The Cascade Identity: 2SLS as a Policy Parameter in Capacity-Constrained Settings

The Cascade Iden tit y: 2SLS as a P olicy P arameter in Capacit y-Constrained Settings Niklas Bengtsson ∗ P er Engström ∗ Marc h 22, 2026 Abstract A gro wing literature shows that t wo-stage least squares (2SLS) with m ultiple treat- men ts yields co eﬃcien ts that are diﬃcult to interpret under heterogeneous treatmen t eﬀects. Bh uller and Sigstad (2024) demonstrate that cross-eﬀects in the ﬁrst stage— where an instrumen t for one treatment shifts enrollmen t in other treatments—are particularly problematic for reco vering meaningful weigh ted a verages of individual- lev el causal eﬀects. W e show that in capacity-constrained allo cation systems, these cross-eﬀects are not a nuisance but the source of a clean p olicy interpretation. When treatmen ts are rationed and the instrumen t op erates on the same margin as the policy of in terest, the 2SLS co eﬃcient β k equals the total so cietal eﬀect of expanding treatmen t k b y one slot, including all cascading reallo cations through the system. The mechanism is general: it applies whenev er ﬁxed supply constrains allo cation, whether through ranked queues, waitlists, or market-clearing prices. This c asc ade identity T = β holds for any ﬁrst-stage matrix, under arbitrary treatmen t eﬀect heterogeneit y , and requires only instrumen t relev ance and that the instrumen t op erates on the same margin as the p olicy . The result applies to universit y admissions, sc ho ol choice, medical residency matching, ∗ Departmen t of Economics, Uppsala Univ ersity . 1 Casc ade Identity 2 public housing, and other rationed allo cation settings. W e pro vide an empirical applica- tion using lottery-based admission to Swedish univ ersity programs and charitable giving as the outcome. JEL co des: C36, D61, I23, I26, I28, D64 Keyw ords: Instrumen tal v ariables, multiple treatments, heterogeneous treatment eﬀects, capacit y constraints, cascade eﬀects, universit y admissions, c haritable giving 1 In tro duction Instrumen tal v ariables (IV) metho ds are widely used to estimate causal eﬀects in settings where individuals self-select into treatments. When there are multiple treatments—for example, diﬀeren t ﬁelds of univ ersity study , diﬀeren t schools in a choice system, or diﬀeren t job training programs—the standard approac h is to estimate a multiv ariate t wo-stage least squares (2SLS) regression with indicators for eac h treatment as endogenous v ariables and treatmen t-sp eciﬁc instruments. A central concern in recent econometric theory is whether 2SLS co eﬃcients can b e given a causal interpretation under heterogeneous treatment eﬀects. Building on the LA TE framew ork of Im b ens and Angrist (1994), a large literature has inv estigated the conditions under which IV estimands reco ver meaningful weigh ted a verages of individual-level causal eﬀects (Angrist and Imbens, 1995; Hec kman and V ytlacil, 2005; Hec kman et al., 2006; Heckman and Pin to, 2018; Mogstad et al., 2021). Bh uller and Sigstad (2024) bring this analysis to 2SLS with m ultiple treatments and demonstrate that t wo conditions are necessary and suﬃcient for the co eﬃcients to b e p ositively weigh ted sums of individual treatmen t eﬀects: a v erage conditional monotonicity and—crucially— no cr oss-eﬀe cts . The no-cross-eﬀects condition requires that each instrument aﬀects only its o wn treatment indicator, ruling out settings where an instrument for treatmen t k shifts enrollment in treatmen t j  = k . In man y empirically imp ortant settings, cross-eﬀects are p erv asiv e. When a studen t Casc ade Identity 3 wins a lottery for universit y program k , this t ypically reduces the probabilit y of enrolling in alternativ e programs j  = k . When a patient receives a slot in a preferred hospital, this frees a slot at the hospital they would otherwise hav e attended. When a price decrease raises demand for one go o d, consumers substitute aw a y from other go o ds. Cross-eﬀects are not ab errations but the deﬁning feature of systems where treatments are capacity-constrained and individuals substitute across alternativ es. This paper shows that in such settings, the cross-eﬀects that Bhuller and Sigstad (2024) ﬂag as problematic for individual-level in terpretation are in fact the source of a clean system-level p olicy interpr etation . W e prov e the following result: In a c ap acity-c onstr aine d al lo c ation system wher e slots ar e ﬁl le d fr om r anke d queues, the 2SLS c o eﬃcient β k e quals the total so cietal eﬀe ct of exp anding tr e atment k by one slot, including al l c asc ading r e al lo c ations thr ough the system. The pro of is remark ably simple. When program k expands by one slot, the new entran t ma y come from another program j , freeing a slot there. Program j reﬁlls the v acancy from its o wn ranking list, p otentially freeing a slot elsewhere. This cascade of reallo cations contin ues un til the system reac hes a new equilibrium. The total so cietal eﬀect T k satisﬁes a recursive equation that, written in matrix form, yields T = β as an algebraic iden tity . The result holds for an y ﬁrst-stage matrix Π (pro vided the diagonal is nonzero), under arbitrary heterogeneity in treatment eﬀects, and without an y monotonicit y conditions. It do es not require exclusive enrollmen t: treatmen t indicators can o v erlap, as long as the system ﬁlls v acancies from ranking lists and the instrument op erates on the same margin as the p olicy . The class of applicable settings is broad: universit y admissions with cen tralized assignmen t (Öc kert, 2010; Ketel et al., 2016; Kirkebøen et al., 2016; Artmann et al., 2021; Altmejd et al., 2021; Bleemer and Meh ta, 2022; Heinesen et al., 2022), school choice with lottery tie-breaking (Ab dulk adiroğlu et al., 2017), medical residency matc hing (Agarwal, 2015), public housing allo cation (Jacob and Ludwig, 2012), early childhoo d and job training programs with capacity Casc ade Identity 4 constrain ts (Kline and W alters, 2016; Gelb er et al., 2016), and more generally any rationed public service where slots are ﬁlled from queues. In all these settings, the prev ailing view— informed by the theoretical literature on heterogeneous treatmen t eﬀects—is that 2SLS co eﬃcien ts are diﬃcult to interpret. Our result shows that they hav e a direct and in tuitive p olicy in terpretation: the eﬀect of expanding capacity by one slot. The cascade identit y makes the 2SLS co eﬃcien t a natural input to cost-b eneﬁt analysis. It i mplemen ts a ceteris paribus condition at the system level: the so cietal eﬀect of expanding one program, holding all others at their curren t capacity . The p olicy maker needs only one additional n umber—the marginal cost of a slot—to ev aluate whether expansion is worth while, with no information required ab out costs at other programs. This stands in con trast to the o wn instrumen t-sp eciﬁc W ald ratio, which captures only the direct eﬀect on the "lottery winner" and w oul d require trac king the ﬁscal consequences of every cross-eﬀect to conv ert in to a cost-b eneﬁt statement. The applicabilit y th us extends far b eyond strictly rationed settings. W e apply the cascade iden tity to the eﬀect of universit y ﬁeld of study on proso cial b eha vior, measured as charitable giving, using administrativ e data from the Swedish cen tralized admissions system o ver 2008–2021. The Swedish design generates true random v ariation through lottery tie-breaking among applicants with iden tical merit scores, pro viding program- sp eciﬁc instruments for eac h of seven broad ﬁelds of study and av oiding the contin uit y assumptions required b y the regression discontin uit y designs common in this literature. The single-instrumen t W ald ratio suggests that teaching, medicine, health, and STEM generate signiﬁcan t p ositive eﬀects on c haritable giving, with estimates ranging from 4–7 p ercen tage p oin ts, while the direct eﬀects of business and so cial science are close to zero and statistically insigniﬁcan t. These W ald ratios, how ev er, measure the eﬀect of attending a ﬁeld relative to a coun terfactual mix of fallback programs rather than relative to not attending univ ersity at all. The cascade decomp osition corrects for this: expanding any comp etitiv e ﬁeld creates do wnstream v acancies that are ﬁlled from outside higher education, generating additional Casc ade Identity 5 proso cial gains. These cascade eﬀects are statistically signiﬁcant for most ﬁelds, including business and so cial science, and materially c hange the p olicy interpretation of the estimates. Our contribution sp eaks to a long-running debate ab out the p olicy relev ance of IV esti- mands. Heckman and V ytlacil (2005) argued that diﬀerent IV estimators reco ver diﬀeren t w eighted a verages of the marginal treatment eﬀect, and that the connection b etw een any par- ticular IV estimate and a p olicy-relev ant parameter is generally opaque. Heckman et al. (2006) sharp ened this concern by sho wing that IV breaks down for estimating in terpretable treatmen t parameters when agents select in to treatments with kno wledge of their idiosyncratic resp onse. Im b ens (2010) countered that LA TE, while limited to compliers, is a credible and informativ e estimand. Mogstad et al. (2018) dev elop ed MTE-based metho ds for extrap olating from LA TE to p olicy-relev ant parameters. Mogstad and T orgovitsky (2024) survey this literature and organize it around tw o strategies: reverse engineering a LA TE interpretation for linear IV, and forward engineering MTE-based estimators to recov er p olicy-relev ant parameters. Our result oﬀers a third resolution that applies sp eciﬁcally in capacity-constrained settings: the standard multi-treatmen t 2SLS co eﬃcien t is alr e ady the p olicy-relev ant parameter, without requiring MTE extrap olation, complier decomp osition, or an y monotonicity condition. In the sp eciﬁc con text of multiple unordered treatments, iden tiﬁcation has required increasingly strong conditions on c hoice b ehavior. Heckman et al. (2008) sho w that lo cal IV metho ds can iden tify the MTE of one option versus the next-b est alternative, but this requires knowing or inferring eac h individual’s next-b est c hoice. Heckman and Pin to (2018) in tro duce an “unordered monotonicity” condition equiv alen t to additive separabilit y of the c hoice mo del—a strong structural restriction. Angrist and Imbens (1995) show ed that 2SLS with an ordered m ulti-v alued treatment identiﬁes a w eighted av erage of p er-unit eﬀects under monotonicit y , whic h Bhuller and Sigstad (2024) generalize to the unordered case. Mogstad et al. (2024) relax standard monotonicity to partial monotonicit y but require estimating marginal treatmen t resp onse functions. Our cascade iden tit y requires none of these conditions: it is a mechanical consequence of ranking-list reﬁll, not a restriction on individual choice Casc ade Identity 6 b eha vior. Our result is most directly related to Bhuller and Sigstad (2024) and Kirk ebøen et al. (2016). Bhuller and Sigstad c haracterize when 2SLS with multiple treatmen ts yields p ositiv ely w eighted av erages of individual-lev el eﬀects; w e show that even when their conditions fail (as they generically do in capacity-constrained systems), the co eﬃcients ha ve a system-level p olicy in terpretation. Kirkebøen et al. decomp ose 2SLS co eﬃcien ts into complier-sp eciﬁc pairwise LA TEs under a “next-b est” restriction derived from Norwegian universit y admissions; w e pro vide a complementary result that do es not require decomp osition in to complier types and applies to any ﬁrst-stage matrix. The approach in Hec kman et al. (2008), which identiﬁes the MTE of one option v ersus the next-b est alternative, faces a similar informational requiremen t. The cascade identit y bypasses this en tirely—the system-level p olicy eﬀect is identiﬁed from the ﬁrst-stage matrix without kno wledge of individual-level coun terfactual choices. Section 2 presen ts the mo del and assumptions. Section 3 derives the cascade iden tit y . Section 4 discus ses the assumptions and their scop e. Section 5 applies the result to Sw edish univ ersity admissions and c haritable giving. Section 6 concludes. 2 Mo del and Assumptions 2.1 Setup Let there b e K treatmen ts (e.g., universit y programs) indexed b y j ∈ { 1 , . . . , K } and an outside option j = 0 . Individual i has treatmen t indicators A ij ∈ { 0 , 1 } for eac h treatment j , an outcome Y i , and is asso ciated with treatmen t-sp eciﬁc instruments Z ik for k = 1 , . . . , K . The second-stage equation is Y i = β 0 + K X j =1 β j A ij + ε i , (1) estimated by 2SLS using instruments ( Z i 1 , . . . , Z iK ) . W e suppress co v ariates and ﬁxed eﬀects Casc ade Identity 7 for clarity; all results extend immediately . Deﬁne the ﬁrst-stage matrix Π with entries π j k = ∂ E [ A ij ] ∂ Z ik , (2) so that π j k is the eﬀect of instrument k on enrollmen t in treatmen t j . Let D = diag ( π 11 , . . . , π K K ) denote the diagonal of Π and P = Π − D the oﬀ-diagonal part. The reduced-form co eﬃcient for instrument k is b y deﬁnition RF k = ∂ E [ Y i ] ∂ Z ik = K X j =1 β j π j k , (3) or in matrix form, RF = Π T β . 2.2 Capacit y-constrained allo cation W e consider settings where each treatmen t k has a ﬁxed num b er of slots, and these slots are allo cated from a ranked queue of applican ts. Assumption 1 (Instrument relev ance). π kk  = 0 for all k = 1 , . . . , K . Assumption 2 (Ranking-list reﬁll). An y change in program j ’s enrollment is mediated b y admitting or not admitting the marginal applican t on program j ’s ranking list. The instrumen t and the p olicy of expanding capacit y b y one slot operate on this same margin. Assumption 1 is the standard relev ance condition. Assumption 2 is an institutional condi- tion satisﬁed by construction in centralized admission systems (Sweden, Norw a y , Denmark, Chile), where b oth the instrument and capacity expansions shift the cutoﬀ along the same rank ed queue. It applies equally to sc ho ol c hoice mec hanisms with waitlists and other settings where programs ﬁll to capacity from rank ed lists. The alignment b et ween the instrumen t and the p olicy margin is not merely a technical con venience—it is precisely the prop ert y that Casc ade Identity 8 applied researchers in vok e when arguing for the external v alidity of instrumen tal v ariable estimates of, e.g., returns to ﬁeld of study . Kirkebø en et al. (2016), for instance, note that their estimates are “informative ab out p olicy that (marginally) c hanges the supply of slots in diﬀerent ﬁelds ,” app ealing implicitly to the same institutional logic that Assumption 2 formalizes. The result requires remark ably little structure b ey ond these tw o assumptions. T reatmen t indicators need not b e mutually exclusive: a student can b e “ever admitted” to multiple programs, and the cascade op erates on net enrollment ﬂows. W e imp ose no monotonicity condition of any form, and w e explicitly allo w cross-eﬀects—the oﬀ-diagonal elements π j k for j  = k can b e nonzero, as they generically are in capacit y-constrained systems. Nor do w e require that ﬂo ws balance: the column sums P j π j k need not equal zero, so that an instrumen t can dra w entran ts from outside the system as w ell as from other treatmen ts. 3 The Cascade Iden tit y 3.1 The p olicy exp erimen t Consider the p olicy of expanding program k b y one slot. The next applicant on k ’s ranking list is admitted. This admission generates t wo t yp es of eﬀects: 1. Direct eﬀect. The new entran t’s outcome changes. The ﬁrst-stage matrix tells us the comp osition of marginal admits: p er unit increase in Z k , there are π kk net admissions to program k . The reduced form RF k captures the direct eﬀect of one lottery win on the fo cal individual’s outcome. Per new slot, this is RF k /π kk . 2. Cascade eﬀect. The lottery also changes enrollment in other programs: π j k is the c hange in program j enrollmen t p er unit increase in Z k . When π j k < 0 (substitution), program j loses − π j k /π kk studen ts p er new k -admit. By ranking-list reﬁll (Assump- tion 2), eac h v acancy is ﬁlled b y the next applicant on j ’s queue, generating so cietal eﬀect T j . When π j k > 0 (complemen tarity), program j gains studen ts, crowding out Casc ade Identity 9 marginal applicants with so cietal eﬀect − T j . 3.2 The cascade equation Let T k denote the total so cietal eﬀect of expanding program k b y one slot, including all cascading reallo cations. The argument ab ov e yields the recursive equation: T k = RF k π kk + X j  = k − π j k π kk T j , k = 1 , . . . , K . (4) In matrix form: ( I + D − 1 P T ) T = D − 1 RF . (5) 3.3 Main result Prop osition 1 (Cascade Identit y) . Under Assumptions 1 and 2, the 2SLS c o eﬃcients e qual the so cietal p olicy eﬀe cts: T = β (6) That is, β k e quals the total so cietal eﬀe ct of exp anding pr o gr am k by one slot, including al l c asc ading r e al lo c ations thr ough the system. Pr o of. Substitute the reduced-form identit y RF = Π T β = ( D + P T ) β in to equation (5): ( I + D − 1 P T ) T = D − 1 ( D + P T ) β (7) = ( I + D − 1 P T ) β . (8) Since I + D − 1 P T is in vertible (it has nonzero diagonal and is diagonally dominan t in typical applications), T = β . R emark 1 (In tuition) . 2SLS computes β = (Π T ) − 1 RF , inv erting the full ﬁrst-stage matrix. Matrix inv ersion automatically accounts for all cross-program reallo cation eﬀects. The cascade iden tity is the dynamic reading of this matrix in version: the Neumann series ( I + D − 1 P T ) − 1 = Casc ade Identity 10 P ∞ n =0 ( − D − 1 P T ) n corresp onds to successive round s of the cascade (ﬁrst reﬁll, second reﬁll, etc.). App endix B.2 works through the t wo-program case in detail, connecting each term in the series to a sp eciﬁc round of the cascade. R emark 2 (Generalit y b eyond queues) . The pro of uses only tw o prop erties: the reduced-form iden tity RF = Π T β and the cascade equation (4) . The ﬁrst is deﬁnitional. The second follo ws from any allo cation mechanism in which (i) total supply of each go o d is ﬁxed and (ii) the instrumen t is the v ariable through which a marginal supply expansion is transmitted to individual allo cations. Rank ed queues are one such mechanism; comp etitiv e mark ets with price instruments are another. App endix B.1 states and prov es the cascade identit y for an arbitrary allo cation mec hanism, s ho wing that the ﬁxed-supply condition—not the institutional form of the mec hanism—is the essential assumption. 3.4 Homogeneous treatment eﬀects: the cascade as a zero-sum game T o build intuition for the general result, consider the b enchmark case where treatment eﬀects are homogeneous: ev ery individual gains ∆ j = Y ( j ) − Y (0) from enrollin g in program j , regardless of type. What happ ens when program k expands by one slot? The new entran t to program k ma y hav e come from another program j . Her net gain is ∆ k − ∆ j : she gains ∆ k from en tering k but loses ∆ j from lea ving j . Her v acated seat in j is ﬁlled by the next p erson on j ’s ranking list, who may in turn hav e come from program m , gaining ∆ j − ∆ m . This p erson’s departure from m is again ﬁlled from m ’s ranking list, and so on. The cascade terminates when a seat is ﬁlled b y someone from the outside option, who gains ∆ ℓ for whatever program ℓ she en ters. Under homogeneity , the in termediate terms telescop e. Summing along the chain: (∆ k − ∆ j ) + (∆ j − ∆ m ) + · · · + ∆ ℓ = ∆ k . Ev ery intermediate program cancels: it app ears once as a gain (for the p erson entering) and Casc ade Identity 11 once as a loss (for the p erson lea ving). The only term that surviv es is ∆ k , the eﬀect at the origin of the chain. The cascade is a zero-sum reallo cation at ev ery in termediate step, and the sole net eﬀect is that one additional p erson—wherev er she sits at the end of the chain—has b een drawn in to the system through program k ’s expansion. 1 Hence β k = ∆ k = Y ( k ) − Y (0) , the standard textb o ok interpretation. This makes precise what heterogeneous treatmen t eﬀects add. When treatmen t eﬀects diﬀer across individuals, the p erson who enters program j at one link of the cascade may gain more or less than the p erson who left j at the previous link. The intermediate terms no longer cancel. The 2SLS co eﬃcien t β k captures the full c hain of non-cancelling gains and losses. The cascade identit y T = β sa ys that 2SLS p erforms this accoun ting automatically: it aggregates the direct eﬀect and all downstream spillo vers into a single p olicy-relev an t n um b er. R emark 3 (What heterogeneit y do es) . Under homogeneous eﬀects, the cascade is inert— ev ery reallo cation is zero-sum—and β k reduces to the individual-lev el causal eﬀect. Under heterogeneous eﬀects, the cascade is active: eac h link con tributes a non-zero net eﬀect b ecause the p erson entering a program diﬀers from the p erson leaving. The magnitude of the cascade correction dep ends on (i) the exten t of cross-program reallo cation (the oﬀ-diagonal elements of Π ) and (ii) the degree of treatment eﬀect heterogeneity across complier types at each margin. When either is small, β k ≈ ∆ k ; when b oth are l arge, the cascade can substantially alter the p olicy-relev ant eﬀect. 3.5 Example using p otential outcomes T o build further in tuition, w e w ork through the simplest non-trivial case in a p oten tial outcomes framework: three m utually exclusive choices indexed j ∈ { 0 , 1 , 2 } , where j = 0 is the outside option (no higher education), j = 1 is a mid-tier program, and j = 2 is a more 1 When treatment indicators are not mutually exclusive, the terminal entran t in the cascade need not come from the outside option—she ma y already be enrolled in other programs. How ever, the second-stage equation is linear and additive in the treatmen t indicators, so under homogeneous eﬀects the incremen tal gain from adding program k is ∆ k regardless of which other programs the individual is already enrolled in. The telescoping argumen t go es through unc hanged. Casc ade Identity 12 comp etitiv e program. Each program has its o wn admission lottery . W e ask: what do es the 2SLS co eﬃcien t on program j actually measure, and do es it equal the so cietal p olicy eﬀect T j ? F ollowing the p oten tial outcomes framework used in e.g. Kirkebøen et al. (2016), a studen t’s outcome is Y = Y 0 + ( Y 1 − Y 0 ) 1 ( j = 1) + ( Y 2 − Y 0 ) 1 ( j = 2) , (9) where Y 0 , Y 1 , Y 2 are the p otential outcomes under eac h choice. The 2SLS second stage is Y = β 0 + β 1 A 1 + β 2 A 2 + ε, (10) with instruments Z 1 and Z 2 for each program’s lottery . T o make the cascade tractable by hand in this three-program example, w e imp ose π 21 = 0 . (11) This restriction says that the program 1 lottery do es not aﬀect program 2 enrollmen t. It is plausible in a one-shot application game where programs can b e ordered by selectivity: a studen t on the margin for the less comp etitive program 1 is t ypically far from the admission threshold for program 2. In our data, which span 2008–2021, the restriction need not hold—a studen t who loses the program 1 lottery could retak e the Sw edish Scholastic Aptitude T est and even tually gain admission to program 2. W e adopt it here purely b ecause it collapses the cascade to a single step and allows the full argumen t to b e traced easily b y hand. The general result (Prop osition 1) do es not require it: the cascade identit y β k = T k holds for an y ﬁrst-stage matrix, including cases where π 21  = 0 . Under the restriction, the only transitions induced by the tw o lotteries are: Casc ade Identity 13 0 → 1 a program 1 lottery win admits a student from outside higher education 0 → 2 a program 2 lottery win admits a student from outside higher education 1 → 2 a program 2 lottery win admits a student who w ould otherwise hav e attended program 1 Solving the moment conditions. The tw o reduced-form equations are RF 1 = β 1 π 11 + β 2 π 21 = β 1 π 11 , (12) RF 2 = β 1 π 12 + β 2 π 22 , (13) where the simpliﬁcation in (12) uses the restriction π 21 = 0 . Since the program 1 lottery mo ves studen ts only b etw een the outside option and program 1, the ﬁrst equation yields a clean W ald ratio: β 1 = RF 1 π 11 = E [ Y 1 − Y 0 | 0 → 1] . (14) This is a standard LA TE: the av erage gain from attending program 1 relative to no higher education, for students at the program 1 admission margin. F or program 2, solving (13) giv es β 2 = RF 2 π 22 − π 12 π 22 β 1 . (15) The ﬁrst term, RF 2 /π 22 , is the instrument-speciﬁc W ald ratio for program 2: the direct eﬀect p er new admit, ignoring cross-eﬀects. The second term is the cascade correction. Because program 2 lottery wins dra w some studen ts aw a y from program 1, we hav e π 12 < 0 , so − π 12 /π 22 > 0 is the rate at whic h new program 2 admits v acate seats in program 1. Each suc h v acancy is ﬁlled from program 1’s ranking list, generating a so cietal gain of β 1 . The cascade correction thus adds the do wnstream eﬀect of reﬁlling the seats left b ehind. Expressing the co eﬃcients in terms of complier t yp es. W e can no w substitute the complier-t yp e expressions for eac h comp onent. The program 2 lottery induces tw o transitions: Casc ade Identity 14 0 → 2 with probability p 02 and 1 → 2 with probabilit y p 12 . These give: RF 2 = p 02 E [ Y 2 − Y 0 | 0 → 2] + p 12 E [ Y 2 − Y 1 | 1 → 2] , (16) π 22 = p 02 + p 12 , (17) π 12 = − p 12 . (18) The last expression reﬂects that each 1 → 2 transition reduces program 1 enrollment by one. Substituting into (15): β 2 = p 02 E [ Y 2 − Y 0 | 0 → 2] + p 12  E [ Y 2 − Y 1 | 1 → 2] + E [ Y 1 − Y 0 | 0 → 1]  p 02 + p 12 . (19) The numerator makes the cascade visible at the level of individual treatmen t eﬀects. When the new program 2 seat is ﬁlled directly from outside higher education (probability p 02 ), so ciet y gains Y 2 − Y 0 for that student. When it is ﬁlled by a studen t previously enrolled in program 1 (probabilit y p 12 ), there are tw o gains: the student who mov es up gains Y 2 − Y 1 , and the p erson who ﬁlls the v acated program 1 seat from outside higher education gains Y 1 − Y 0 . Expression (19) is the cascade identit y . Expanding program 2 by one seat sets oﬀ the follo wing chain: 1. With probability p 02 / ( p 02 + p 12 ) , the new seat is ﬁlled directly from outside higher education. So ciet y gains Y 2 − Y 0 for that student. 2. With probabilit y p 12 / ( p 02 + p 12 ) , the new seat is ﬁlled b y a student previously enrolled in program 1. So ciety gains Y 2 − Y 1 for that student. But the v acated program 1 seat is then ﬁlled from outside higher education, generating a further gain of Y 1 − Y 0 . The cascade terminates after one step b ecause restriction (11) rules out an y further displace- men t: the program 1 lottery do es not aﬀect program 2 enrollment, so the reﬁlled program 1 seat cannot trigger further substitution. The 2SLS co eﬃcient β 2 is exactly the p er-seat so cietal eﬀect of this expansion, conﬁrming Prop osition 1 in this setting. Casc ade Identity 15 Without restriction (11) , a program 1 lottery win could dra w studen ts aw ay from pro- gram 2, generating a further round of substitution when program 2 reﬁlls its v acancy . The cascade would then require summing an inﬁnite series—which is what the matrix in version in Prop osition 1 accomplishes in general. The rank ed restriction collapses this to a one-step cascade and allows the full argumen t to b e traced b y hand. 4 Discussion 4.1 Relation to Bhuller and Sigstad (2024) Bh uller and Sigstad sho w that t wo conditions are necessary and suﬃcient for 2SLS with m ultiple treatments to yield p ositiv ely weigh ted av erages of individual-lev el eﬀects: av erage conditional monotonicit y and no cross-eﬀects. In capacity-constrained systems, the no- cross-eﬀects condition generically fails: winning a lottery for program k necessarily changes enrollmen t in alternativ es. Our result resolv es this apparent impasse by changing the target parameter. Rather than asking whether β k decomp oses in to a w ell-b eha ved av erage of individual β ki , w e ask whether β k answ ers a natural p olicy question: what is the so cietal eﬀect of adding one slot to program k ? The answer is y es, and it holds precisely b e c ause cross-eﬀects transmit the cascade through the system. 4.2 Relation to Kirkebøen et al. (2016) Kirk ebøen et al. decomp ose 2SLS into pairwise LA TEs using information ab out applican ts’ next-b est alternativ es. Their “irrelev ance” and “next-b est” conditions restrict whic h complier t yp es can arise. Our result is complemen tary: it pro vides a p olicy in terpretation of the aggr e gate β k without requiring decomp osition into complier-sp eciﬁc eﬀects, and applies to an y ﬁrst-stage matrix without institutional restrictions b eyond ranking-list reﬁll. One notable diﬀerence in interpretation regards the outside option. Kirk ebøen et al. argue Casc ade Identity 16 that pairwise ﬁeld-vs-ﬁeld comparisons are more p olicy-relev an t than comparisons against no enrollmen t, since most studen ts choose b etw een ﬁelds rather than b et w een a ﬁeld and nothing. The cascade identit y reframes this concern: β k is alwa ys ultimately anc hored to the outside option, but the Y (0) comparison falls not on the complier at the top of the cascade but on the terminal entran t — who ever ﬁlls the v acancy at the end of the reallo cation chain, typically in a less selectiv e program. The outside option is alwa ys relev ant; it is simply displaced to a diﬀeren t p erson. 4.3 Reduced form, W ald ratio, or m ulti-treatment 2SLS? A recurring theme in applied econometrics is that the reduced form — the direct eﬀect of the instrument on the outcome — is often more transparent and robust than the structural IV estimate. Angrist and Pisc hke (2009) adv o cate rep orting the reduced form alongside or instead of 2SLS, noting that it av oids w eak-instrument bias and do es not require monotonicity for a causal interpretation. Im b ens (2014) argues that in many applications “inten tion-to- treat or reduced-form estimates are often of greater in terest” than structural parameters. Chernozh uko v and Hansen (2008) show that inference based on the reduced form is v alid ev en under w eak identiﬁcation. With a single treatment and a single instrument, the W ald ratio RF k /π kk and the 2SLS co eﬃcien t β k are iden tical. The only choice is whether to rep ort the reduced form RF k or to rescale it b y the ﬁrst stage — a matter of scaling. With multiple treatments, how ev er, three distinct ob jects emerge: 1. The r e duc e d form RF k : the causal eﬀect of instrumen t Z k on Y , with no rescaling. 2. The instrument-sp e ciﬁc W ald r atio RF k /π kk : the reduced form rescaled b y the o wn ﬁrst stage. This gives the eﬀect p er unit of treatment k induced, but ignores an y cross-eﬀects of Z k on other treatments. 3. The multi-tr e atment 2SLS c o eﬃcient β k : obtained by in verting the full ﬁrst-stage matrix. This accounts for all cross-eﬀects across treatments. Casc ade Identity 17 The traditional “reduced form v ersus 2SLS” debate concerns the choice b et ween (i) and (ii)/(iii), which coincide when there is a single treatmen t. Our result concerns the choice b et w een (ii) and (iii), which diverge whenever multiple treatments generate cross-eﬀects in the ﬁrst stage. W e argue that this c hoice dep ends on whether treatmen ts are capacit y-constrained. When treatments are in unlimited supply , the p olicy eﬀect of inducing one more p erson in to treatment k through instrumen t Z k is T k = RF k π kk . (20) No one else is aﬀected: one more p erson taking medicine A do es not displace an yone from taking medicine B. The instrument-speciﬁc W ald ratio captures the full causal eﬀect, including all mediators (e.g. reduced/increased consumption of medicine B). The multi-treatmen t 2SLS co eﬃcien t β k answ ers a diﬀerent question in this setting — the eﬀect of treatment k holding all other treatments constant on system level — whic h may or may not corresp ond to a natural p olicy (see discussion b elow). The preference for simpler estimands is well-founded when treatments are uncapp ed. When treatments are slots allo cated from rank ed queues, the W ald ratio has t w o short- comings. First, it has no clear coun terfactual. The ratio RF k /π kk giv es the causal eﬀect of en tering programme k v ersus a mixture of alternatives — a fraction − π j k /π kk from each programme j , the remainder from the outside option — where the mixture is determined by the ﬁrst-stage cross-eﬀects. This is a w ell-deﬁned causal parameter, but the counterfactual is dep ending on cross-eﬀects and diﬃcult to communicate. Even under homogeneous treatmen t eﬀects, the W ald ratio conﬂates the eﬀect of programme k with the eﬀects of the programmes that the entran t w ould otherwise hav e attended. Second, the W ald ratio misses the cascade: the outcome c hanges exp erienced by every one else in the system who is reallo cated as a consequence of the new admission. The m ulti-treatmen t 2SLS co eﬃcient β k resolv es b oth problems. The cascade iden tity Casc ade Identity 18 (Prop osition 1) sho ws that β k = T k , the total so cietal eﬀect of expanding programme k b y one slot while the rest of the system con tinues at capacity . The estimate thus has a clean and simple causal p olicy interpretation. In general this system treatmen t eﬀect is very hard to translate to individual treatment eﬀects. How ever, under homogeneous treatment eﬀects, β k collapses further to Y ( k ) − Y (0) , the individual-level eﬀect of programme k relativ e to the outside option — precisely the parameter most researc hers hav e in mind. Rep orting only the W ald ratio in a capacit y-constrained setting b oth misrepresents the magnitude of the so cietal p olicy eﬀect and obscures the coun terfactual against which it is measured. Man y of the empirical settings where multi-treatmen t IV is used — universit y admissions (Kirk ebøen et al., 2016), sc ho ol choice (Ab dulk adiroğlu et al., 2017; W alters, 2018), medical residency matc hing, job training programmes with limited capacity — are precisely the settings where treatmen ts are rationed. In these contexts, m ulti-treatment 2SLS is not merely a conv enience for combining instrumen ts; it is the estimator that answ ers the p olicy question. The W ald ratio/reduced form tells you what happ ens to the p erson who gets luc ky in the lottery . The 2SLS co eﬃcient tells you what happ ens to so ciety when one more seat is created. These are diﬀerent questions, and the diﬀerence is the cascade. In eﬀect, β k implemen ts a c eteris p aribus condition at the system lev el: the so cietal eﬀect of expanding programme k , holding the rest of the system constant. In basic price theory , ceteris paribus is a useful abstraction that corresp onds to reality when other mark ets are appro ximately unaﬀected. Here, the condition holds not b y approximation but b y institutional design: capacity constraints, administrativ e budget separation, or p olicy c hoice ensure that other programmes contin ue at their current scale. Ho wev er, the relev ance of β k extends b ey ond strictly rationed settings. F ew real systems are purely capacity-constrained or purely uncapp ed. A hospital w ard has ﬁxed b eds in the short run but ma y expand ov er time in resp onse to demand. A univ ersity programme admits a ﬁxed cohort eac h year but adjusts its intak e p erio dically . What matters for the in terpretation of β k is not whether capacity is literally ﬁxed, but that expanding programme k do es not Casc ade Identity 19 c hange the resources av ailable to other programmes. This condition can b e guaran teed b y ph ysical capacity constraints, by administrative budget separation, or simply by p olicy choice — and in practice, most publicly funded systems exhibit some com bination of all three. Consider a p olicy maker ev aluating whether to expand a particular healthcare facilit y . Using the W ald ratio RF k /π kk , she ob tains the health improv emen t for the individual induced in to facility k , including an y substitution a w ay from other facilities. T o conv ert this in to a cost-b eneﬁt statemen t, she w ould need to account for the ﬁscal consequences of all cross- eﬀects: each patient who leav es another facilit y j reduces public sp ending there, while eac h patien t drawn into a complemen tary service increases it. In a realistic system, b oth patterns co exist — expanding a screening programme ma y reduce demand at comp eting diagnostic facilities while sim ultaneously increasing demand at do wnstream treatmen t facilities. A complete cost-b eneﬁt analysis based on the W ald ratio therefore requires knowledge of the marginal cost at every facility aﬀected by the cross-eﬀects, and the direction of the bias from ignoring these costs dep ends on whether substitution or complementarit y dominates. The 2SLS co eﬃcien t β k sidesteps this entirely . It gives the so cietal health improv emen t from expanding facility k b y one slot while other facilities contin ue at their curren t scale. The cost-b eneﬁt calculation then requires only one additional num b er: the marginal cost of a slot in facilit y k . If β k exceeds this cost, expansion is worth while. The p olicy maker needs no information ab out costs at other facilities, b ecause those budgets are held constant — whether b y binding capacity , b y institutional design, or b y choice. This makes β k a practical cost-b eneﬁt parameter across a wide range of publicly managed systems, not only those with strictly ﬁxed capacity . 4.4 Heterogeneous eﬀects in a capacit y-constrained system A natural question in an y treatment ev aluation is whether eﬀects diﬀer across subgroups — for example, b y gender, age, or family bac kground. In the standard LA TE framework with a single uncapp ed treatment, the approac h is straigh tforward: estimate the mo del separately Casc ade Identity 20 on eac h subgroup. The resulting co eﬃcien t identiﬁes the LA TE for compliers within that subgroup. The cascade identit y complicates this, b ecause the 2SLS co eﬃcient is not an individual-lev el parameter but a system-level one. The system do es not hav e a gender. 4.4.1 Why subsample estimation answers the wr ong question Consider estimating β k on women only — dropping all men from the sample and running 2SLS with the female subsample. The resulting co eﬃcient β f k in verts the women-only ﬁrst-stage matrix Π f and uses the women-only reduced form. By the cascade iden tity applied to this subsample, β f k equals the so cietal eﬀect of expanding programme k b y one women-only slot, where the entire downstream cascade also op erates exclusively through w omen. This is a ﬁctional p olicy . In the real admission system, when a w oman v acates a slot in programme j , the next p erson on j ’s ranking list ma y b e a man. The cascade is gender-blind from the second step onw ards. The women-only estimate imp oses a single-gender system that do es not exist, pro ducing a parameter that do es not corresp ond to an y implementable p olicy . T wo well-deﬁned heterogeneity questions can b e ask ed instead. W e describ e each in turn. 4.4.2 De c omp osing the p olicy eﬀe ct by sub gr oup The ﬁrst question is: when pr o gr amme k exp ands by one slot, how much of the total so cietal eﬀe ct ac crues to women versus men? This is answ ered by running full-sample 2SLS with group-sp eciﬁc outcomes as dep enden t v ariables. Deﬁne Y f i = f i · Y i and Y m i = (1 − f i ) · Y i , where f i is a female indicator. Estimate Y f i = α f 0 + K X j =1 β f ∗ j A ij + ε f i (21) b y 2SLS on the full sample, using the same instruments and endogenous v ariables as the main sp eciﬁcation. Deﬁne β m ∗ j analogously from the regression with Y m i as the dep endent v ariable. Casc ade Identity 21 Since Y f i + Y m i = Y i , linearity of 2SLS implies β f ∗ k + β m ∗ k = β k (22) for each k . The decomp osition is exact and additiv e. Both β f ∗ k and β m ∗ k inherit the cascade in terpretation from the full-sample sp eciﬁcation: the ﬁrst-stage matrix Π is the actual mixed- gender system, and the cascade op erates through mixed-gender queues at every step. W e are simply partitioning the outcome changes at eac h step into those exp erienced by w omen and those exp erienced by men. The co eﬃcient β f ∗ k answ ers: “of the total so cietal eﬀect of expanding programme k b y one slot, how muc h is due to changes in women’s outcomes?” This includes women who en ter programme k directly , women who enter other programmes through the cascade, and w omen who are displaced. It is a w ell-deﬁned p olicy parameter that requires no ﬁctional single-gender system. This approac h generalises immediately to any partition of the p opulation: age groups, paren tal income quartiles, prior education lev els. F or any exhaustive set of group indicators g ∈ { 1 , . . . , G } with P g 1 [ g i = g ] = 1 , deﬁne Y ( g ) i = 1 [ g i = g ] · Y i and estimate full-sample 2SLS for each group outcome. The co eﬃcien ts sum to the total: P g β ( g ) ∗ k = β k . 4.4.3 The eﬀe ct of c onditioning the mar ginal entr ant The second question is: what is the total so cietal eﬀe ct of admitting one mor e woman (sp e ciﬁc al ly) to pr o gr amme k , r ather than one mor e p erson of unsp e ciﬁe d gender? This question asks whether it matters for so ciety who the marginal en trant is. It is the closest analogue to the standard heterogeneous LA TE and requires a diﬀerent construction. When programme k admits one more w oman: • The direct eﬀect is on a woman. Her outcome c hanges b y the female-sp eciﬁc treatmen t eﬀect at the margin, captured b y the w omen-only reduced form and ﬁrst stage: RF f k /π f kk . Casc ade Identity 22 • She may v acate a slot in programme j . The probabilit y of this is determined by the w omen-only ﬁrst stage: − π f j k /π f kk . • The v acated slot in j is reﬁlled from j ’s mixed-gender ranking list. F rom this p oint on, the cascade is gender-blind. The so cietal eﬀect of one additional slot in j — regardless of who ﬁlls it — is the full-sample β j . The total eﬀect of admitting one more woman to programme k is therefore T | f k = RF f k π f kk + X j  = k − π f j k π f kk β j , (23) where RF f k and π f j k are the reduced form and ﬁrst stage estimated on women only , and β j is the full-sample 2SLS co eﬃcient. The ﬁrst term is female-sp eciﬁc: the direct eﬀect on the marginal w oman. The cascade correction uses female-sp eciﬁc v acancy creation rates (since it is a woman who p oten tially departs programme j ) but full-p opulation cascade eﬀects (since the reﬁll draws from the mixed-gender queue). All ingredien ts — the women-only ﬁrst stage and reduced form, and the full-sample β j — are standard output from t wo sets of 2SLS regressions. Comparing T | f k with T | m k (deﬁned analogously for men) rev eals whether the gender of the marginal en trant matters for the total so cietal eﬀect. The tw o can diﬀer for tw o reasons. First, the direct eﬀect may diﬀer: the marginal woman admitted to programme k ma y gain more or less than the marginal man w ould ( RF f k /π f kk  = RF m k /π m kk ). Second, the cascade triggered by a w oman may diﬀer from the cascade triggered b y a man, b ecause women and men tend to v acate diﬀeren t programmes when admitted to k ( π f j k  = π m j k ). F or example, if w omen admitted to a STEM programme disprop ortionately v acate slots in health sciences while men disprop ortionately v acate slots in business, the do wnstream reallo cation — and hence its so cietal eﬀect — diﬀers by gender ev en though the reﬁll at each step draws from the same mixed-gender queue. If neither c hannel is op erativ e ( T | f k = T | m k ), the p olicy maker expanding programme k need not b e concerned with the gender comp osition of the marginal Casc ade Identity 23 en trants. 5 Application: Proso cialit y and Field of Study Univ ersities ha ve long claimed that their purpose extends beyond the transmission of tec hnical skills to the cultiv ation of proso cial v alues, civic resp onsibility , and moral character. 2 The con tent of these commitments diﬀers systematically across ﬁelds. Programs in health care, so cial work, and education frame their mission around public service and human dev elopment, while programs in economics, business, and engineering emphasize comp etition, innov ation, and market-based problem solving. Whether these diﬀerences in mission translate into diﬀerences in the proso cial b eha vior of graduates is an op en empirical question with direct implications for education p olicy . A long-standing literature do cuments that economics and business students exhibit low er lev els of co op erativ e b eha vior, charitable giving, and trust relative to students in other ﬁelds (see, e.g., F rank et al., 1993; F rey and Meier, 2003; Bauman and Rose, 2011; Carter and Irons, 1991; Sundemo and Löfgren, 2025). These studies employ a range of iden tiﬁcation strategies— lab oratory exp erimen ts comparing economics and non-economics studen ts (Marw ell and Ames, 1981; F rank et al., 1993; Carter and Irons, 1991), natural exp erimen ts exploiting institutional features of universit y donation sc hemes (F rey and Meier, 2003), cross-sectional comparisons con trolling for selection on observ ables (Bauman and Rose, 2011), and panel designs trac king the same studen ts o ver time to separate within-individual c hange from cross-cohort diﬀerences (Sundemo and Löfgren, 2025). T wo comp eting explanations hav e nonetheless pro ved diﬃcult to disentangle. The ﬁrst is tr e atment : economics education itself shap es v alues b y reinforcing a mo del of human behavior premised on narrow self-in terest—the 2 Classical and progressive educational theory emphasizes education as character formation and preparation for democratic citizenship; see, e.g., Aristotle (1984, Bo ok VI I I) and Dew ey (1916). A large empirical literature studies the causal impact of education in general on proso ciality and civic engagement, typically exploiting compulsory schooling reforms or other sources of exogenous v ariation in years of schooling (Dee, 2004; Milligan et al., 2004; Glaeser et al., 2002; Helliwell and Putnam, 2007; Persson, 2015; Almén et al., 2025). Dee (2020) and Willec k and Mendelb erg (2022) pro vide recent surveys. Casc ade Identity 24 homo e c onomicus assumption—or by legitimizing self-interested action through the language of incentiv es and eﬃciency (Marwell and Ames, 1981; F rank et al., 1993). The second is sele ction : studen ts who choose economics are already less proso cially oriented before enrollmen t, and the ﬁeld attracts rather than creates self-interested individuals (Bauman and Rose, 2011; Girardi et al., 2024). What these approac hes hav e in common is that ﬁeld of study is either self-selected or, in the exp erimental designs, compared across studen ts who diﬀer on unobserv ed dimensions. F ully resolving the selection-v ersus-treatment debate requires exogenous v ariation in ﬁeld assignmen t itself—precisely what the Swedish admissions lottery provides. Our application addresses this question using the Sw edish centralized admissions lottery as a source of exogenous v ariation, and the cascade framework developed in Section 3 as the estimand. The cascade estimator is particularly w ell-suited to this setting b ecause the coun terfactual for a studen t marginally admitted to, sa y , economics is not “no higher education” but rather enrollmen t in whatever program they would hav e attended instead. A naiv e W ald ratio that ignores this substitution confounds the eﬀect of attending economics with the eﬀect of not attending a program that ma y itself increase proso cial b eha vior. The cascade iden tity resolves this b y estimating the full so cietal eﬀect of expanding each ﬁeld by one slot, including the downstream reallo cation of students across substitute programs that suc h an expansion triggers. W e measure proso ciality using annual data on c haritable giving dra wn from Swedish tax registers, whic h record all donations qualifying for tax deductions, 3 for the full p opulation of univ ersity applicants o ver 2008–2021. Giving is observ ed b oth b efore and after the admission decision, allo wing us to control for pre-existing levels and trends and isolate the causal eﬀect of ﬁeld attendance. As a rev ealed-preference measure—priv ate, v olun tary , and ﬁnancially 3 The tax deduction for charitable donations was a v ailable during 2012–2015 and reintroduced from 2019 on ward. Giving data are therefore observed for these years only . Since our admission sample co v ers 2008–2021, nearly all applicants hav e at least one post-admission observ ation of giving. How ev er, for applican ts admitted b efore 2012, we lack a pre-admission measure of giving. T o maintain a common sample across all sp eciﬁcations, w e set prior giving equal to zero for these applicants. Section A.3 shows that the main estimates are robust to alternativ e treatments of this v ariable. Casc ade Identity 25 costly—it is free from the so cial desirabilit y bias that plagues survey-based attitude measures. These data are linked to the full administrativ e record of applications and admission decisions main tained by Univ ersitets- o ch högskolerådet (UHR). In the Sw edish universit y admissions system, applicants submit a single rank ed list of programs and are admitted through a national clearing pro cess administered by UHR. A dmission is determined b y a merit score ( meritvär de ) based on upp er secondary grades, results from the Swedish Scholastic Aptitude T est ( Hö gskolepr ovet ), or prior undergraduate credits. Each program allo cates seats across parallel tracks corresp onding to these criteria. When qualiﬁed applican ts exceed a v ailable seats, the admission cutoﬀ falls at the low est merit score among admitted students within eac h trac k. Ties at the cutoﬀ are resolv ed by lottery , conducted indep enden tly within each program and admission round. The lottery pro duces a complete ranking of applican ts within each merit-score brack et, with p osition determined b y c hance alone. The lottery tie-breaking rule deﬁnes a natural set of pivotal gr oups : for eac h program and admission round, the pivotal group consists of all applicants whose merit score equals the admission cutoﬀ. Withi n this group, admission is determined by the lottery dra w alone, while applicants ab ov e the cutoﬀ are admitted with certain t y and those b elo w are rejected with certain ty . W e construct a luck v ariable equal to the applican t’s normalized rank within the pivotal group, which is uniformly distributed b et ween 0 and 1 and indep endent of all pre-determined c haracteristics conditional on group membership. Crucially , the lottery ranking also determines the marginal applican t in the natural p olicy exp eriment of expanding a program b y one seat: it is the highest-rank ed rejected applican t—the next in the lottery queue—who would b e admitted under a marginal capacit y expansion. This alignment b etw een the instrument and the p olicy margin is what gives the cascade estimator its institutional in terpretation. This design is closely related to the regression discontin uit y approach used in related w ork on Scandinavian higher education (Altmejd et al., 2021; Kirk ebøen et al., 2016), which Casc ade Identity 26 iden tiﬁes causal eﬀects by comparing applican ts just ab ov e and just b elo w the admission cutoﬀ. Our setting diﬀers in that applican ts within the same merit-score brack et are identic al on the running v ariable by construction, and admission among them is determined by lottery . Con tinuit y assumptions and bandwidth c hoices are therefore not required, and the instrument can b e treated as the outcome of a randomization within eac h pivotal group. Not all programs use pure lottery tie-breaking. Un til 2012, a small num b er of programs ap- plied gender quotas at the admission margin. Other programs resolve ties using Hö gskolepr ovet scores rather than random dra ws, generating a deterministic rather than sto c hastic margin. Both deviations are ﬂagged in the administrativ e data. W e exclude all aﬀected programs from the analysis, retaining only programs for which the within-pivotal-group luc k v ariable is consisten t with random assignment. The ﬁnal analytical sample comprises 11,604 piv otal groups across 28 admission rounds b etw een 2008 and 2021, with an av erage piv otal group size of approximately 12 applican ts. 5.1 Estimation W e group the 11,604 pivotal groups in our sample into seven mutually exclusiv e ﬁelds u sing the standard Swedish SUN classiﬁcation ( Standar d för svensk utbildningsnomenklatur ). The ﬁelds are: business including economics (SUN 34 and 314), so cial science (SUN 2–3 excluding business), p edagogy (SUN 1), medicine (SUN 721, 724, 727), health (remaining SUN 7), STEM (SUN 4–5), and a residual other category . App endix A.2 discusses the aggregation w eights and the conditions under whic h ﬁeld-level estimates inherit the cascade in terpretation from the underlying program-level eﬀects. The structural equation is Y i = β 0 + F X j =1 β j A ij + X ′ i γ + ε i , (24) where Y i is an indicator for charitable giving in the y ears following the admission decision, A ij Casc ade Identity 27 indicates enrollmen t in ﬁeld j , and X i con tains ﬁeld of application, prior giving, gender, age, y ear of admission, and application priority . W e estimate the system b y 2SLS on the sample of piv otal-group mem b ers, using ﬁeld-sp eciﬁc lottery instrumen ts Z ij = L i × 1 ( applied to ﬁeld j ) as excluded instruments, where L i = 1 − lottery rank i (1 + n g ) (25) is the applican t’s normalized lottery rank within pivotal group g of size n g . By construction, L i is symmetric and uniformly distributed on (0 , 1) within each piv otal group and therefore orthogonal to all v ariables ﬁxed within groups—including ﬁeld of application, year of admission, and merit score. Pivotal group ﬁxed eﬀects are consequen tly not required for iden tiﬁcation. Section A.3 v eriﬁes balance on predetermined individual c haracteristics and demonstrates robustness to alternative control sp eciﬁcations (including pivotal group ﬁxed eﬀects). By the cascade iden tity of Section 3, the co eﬃcients ˆ β j reco ver the full so cietal eﬀect of expanding ﬁeld j b y one seat, inclusive of all downstream student reallo cations across substitute ﬁelds. A central feature of the cascade framework is that the cascade eﬀect itself requires no additional estimation b eyond what is already standard practice. As established in Section 3, the diﬀerence b etw een the full 2SLS co eﬃcien t T k and the single-instrument W ald ratio W k = RF k /π kk reco vers exactly the do wnstream reallo cation eﬀect: T k − W k = X j  = k − π j k π kk T j . (26) Both ob jects are standard regression output: T k is the co eﬃcient from the full 2SLS system and W k is the co eﬃcien t from a single-instrument IV regression restricted to ﬁeld k ’s own lottery . Their diﬀerence, rep orted in column (5) of T able 1, measures the so cietal eﬀect of the reallo cation triggered by a marginal expansion of ﬁeld k —the eﬀect on every one else in the system, net of the direct eﬀect on the lottery winner. Casc ade Identity 28 5.2 Results Figure 1 displays the estimated ﬁrst-stage matrix. Eac h cell rep orts the eﬀect of a high lottery score in one ﬁeld (column) on the probability of ev entual admission (i.e. ever admitted during 2008-2021) to the same or another ﬁeld (row), measured ov er a tw o-to-t welv e year windo w following the lottery draw. The diagonal elements are large, p ositive, and uniformly signiﬁcan t: a high lottery score for a giv en ﬁeld substan tially raises the probabilit y of even tual admission to that same ﬁeld, conﬁrming strong compliance. The oﬀ-diagonal elemen ts are negativ e throughout, consisten t with the capacity-constrain t mechanism underlying the cascade identit y: a studen t who secures admission to one ﬁeld is displaced from others. The pattern of cross-eﬀects rev eals the substitution structure of the Sw edish higher education mark et. Business, so cial science, STEM, and medicine exhibit substantial mutual substitution, indicating that applicants to these comp etitiv e ﬁelds hold ov erlapping preferences and treat them as close alternatives. T eac hing and health, b y contrast, show negligible cross- eﬀects with other ﬁelds. Their applicant p o ols are largely distinct, and the coun terfactual for a rejected applicant is more likely to b e n o higher education at all rather than enrollmen t in a substitute ﬁeld. This reﬂects the low er a verage merit-score cutoﬀs for teaching and non-sp ecialist health programs, whic h dra w from a diﬀerent part of the applican t distribution than the more comp etitiv e ﬁelds. App endix A.1 shows a more granular table at the 3-digit SUN-lev el, revealing that there is also some substitution within ﬁelds. These cross-eﬀects are the structural inputs to the cascade estimator. The oﬀ-diagonal elemen ts of the ﬁrst-stage matrix determine b oth the aggregation weigh ts—ho w the p o oled IV estimate distributes across ﬁelds—and the cascade correction that separates the direct eﬀect of enrolling in a ﬁeld from the indirect eﬀect of displacing studen ts from substitute ﬁelds. The heatmap thus summarizes the identiﬁcation structure of the entire analysis: strong diagonals conﬁrm that each ﬁeld’s o wn lottery instrumen t provides credible v ariation, while the pattern of oﬀ-diagonal substitution maps the reallo cation mechanism through which a marginal capacity expansion propagates across the system. Casc ade Identity 29 Business So cial science T eaching Medicine Other health STEM Other Business So cial science T eaching Medicine Other health STEM Other Field applied to Field admitted to 0.267 (19.71) -0.038 (-7.07) -0.002 (-0.26) -0.009 (-1.55) 0.004 (1.05) -0.042 (-4.56) -0.014 (-0.71) -0.024 (-2.35) 0.200 (24.62) 0.003 (0.25) -0.016 (-1.85) -0.007 (-1.04) -0.022 (-2.37) -0.063 (-2.37) 0.000 (-0.07) -0.020 (-4.23) 0.326 (17.32) -0.003 (-0.78) -0.011 (-1.83) -0.007 (-1.28) -0.037 (-1.98) -0.008 (-2.22) 0.000 (0.09) -0.001 (-0.17) 0.183 (12.99) -0.012 (-1.72) -0.007 (-1.32) 0.009 (0.49) -0.010 (-1.76) -0.028 (-5.36) -0.013 (-1.66) -0.018 (-1.89) 0.215 (22.05) -0.012 (-1.98) -0.045 (-1.66) -0.061 (-5.39) -0.017 (-2.89) -0.011 (-1.25) -0.037 (-3.81) -0.009 (-1.35) 0.203 (18.15) -0.009 (-0.34) -0.011 (-2.28) -0.006 (-1.85) -0.008 (-1.50) -0.012 (-2.82) 0.001 (0.13) -0.005 (-1.01) 0.475 (14.72) Figure 1: First-stage matrix. Eﬀect of the instrument (lottery admission) on b eing admitted to a sp eciﬁc ﬁeld. Rows indicate ﬁelds admitted to ( D k ) and columns indicate ﬁeld applied to. Red = p ositiv e co eﬃcien t, blue = negativ e. F aded color = not signiﬁcant ( | t | < 1 . 96 ). t -v alues in parentheses. 149579 observ ations, t-statistics computed using clustered standard errors on 11,604 pivotal groups. T able 1 rep orts the main results. Column (1) reports the full 2SLS estimate T k , in terpreted as the total so cietal eﬀect of admitting one additional studen t to ﬁeld k —including b oth the direct eﬀect on the marginal entran t and all downstream reallo cation eﬀects through the capacit y-constrained system. Column (2) rep orts the o wn-instrument W ald ratio W k , which captures onl y the direct eﬀect on the lottery winner, ignoring what happ ens to displaced studen ts. Column (5) rep orts the cascade T k − W k . The results reveal substantial heterogeneity across ﬁelds. T eaching, medicine, health, and STEM all generate large and statistically signiﬁcant total eﬀects on charitable giving, with p oin t estimates ranging from 0.063 to 0.086. F or teaching, virtually the entire eﬀect is Casc ade Identity 30 direct: the cascade is small and insigniﬁcant (0.003), indicating that students displaced when a teaching seat op ens do not systematically sort in to ﬁelds with strong proso cial eﬀects. This is consisten t with the ﬁrst-stage evidence that teaching draws from a distinct applican t p o ol with few close substitutes—the counterfactual for a rejected teaching applicant is more likely no higher education than enrollment in another proso cial ﬁeld. F or medicine, health, and STEM, b y contrast, the cascade accounts for a meaningful share of the total eﬀect, ranging from 0.012 to 0.026, all statistically signiﬁcan t. Expanding these comp etitiv e ﬁelds displaces applican ts who w ould otherwise ha v e enrolled in other programs with positive proso cial eﬀects, and these programs recruit, in turn, individuals from outside higher education. This reallo cation adds to the total so cietal b eneﬁt of expanding these disciplines. The results for business and so cial science are particularly instructiv e for the debate on economics and proso ciality . The o wn-instrument W ald ratio for business ( W k = 0 . 004 ) is precisely estimated and statistically indistinguishable from zero. This is the cleanest a v ailable estimate of what attending business do es to a student’s proso cial b eha vior relativ e to their coun terfactual: it ﬁnds no eﬀect whatsoever. Studen ts who are marginally admitted to business programs give no more or less to c harity than they would ha ve had they attended their next-b est alternative. The total p olicy eﬀect of expanding business with one seat is, how ev er, p ositive ( T k = 0 . 028 ), and the diﬀerence is the cascade ( T k − W k = 0 . 023 , p < 0 . 01 ). Op ening one additional seat in business draws a studen t aw a y from their next-b est alternativ e—so cial science, medicine, or another ﬁeld with a p ositive proso cial eﬀect—and ﬁlling that v acancy generates a so cietal gain that the own-instrumen t estimate misses entirely . So cial science presents a similar but empirically sharp er pattern. The total p olicy eﬀect is p ositiv e and signiﬁcant ( T k = 0 . 047 , p < 0 . 01 ), y et the own-instrumen t W ald ratio is less than half as large and statistically insigniﬁcant ( W k = 0 . 021 ). The signiﬁcan t cascade ( T k − W k = 0 . 027 , p < 0 . 01 ) accoun ts for the diﬀerence. A researc her rep orting only T k w ould conclude that expanding so cial science raises c haritable giving; a researcher rep orting only W k w ould conclude it has no eﬀect. Both conclusions are correct—they answer diﬀerent Casc ade Identity 31 T able 1: The eﬀects of expanding ﬁelds of study on charitable giving β k W k ∆ b k F -stat N (Cascade) (1) (2) (3) (4) (5) Business .0278 .00449 391 19882 .0233 ∗∗∗ (.0175) (.0143) (.00829) So cial science .0476 ∗∗∗ .0207 606 47011 .0269 ∗∗∗ (.018) (.0152) (.00881) T eaching .0749 ∗∗∗ .0713 ∗∗∗ 300 11102 .00358 (.0284) (.0274) (.00545) Medicine .0854 ∗∗∗ .0598 ∗∗ 166 21169 .0256 ∗∗ (.0287) (.0258) (.0102) Health .0618 ∗∗∗ .0495 ∗∗ 488 26867 .0123 ∗∗ (.0224) (.0214) (.00545) STEM .064 ∗∗∗ .0453 ∗∗ 332 20358 .0187 ∗∗ (.0229) (.0195) (.00848) Other -.0185 -.0376 ∗ 214 3190 .0191 ∗ (.0226) (.0205) (.00987) ¯ β 0.0490 SE( ¯ β ) 0.0127 KP F-stat 34.042 Join t χ 2 (b etas=0) 26.011 Observ ations 149579 Notes: Outcome v ariable is a binary indicator equal to one if studen t donated a p ositive amount to c harity in the past year, zero otherwise. Standard errors in parentheses. Standard errors for β k and b k are clustered at the pivotal group lev el. Standard errors for the cascade are computed b y cluster b o otstrap (1000 reps), resampling at the level of pivotal groups. The F-statistic refers to the Kleib ergen–P aap ﬁrst-stage F-statistic, testing the join t signiﬁcance of the excluded instrumen ts. ∗ p < 0 . 10 , ∗∗ p < 0 . 05 , ∗∗∗ p < 0 . 01 . Casc ade Identity 32 questions. The cascade decomp osition rev eals that the p olicy eﬀect of expanding so cial science op erates primarily through the reallo cation of displaced studen ts into ﬁelds with stronger proso cial eﬀects, rather than through the direct eﬀect of so cial science attendance itself. This is precisely the confound that a single-instrument W ald estimate cannot disen tangle, and that the cascade framework is designed to resolve. The broader implication for the literature is that studies estimating the cost of economics or business education in proso cial terms—t ypically b y comparing economics studen ts to studen ts in other ﬁelds—are implicitly measuring a comparison that includes not just the eﬀect of attending business but the eﬀect of not attending the alternative. Our results suggest the latter is doing most of the w ork in these studies, with large proso cial eﬀects in particularly p edagogics, health and medicine. 5.3 The ov erall eﬀect of higher education The ﬁeld-level estimates in T able 1 are uniformly p ositive (except for “other”), suggesting that higher education broadly increases proso cial b ehavior. The a verage ˆ β k of 0.049 is statistically signiﬁcan t, providing causal evidence in fav or of the long-standing view that expanding univ ersity education cultiv ates moral character b eyond the transmission of technical skills. A natural question is ho w this av erage—which reﬂects the mean eﬀect of expanding each ﬁeld b y one slot—relates to what a researcher would typically estimate when asking whether higher education in general increases proso cial b ehavior: regressing the outcome on a single indicator for an y enrollment, instrumen ted by L i or a set of ﬁeld-sp eciﬁc interactions. This is the standard single-regressor lo cal av erage treatment eﬀect (LA TE). App endix A.2 discusses the relationship betw een the a v erage of β 1 , . . . , β K and this po oled 2SLS estimate, and T able 2 rep orts the results. Casc ade Identity 33 5.4 Heterogeneit y b y gender The conditional heterogeneit y formula in equation (23) allo ws the cascade to b e decomp osed b y the gender of the marginal en trant. T able 5 in App endix A.4 rep orts T | f k and T | m k for eac h ﬁeld. The ﬁeld distribution of signiﬁcant eﬀects diﬀers substantially b y gender: for w omen, signiﬁcant eﬀects are concen trated in so cial science and health; for men, in STEM and teaching. The gender diﬀerences in total p olicy eﬀects are statistically signiﬁcan t for STEM (fa voring men) and health (fav oring women). In b oth cases the cascade correction is quantitativ ely imp ortan t, illustrating that the gender-sp eciﬁc p olicy eﬀect dep ends b oth on the direct eﬀect on the marginal en trant and on the reallo cation their admission triggers do wnstream. 6 Conclusion The prev ailing view in the econometric literature is that 2SLS with m ultiple treatments yields co eﬃcien ts that are diﬃcult to interpret under heterogeneous treatment eﬀects. W e show that in capacit y-constrained allo cation systems—a class that includes universit y admissions, sc ho ol choice, medical residency matching, public housing, and other rationed settings—the 2SLS co eﬃcien t β k has a direct p olicy in terpretation: the total so cietal eﬀect of expanding treatmen t k b y one slot, including all cascading reallo cations through th e system. The result is an algebraic iden tity that holds for an y ﬁrst-stage matrix, requires only instrumen t relev ance and that the allo cation mechanism transmits supply expansions through the instruments, and imp oses no restrictions on treatment eﬀect heterogeneity or individual c hoice b ehavior. The iden tity applies not only to queue-based systems but to an y allo cation mechanism op erating o ver go o ds with ﬁxed supply , including comp etitive markets with price instruments. F or applied researc hers working in capacit y-constrained settings, the practical implication is straigh tforward. Multi-treatment 2SLS is not merely a conv enience for combining instruments— it is the estimator that answers the p olicy question of whether to expand capacit y . The Casc ade Identity 34 cascade decomp osition into o wn eﬀects and downstream reallo cation eﬀects requires no additional estimation b eyond what is already standard practice: the diﬀerence b etw een the m ulti-treatment 2SLS co eﬃcient and the single-instrument W ald ratio recov ers the cascade directly . W e recommend that applied pap ers in these settings rep ort b oth ob jects alongside the reduced form, making the cascade contribution transparent. The result has limitations. Assumption 2 requires that the instrument and the p olicy op erate on the same margin—a condition satisﬁed by construction in centralized systems but p oten tially violated in decentralized settings where v acancies are ﬁlled through diﬀerent c hannels than those generating the instrument v ariation. The ﬁxed-supply condition is essen tial: if competing programs can expand endogenously in resp onse to the demand pressure created b y the cascade, the identit y breaks do wn. In practice, most publicly funded systems enforce ﬁxed supply through administrative budget separation, regulatory constrain ts, or p olicy choice, but the assumption should b e assessed case by case. Finally , the cascade iden tity c haracterizes a system-level p olicy eﬀect that is in general diﬃcult to decomp ose in to individual-lev el treatment eﬀects. Under homogeneous eﬀects, the system-lev el and individual-lev el parameters coincide; under heterogeneit y , they diverge, and the researcher m ust b e clear ab out whic h question is b eing answered. Our empirical application demonstrates the cascade framework using Sw edish universit y admissions and c haritable giving. The results show that the cascade correction is quan titativ ely imp ortan t: for comp etitive ﬁelds like business and so cial science, the entire p olicy eﬀect of expansion op erates mainly through the do wnstream reallo cation of displaced studen ts rather than through the direct eﬀect on the marginal en trant. This ﬁnding reframes the long-standing debate on whether economics education ero des proso cial v alues: the apparen t proso cialit y gap b et ween economics studen ts and others is driv en primarily by the proso cial eﬀects of the ﬁelds that economics students w ould otherwise hav e attended, not b y an y negativ e eﬀect of economics itself. The Swedish admissions lottery pro vides a uniquely clean setting for applying the cascade Casc ade Identity 35 framew ork, and sev eral natural extensions suggest themselves. Extending the data back to the mid-1990s would allow the cascade identit y to b e applied to long-run outcomes such as p ermanen t income, entrepreneurship, paten ting, and fertilit y . Since the resulting estimates are directly in terpretable as p olicy parameters—the total so cietal eﬀect of expanding each ﬁeld by one slot—they pro vide exactly the empirical inputs needed to ev aluate which ﬁelds should b e expanded to promote growth and innov ation, questions that are central to higher education p olicy but hav e so far lack ed a credible causal framew ork. Casc ade Identity 36 References Ab dulk adiroğlu, Atila, Josh ua D. Angrist, Y usuk e Narita, and P arag A. P athak (2017) “Re- searc h Design Meets Mark et Design: Using Cen tralized Assignment for Impact Ev aluation,” Ec onometric a , 85 (5), 1373–1432, 10.3982/ECT A13925. Agarw al, Nikhil (2015) “An Empirical Mo del of the Medical Match,” Americ an Ec onomic R eview , 105 (7), 1939–1978. 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Casc ade Identity 41 A Application app endix A.1 Heatmap with more gran ular programs T ech. intro. Educ. (gen.) Pedagogy Early child. tchg Primary tchg Secondary tchg V ocational tchg Educ. (other) Arts & Media Humanities Soc. & Behav. Sci. Psychology Sociology Pol. Science Economics Journalism Business Law Biology & Env. Physics & Chem. Mathematics ICT Engineering (gen.) Mech. Engineering Energy & Electr. Electronics & IT Chem. & Biotech. V ehicle Eng. Industrial Econ. Env. Engineering Materials Civil Engineering Agriculture V eterinary Healthcare (gen.) Medicine Nursing Dentistry Health T echnology Therapy & Rehab. Pharmacy Healthcare (other) Social W ork Personal Services T ransport Occup. Safety Security Services T ech. intro. Educ. (gen.) Pedagogy Early child. tchg Primary tchg Secondary tchg V ocational tchg Educ. (other) Arts & Media Humanities Soc. & Behav. Sci. Psychology Sociology Pol. Science Economics Journalism Business Law Biology & Env. Physics & Chem. Mathematics ICT Engineering (gen.) Mech. Engineering Energy & Electr. Electronics & IT Chem. & Biotech. V ehicle Eng. Industrial Econ. Env. Engineering Materials Civil Engineering Agriculture V eterinary Healthcare (gen.) Medicine Nursing Dentistry Health T echnology Therapy & Rehab. Pharmacy Healthcare (other) Social W ork Personal Services T ransport Occup. Safety Security Services Lottery instrument (column) → ﬁrst-stage outcome (row) Figure 2: First-stage matrix for more granular ﬁeld deﬁnitions. E ac h cell depicts the sign and strength of the lottery instrumen t on b eing admitted to each program (ﬁrst-stage co eﬃcien ts). Red = p ositive, blue = negativ e. F aded = not signiﬁcant ( | t | < 1 . 96 ). Casc ade Identity 42 A.2 Aggregation across blo cks of programs Programs are often group ed in to coarser blo cks —for example ﬁelds of study—either b ecause program-lev el instruments are unav ailable or b ecause the researc h question concerns broader categories. In practice, grouping is also necessary for inference, computation, and in terpreta- tion: with thousands of programs, estimating a fully disaggregated system is infeasible, and ev en where feasible, the resulting program-level estimates are to o noisy to b e informative. The standard approach is to assume treatmen t homogeneity within blo cks–that is, β m = β B for all m ∈ J B —under which the blo c k co eﬃcient inherits the T-in terpretation directly: it equals the so cietal eﬀect of expanding the blo c k by one slot, since all programs within the blo c k hav e the same p olicy eﬀect by assumption. Let A im denote enrollment in program m for m = 1 , . . . , M . F or a blo c k B consisting of programs J B ⊆ { 1 , . . . , M } deﬁne the blo ck treatment as A iB = X m ∈J B A im , (27) whic h is binary when programs are mutually exclusiv e. The most gran ular sp eciﬁcation is Y i = β 0 + M X m =1 β m A im + u i , (28) while the blo c k-level sp eciﬁcation replaces the program indicators with the blo ck treatment, Y i = β 0 + β B A iB + v i . (29) When b oth sp eciﬁcations are estimated on the same sample with the same controls, the blo c k co eﬃcien t can b e written as a w eighted av erage of the underlying program co eﬃcien ts, β B = X m ∈J B ω m | B β m , ω m | B = Co v ( ˜ Z B , ˜ A im ) P j ∈J B Co v ( ˜ Z B , ˜ A ij ) , (30) Casc ade Identity 43 where tildes denote residuals after partialling out included exogenous v ariables. Under treatmen t homogeneit y within blo c ks, β B = β m for all m ∈ J B , and the w eighted a verage collapses to the common v alue regardless of the w eights. More generally , without homogeneit y , the blo ck co eﬃcient aggregates program-sp eciﬁc p olicy eﬀects with w eights determined by ho w strongly the blo c k-lev el instrument shifts enrollmen t into eac h constituent program. Without program-sp eciﬁc instruments—when a single p o oled instrumen t is used—the w eights in equation (30) can b e negativ e. This o ccurs when monotonicity fails: if applican ts winning a comp etitiv e program v acate a secondary program within the same blo ck at a higher rate than the instrument directly ﬁlls it, the blo ck co eﬃcien t remains a well-deﬁned linear com bination of the underlying p olicy eﬀects but need not lie b et ween them, and its p olicy in terpretation b ecomes diﬃcult to c haracterize. A natural c hoice of blo ck instrumen t is the v ector of program-sp eciﬁc instruments Z im = luc k i × 1 ( applied to m ) , one for each program in the blo ck. Using these interacted instrumen ts and including program-of-application dummies 1 ( applied to m ) as exogenous con trols eliminates cross-program con tamination and ensures strictly p ositive weigh ts as in equation (32). The blo ck IV estimator is β B = Co v ( ˜ Z eﬀ B , ˜ Y ) Co v ( ˜ Z eﬀ B , ˜ A iB ) , (31) where ˜ Z eﬀ B is the ﬁtted v alue from pro jecting A iB on to the in teracted instrumen ts and con trols. Because Z im is zero for all applicants outside program m ’s piv otal group, the cross-program co v ariances Co v ( ˜ Z im , ˜ A ij ) are zero for j  = m after partialling out the program-of-application dummies. The weigh ts in equation (30) therefore reduce to ω m | B = π mm P j ∈J B π j j , (32) where π mm = Co v ( ˜ Z im , ˜ A im ) is the direct ﬁrst-stage eﬀect of program m ’s o wn instrument on enrollment in program m . These weigh ts are strictly p ositiv e and sum to one, so β B is a Casc ade Identity 44 prop er w eighted av erage of the program-sp eciﬁc p olicy eﬀects β m . When the blo ck cov ers all programs in the system, the p o oled IV estimator with program- sp eciﬁc instruments has a particularly clean interpretation. The ﬁrst stage of each interacted instrumen t Z im on total enrollmen t A iB is zero for inframarginal studen ts—those who would enroll somewhere regardless of any individual lottery outcome—and p ositive only for studen ts on the extensive margin of higher education. The blo ck co eﬃcien t therefore measures the so cietal eﬀect of expanding univ ersity capacity for studen ts who w ould not otherwise attend, with weigh ts prop ortional to ho w man y such studen ts eac h program dra ws from outside higher education altogether. A program that attracts only studen ts who w ould otherwise ha ve enrolled elsewhere receiv es zero w eight, even if its program-sp eciﬁc p olicy eﬀect β k is large: it con tributes to the cascade within higher edu cation but not to the extensive margin that this estimator identiﬁes. T able 2 rep orts this estimate under three instrument sets of increasing granularit y . The co eﬃcien t on an y higher education enrollment is stable across columns—0.055, 0.050, and 0.044—and remains highly signiﬁcant throughout, despite the Kleib ergen-Paap F-statistic falling from 299 with a single p o oled instrument to 10 with 47 subﬁeld instrumen ts. This stabilit y is informativ e b eyond mere robustness: under treatment homogeneity , all three sp eciﬁcations iden tify the same parameter, and disagreemen t across instrumen t sets w ould signal either instrument inv alidit y or heterogeneous treatmen t eﬀects at diﬀerent margins of the distribution—the o veriden tiﬁcation logic of Angrist and Pisc hk e (2009) and the monotonicit y diagnostics of Hec kman and V ytlacil (2005). The consistency of the estimates across instrumen t designs with v ery diﬀerent ﬁrst-stage structures is therefore evidence that monotonicit y holds within the full blo ck of higher education and that the estimated eﬀect is not an artifact of a particular instrument’s identifying margin. Expanding higher education capacit y along the extensive margin increases charitable giving by approximately 0.05, an eﬀect that is robust to the choice of instrument gran ularity . Casc ade Identity 45 T able 2: Aggregation: eﬀect of any higher education (1) (2) (3) An y higher education 0.0551 ∗∗∗ 0.0501 ∗∗∗ 0.0440 ∗∗∗ (0.0134) (0.0130) (0.0117) Prior giving 0.461 ∗∗∗ 0.461 ∗∗∗ 0.456 ∗∗∗ (0.00898) (0.00898) (0.00895) F emale 0.0332 ∗∗∗ 0.0330 ∗∗∗ 0.0335 ∗∗∗ (0.00109) (0.00108) (0.00108) Age 0.00301 ∗∗∗ 0.00299 ∗∗∗ 0.00267 ∗∗∗ (0.000122) (0.000120) (0.000119) Instrumen ts Single (luck) 7 ﬁelds 47 subﬁelds Field FE Y es Y es Y es Y ear FE Y es Y es Y es KP F-statistic 298.843 57.733 10.077 Observ ations 149579 149579 149579 Notes: The dependent v ariable is an indicator for charitable giving in the years follo wing the admission decision. The endogenous v ariable is enrollmen t in an y higher education program. Column (1) uses a single po oled lottery instrument; column (2) uses sev en ﬁeld-sp eciﬁc lottery instruments interacted with ﬁeld-of- application dummies; column (3) uses 47 subﬁeld-sp eciﬁc lottery instrumen ts in teracted with subﬁeld-of-application dummies (see App endix A.1 for the ﬁrst-stage matrix). All sp eciﬁcations include ﬁeld-of-application dummies, y ear ﬁxed eﬀects, prior giving, gender, age, and application priorit y as controls. Standard errors clustered by pivotal group. KP F-statistic is the Kleib ergen- P aap rk W ald F-statistic for w eak identiﬁcation. Casc ade Identity 46 A.3 Balance and Robustness The balance test in T able 3 v eriﬁes that that the instrument is orthogonal to individual-level predetermined characteristics. W e regress eac h con trol v ariable on L i without ﬁxed eﬀects, whic h is the appropriate sp eciﬁcation giv en that the instrument is already normalized within groups. All four co eﬃcients are small and statistically indistinguishable from zero, and the join t F-statistic of 0.63 ( p = 0 . 64 ) conﬁrms that the lottery is as go o d as randomly assigned. T able 3: Balance test Co eﬃcien t on L i Prior giving 0.000115 (0.00117) F emale -0.00317 (0.00425) Age -0.0480 (0.0484) Application priority 0.0252 (0.0257) Join t F-statistic 0.630 Join t p-v alue 0.641 Observ ations 149579 Notes: Each row rep orts the coeﬃcient from a separate OLS regression of the row v ariable on the lottery instrumen t L i , clus- tered b y pivotal group. The join t F-statistic tests the n ull that all coeﬃcients are simultaneously zero. T able 4 examines the sensitivit y of the main estimates (ﬁrst column of T able 1) to the inclusion of controls and ﬁxed eﬀects. Column (1) includes ﬁeld-of-application dummies only . Columns (2) and (3) add prior giving and then the full set of demographic controls. Column (4) adds year ﬁxed eﬀects and column (5) absorbs piv otal group ﬁxed eﬀects via within-group demeaning. The ﬁeld-lev el co eﬃcients are stable throughout: p oin t estimates shift by at most a few p ercen tage p oints across sp eciﬁcations and signiﬁcance patterns are Casc ade Identity 47 unc hanged. The Kleib ergen-Paap F-statistic remains close to 34 in all columns, conﬁrming that the instrument strength is unaﬀected by the inclusion of controls—as exp ected giv en the orthogonalit y established in T able 3. Among the con trol v ariables, prior giving is strongly p ersisten t, w omen give more than men, giving increases with age, and applican ts who ranked the piv otal program lo wer in their preference list giv e slightly less—a pattern consisten t with w eaker attachmen t to the ﬁeld rather than a threat to iden tiﬁcation. Casc ade Identity 48 T able 4: Robustness to con trol function (1) (2) (3) (4) (5) Business 0.0317 ∗ 0.0274 0.0280 0.0278 0.0281 (0.0179) (0.0173) (0.0175) (0.0175) (0.0174) So cial science 0.0454 ∗∗ 0.0483 ∗∗∗ 0.0475 ∗∗∗ 0.0476 ∗∗∗ 0.0479 ∗∗∗ (0.0189) (0.0182) (0.0180) (0.0180) (0.0181) T eaching 0.0722 ∗∗ 0.0722 ∗∗ 0.0747 ∗∗∗ 0.0749 ∗∗∗ 0.0733 ∗∗∗ (0.0331) (0.0285) (0.0284) (0.0284) (0.0284) Medicine 0.0782 ∗∗∗ 0.0767 ∗∗∗ 0.0853 ∗∗∗ 0.0854 ∗∗∗ 0.0829 ∗∗∗ (0.0290) (0.0285) (0.0286) (0.0287) (0.0287) Health 0.0606 ∗∗∗ 0.0580 ∗∗ 0.0618 ∗∗∗ 0.0618 ∗∗∗ 0.0611 ∗∗∗ (0.0231) (0.0225) (0.0224) (0.0224) (0.0224) STEM 0.0739 ∗∗∗ 0.0638 ∗∗∗ 0.0644 ∗∗∗ 0.0640 ∗∗∗ 0.0646 ∗∗∗ (0.0231) (0.0226) (0.0229) (0.0229) (0.0229) Other -0.0191 -0.0248 -0.0184 -0.0185 -0.0199 (0.0255) (0.0228) (0.0226) (0.0226) (0.0226) Prior giving 0.482 ∗∗∗ 0.447 ∗∗∗ 0.461 ∗∗∗ 0.451 ∗∗∗ (0.00937) (0.00911) (0.00901) (0.00916) F emale 0.0330 ∗∗∗ 0.0321 ∗∗∗ 0.0324 ∗∗∗ (0.00298) (0.00295) (0.00214) Age 0.00277 ∗∗∗ 0.00293 ∗∗∗ 0.00172 ∗∗∗ (0.000180) (0.000178) (0.000157) Application priority -0.00291 ∗∗∗ -0.00277 ∗∗∗ -0.00207 ∗∗∗ (0.000289) (0.000277) (0.000354) Piv otal group FE No No No No Y es Y ear FE No No No Y es No KP F-statistic 34.687 34.766 34.036 34.042 34.529 Observ ations 149579 149579 149579 149579 149579 Notes: All sp eciﬁcations include ﬁeld-of-application dummies. Standard errors clustered b y pivotal group. Y ear ﬁxed eﬀects are subsumed b y pivotal group ﬁxed eﬀects in column (5). Casc ade Identity 49 A.4 Estimates across gender T able 5 rep orts gender-sp eciﬁc p olicy eﬀects computed follo wing equation (23) . Columns (1) and (2) rep ort T | f k and W | f k for women; columns (3) and (4) rep ort the corresp onding estimates for men; column (5) rep orts the diﬀerence. The most notable pattern is that the ﬁelds generating signiﬁcan t total p olicy eﬀects diﬀer substantially b y gender. F or women, the signiﬁcan t eﬀects are concentrated in so cial science ( T | f k = 0 . 072 ) and health ( T | f k = 0 . 082 ), with medicine also p ositive and signiﬁcant ( T | f k = 0 . 076 ). F or men, the signiﬁcant eﬀects app ear in STEM ( T | m k = 0 . 096 ) and teac hing ( T | m k = 0 . 153 ), though the latter estimate is based on a smaller male subsample and should b e in terpreted with caution. Business is indistinguishable from zero for b oth genders. The gender diﬀerences in column (5) are statistically signiﬁcan t for STEM ( − 0 . 089 , fa voring men) and health ( 0 . 091 , fa voring women), with so cial science also showing a p ositiv e diﬀerence fa v oring women ( 0 . 065 ). Within the cascade framework, these estimates capture the full so cietal eﬀect of admitting one more w oman or man to each ﬁeld, inclusive of downstream reallo cations. The decomp osi- tion in to own eﬀects (columns 2 and 4) and cascade contributions is informative. F or women in so cial science, the own eﬀect ( W | f k = 0 . 045 ) is smaller than the total ( T | f k = 0 . 072 ), with the diﬀerence of 0.027 reﬂecting the cascade: a w oman admitted to so cial science displaces studen ts into other ﬁelds whose admission generates additional proso cial gains, w eigh ted by the full-p opulation β j . F or men in STEM, the own eﬀect ( W | m k = 0 . 067 ) likewise falls short of the total ( T | m k = 0 . 096 ), with a cascade con tribution of 0.029. The cascade correction uses full-p opulation p olicy eﬀects for the displaced students regardless of gender, since the reﬁll dra ws from the mixed-gender queue. These results illustrate that the gender-sp eciﬁc p olicy eﬀect of expanding a ﬁeld dep ends b oth on what the marginal entran t of that gender gains directly and on the comp osition of the reallo cation their admission sets in motion. Casc ade Identity 50 T able 5: Gender-sp eciﬁc p olicy eﬀects W omen Men T | f k W | f k T | m k W | m k T | f k − T | m k (1) (2) (3) (4) (5) Business 0.0255 0.00326 0.0334 0.00942 -0.00790 (0.0276) (0.0251) (0.0206) (0.0177) (0.0316) So cial science 0.0719 ∗∗∗ 0.0445 ∗∗ 0.00686 -0.0182 0.0650 ∗∗ (0.0246) (0.0217) (0.0231) (0.0202) (0.0313) T eaching 0.0547 ∗ 0.0502 0.153 ∗∗ 0.148 ∗∗ -0.0988 (0.0323) (0.0318) (0.0701) (0.0677) (0.0789) Medicine 0.0761 ∗∗ 0.0532 ∗ 0.100 ∗ 0.0734 -0.0243 (0.0338) (0.0315) (0.0538) (0.0478) (0.0600) Health 0.0819 ∗∗∗ 0.0707 ∗∗∗ -0.00918 -0.0262 0.0911 ∗∗ (0.0277) (0.0265) (0.0356) (0.0333) (0.0447) STEM 0.00719 0.00347 0.0957 ∗∗∗ 0.0669 ∗∗∗ -0.0885 ∗ (0.0371) (0.0349) (0.0293) (0.0247) (0.0462) Other -0.0222 -0.0489 ∗ -0.0107 -0.0121 -0.0115 (0.0324) (0.0285) (0.0314) (0.0280) (0.0479) Observ ations Notes: Bo otstrap standard errors in paren theses, 1000 replications clustered b y pivotal group. T | g k is the total so cietal eﬀect of admitting one more woman (men) to ﬁeld k , computed using gender-sp eciﬁc ﬁrst stages and reduced forms with full-p opulation cascade w eights β j . W | g k = RF g k /π g kk is the gender-speciﬁc own-instrumen t W ald ratio. The ﬁnal column rep orts the gender diﬀerence in total policy eﬀects. Casc ade Identity 51 B Theory app endix B.1 The cascade identit y under general allo cation mechanisms The cascade identit y deriv ed in Section 3 is stated in the language of ranked queues, but its logic do es not dep end on the allo cation mechanism. This app endix prov es the result for an arbitrary mechanism, clarifying what is essen tial and what is institutional detail. Setup. There are K go o ds with ﬁxed total supply ¯ Q 1 , . . . , ¯ Q K and an outside option with unrestricted supply . Individuals are allo cated quantities Q ik through some mechanism parameterized by v ariables Z = ( Z 1 , . . . , Z K ) . In a queue system, Z k is the lottery rank or cutoﬀ for program k . In a mark et, Z k is the price of go o d k . In a matc hing algorithm, Z k ma y b e a priorit y score. The mechanism is otherwise unrestricted. The ﬁrst-stage matrix has en tries π j k = ∂ E [ Q ij ] /∂ Z k , the reduced form is RF k = ∂ E [ Y i ] /∂ Z k , and the 2SLS co eﬃcients satisfy RF = Π T β b y construction. The key restriction. W e require tw o conditions: Assumption 1 ′ (Relev ance). π kk  = 0 for all k . Assumption 2 ′ (Supply–instrumen t alignmen t). When the supply of go o d k increases b y one unit, holding all other supplies ﬁxed, the mechanism restores equilibrium b y adjusting Z k . That is, Z k is the v ariable through which a marginal supply expansion of go o d k is transmitted to individual allo cations. Assumption 2 ′ is the general form of ranking-list reﬁll. In a queue system, adding a slot to program k shifts the admission cutoﬀ—a c hange in Z k —and the ranking list determines who is aﬀected. In a market, adding a unit of go o d k lo wers its price—again a change in Z k —and the demand system determines who is aﬀected. In b oth cases, the ﬁrst-stage matrix Π captures how individual allo cations resp ond to the mechanism v ariable, and the same Casc ade Identity 52 matrix gov erns the resp onse to a supply expansion. The sp eciﬁc institution determines the magnitudes of π j k and RF k , but the algebraic structure is the same. Deriv ation. Let T k = d Y /d ¯ Q k denote the total so cietal eﬀect of increasing the supply of go o d k b y one unit, holding all other supplies ﬁxed and letting the mec hanism adjust to restore equilibrium. By Assumption 2 ′ , the supply expansion op erates through Z k . The direct eﬀect p er unit of new supply is RF k /π kk : the reduced-form outcome change p er unit of Z k , rescaled b y how man y units of go o d k a one-unit c hange in Z k deliv ers. But the change in Z k also shifts allo cations of other go o ds: p er unit increase in Z k , the total allo cation to go o d j c hanges by π j k . Since the supply of go o d j is ﬁxed, the mechanism m ust adjust Z j to undo this change, restoring P i Q ij = ¯ Q j . This oﬀsetting adjustment in Z j is itself a marginal supply-neutral reallo cation of go o d j across individuals. P er unit of go o d k expanded, the magnitude of the reallo cation of go o d j is | π j k | /π kk . When π j k < 0 (the typical substitution case), the initial sho c k reduces total demand for go o d j , so the mec hanism must push Z j in the direction that increases go o d- j allo cations— exactly as if go o d j ’s supply had expanded. The so cietal eﬀect of this reallo cation is T j p er unit. When π j k > 0 (complementarit y), the mechanism must contract go o d- j allo cations, with eﬀect − T j p er unit. In b oth cases, the con tribution to T k is ( − π j k /π kk ) · T j . Summing: T k = RF k π kk + X j  = k − π j k π kk T j , k = 1 , . . . , K . (33) This is identical to equation (4) . The remainder of the pro of follows Prop osition 1: substitute RF = Π T β and cancel. Prop osition 2 (General Cascade Identit y) . Under Assumptions 1 ′ and 2 ′ , for any al lo c ation me chanism with K ﬁxe d-supply go o ds, T = β . (34) Casc ade Identity 53 The 2SLS c o eﬃcient β k e quals the total so cietal eﬀe ct of exp anding the supply of go o d k by one unit, including al l e quilibrium r e al lo c ations thr ough the me chanism. The result holds b ecause the 2SLS estimand inv erts the full ﬁrst-stage matrix, and the ﬁrst-stage matrix is exactly the ob ject that gov erns ho w the mechanism reallo cates go o ds when supply c hanges. In a queue system, Π enco des ho w lottery outcomes redistribute studen ts across programs. In a mark et, Π enco des how price changes redistribute go o ds across consumers. In either case, inv erting Π traces out the full chain of adjustmen ts that a supply expansion triggers—the cascade. The institution determines the conten t of Π ; the algebra that conv erts Π and the reduced form in to a p olicy eﬀect is the same regardless. The iden tit y fails when the supply of some go o d j inside the system resp onds endogenously to the expansion of go o d k . If ¯ Q j is not ﬁxed, then the mec hanism need not fully oﬀset the cross-eﬀect π j k through reallo cation: part of the adjustmen t is absorb ed by a c hange in total quantit y rather than a redistribution among individuals. The cascade equation then o verstates the reallo cation comp onent and understates the quan tity-adjustmen t comp onen t, and T  = β . This is wh y the ﬁxed-supply condition—whether enforced by administrative capacit y constrain ts, regulatory quotas, ph ysical limits, or short-run inelasticit y—is the essen tial assumption, not the details of the allo cation mec hanism. Casc ade Identity 54 B.2 The Neumann series and the inﬁnite cascade: a tw o-program illustration T o make the connection b et ween the matrix inv ersion in Prop osition 1 and the dynamic cascade concrete, w e sp ell out the Neumann series for the simplest non-trivial case: tw o programs with mutual substitution. Setup. With K = 2 , the cascade equation (4) b ecomes T 1 = RF 1 π 11 + − π 21 π 11 T 2 , T 2 = RF 2 π 22 + − π 12 π 22 T 1 . (35) Deﬁne W k = RF k /π kk , the instrument-speciﬁc W ald ratio for program k , and r 21 = − π 21 π 11 , r 12 = − π 12 π 22 , (36) the vac ancy cr e ation r ates : r 21 is the n umber of v acancies created in program 2 p er new admission to program 1, and r 12 vice versa. Under substitution, π 21 < 0 and π 12 < 0 , so b oth rates are p ositive. The system simpliﬁes to T 1 = W 1 + r 21 T 2 , T 2 = W 2 + r 12 T 1 . (37) Iterated substitution. Rather than solving the system directly , substitute rep eatedly to trace the cascade round b y round. Starting from T 1 : T 1 = W 1 + r 21  W 2 + r 12 T 1  = W 1 + r 21 W 2 + r 21 r 12 T 1 . (38) Instead of solving, substitute once more: T 1 = W 1 + r 21 W 2 + r 21 r 12  W 1 + r 21 T 2  = W 1 + r 21 W 2 + r 21 r 12 W 1 + ( r 21 r 12 ) r 21 T 2 . (39) Casc ade Identity 55 Con tinuing this pro cess, a pattern emerges. Each pair of substitutions generates a factor of ρ = r 21 r 12 —one round trip through b oth programs—and the cascade alternates b etw een reﬁlling program 1 and program 2: T 1 = W 1 + ρ W 1 + ρ 2 W 1 + · · · | {z } reﬁlls in program 1 + r 21 W 2 + ρ r 21 W 2 + ρ 2 r 21 W 2 + · · · | {z } reﬁlls in program 2 = W 1 ∞ X n =0 ρ n + r 21 W 2 ∞ X n =0 ρ n = W 1 + r 21 W 2 1 − ρ . (40) The cascade for program 2 follo ws by symmetry: T 2 = W 2 + r 12 W 1 1 − ρ . (41) Both series conv erge when ρ = r 21 r 12 < 1 : each round trip through the system displaces strictly few er p eople than the previous one, so the cascade attenuates and terminates in the limit. The cascade reads round b y round as follo ws. Consider the expansion of program 1: R ound 0. Program 1 admits one new student. Direct eﬀect: W 1 . This creates r 21 v acancies in program 2. R ound 1. Program 2 reﬁlls. Eﬀect: r 21 W 2 . This creates r 21 r 12 = ρ v acancies back in program 1. R ound 2. Program 1 reﬁlls. Eﬀect: ρ W 1 . This creates ρ r 21 v acancies in program 2. R ound 3. Program 2 reﬁlls. Eﬀect: ρ r 21 W 2 . This creates ρ 2 v acancies in program 1. Eac h round trip attenuates by the factor ρ , and the total eﬀect sums the contributions from all rounds. Connection to the matrix form ulation. Deﬁne the v acancy creation matrix M = − D − 1 P T =    0 r 21 r 12 0    , (42) Casc ade Identity 56 whic h collects the v acancy creation rates with their natural (p ositive) signs. The cascade equation (5) can b e rewritten as ( I − M ) T = D − 1 RF , so that T = ( I − M ) − 1 D − 1 RF . (43) The Neumann series expands the in verse as ( I − M ) − 1 = ∞ X n =0 M n = I + M + M 2 + M 3 + · · · (44) pro vided the sp ectral radius of M is less than one, whic h in the 2 × 2 case reduces to ρ = r 21 r 12 < 1 . Computing the ﬁrst few p o wers: M 0 =    1 0 0 1    , M 1 =    0 r 21 r 12 0    , (45) M 2 =    r 21 r 12 0 0 r 12 r 21    , M 3 =    0 ( r 21 r 12 ) r 21 ( r 12 r 21 ) r 12 0    . (46) Ev en p o wers of M are diagonal: the cascade has completed a round trip and returned to the program where it started. Odd p ow ers are oﬀ-diagonal: the cascade is currently on the other side. All en tries are non-negative (under substitution), so every term in the Neumann series con tributes p ositively—there are no alternating signs. Summing the geometric matrix series: ∞ X n =0 M n = 1 1 − ρ    1 r 21 r 12 1    . (47) Multiplying by D − 1 RF = ( W 1 , W 2 ) T reco vers exactly the scalar expressions (40)–(41).

The Cascade Identity: 2SLS as a Policy Parameter in Capacity-Constrained Settings

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