0 , 1 } J , deﬁne the even t F ( z 1 , . . . , z J ) := J \ j =1 n W ( a j ) = · · · = W ( b j ) = z j o . The even ts { F ( z 1 , . . . , z J ) } are disjoin t and partition T J j =1 E j . Using indep endence across disjoin t sections, P ( F ( z 1 , . . . , z J )) = J Y j =1     b j Y k = a j q k   z j   b j Y k = a j (1 − q k )   1 − z j   . Moreo ver, P   J \ j =1 E j   = J Y j =1   b j Y k = a j q k + b j Y k = a j (1 − q k )   . Therefore, P ( Z 1 = z 1 , . . . , Z J = z J | ∩ J j =1 E j ) = P ( F ( z 1 , . . . , z J )) P ( ∩ J j =1 E j ) 50 = J Y j =1  Q b j k = a j q k  z j  Q b j k = a j (1 − q k )  1 − z j  Q b j k = a j q k  +  Q b j k = a j (1 − q k )  . This factorization shows that, conditional on ∩ J j =1 E j , the common v alues Z 1 , . . . , Z J are indepen- den t, with Z j ∼ Bernoulli( p j ) where p j = Q b j k = a j q k Q b j k = a j q k + Q b j k = a j (1 − q k ) . This completes the pro of. Connection to Algorithm 1 . In Algorithm 1 , each realized section [ s j , e j ] corresponds to merg- ing blocks a j , . . . , b j and imposing the constrain t W ( a j ) = · · · = W ( b j ) . The lemma abov e sho ws that, conditional on this constraint, the section’s common v alue is Bernoulli( p j ) with p j as com- puted in the algorithm, and that these s ection-lev el v alues are indep enden t across disjoint sections. Since p eriods outside the union of realized sections are held ﬁxed to W obs in the conditional ran- domization space ( 6 ), Algorithm 1 samples exactly from the conditional assignment distribution used by the conditional randomization test. C Pro of for Section 3.3 C.1 Pro of of Prop osition 3.1 Proof . Fix an y m ≥ ¯ m . T ake any time t ≥ m + 1, any assignment vector w t − m : t for the most recen t m + 1 p erio ds, and an y tw o histories w ′ 1: t − m − 1 and w ′′ 1: t − m − 1 that share this same w t − m : t . Since m ≥ ¯ m , agreemen t on the last m + 1 assignmen ts implies agreement on the last ¯ m + 1 assignmen ts. Because H ¯ m 0 is true, Y t ( w ) depends only on the last ¯ m + 1 treatmen ts. Therefore, Y t ( w ′ 1: t − m − 1 , w t − m : t ) = Y t ( w ′′ 1: t − m − 1 , w t − m : t ) , 51 whic h is the deﬁning condition for H m 0 . C.2 Pro of of Theorem 3.2 Proof . Let ¯ m b e the smallest index for which H ¯ m 0 is true. By Proposition 3.1 , all H m 0 with m < ¯ m are false and all H m 0 with m ≥ ¯ m are true. A family-wise error o ccurs if and only if the algorithm rejects at least one true null. Under nestedness, if any true n ull is rejected, then the ﬁrst true null H ¯ m 0 is rejected. Moreov er, Algorithm 3 tests H ¯ m 0 only after rejecting all H 0 0 , . . . , H ¯ m − 1 0 . Therefore, FWER = Pr(reject some true H m 0 ) = Pr(reject H ¯ m 0 ) ≤ α, where the last inequality follo ws because H ¯ m 0 is tested at level α . D W eak-n ull theory for ﬁxed-length sessions: pro ofs Throughout this appendix, we work under the ﬁxed-length session setup of Section 4 . Fix L > m and T = J L . Sessions are [ s j , e j ] = { ( j − 1) L + 1 , . . . , j L } , and the focal window within session j is U j = { t : s j + m ≤ t ≤ e j } with n = |U j | = L − m . Let Z j ∈ { 0 , 1 } be the (constan t) treatment assignment on session j , with design probability p j := P ( Z j = 1) ∈ (0 , 1). F or eac h z ∈ { 0 , 1 } deﬁne the session-lev el mean p oten tial outcome ¯ Y j ( z ) := 1 n X t ∈U j Y t ( z T ) , ¯ Y obs j := ¯ Y j ( Z j ) . Deﬁne the HT estimator and upp er-bound v ariance estimator as in ( 12 )–( 13 ): ˆ τ HT = 1 J J X j =1 Z j ¯ Y obs j p j − (1 − Z j ) ¯ Y obs j 1 − p j ! , b V up = 1 J 2 J X j =1 ( ¯ Y obs j ) 2 Z j p 2 j + 1 − Z j (1 − p j ) 2 ! , 52 and T stud = ˆ τ HT / q b V up . D.1 Session-lev el randomization Lemma D.1 (Independent Bernoulli assignment on sessions) . Under Deﬁnition 2.1 with switch times aligne d to the session b oundaries ( [ s j , e j ] = B j − 1 and | B j − 1 | = L ), the session indic ators Z 1 , . . . , Z J ar e indep endent with Z j ∼ Bernoulli( p j ) , wher e p j e quals the c orr esp onding blo ck pr ob- ability (i.e. p j = q j − 1 ). Proof . With switc h times aligned to session b oundaries, each session is exactly one design blo c k. Under Deﬁnition 2.1 , blo c k-level assignmen ts are independent Bernoulli with the speciﬁed block probabilities, and Z j is the block assignment for session j . D.2 Randomization CL T for the studen tized statistic F or the randomization distribution, condition on the observ ed fo cal means { ¯ Y obs j } J j =1 and dra w Z ∗ 1 , . . . , Z ∗ J indep enden tly with Z ∗ j ∼ Bernoulli( p j ). Deﬁne ˆ τ ∗ HT := 1 J J X j =1 Z ∗ j ¯ Y obs j p j − (1 − Z ∗ j ) ¯ Y obs j 1 − p j ! , b V ∗ up := 1 J 2 J X j =1 ( ¯ Y obs j ) 2 Z ∗ j p 2 j + 1 − Z ∗ j (1 − p j ) 2 ! , and T ∗ stud := ˆ τ ∗ HT / q b V ∗ up . Lemma D.2 (Randomization CL T) . Under Assumption 4.1 , c onditional on { ¯ Y obs j } J j =1 , T ∗ stud ⇒ N (0 , 1) , in P -pr ob ability (wher e P r efers to the original assignment me chanism gener ating { ¯ Y obs j } ). Proof . Condition on the realized fo cal means { ¯ Y obs j } J j =1 and draw Z ∗ 1 , . . . , Z ∗ J indep enden tly with 53 Z ∗ j ∼ Bernoulli( p j ). W rite ˆ τ ∗ HT = 1 J J X j =1 ξ ∗ j,J , ξ ∗ j,J := ¯ Y obs j  Z ∗ j p j − 1 − Z ∗ j 1 − p j  . Then { ξ ∗ j,J } are independent and mean zero conditional on { ¯ Y obs j } , with V ar  ξ ∗ j,J    { ¯ Y obs j }  = ( ¯ Y obs j ) 2 p j (1 − p j ) . Let s 2 J := J X j =1 ( ¯ Y obs j ) 2 p j (1 − p j ) . Step 1: sho w s 2 J /J → σ 2 R in probability for some σ 2 R ∈ (0 , ∞ ) . Under the original assignment, ¯ Y obs j = ¯ Y j ( Z j ) with Z j ∼ Bernoulli( p j ) indep enden t. Hence E " ( ¯ Y obs j ) 2 p j (1 − p j ) # = ¯ Y j (1) 2 1 − p j + ¯ Y j (0) 2 p j . Deﬁne the (deterministic) randomization-v ariance pro xy σ 2 R ( J ) := 1 J J X j =1  ¯ Y j (1) 2 1 − p j + ¯ Y j (0) 2 p j  . By Assumption 4.1 (ii) and the fact that the in tegrand is a con tinuous function of ( p, y 1 , y 0 ) with at most quadratic growth (and p is b ounded aw a y from 0 and 1), σ 2 R ( J ) → σ 2 R := E ν  Y 2 1 1 − P + Y 2 0 P  ∈ [0 , ∞ ) . Moreo ver, Assumption 4.1 (iii) implies ν is not supp orted on ( Y 1 , Y 0 ) = (0 , 0), hence σ 2 R > 0 (since the integrand is strictly p ositiv e whenever ( Y 1 , Y 0 )  = (0 , 0)). 54 Next, since p j ∈ [ p, 1 − p ], ( ¯ Y obs j ) 2 p j (1 − p j ) ! 2 ≤ C p ( ¯ Y obs j ) 4 ≤ C p M 4 j for a constan t C p < ∞ . Therefore, V ar  s 2 J J  = 1 J 2 J X j =1 V ar ( ¯ Y obs j ) 2 p j (1 − p j ) ! ≤ 1 J 2 J X j =1 E   ( ¯ Y obs j ) 2 p j (1 − p j ) ! 2   ≤ C p J 2 J X j =1 M 4 j = O (1 /J ) → 0 , b y Assumption 4.1 (i). Hence s 2 J /J − E [ s 2 J /J ] → 0 in probabilit y , and since E [ s 2 J /J ] = σ 2 R ( J ) → σ 2 R > 0, w e conclude s 2 J /J → σ 2 R in probability . In particular, s 2 J = Θ P ( J ). Step 2: Lyapuno v condition (conditional) and CL T for P ξ ∗ j,J . Conditional on { ¯ Y obs j } , E h ( ξ ∗ j,J ) 4    { ¯ Y obs j } i = ( ¯ Y obs j ) 4 1 p 3 j + 1 (1 − p j ) 3 ! ≤ C ′ p ( ¯ Y obs j ) 4 for some ﬁnite constant C ′ p . Also, 1 J J X j =1 ( ¯ Y obs j ) 4 ≤ 1 J J X j =1 M 4 j = O (1) b y Assumption 4.1 (i), so P J j =1 ( ¯ Y obs j ) 4 = O P ( J ). Therefore the Ly apunov ratio satisﬁes 1 s 4 J J X j =1 E h ( ξ ∗ j,J ) 4    { ¯ Y obs j } i ≤ C ′ p s 4 J J X j =1 ( ¯ Y obs j ) 4 = O P (1 /J ) → 0 . Th us, with probability tending to one, the Lyapuno v condition holds and the Lyapuno v CL T yields P J j =1 ξ ∗ j,J s J ⇒ N (0 , 1) (conditionally on { ¯ Y obs j } ) . 55 Step 3: studentization by b V ∗ up . Recall b V ∗ up = 1 J 2 J X j =1 ( ¯ Y obs j ) 2 Z ∗ j p 2 j + 1 − Z ∗ j (1 − p j ) 2 ! . Conditional on { ¯ Y obs j } , E h J b V ∗ up    { ¯ Y obs j } i = 1 J J X j =1 ( ¯ Y obs j ) 2 p j (1 − p j ) = s 2 J J . A direct v ariance b ound (using p j ∈ [ p, 1 − p ]) gives V ar  J b V ∗ up    { ¯ Y obs j }  ≤ C ′′ p J 2 J X j =1 ( ¯ Y obs j ) 4 , so V ar J b V ∗ up s 2 J /J      { ¯ Y obs j } ! ≤ C · P J j =1 ( ¯ Y obs j ) 4 s 4 J → 0 in P -probabilit y . Hence J b V ∗ up / ( s 2 J /J ) → 1 in probability , and Slutsky’s theorem yields T ∗ stud ⇒ N (0 , 1) in P - probabilit y . D.3 Sampling CL T for the observ ed statistic Deﬁne the session-a veraged target τ C := 1 J J X j =1 { ¯ Y j (1) − ¯ Y j (0) } . Under Assumptions 2.1 and 2.2 , fo cal outcomes dep end only on the constant within-session assign- men t, so ¯ Y obs j = ¯ Y j ( Z j ). 56 Lemma D.3 (Sampling CL T for the HT estimator) . Under Assumption 4.1 , √ J  ˆ τ HT − τ C  ⇒ N (0 , σ 2 S ) , wher e σ 2 S is deﬁne d in Assumption 4.1 (iii). Proof . W rite ˆ τ HT − τ C = 1 J J X j =1 ψ j,J , where ψ j,J :=  Z j p j − 1  ¯ Y j (1) −  1 − Z j 1 − p j − 1  ¯ Y j (0) . Then { ψ j,J } J j =1 are indep enden t and mean zero (conditioning on the ﬁxed potential outcomes). A direct computation gives V ar( ψ j,J ) = 1 − p j p j ¯ Y j (1) 2 + p j 1 − p j ¯ Y j (0) 2 + 2 ¯ Y j (1) ¯ Y j (0) = s 1 − p j p j ¯ Y j (1) + r p j 1 − p j ¯ Y j (0) ! 2 . Let S 2 J := P J j =1 V ar( ψ j,J ) and deﬁne σ 2 S ( J ) := S 2 J J = 1 J J X j =1 s 1 − p j p j ¯ Y j (1) + r p j 1 − p j ¯ Y j (0) ! 2 . By Assumption 4.1 (ii) and con tin uity/quadratic gro wth of the in tegrand, σ 2 S ( J ) → E ν   r 1 − P P Y 1 + r P 1 − P Y 0 ! 2   = σ 2 S . By Assumption 4.1 (iii), σ 2 S > 0, hence S 2 J = Θ( J ). 57 Ly apuno v condition. Since p j ∈ [ p, 1 − p ], there exists C p < ∞ such that | ψ j,J | ≤ C p M j , hence E [ ψ 4 j,J ] ≤ C 4 p M 4 j . Therefore 1 S 4 J J X j =1 E [ ψ 4 j,J ] ≤ C 4 p S 4 J J X j =1 M 4 j . By Assumption 4.1 (i), P J j =1 M 4 j = O ( J ), while S 4 J = Θ( J 2 ), so the right-hand side is O (1 /J ) → 0. Th us the Lyapuno v condition (with δ = 2) holds and the Ly apunov CL T giv es P J j =1 ψ j,J S J ⇒ N (0 , 1) . Finally , since √ J ( ˆ τ HT − τ C ) = ( P J j =1 ψ j,J ) / √ J and S 2 J /J → σ 2 S , we obtain √ J  ˆ τ HT − τ C  = P J j =1 ψ j,J S J !  S J √ J  ⇒ N (0 , σ 2 S ) . D.4 Upp er-b ound v ariance and consistency Lemma D.4 (Upp er bound on the session-level sampling v ariance) . F or e ach j and any p j ∈ (0 , 1) , 1 − p j p j ¯ Y j (1) 2 + p j 1 − p j ¯ Y j (0) 2 + 2 ¯ Y j (1) ¯ Y j (0) ≤ 1 p j ¯ Y j (1) 2 + 1 1 − p j ¯ Y j (0) 2 . Conse quently, σ 2 S ≤ σ 2 up , wher e σ 2 up := lim J →∞ 1 J J X j =1  1 p j ¯ Y j (1) 2 + 1 1 − p j ¯ Y j (0) 2  . Proof . Apply 2 ab ≤ a 2 + b 2 with a = ¯ Y j (1) and b = ¯ Y j (0), then simplify . 58 Lemma D.5 (Consistency of the upp er-b ound v ariance estimator) . Under Assumption 4.1 , J b V up P − − → σ 2 up , wher e σ 2 up := E ν  Y 2 1 P + Y 2 0 1 − P  ∈ (0 , ∞ ) . Proof . Recall J b V up = 1 J J X j =1 U j,J , U j,J := ( ¯ Y obs j ) 2 Z j p 2 j + 1 − Z j (1 − p j ) 2 ! , and ¯ Y obs j = ¯ Y j ( Z j ). Step 1: conv ergence of the mean. By iterated exp ectation, E [ U j,J ] = ¯ Y j (1) 2 p j + ¯ Y j (0) 2 1 − p j . Deﬁne the deterministic proxy σ 2 up ( J ) := 1 J J X j =1  ¯ Y j (1) 2 p j + ¯ Y j (0) 2 1 − p j  . By Assumption 4.1 (ii) and con tin uity/quadratic gro wth of the in tegrand, σ 2 up ( J ) → E ν  Y 2 1 P + Y 2 0 1 − P  =: σ 2 up < ∞ . Moreo ver, Assumption 4.1 (iii) implies ν is not supp orted on ( Y 1 , Y 0 ) = (0 , 0), so σ 2 up > 0. Th us E [ J b V up ] = σ 2 up ( J ) → σ 2 up . 59 Step 2: v anishing v ariance. The { U j,J } are indep enden t across j because the Z j ’s are inde- p enden t. Also, since p j ∈ [ p, 1 − p ], there exists C p < ∞ suc h that U 2 j,J ≤ C p ( ¯ Y obs j ) 4 ≤ C p M 4 j . Therefore V ar( U j,J ) ≤ E [ U 2 j,J ] ≤ C p M 4 j , and V ar  J b V up  = 1 J 2 J X j =1 V ar( U j,J ) ≤ C p J 2 J X j =1 M 4 j = O (1 /J ) → 0 , b y Assumption 4.1 (i). Hence J b V up − E [ J b V up ] → 0 in probability , and combining with Step 1 yields J b V up P − → σ 2 up . D.5 The session-wise weak n ull implies a zero target F or eac h session j and within-session fo cal index ℓ = 1 , . . . , n , deﬁne Y j,ℓ ( z ) := Y s j + m + ℓ − 1 ( z T ) , z ∈ { 0 , 1 } . Deﬁne τ ℓ := 1 J J X j =1 { Y j,ℓ (1) − Y j,ℓ (0) } , ℓ = 1 , . . . , n, and recall the session-wise w eak null H sw 0 is the collection of restrictions τ ℓ = 0 for all ℓ . Lemma D.6 (Session-wise n ull implies zero target) . Under H sw 0 , we have τ C = 0 , wher e τ C := 1 J J X j =1 { ¯ Y j (1) − ¯ Y j (0) } . 60 Proof . Using ¯ Y j ( z ) = n − 1 P n ℓ =1 Y j,ℓ ( z ), τ C = 1 J J X j =1 1 n n X ℓ =1 { Y j,ℓ (1) − Y j,ℓ (0) } = 1 n n X ℓ =1   1 J J X j =1 { Y j,ℓ (1) − Y j,ℓ (0) }   = 1 n n X ℓ =1 τ ℓ . Under H sw 0 , τ ℓ = 0 for all ℓ , hence τ C = 0. D.6 Pro of of Theorem 4.1 Pr o of of The or em 4.1 . Fix α ∈ (0 , 1). Let c 1 − α b e the (1 − α ) quantile of the conditional ran- domization distribution of T ∗ stud giv en { ¯ Y obs j } J j =1 (appro ximated by Monte Carlo). By Lemma D.2 , c 1 − α → z 1 − α in probability . Under the session-wise weak n ull H sw 0 , Lemma D.6 gives τ C = 0. By Lemma D.3 , √ J ˆ τ HT ⇒ N (0 , σ 2 S ) . By Lemma D.5 , J b V up → σ 2 up in probabilit y , and b y Lemma D.4 , σ 2 S ≤ σ 2 up . Therefore, Slutsky’s theorem yields T stud = ˆ τ HT q b V up = √ J ˆ τ HT q J b V up ⇒ N  0 , σ 2 S σ 2 up  , σ 2 S σ 2 up ≤ 1 . Hence, lim sup J →∞ P ( T stud ≥ z 1 − α ) ≤ α. Since c 1 − α → z 1 − α in probability , the same bound holds with c 1 − α in place of z 1 − α , which is equiv alent to lim sup T →∞ P ( ˆ p w ≤ α ) ≤ α. 61 D.7 V alidity of p osition-wise and join t studen tized CR Ts This subsection formalizes the asymptotic v alidity claims in Section 4.3 for (i) the position-wise studen tized CR Ts and (ii) the quadratic-form (F-type) join t test across within-session fo cal p osi- tions. Recall the ﬁxed-length session setup: L > m , T = J L → ∞ with J → ∞ , and n := L − m fo cal p ositions p er session. F or eac h session j and fo cal p osition ℓ ∈ { 1 , . . . , n } , let Y j,ℓ ( z ) := Y s j + m + ℓ − 1 ( z T ) for z ∈ { 0 , 1 } and deﬁne the within-session fo cal outcome v ector Y j ( z ) :=  Y j, 1 ( z ) , . . . , Y j,n ( z )  ⊤ ∈ R n , z ∈ { 0 , 1 } . Let Z j ∼ Bernoulli( p j ) denote the session-level assignmen t with known p j ∈ (0 , 1) and write Y obs j,ℓ := Y j,ℓ ( Z j ) , Y obs j := Y j ( Z j ) . Deﬁne the a verage eﬀect at focal position ℓ as τ ℓ := 1 J J X j =1 { Y j,ℓ (1) − Y j,ℓ (0) } , ℓ = 1 , . . . , n, and let τ := ( τ 1 , . . . , τ n ) ⊤ . Assumption D.1 (V ector weak-n ull asymptotics under ﬁxed-length sessions) . The carry ov er length m and session length L are ﬁxed with L > m , and T = J L → ∞ so that J → ∞ . There exists p ∈ (0 , 1 / 2) such that p ≤ p j ≤ 1 − p for all j and all T . Let M j := max {∥ Y j (1) ∥ , ∥ Y j (0) ∥} . Assume: (i) ( Uniform fourth-moment b ound ) There exists C < ∞ suc h that sup J ≥ 1 1 J P J j =1 M 4 j ≤ C . (ii) ( Stabilization ) The empirical measures ν J := 1 J P J j =1 δ ( p j , Y j (1) , Y j (0)) con verge w eakly to some probabilit y measure ν on [ p, 1 − p ] × R 2 n . 62 (iii) ( Nonde gener acy ) If ( P , Y 1 , Y 0 ) ∼ ν , then Σ R := E ν  Y 1 Y ⊤ 1 1 − P + Y 0 Y ⊤ 0 P  is p ositiv e deﬁnite. F or ℓ ∈ { 1 , . . . , n } , deﬁne the p osition-wise HT estimator and upp er-b ound v ariance estimator as in Section 4.3 : ˆ τ ℓ, HT := 1 J J X j =1 Z j Y obs j,ℓ p j − (1 − Z j ) Y obs j,ℓ 1 − p j ! , b V ℓ, up := 1 J 2 J X j =1 ( Y obs j,ℓ ) 2 Z j p 2 j + 1 − Z j (1 − p j ) 2 ! , T ℓ := ˆ τ ℓ, HT q b V ℓ, up . The position-wise studentized CR T draws Z ∗ 1 , . . . , Z ∗ J indep enden tly with Z ∗ j ∼ Bernoulli( p j ), holds { Y obs j,ℓ } J j =1 ﬁxed, recomputes T ∗ ℓ , and returns the one-sided Mon te Carlo p -v alue ˆ p ℓ . F or the join t test, deﬁne the v ector HT estimator and its matrix upper-b ound analogue: b τ HT := 1 J J X j =1 Z j Y obs j p j − (1 − Z j ) Y obs j 1 − p j ! , b Σ up := 1 J 2 J X j =1 Y obs j ( Y obs j ) ⊤ Z j p 2 j + 1 − Z j (1 − p j ) 2 ! , and the quadratic-form statistic T F := b τ ⊤ HT b Σ − 1 up b τ HT , (where, as in the main text, a generalized inv erse ma y be used if needed). The join t CR T draws Z ∗ 1 , . . . , Z ∗ J indep enden tly with Z ∗ j ∼ Bernoulli( p j ), holds { Y obs j } J j =1 ﬁxed, recomputes T ∗ F , and returns the Mon te Carlo p -v alue ˆ p F . Theorem D.1 (Asymptotic v alidity for position-wise and joint studentized CR Ts) . Work under the ﬁxe d-length session setup of Se ction 4 and supp ose Assumptions 2.1 and 2.2 hold. Assume Assumption D.1 . 63 (i) ( Position-wise v alidity ) F or any fo c al p osition ℓ ∈ { 1 , . . . , n } , under the p osition-wise we ak nul l H 0 ,ℓ : τ ℓ = 0 , lim sup T →∞ P ( ˆ p ℓ ≤ α ) ≤ α for al l α ∈ (0 , 1) . (ii) ( Joint v alidity ) Under the str ong joint nul l H sw 0 in ( 10 ) (e quivalently, τ = 0 ), lim sup T →∞ P ( ˆ p F ≤ α ) ≤ α for al l α ∈ (0 , 1) . The same c onclusions hold for testing any subset of fo c al p ositions by r estricting ℓ (for (i)) or by r eplacing b τ HT and b Σ up with the c orr esp onding subve ctor and princip al submatrix (for (ii)). Proof . W e pro ve (i) and (ii) using the same t w o-step template as in the proof of Theorem 4.1 : (a) a randomization limit for the resampled statistic (hence critical v alue conv ergence) and (b) a sampling limit for the observ ed statistic under the n ull, together with conserv ativeness of the upp er-bound studen tizer. Step 1: randomization limits (conditional on observ ed focal outcomes). (i) Position- wise. Fix ℓ and condition on { Y obs j,ℓ } J j =1 . W rite y j,ℓ := Y obs j,ℓ and deﬁne ξ ∗ j,ℓ := y j,ℓ  Z ∗ j p j − 1 − Z ∗ j 1 − p j  , ˆ τ ∗ ℓ, HT = 1 J J X j =1 ξ ∗ j,ℓ . Conditional on { y j,ℓ } , the { ξ ∗ j,ℓ } are independent and mean zero with V ar( ξ ∗ j,ℓ | { y j,ℓ } ) = y 2 j,ℓ / ( p j (1 − p j )). Assumption D.1 (i)–(iii) implies the conditional Ly apunov condition and yields a CL T for √ J ˆ τ ∗ ℓ, HT , and the same moment b ounds imply J b V ∗ ℓ, up concen trates around its conditional mean J − 1 P J j =1 y 2 j,ℓ / ( p j (1 − p j )). Therefore, by Slutsky’s theorem, T ∗ ℓ = ˆ τ ∗ ℓ, HT q b V ∗ ℓ, up ⇒ N (0 , 1) conditional on { Y obs j,ℓ } , 64 in P -probability . Let c ℓ, 1 − α b e the (1 − α ) quantile of the conditional randomization distribution of T ∗ ℓ . Then c ℓ, 1 − α → z 1 − α in probability . (ii) Joint statistic. Condition on { Y obs j } J j =1 and write y j := Y obs j . Deﬁne ξ ∗ j := y j  Z ∗ j p j − 1 − Z ∗ j 1 − p j  , b τ ∗ HT = 1 J J X j =1 ξ ∗ j . Conditional on { y j } , the { ξ ∗ j } are independent, mean zero, and V ar  ξ ∗ j   { y j }  = y j y ⊤ j p j (1 − p j ) . Let Σ obs R ,J := 1 J J X j =1 y j y ⊤ j p j (1 − p j ) . Assumption D.1 (i)–(iii) implies a conditional m ultiv ariate Lyapuno v CL T, so √ J b τ ∗ HT ⇒ N ( 0 , Σ obs R ,J ) conditionally on { y j } , in P -probability . Moreo ver, J b Σ ∗ up = 1 J J X j =1 y j y ⊤ j Z ∗ j p 2 j + 1 − Z ∗ j (1 − p j ) 2 ! has conditional mean Σ obs R ,J and (entrywise) conditional v ariance of order O P (1 /J ), so J b Σ ∗ up − Σ obs R ,J → 0 in conditional probability . By Assumption D.1 (iii), Σ obs R ,J is in vertible with probability tending to one, and hence so is J b Σ ∗ up . Therefore, by Slutsky’s theorem and the con tinuous mapping theorem, T ∗ F = ( √ J b τ ∗ HT ) ⊤ ( J b Σ ∗ up ) − 1 ( √ J b τ ∗ HT ) ⇒ χ 2 n conditional on { Y obs j } , 65 in P -probability . Let c F, 1 − α b e the (1 − α ) quan tile of the conditional randomization distribution of T ∗ F . Since the χ 2 n CDF is con tinuous and strictly increasing, w e ha v e c F, 1 − α → χ 2 n, 1 − α in probabilit y . Step 2: sampling limits for the observ ed statistics under the n ull. (i) Position-wise. Fix ℓ and consider H 0 ,ℓ : τ ℓ = 0. By indep endence of { Z j } and m -carry ov er, we may treat { Y j,ℓ (1) , Y j,ℓ (0) } J j =1 as ﬁxed potential outcomes for the scalar “units” j = 1 , . . . , J . Under As- sumption D.1 , the scalar array { ( p j , Y j,ℓ (1) , Y j,ℓ (0)) } satisﬁes the same momen t and stabiliza- tion conditions as in Assumption 4.1 (by coordinate pro jection), so the Lyapuno v CL T yields √ J ˆ τ ℓ, HT ⇒ N (0 , σ 2 S ,ℓ ) for some σ 2 S ,ℓ ∈ (0 , ∞ ), and the LLN yields J b V ℓ, up → σ 2 up ,ℓ . Moreov er, the same algebra as Lemma D.4 giv es σ 2 S ,ℓ ≤ σ 2 up ,ℓ . Hence T ℓ = √ J ˆ τ ℓ, HT q J b V ℓ, up ⇒ N 0 , σ 2 S ,ℓ σ 2 up ,ℓ ! , σ 2 S ,ℓ σ 2 up ,ℓ ≤ 1 , and therefore lim sup T →∞ P ( T ℓ ≥ z 1 − α ) ≤ α . Since c ℓ, 1 − α → z 1 − α in probabilit y , w e also hav e lim sup T →∞ P ( T ℓ ≥ c ℓ, 1 − α ) ≤ α , whic h is equiv alent to lim sup T →∞ P ( ˆ p ℓ ≤ α ) ≤ α . (ii) Joint statistic. Under H sw 0 , w e ha ve τ = 0 . Using Cram ´ er–W old, for an y ﬁxed a ∈ R n the scalar quantit y a ⊤ b τ HT is an HT estimator formed from the scalar p oten tial outcomes a ⊤ Y j (1) and a ⊤ Y j (0). Assumption D.1 implies the required moment and stabilization conditions for every ﬁxed a , so the scalar Ly apuno v CL T applies to each a ⊤ b τ HT . Hence √ J b τ HT ⇒ N ( 0 , Σ S ) , for some co v ariance matrix Σ S . Similarly , J b Σ up → Σ up in probabilit y for a deterministic p ositiv e deﬁnite matrix Σ up (en trywise LLN under the momen t bound). Moreov er, the (matrix) analogue of Lemma D.4 holds p oin t wise: 66 for each j , Y j (1) Y j (1) ⊤ p j + Y j (0) Y j (0) ⊤ 1 − p j − V ar( ψ j ) =  Y j (1) − Y j (0)  Y j (1) − Y j (0)  ⊤ ⪰ 0 , where ψ j is the cen tered summand in b τ HT . Av eraging o ver j and taking limits giv es Σ S ⪯ Σ up . No w write T F = ( √ J b τ HT ) ⊤ ( J b Σ up ) − 1 ( √ J b τ HT ) . By Slutsky and the contin uous mapping theorem, T F ⇒ Q , where if Z ∼ N ( 0 , I n ) and A := Σ − 1 / 2 up Σ S Σ − 1 / 2 up ⪯ I n , then Q d = Z ⊤ A Z . Diagonalize A = U diag( λ 1 , . . . , λ n ) U ⊤ with λ i ∈ [0 , 1]. Since U ⊤ Z d = Z , we hav e the coupling Q d = n X i =1 λ i Z 2 i ≤ n X i =1 Z 2 i a.s. , and th us for every t ≥ 0, P ( Q ≥ t ) ≤ P ( χ 2 n ≥ t ). T aking t = χ 2 n, 1 − α yields lim sup T →∞ P ( T F ≥ χ 2 n, 1 − α ) ≤ α . Since c F, 1 − α → χ 2 n, 1 − α in probabilit y , we also ha ve lim sup T →∞ P ( T F ≥ c F, 1 − α ) ≤ α , equiv alently lim sup T →∞ P ( ˆ p F ≤ α ) ≤ α . D.8 Pro of of Corollary A.1 Recall that Z 1 , . . . , Z J are indep enden t with Z j ∼ Bernoulli( p j ), where p ≤ p j ≤ 1 − p for all j and all J . Let ¯ Y j ( z ) b e the session-level fo cal mean p oten tial outcome under the constan t path z T and ¯ Y obs j = ¯ Y j ( Z j ). 67 Regression-based statistic. The regression coeﬃcient ˆ τ reg deﬁned by the w eigh ted regression ( 32 ) equals the diﬀerence of the IPW-normalized means in ( 33 ): ˆ τ reg = ˆ µ 1 − ˆ µ 0 , ˆ µ 1 = P J j =1 Z j ¯ Y j (1) p j P J j =1 Z j p j , ˆ µ 0 = P J j =1 (1 − Z j ) ¯ Y j (0) 1 − p j P J j =1 1 − Z j 1 − p j . Let b V reg b e the HC0 v ariance estimator in ( 34 ) and T reg = ˆ τ reg / q b V reg as in ( 35 ). F or the randomiza- tion distribution, conditional on { ¯ Y obs j } J j =1 dra w Z ∗ 1 , . . . , Z ∗ J indep enden tly with Z ∗ j ∼ Bernoulli( p j ) and deﬁne T ∗ reg b y recomputing ( 33 )–( 35 ) with Z ∗ in place of Z . Additional nondegeneracy . Because b V reg is based on within-arm residual v ariation, w e imp ose the follo wing mild nondegeneracy condition in addition to Assumption 4.1 . Let ( P, Y 1 , Y 0 ) ∼ ν b e the limit la w from Assumption 4.1 (ii), and deﬁne µ 1 := E ν [ Y 1 ] and µ 0 := E ν [ Y 0 ]. Assume σ 2 up , reg := E ν  ( Y 1 − µ 1 ) 2 P + ( Y 0 − µ 0 ) 2 1 − P  > 0 . (36) This rules out the degenerate case where b oth { ¯ Y j (1) } and { ¯ Y j (0) } are asymptotically constant across sessions. Step 1: a randomization CL T for T ∗ reg and quan tile con vergence Lemma D.7 (Randomization CL T) . Under Assumption 4.1 and ( 36 ) , c onditional on { ¯ Y obs j } J j =1 we have T ∗ reg ⇒ N (0 , 1) , in P -pr ob ability (wher e P is the original assignment law gener ating { ¯ Y obs j } ). Conse quently, if c 1 − α denotes the (1 − α ) quantile of the c onditional r andomization distribution of T ∗ reg given { ¯ Y obs j } , then c 1 − α → z 1 − α in pr ob ability. Proof . Fix J and condition on the realized fo cal means { ¯ Y obs j } J j =1 ; write y j := ¯ Y obs j and ¯ y := 68 J − 1 P J j =1 y j . Deﬁne the resampled IPW means ˆ µ ∗ 1 = A ∗ 1 ,J B ∗ 1 ,J , A ∗ 1 ,J := 1 J J X j =1 Z ∗ j y j p j , B ∗ 1 ,J := 1 J J X j =1 Z ∗ j p j , ˆ µ ∗ 0 = A ∗ 0 ,J B ∗ 0 ,J , A ∗ 0 ,J := 1 J J X j =1 (1 − Z ∗ j ) y j 1 − p j , B ∗ 0 ,J := 1 J J X j =1 1 − Z ∗ j 1 − p j , and ˆ τ ∗ reg := ˆ µ ∗ 1 − ˆ µ ∗ 0 . Step 1a: linearization of ˆ τ ∗ reg . Since E [ Z ∗ j /p j ] = 1 and E [(1 − Z ∗ j ) / (1 − p j )] = 1, E  A ∗ 1 ,J   { y j }  = ¯ y , E  B ∗ 1 ,J   { y j }  = 1 , E  A ∗ 0 ,J   { y j }  = ¯ y , E  B ∗ 0 ,J   { y j }  = 1 . W rite ˆ µ ∗ 1 − ¯ y = ( A ∗ 1 ,J − ¯ y ) − ¯ y ( B ∗ 1 ,J − 1) B ∗ 1 ,J , ˆ µ ∗ 0 − ¯ y = ( A ∗ 0 ,J − ¯ y ) − ¯ y ( B ∗ 0 ,J − 1) B ∗ 0 ,J . A direct simpliﬁcation yields ( A ∗ 1 ,J − ¯ y ) − ¯ y ( B ∗ 1 ,J − 1) = 1 J J X j =1  Z ∗ j p j − 1  ( y j − ¯ y ) , ( A ∗ 0 ,J − ¯ y ) − ¯ y ( B ∗ 0 ,J − 1) = 1 J J X j =1  1 − Z ∗ j 1 − p j − 1  ( y j − ¯ y ) . Moreo ver, conditional on { y j } , V ar  B ∗ 1 ,J   { y j }  = 1 J 2 J X j =1 1 − p j p j ≤ C J , V ar  B ∗ 0 ,J   { y j }  = 1 J 2 J X j =1 p j 1 − p j ≤ C J , so B ∗ 1 ,J → 1 and B ∗ 0 ,J → 1 in conditional probabilit y . Hence, ˆ τ ∗ reg = 1 J J X j =1 ξ ∗ j,J + r ∗ J , ξ ∗ j,J := ( y j − ¯ y )  Z ∗ j p j − 1 − Z ∗ j 1 − p j  , (37) 69 where r ∗ J = o P ( J − 1 / 2 ) conditionally on { y j } . Step 1b: conditional CL T for P ξ ∗ j,J . Conditional on { y j } , the { ξ ∗ j,J } J j =1 are independent, mean zero, and satisfy V ar  ξ ∗ j,J   { y j }  = ( y j − ¯ y ) 2 p j (1 − p j ) . Let s 2 J := J X j =1 ( y j − ¯ y ) 2 p j (1 − p j ) . Because p j ∈ [ p, 1 − p ], we ha v e s 2 J ≍ P J j =1 ( y j − ¯ y ) 2 . Assumption 4.1 (i) implies J − 1 P J j =1 y 4 j = O P (1), hence J − 1 P J j =1 ( y j − ¯ y ) 4 = O P (1), and therefore J X j =1 ( y j − ¯ y ) 4 = O P ( J ) . In addition, ( 36 ) implies that lim inf J →∞ P ( s 2 J /J > ε ) = 1 for some ε > 0 (otherwise y j w ould b e asymptotically constant across j , forcing σ 2 up , reg = 0). Th us s 2 J = Θ P ( J ). F urthermore, conditional on { y j } , E  ( ξ ∗ j,J ) 4   { y j }  = ( y j − ¯ y ) 4 1 p 3 j + 1 (1 − p j ) 3 ! ≤ C p ( y j − ¯ y ) 4 . Hence, with P -probabilit y tending to one, 1 s 4 J J X j =1 E  ( ξ ∗ j,J ) 4   { y j }  ≤ C p P J j =1 ( y j − ¯ y ) 4 s 4 J = O P  J J 2  → 0 . By the conditional Lyapuno v CL T, P J j =1 ξ ∗ j,J s J ⇒ N (0 , 1) conditionally on { y j } . (38) 70 Com bining ( 37 ) and ( 38 ) yields √ J ˆ τ ∗ reg q s 2 J /J ⇒ N (0 , 1) conditionally on { y j } . Step 1c: consistency of J b V ∗ reg for s 2 J /J . Deﬁne the resampled residuals ˆ r ∗ j, 1 := y j − ˆ µ ∗ 1 for Z ∗ j = 1 and ˆ r ∗ j, 0 := y j − ˆ µ ∗ 0 for Z ∗ j = 0, and let b V ∗ reg b e the HC0 v ariance estimator obtained b y substituting ( Z ∗ , ˆ µ ∗ 1 , ˆ µ ∗ 0 ) into ( 34 ). W rite J b V ∗ reg = N ∗ 1 ,J ( B ∗ 1 ,J ) 2 + N ∗ 0 ,J ( B ∗ 0 ,J ) 2 , where N ∗ 1 ,J := 1 J J X j =1 Z ∗ j ( ˆ r ∗ j, 1 ) 2 p 2 j , N ∗ 0 ,J := 1 J J X j =1 (1 − Z ∗ j )( ˆ r ∗ j, 0 ) 2 (1 − p j ) 2 . Since B ∗ 1 ,J → 1 and B ∗ 0 ,J → 1 in conditional probabilit y , it suﬃces to sho w N ∗ 1 ,J + N ∗ 0 ,J → s 2 J /J in conditional probability . First replace ˆ µ ∗ 1 , ˆ µ ∗ 0 b y ¯ y . Because ˆ µ ∗ 1 − ¯ y = O P ( J − 1 / 2 ) and ˆ µ ∗ 0 − ¯ y = O P ( J − 1 / 2 ) conditionally on { y j } (by the preceding CL T), a direct expansion giv es    N ∗ 1 ,J − ˜ N ∗ 1 ,J    +    N ∗ 0 ,J − ˜ N ∗ 0 ,J    = o P (1) conditionally on { y j } , where ˜ N ∗ 1 ,J := 1 J J X j =1 Z ∗ j ( y j − ¯ y ) 2 p 2 j , ˜ N ∗ 0 ,J := 1 J J X j =1 (1 − Z ∗ j )( y j − ¯ y ) 2 (1 − p j ) 2 . Next, conditional on { y j } , E [ ˜ N ∗ 1 ,J + ˜ N ∗ 0 ,J | { y j } ] = 1 J J X j =1 ( y j − ¯ y ) 2  1 p j + 1 1 − p j  = s 2 J J . 71 Moreo ver, using p j ∈ [ p, 1 − p ] and independence, V ar  ˜ N ∗ 1 ,J + ˜ N ∗ 0 ,J    { y j }  ≤ C p J 2 J X j =1 ( y j − ¯ y ) 4 = O P (1 /J ) → 0 . Th us ˜ N ∗ 1 ,J + ˜ N ∗ 0 ,J → s 2 J /J in conditional probabilit y , and hence N ∗ 1 ,J + N ∗ 0 ,J → s 2 J /J as w ell. Therefore J b V ∗ reg s 2 J /J P − − → 1 conditionally on { y j } . Com bining with ( 38 ) and Slutsky’s theorem yields T ∗ reg ⇒ N (0 , 1) conditionally on { y j } , in P - probabilit y . Quan tile conv ergence. Let F ∗ J ( · ) := P ( T ∗ reg ≤ · | { y j } ) b e the conditional CDF. The ab o ve sho ws F ∗ J ( t ) → Φ( t ) in probability for each t ∈ R , and Φ is con tinuous and strictly increasing. Therefore the conditional (1 − α ) quantile c 1 − α := inf { t : F ∗ J ( t ) ≥ 1 − α } satisﬁes c 1 − α → z 1 − α in probabilit y . Step 2: a sampling CL T for T reg under H sw 0 Deﬁne the (deterministic) session-level means µ 1 ,J := 1 J J X j =1 ¯ Y j (1) , µ 0 ,J := 1 J J X j =1 ¯ Y j (0) , τ C = µ 1 ,J − µ 0 ,J . Under H sw 0 , Lemma D.6 implies τ C = 0. Lemma D.8 (Sampling CL T for ˆ τ reg ) . Under Assumption 4.1 and ( 36 ) , √ J  ˆ τ reg − τ C  ⇒ N (0 , σ 2 reg ) , 72 wher e σ 2 reg := E ν   r 1 − P P ( Y 1 − µ 1 ) + r P 1 − P ( Y 0 − µ 0 ) ! 2   . Proof . W rite ˆ τ reg = ˆ µ 1 − ˆ µ 0 with ˆ µ 1 = A 1 ,J B 1 ,J , A 1 ,J := 1 J J X j =1 Z j ¯ Y j (1) p j , B 1 ,J := 1 J J X j =1 Z j p j , ˆ µ 0 = A 0 ,J B 0 ,J , A 0 ,J := 1 J J X j =1 (1 − Z j ) ¯ Y j (0) 1 − p j , B 0 ,J := 1 J J X j =1 1 − Z j 1 − p j . Because E [ Z j /p j ] = 1 and E [(1 − Z j ) / (1 − p j )] = 1, we hav e E [ B 1 ,J ] = E [ B 0 ,J ] = 1 and V ar( B 1 ,J ) = 1 J 2 J X j =1 1 − p j p j ≤ C J , V ar( B 0 ,J ) = 1 J 2 J X j =1 p j 1 − p j ≤ C J , so B 1 ,J → 1 and B 0 ,J → 1 in probability . Next, note the exact iden tit y ˆ µ 1 − µ 1 ,J = ( A 1 ,J − µ 1 ,J ) − µ 1 ,J ( B 1 ,J − 1) B 1 ,J = 1 B 1 ,J · 1 J J X j =1  Z j p j − 1   ¯ Y j (1) − µ 1 ,J  . Since B 1 ,J → 1, ˆ µ 1 − µ 1 ,J = 1 J J X j =1  Z j p j − 1   ¯ Y j (1) − µ 1 ,J  + o P ( J − 1 / 2 ) . (39) Similarly , ˆ µ 0 − µ 0 ,J = 1 J J X j =1  1 − Z j 1 − p j − 1   ¯ Y j (0) − µ 0 ,J  + o P ( J − 1 / 2 ) . (40) Subtracting ( 40 ) from ( 39 ) yields ˆ τ reg − τ C = 1 J J X j =1 ψ j,J + o P ( J − 1 / 2 ) , 73 where the summands are independent and mean zero: ψ j,J :=  Z j p j − 1   ¯ Y j (1) − µ 1 ,J  −  1 − Z j 1 − p j − 1   ¯ Y j (0) − µ 0 ,J  . A direct t wo-point calculation in Z j ∈ { 0 , 1 } gives V ar( ψ j,J ) = s 1 − p j p j ( ¯ Y j (1) − µ 1 ,J ) + r p j 1 − p j ( ¯ Y j (0) − µ 0 ,J ) ! 2 =: v j,J . (41) Let S 2 J := P J j =1 v j,J and σ 2 reg ( J ) := S 2 J /J . By Assumption 4.1 (ii), µ 1 ,J → µ 1 and µ 0 ,J → µ 0 , and the empirical measures ν J ⇒ ν imply σ 2 reg ( J ) → E ν   r 1 − P P ( Y 1 − µ 1 ) + r P 1 − P ( Y 0 − µ 0 ) ! 2   =: σ 2 reg . Moreo ver, ( 36 ) implies σ 2 reg > 0, hence S 2 J = Θ( J ). T o apply Ly apunov’s CL T, note that p j ∈ [ p, 1 − p ] implies | ψ j,J | ≤ C p ( | ¯ Y j (1) − µ 1 ,J | + | ¯ Y j (0) − µ 0 ,J | ) ≤ C ′ p M j , so E [ ψ 4 j,J ] ≤ C ′′ p M 4 j and therefore 1 S 4 J J X j =1 E [ ψ 4 j,J ] ≤ C ′′ p P J j =1 M 4 j S 4 J = O (1 /J ) → 0 , b y Assumption 4.1 (i) and S 4 J = Θ( J 2 ). The Lyapuno v CL T gives P J j =1 ψ j,J /S J ⇒ N (0 , 1), and since √ J ( ˆ τ reg − τ C ) = ( P J j =1 ψ j,J ) / √ J + o P (1) with S 2 J /J → σ 2 reg , the claim follows. Lemma D.9 (Consistency and conserv ativeness of b V reg ) . Under Assumption 4.1 and ( 36 ) , J b V reg P − − → σ 2 up , reg , and σ 2 reg ≤ σ 2 up , reg . Proof . W rite B 1 := P J j =1 Z j /p j = J B 1 ,J and B 0 := P J j =1 (1 − Z j ) / (1 − p j ) = J B 0 ,J . F rom ab o ve, 74 B 1 ,J → 1 and B 0 ,J → 1 in probability . Using ( 34 ) w e can write J b V reg = N 1 ,J ( B 1 ,J ) 2 + N 0 ,J ( B 0 ,J ) 2 , where N 1 ,J := 1 J J X j =1 Z j ( ¯ Y j (1) − ˆ µ 1 ) 2 p 2 j , N 0 ,J := 1 J J X j =1 (1 − Z j )( ¯ Y j (0) − ˆ µ 0 ) 2 (1 − p j ) 2 . Since ( B 1 ,J ) − 2 = 1 + o P (1) and ( B 0 ,J ) − 2 = 1 + o P (1), it suﬃces to pro v e N 1 ,J + N 0 ,J P − → σ 2 up , reg . Deﬁne the deterministic means µ 1 ,J , µ 0 ,J as ab o ve and write ∆ 1 := ˆ µ 1 − µ 1 ,J and ∆ 0 := ˆ µ 0 − µ 0 ,J . By Lemma D.8 , ∆ 1 = O P ( J − 1 / 2 ) and ∆ 0 = O P ( J − 1 / 2 ). Expand ( ¯ Y j (1) − ˆ µ 1 ) 2 − ( ¯ Y j (1) − µ 1 ,J ) 2 = ∆ 2 1 − 2( ¯ Y j (1) − µ 1 ,J )∆ 1 . Therefore N 1 ,J = N ◦ 1 ,J + ∆ 2 1 · 1 J J X j =1 Z j p 2 j − 2∆ 1 · 1 J J X j =1 Z j ( ¯ Y j (1) − µ 1 ,J ) p 2 j , where N ◦ 1 ,J := 1 J P J j =1 Z j ( ¯ Y j (1) − µ 1 ,J ) 2 p 2 j . Since ∆ 2 1 = O P (1 /J ) and J − 1 P Z j /p 2 j = O P (1), the second term is o P (1). F or the third term, J − 1 P Z j ( ¯ Y j (1) − µ 1 ,J ) /p 2 j = O P (1) b y a v ariance b ound (using Assumption 4.1 (i)), hence it is also o P (1) b ecause ∆ 1 = O P ( J − 1 / 2 ). Th us N 1 ,J − N ◦ 1 ,J = o P (1). The same argument yields N 0 ,J − N ◦ 0 ,J = o P (1), where N ◦ 0 ,J := 1 J P J j =1 (1 − Z j )( ¯ Y j (0) − µ 0 ,J ) 2 (1 − p j ) 2 . Next compute expectations: E [ N ◦ 1 ,J ] = 1 J J X j =1 ( ¯ Y j (1) − µ 1 ,J ) 2 p j , E [ N ◦ 0 ,J ] = 1 J J X j =1 ( ¯ Y j (0) − µ 0 ,J ) 2 1 − p j . Moreo ver, using p j ∈ [ p, 1 − p ] and Assumption 4.1 (i), V ar( N ◦ 1 ,J ) + V ar( N ◦ 0 ,J ) ≤ C p J 2 J X j =1 M 4 j = O (1 /J ) → 0 . 75 Hence N ◦ 1 ,J + N ◦ 0 ,J − 1 J J X j =1  ( ¯ Y j (1) − µ 1 ,J ) 2 p j + ( ¯ Y j (0) − µ 0 ,J ) 2 1 − p j  P − − → 0 . By Assumption 4.1 (ii) and µ 1 ,J → µ 1 , µ 0 ,J → µ 0 , the displa y ed deterministic a verage con verges to σ 2 up , reg in ( 36 ). Combining these steps giv es N 1 ,J + N 0 ,J P − → σ 2 up , reg , and hence J b V reg P − → σ 2 up , reg . Finally , the p oin t wise inequalit y σ 2 reg ≤ σ 2 up , reg follo ws from ( Y 1 − µ 1 ) 2 P + ( Y 0 − µ 0 ) 2 1 − P − r 1 − P P ( Y 1 − µ 1 ) + r P 1 − P ( Y 0 − µ 0 ) ! 2 =  ( Y 1 − µ 1 ) − ( Y 0 − µ 0 )  2 ≥ 0 , and taking expectations under ν . Consequence for T reg under H sw 0 . Under H sw 0 , τ C = 0 (Lemma D.6 ), so Lemma D.8 and Lemma D.9 imply by Slutsky that T reg = ˆ τ reg q b V reg = √ J ˆ τ reg q J b V reg ⇒ N 0 , σ 2 reg σ 2 up , reg ! , σ 2 reg σ 2 up , reg ≤ 1 . In particular, lim sup J →∞ P ( T reg ≥ z 1 − α ) ≤ α. (42) Step 3: conclude Corollary A.1 Pr o of of Cor ol lary A.1 . Let c 1 − α b e the (1 − α ) quan tile of the conditional randomization distri- bution of T ∗ reg giv en { ¯ Y obs j } J j =1 . By Lemma D.7 , c 1 − α → z 1 − α in probability . The randomization test rejects at level α whenever T reg ≥ c 1 − α , whic h is equiv alent to ˆ p reg ≤ α for the corresponding one-sided randomization p -v alue. Fix ε > 0. Then P ( T reg ≥ c 1 − α ) ≤ P ( T reg ≥ z 1 − α − ε ) + P ( | c 1 − α − z 1 − α | > ε ) . 76 T aking lim sup and using c 1 − α → z 1 − α in probability gives lim sup J →∞ P ( T reg ≥ c 1 − α ) ≤ lim sup J →∞ P ( T reg ≥ z 1 − α − ε ) . By the weak con vergence of T reg established ab o ve, the righ t-hand side equals 1 − Φ  ( z 1 − α − ε ) /ρ  where ρ 2 = σ 2 reg /σ 2 up , reg ≤ 1. Letting ε ↓ 0 and using contin uit y of Φ yields lim sup J →∞ P ( T reg ≥ c 1 − α ) ≤ 1 − Φ  z 1 − α ρ  ≤ 1 − Φ( z 1 − α ) = α, whic h pro v es lim sup T →∞ P ( ˆ p reg ≤ α ) ≤ α under H sw 0 . E Pro of for P o w er Analysis This app endix provides supporting deriv ations for Section 5 . Throughout, we w ork with the prede- termined p o oled-section construction: T = M L , p o oled size r | M , J = M /r , predetermined p o oled sections [ s j , e j ] = { ( j − 1) r L + 1 , . . . , j r L } , and the constancy indicators E j = { W (( j − 1) r +1) = · · · = W ( j r ) } . F or the total-eﬀect test, the selected set is J ( W ) = { j : E j } with J tot = |J ( W ) | . W e write p = p ( r ; q ) = q r / ( q r + (1 − q ) r ) and π r ( q ) = q r + (1 − q ) r , and n = r L − m . E.1 Conditional assignmen t la w under predetermined p o oling Lemma E.1 (Constancy probability and p ooled-section coun t) . Under ( 14 ) , the c onstancy indi- c ators 1 { E j } ar e i.i.d. Bernoul li with P ( E j ) = π r ( q ) = q r + (1 − q ) r , so J tot ∼ Binomial( J, π r ( q )) and E [ J tot ] = J π r ( q ) . Mor e over, J tot /J → π r ( q ) almost sur ely, and henc e J tot → ∞ almost sur ely as J → ∞ . 77 Proof . Eac h p ooled section j depends on the disjoint blo c k set { W (( j − 1) r +1) , . . . , W ( j r ) } . The ev ent E j o ccurs iﬀ all r blo cks equal 1 or all equal 0, hence P ( E j ) = q r + (1 − q ) r = π r ( q ). Disjoin tness implies indep endence across j , so J tot is binomial. The almost sure limit follo ws from the strong la w of large num b ers applied to P J j =1 1 { E j } . Pr o of of L emma 5.1 . Fix p o oled section j and deﬁne A j := { W (( j − 1) r +1) = · · · = W ( j r ) = 1 } and B j := { W (( j − 1) r +1) = · · · = W ( j r ) = 0 } . Then E j = A j ˙ ∪ B j and b y indep endence, P ( A j ) = q r , P ( B j ) = (1 − q ) r , P ( E j ) = q r + (1 − q ) r . Let Z j b e the common v alue on E j . Then P ( Z j = 1 | E j ) = P ( A j ) P ( E j ) = q r q r + (1 − q ) r = p ( r ; q ) . Indep endence across disjoint sections follo ws b ecause the underlying block sets are disjoin t across j . E.2 Pro of of Prop osition 5.1 W e work on a product probabilit y space (Ω , F , P ) supp orting tw o indep enden t sources of random- ness: (i) the assignment mec hanism W and (ii) the sup erp opulation error pro cess { ε t } in ( 17 ). F ormally , let (Ω , F , P ) = (Ω ε × Ω W , F ε ⊗ F W , P ε ⊗ P W ) , where P W generates the blo ck assignments in ( 14 ) and P ε generates the stationary AR(1) pro cess ( 17 ). Throughout, w e condition on the realized p ooled-section structure used b y the CR T, i.e., on J ( W ) and hence on J tot = |J ( W ) | . 78 Notation. F or eac h selected p ooled section j ∈ J ( W ), let U j denote its n = r L − m fo cal time p oin ts and deﬁne ¯ Y j ( z ) := 1 n X t ∈U j Y t ( z T ) , z ∈ { 0 , 1 } , ¯ Y obs j = ¯ Y j ( Z j ) , where Z j ∈ { 0 , 1 } is the po oled-section lab el (constan t on the section when j ∈ J ( W )). Conditional on J ( W ), the labels { Z j } j ∈J ( W ) are indep enden t Bernoulli( p ) with p = p ( r ; q ) (Lemma 5.1 ). Deﬁne the ﬁnite-population p ooled-section estimand τ J tot := 1 J tot X j ∈J ( W ) { ¯ Y j (1) − ¯ Y j (0) } . (43) Step 1: Identify the ﬁnite-p opulation target under the DGP Assume m ≥ m 0 . Fix any selected p ooled section j and any fo cal time t ∈ U j . By construction of U j , the last m 0 lags of t remain within the same po oled section, and b ecause the section is constant in time under z T w e ha v e w t − ℓ = z for all ℓ ∈ { 0 , . . . , m 0 } . Therefore, under ( 16 ), Y t ( 0 T ) = µ + ε t , Y t ( 1 T ) = µ + τ tot + ε t , τ tot = m 0 X ℓ =0 β ℓ . Av eraging o v er t ∈ U j yields ¯ Y j (0) = µ + ¯ ε j , ¯ Y j (1) = µ + τ tot + ¯ ε j , ¯ ε j := 1 n X t ∈U j ε t . (44) In particular, ¯ Y j (1) − ¯ Y j (0) = τ tot for every selected section j , and hence τ J tot = τ tot deterministically . (45) 79 Step 2: Conditional (randomization) CL T giv en realized p oten tial outcomes Let H b e the σ -ﬁeld generated b y (i) the realized p oten tial outcomes { ( ¯ Y j (1) , ¯ Y j (0)) } j ∈J ( W ) and (ii) the realized p o oled-section structure J ( W ). Conditional on H , the only randomness in ( ˆ τ HT , b V up ) comes from the indep enden t Bernoulli lab els { Z j } j ∈J ( W ) . W rite the HT estimator ( 18 ) as ˆ τ HT = 1 J tot X j ∈J ( W ) ξ j,J tot , ξ j,J tot := Z j ¯ Y j (1) p − (1 − Z j ) ¯ Y j (0) 1 − p . Then, conditional on H , the summands { ξ j,J tot } are indep endent with E [ ξ j,J tot | H ] = ¯ Y j (1) − ¯ Y j (0) = τ tot , so E [ ˆ τ HT | H ] = τ tot . Deﬁne centered summands ψ j,J tot := ξ j,J tot − { ¯ Y j (1) − ¯ Y j (0) } . A direct computation (as in App endix D.3 ) giv es V ar ( ψ j,J tot ) = 1 − p j p j ¯ Y j (1) 2 + pj 1 − p j ¯ Y j (0) 2 + 2 ¯ Y j (1) ¯ Y j (0) = s 1 − pj p j ¯ Y j (1) + s pj 1 − p j ¯ Y j (0) ! 2 Let S 2 J tot := X j ∈J ( W ) V ar( ψ j,J tot | H ) , σ 2 τ ( J tot ) := S 2 J tot J tot . Conditional CL T for ˆ τ HT . Because Z j are indep endent giv en H and the section lengths are ﬁxed, a Lyapuno v condition holds under ﬁnite fourth moments of ¯ Y j (1) and ¯ Y j (0), implying the conditional CL T √ J tot ( ˆ τ HT − τ tot ) p σ 2 τ ( J tot ) ⇒ N (0 , 1) , conditional on H , (46) 80 where the conv ergence holds in P -probabilit y (and in fact almost surely along typical realizations under the superp opulation model). Conditional LLN for b V up . Recall b V up = 1 J 2 tot X j ∈J ( W ) ( ¯ Y obs j ) 2  Z j p 2 + 1 − Z j (1 − p ) 2  , ¯ Y obs j = ¯ Y j ( Z j ) . Deﬁne the ﬁnite-population upp er-bound functional v up ( J tot ) := 1 J tot X j ∈J ( W )  ¯ Y j (1) 2 p + ¯ Y j (0) 2 1 − p  . (47) Then a conditional law of large n umbers yields J tot b V up P − − → v up ( J tot ) , conditional on H . (48) Conditional limit for T tot . Com bining ( 46 ) and ( 48 ) with Slutsky’s theorem (conditionally on H ) giv es T tot = ˆ τ HT q b V up = √ J tot ˆ τ HT q J tot b V up ⇒ N τ tot √ J tot p v up ( J tot ) , σ 2 τ ( J tot ) v up ( J tot ) ! , conditional on H . (49) Step 3: Sup erp opulation limits of random v ariance functionals W e no w show that the random ﬁnite-p opulation functionals σ 2 τ ( J tot ) and v up ( J tot ) con verge in probabilit y to deterministic limits under the superp opulation model (indeed, almost surely). Lemma E.2 (Ergo dic ratio LLN under indep enden t thinning) . L et { X j } j ≥ 1 b e a stationary er go dic se quenc e with E | X 1 | < ∞ , and let { E j } j ≥ 1 b e i.i.d. Bernoulli( π ) , indep endent of { X j } , with π ∈ 81 (0 , 1) . Deﬁne J N := P N j =1 E j . Then on the event { J N → ∞} , 1 J N N X j =1 E j X j a.s. − − − → E [ X 1 ] . Proof . By ergo dicity , 1 N P N j =1 E j X j → E [ E 1 X 1 ] = π E [ X 1 ] a.s. and 1 N P N j =1 E j → E [ E 1 ] = π a.s. T aking ratios yields the claim. Apply Lemma E.2 to section means. Let ¯ ε j b e the within-section focal mean error in ( 44 ), computed on the predetermined po oled partition. The sequence { ¯ ε j } is stationary ergo dic under the stationary AR(1) model and has ﬁnite moments. The selection indicators { E j } are i.i.d. Bernoulli with π = π r ( q ) > 0 and are indep endent of { ¯ ε j } . Therefore, by Lemma E.2 and Lemma E.1 , 1 J tot X j ∈J ( W ) ¯ ε j a.s. − − − → 0 , 1 J tot X j ∈J ( W ) ¯ ε 2 j a.s. − − − → E [ ¯ ε 2 1 ] = σ 2 ¯ ε ( n, ρ ) , and similarly for higher momen ts. Limits of σ 2 τ ( J tot ) and v up ( J tot ) . Using ( 44 ) and the abov e ratio LLNs, we obtain (in probabilit y , indeed a.s.) 1 J tot X j ∈J ( W ) ¯ Y j (0) 2 → µ 2 + σ 2 ¯ ε ( n, ρ ) , 1 J tot X j ∈J ( W ) ¯ Y j (1) 2 → ( µ + τ tot ) 2 + σ 2 ¯ ε ( n, ρ ) , and 1 J tot X j ∈J ( W ) ¯ Y j (1) ¯ Y j (0) → µ ( µ + τ tot ) + σ 2 ¯ ε ( n, ρ ) . Substituting these limits yields the deterministic quantities in ( 24 ). Step 4: F rom conditional CL T to (appro ximate) unconditional pow er W e use the following general device. 82 Lemma E.3 (Conditional-to-unconditional normal limit) . L et X J b e a se quenc e of r andom vari- ables and let H J b e σ -ﬁelds. Supp ose that c onditional on H J , X J ⇒ N ( m J , s 2 J ) in P -pr ob ability, wher e ( m J , s 2 J ) ar e H J -me asur able r andom p ar ameters, and supp ose m J P − − → m, s 2 J P − − → s 2 ∈ (0 , ∞ ) . Then X J ⇒ N ( m, s 2 ) unc onditional ly. Proof . Fix t ∈ R . Let ϕ J ( t ) := E [ e itX J | H J ] b e the conditional characteristic function. By assumption, ϕ J ( t ) → exp( itm J − 1 2 t 2 s 2 J ) in probabilit y and | ϕ J ( t ) | ≤ 1. Since m J → m and s 2 J → s 2 in probability , w e ha ve exp( itm J − 1 2 t 2 s 2 J ) → exp( itm − 1 2 t 2 s 2 ) in probability . By b ounded con vergence along subsequences (or uniform in tegrabilit y using | ϕ J ( t ) | ≤ 1), E [ e itX J ] = E [ ϕ J ( t )] → exp( itm − 1 2 t 2 s 2 ) . Hence X J ⇒ N ( m, s 2 ). Apply Lemma E.3 to T tot . Equation ( 49 ) giv es a conditional normal limit for T tot with random mean τ tot √ J tot / p v up ( J tot ) and random v ariance σ 2 τ ( J tot ) /v up ( J tot ). Step 3 sho ws these parame- ters con v erge in probabilit y (indeed a.s.) to the deterministic quantities in ( 24 ). Therefore, by Lemma E.3 , T tot ≈ N  µ tot ( J tot ) , σ 2 tot  , conditionally on J tot . Randomization critical v alue and pow er. Let T ∗ tot b e the statistic recomputed after resam- pling Z ∗ j ind ∼ Bernoulli( p ) on J ( W ) while holding { ¯ Y obs j } j ∈J ( W ) ﬁxed. Conditional on { ¯ Y obs j } , 83 a randomization CL T yields T ∗ tot ⇒ N (0 , 1), so the conditional (1 − α ) quantile c 1 − α satisﬁes c 1 − α → z 1 − α in probability . Rejecting when T tot ≥ c 1 − α yields the appro ximation ( 23 ). E.3 T ail-signal deriv ation and pro of of Prop osition 5.2 W e analyze the paired predetermined po oled-section statistic ( 20 )–( 21 ). Let J e = ⌊ J / 2 ⌋ and index adjacen t odd–even p ooled-section pairs by j = 1 , . . . , J e . Step 1: T ail-signal under m 0 ≤ rL (last-blo c k lab el). Fix an ev en po oled section 2 j with start time s and consider a fo cal time t = s + a with oﬀset a ∈ { m, m + 1 , . . . , r L − 1 } . The randomized lab el in ( 20 ) is the assignment on the last design blo ck of the preceding odd p o oled section, which applies to the L time p oints { s − L, . . . , s − 1 } . F or a giv en lag ℓ ∈ { 0 , . . . , m 0 } , the lagged exp osure w t − ℓ equals this label if and only if t − ℓ ∈ { s − L, . . . , s − 1 } ⇐ ⇒ s − L ≤ s + a − ℓ ≤ s − 1 ⇐ ⇒ a + 1 ≤ ℓ ≤ a + L. Fix ℓ > m . The n umber of fo cal oﬀsets a ∈ { m, . . . , r L − 1 } suc h that a + 1 ≤ ℓ ≤ a + L is w ℓ := | [ m, r L − 1] ∩ [ ℓ − L, ℓ − 1] | . Under ( 25 ) w e ha ve ℓ ≤ m 0 ≤ r L , so the upp er truncation does not bind and w ℓ = ℓ − max { m, ℓ − L } = min { L, ℓ − m } , ℓ = m + 1 , . . . , m 0 . Therefore, toggling the last-blo c k label from 0 to 1 shifts the even-section fo cal mean b y ¯ Y 2 j (1) − ¯ Y 2 j (0) = 1 n m 0 X ℓ = m +1 w ℓ β ℓ = 1 n m 0 X ℓ = m +1 min { L, ℓ − m } β ℓ =: δ m ( n ) , n = r L − m, 84 whic h is ( 26 ). Here ¯ Y 2 j ( z ) denotes the even-section fo cal mean under the coun terfactual in terven- tion setting the last design blo c k of the preceding o dd p ooled section to z , with all other block assignmen ts and the error pro cess held ﬁxed. Step 2: Conditional CL T for the HT estimat or. Let H e b e the σ -ﬁeld generated b y the collection of ev en-section p otential means { ( ¯ Y 2 j (1) , ¯ Y 2 j (0)) } J e j =1 (and any additional randomness unrelated to the last-block lab els, suc h as the remaining blo c k assignments and the error pro cess). Under ( 14 ), the last-blo c k lab els { Z 2 j − 1 } are independent across j with Z 2 j − 1 ∼ Bernoulli( q ), and are indep enden t of H e . Deﬁne the per-pair HT summand ξ j,J e := Z 2 j − 1 ¯ Y 2 j (1) q − (1 − Z 2 j − 1 ) ¯ Y 2 j (0) 1 − q . Then ˆ δ m = J − 1 e P J e j =1 ξ j,J e and, conditional on H e , E [ ξ j,J e | H e ] = ¯ Y 2 j (1) − ¯ Y 2 j (0) = δ m ( n ) . Moreo ver, conditional on H e , V ar( ξ j,J e | H e ) =  1 q − 1  ¯ Y 2 j (1) 2 +  1 1 − q − 1  ¯ Y 2 j (0) 2 + 2 ¯ Y 2 j (1) ¯ Y 2 j (0) . Let σ 2 δ ( J e ) := J − 1 e P J e j =1 V ar( ξ j,J e | H e ). Under ﬁxed n and the AR(1) mo del, a conditional Ly a- puno v CL T yields √ J e  ˆ δ m − δ m ( n )  q σ 2 δ ( J e ) ⇒ N (0 , 1) , conditional on H e , with conv ergence in P -probabilit y . 85 Step 3: Studentization. Deﬁne the ﬁnite-p opulation upper-b ound functional v ( m ) up ( J e ) := 1 J e J e X j =1  ¯ Y 2 j (1) 2 q + ¯ Y 2 j (0) 2 1 − q  . A conditional LLN gives J e b V ( m ) up P − − → v ( m ) up ( J e ) , conditional on H e , where b V ( m ) up is deﬁned in ( 21 ). Therefore, b y Slutsky’s theorem, T m = ˆ δ m q b V ( m ) up ⇒ N   δ m ( n ) √ J e q v ( m ) up ( J e ) , σ 2 δ ( J e ) v ( m ) up ( J e )   , conditional on H e . Step 4: Deterministic limits and p o wer. Because ( L, r, m ) are ﬁxed, the ev en-s ection focal means form a stationary ergo dic sequence under stationary AR(1) errors and i.i.d. block assign- men ts, so v ( m ) up ( J e ) P − − → v ( m ) up , σ 2 δ ( J e ) P − − → σ 2 δ , where v ( m ) up and σ 2 δ are the expectations in ( 29 )–( 30 ) (with π 1 = q and π 0 = 1 − q ). Under the CR T resampling sc heme, o dd po oled sections are redra wn according to ( 14 ) while holding the even-section outcomes ﬁxed; under the null δ m ( n ) = 0, the corresp onding resampled statistic satisﬁes T ∗ m ⇒ N (0 , 1), so the (1 − α ) critical v alue is asymptotically z 1 − α . Substituting the ab o ve normal appro ximation for T m yields ( 27 ). 86

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