JEL: C36, C38, I23
Introduction
Many policy evaluations involve multiple unordered treatments, such as location decisions by firms or families [17 ] or student fields of study choices [1 ] . Researchers often aggregate these treatments into clusters. For instance, STEM and non-STEM programs (Ng and Riehl, 2024), Business and non-Business majors [3 ] , or broad study directions [18 ] , all consist of fields of study such as Physics or Finance. When treatment choice is endogenous, instrumental variable (IV) methods may be used to estimate a local average treatment effect (LATE) of a cluster of treatments. A STEM degree is then considered as a binary treatment choice, and the IV assumptions are invoked at the cluster level. However, individuals choose among treatments within the clusters, and therefore the instruments may induce changes in treatment take up across and within clusters. Hence, treatments have to be considered at the unclustered level as otherwise violations of the IV assumptions may stay undetected [34 ] . This raises the question of what parameter IV estimation at the cluster level is targeting, and under what conditions this parameter has a causal interpretation.
This paper studies IV estimation with treatments aggregated into clusters in a setting where both treatment and instrument are multi-valued unordered variables defined over the same support. We show that IV with clustered treatments identifies a weighted average of LATEs under a set of assumptions imposed at the unclustered treatment level. This weighted average includes the LATEs of all pairs of treatments across two clusters, and all instrument values associated with these clusters. The assumptions clarify which potential treatment choices have to be excluded, which does not directly follow from the IV assumptions of Imbens and Angrist [32 ] invoked at the cluster level. The assumptions are less strict than the IV assumptions required for identifying LATEs between the unclustered treatments [9 ] . We show that this result applies to settings with an arbitrary number of treatment clusters and a corresponding number of instruments. We also derive necessary conditions for a causal interpretation of the IV estimand with treatment clusters that are testable using data at the unclustered treatment level.
Treatment evaluation often considers clusters of treatments instead of the actual treatments for three reasons. First, administrative data on treatments with small enrollment numbers may not be disclosed due to privacy concerns. Second, the estimation of a large number of pairwise treatment effects may suffer from a lack of accuracy. Third, a large number of estimates may be less relevant if the policy is designed at a less fine-grained level. Our results are applicable to these settings, and help practitioners interpret the parameter targeted by an IV analysis at the cluster level, and justify the IV assumptions required for a causal interpretation of this parameter.
To illustrate the empirical relevance of our methods, we study the causal effect of clusters of fields of study on the number of study credit points. We have access to an administrative data set that includes four cohorts in the Portuguese public higher education system, from which we exploit a natural experiment to construct instruments in a fuzzy regression discontinuity design (RDD). We estimate treatment cluster effects using the STEM versus non-STEM classification and eight broad academic clusters defined according to the International Standard Classification of Education (ISCED).
To the best of our knowledge, we are the first to analyze the causal interpretation of the IV estimand with clusters of multiple unordered treatments. In a related setting, but with an ordered multivalued treatment, Angrist and Imbens [5 ] ; Marshall [41 ] ; Andresen and Huber [2 ] discuss the implications of binarising the multivalued treatment. Treatment clusters have also been discussed in combination with conditional independence assumptions. For instance, Lechner [38 ] discusses how to estimate the effect of treatment clusters in this setting. Heiler and Knaus [29 ] decompose treatment effect heterogeneity with two treatment clusters into heterogeneous treatment responses and heterogeneity in the actual treatment take up.
There is an extensive literature that aims to identify pairwise unclustered treatment effects in the presence of multiple unordered treatments. When using multiple instrumental variables, Behaghel et al. [7 ] ; Kirkebøen et al. [36 ] ; Dahl et al. [18 ] ; Bhuller and Sigstad [9 ] ; Heinesen et al. [30 ] allow each instrument to induce the choice of only one particular treatment, which is more restrictive than the assumptions required for an analysis at the treatment cluster level. Heckman and Pinto [28 ] and Pinto [47 ] discuss alternative assumptions for identifying multiple (local) average treatment effects with multiple discrete instruments. Lee and Salanié [40 ] and Mountjoy [43 ] discuss this setting with continuous instruments. Blackwell [10 ] discusses the identification of interaction effects between multiple treatments with binary instruments. Kline and Walters [37 ] consider the multiple treatment case with one instrument.
The paper proceeds as follows. Section 2 introduces the setting with IVs for multiple unordered treatments. Section 3 analyses IV estimation with treatment clusters. Section 4 derives testable necessary conditions for the identifying assumptions, and Section 5 extends the results to a fuzzy RDD. Section 6 illustrates the empirical relevance of our methods with fields of study as multiple unordered treatments. Section 7 concludes.
IV with multiple unordered treatments
Suppose we have a multi-valued unordered treatment D ∈ { 1 , … , J } D \in\{1,\ldots,J\} D ∈ { 1 , … , J } Z ∈ { 1 , … , J } Z \in\{1,\ldots,J\} Z ∈ { 1 , … , J } D j = 1 [ D = j ] D_j = \mathbb{1}[D=j] D j = 1 [ D = j ] Z j = 1 [ Z = j ] Z_j = \mathbb{1}[Z=j] Z j = 1 [ Z = j ] D = ∑ j = 1 J j × D j D = \sum_{j=1}^{J} j \times D_j D = ∑ j = 1 J j × D j Z = ∑ j = 1 J j × Z j Z = \sum_{j=1}^{J} j \times Z_j Z = ∑ j = 1 J j × Z j [9 ] . Section 3.1 discusses how distinct treatment and instrument supports can be accommodated in our treatment cluster framework.
Treatments that are not mutually exclusive can always be made into mutually exclusive treatments by defining new categories for each combination of treatments.
Define Y Y Y Y Y Y j j j k k k Z j = 1 Z_j=1 Z j = 1 j j j l ≠ j l \ne j l = j D ( z ) D(z) D ( z ) Z Z Z z z z z = 1 , … , J z = 1, \ldots, J z = 1 , … , J D j ( z ) D_j(z) D j ( z ) j j j Z = z Z = z Z = z D j = ∑ l = 1 J D j ( l ) × Z l D_j = \sum_{l=1}^{J} D_j(l) \times Z_l D j = ∑ l = 1 J D j ( l ) × Z l j j j Y ( d , z ) Y(d,z) Y ( d , z ) D = d D = d D = d Z = z Z = z Z = z d = 1 , … , J d = 1, \ldots, J d = 1 , … , J z = 1 , … , J z = 1, \ldots, J z = 1 , … , J
Assumption 1 (IV assumptions).
a. (Independence) Y ( d , z ) , D ( z ) ⊥ Z Y(d,z), D(z) \perp Z Y ( d , z ) , D ( z ) ⊥ Z d , z d,z d , z
b. (Exclusion) Y ( d , z ) = Y ( d ) Y(d,z) = Y(d) Y ( d , z ) = Y ( d ) d , z d,z d , z
Analyzing treatment choice at the unclustered treatment level is not always possible. For instance, consider fields of study as the multiple unordered treatments. Obtaining administrative data on study enrollment at the field of study level may be infeasible, in particular due to privacy constraints with the emergence of smaller degrees. With many small degrees it is challenging to obtain accurate estimates for the pairwise treatment effects between the degrees. Moreover, a large number of treatment effect estimates may be less relevant if the policy is designed at a less granular level, such as the promotion of gender equality in STEM enrollment [35 ] . In such cases, practitioners often aggregate treatments into clusters and analyze the causal effects between these treatment clusters.
IV with clustered treatment data
This section analyses IV estimation with treatment clusters. First, it shows under which assumptions the IV estimand with two treatment clusters identifies a causal parameter, and discusses the interpretation of this parameter. Second, it shows that the results extend to the two-stage least squares (TSLS) estimand with an arbitrary number of clusters.
The IV estimand with two treatment clusters
Suppose we have two mutually exclusive clusters of treatments S 1 S_1 S 1 S 2 S_2 S 2 D ~ 1 = ∑ n ∈ S 1 D n \widetilde{D}_1 = \sum_{n\in S_1} D_n D 1 = ∑ n ∈ S 1 D n S 1 S_1 S 1 Z ~ 1 = ∑ n ∈ S 1 Z n \widetilde{Z}_1 = \sum_{n\in S_1} Z_n Z 1 = ∑ n ∈ S 1 Z n S 1 S_1 S 1 D n D_n D n Z n Z_n Z n D ~ 1 \widetilde{D}_1 D 1 Z ~ 1 \widetilde{Z}_1 Z 1
β = Δ E [ Y ∣ Z ~ 1 ] Δ E [ D ~ 1 ∣ Z ~ 1 ] , (1) \beta= \frac{\Delta\mathbb{E}[Y \mid\widetilde{Z}_1]}{\Delta\mathbb{E}[\widetilde{D}_1 \mid\widetilde{Z}_1]}, \tag*{(1)} β = Δ E [ D 1 ∣ Z 1 ] Δ E [ Y ∣ Z 1 ] , (1) where Δ E [ A ∣ Z ~ 1 ] = E [ A ∣ Z ~ 1 = 1 ] − E [ A ∣ Z ~ 1 = 0 ] \Delta\mathbb{E}[A \mid\widetilde{Z}_1] = \mathbb{E}[A \mid\widetilde{Z}_1 = 1] - \mathbb{E}[A \mid\widetilde{Z}_1 = 0] Δ E [ A ∣ Z 1 ] = E [ A ∣ Z 1 = 1 ] − E [ A ∣ Z 1 = 0 ] Y Y Y D ~ 1 = 1 \widetilde{D}_1 = 1 D 1 = 1 Z ~ 1 = 1 \widetilde{Z}_1 = 1 Z 1 = 1
Assumption 2 (Relevance). Δ E [ D ~ 1 ∣ Z ~ 1 ] > 0 \Delta\mathbb{E}[\widetilde{D}_1 \mid\widetilde{Z}_1] > 0 Δ E [ D 1 ∣ Z 1 ] > 0
This imposes relevance only at the treatment cluster level. Hence, the instrument may have no effect on the probability of taking treatments at the unclustered level, as long as it has a nonzero effect on the probability of taking treatment from cluster S 1 S_1 S 1
We assume that the estimand in (1) is considered when the effect of taking treatment from cluster S 1 S_1 S 1 S 2 S_2 S 2 [28 ] and define a response type as a vector of potential treatment choices for Z Z Z ( D ( 1 ) , D ( 2 ) , … , D ( J ) ) (D(1), D(2), \ldots, D(J)) ( D ( 1 ) , D ( 2 ) , … , D ( J )) l l l Z = j Z = j Z = j m m m Z = k Z = k Z = k ( D ( j ) = l , D ( k ) = m ) (D(j) = l, D(k) = m) ( D ( j ) = l , D ( k ) = m ) ( D ( j ) ∈ S 1 , D ( k ) ∈ S 2 ) (D(j) \in S_1, D(k) \in S_2) ( D ( j ) ∈ S 1 , D ( k ) ∈ S 2 ) S 1 S_1 S 1 Z = j Z = j Z = j S 2 S_2 S 2 Z = k Z = k Z = k
Table 1 shows four different response types if we both consider the treatments and the instruments at the cluster level. These include all possible types, if we exclude individuals Notes: For j ∈ S 1 j \in S_1 j ∈ S 1 k ∈ S 2 k \in S_2 k ∈ S 2 D ( j ) ≠ D ( k ) D(j) \ne D(k) D ( j ) = D ( k )
Table 1. Response types with two treatment clusters
with D ( j ) = D ( k ) D(j) = D(k) D ( j ) = D ( k ) j ∈ S 1 j \in S_1 j ∈ S 1 k ∈ S 2 k \in S_2 k ∈ S 2 S 1 S_1 S 1 S 2 S_2 S 2 ( D ( j ) ∈ S 1 , D ( k ) ∈ S 2 ) (D(j) \in S_1, D(k) \in S_2) ( D ( j ) ∈ S 1 , D ( k ) ∈ S 2 ) j ∈ S 1 j \in S_1 j ∈ S 1 k ∈ S 2 k \in S_2 k ∈ S 2
Lemma 1 (Decomposition IV estimand).
Under Assumptions 1 and 2, it holds for β \beta β
β = ∑ j ∈ S 1 ∑ k ∈ S 2 ω j k [ ∑ l ∈ S 1 ∑ m ∈ S 2 Δ j k l m P j k l m − Δ j k m l P j k m l ] + B 1 + B 2 ∑ j ∈ S 1 ∑ k ∈ S 2 ω j k [ ∑ l ∈ S 1 ∑ m ∈ S 2 P j k l m − P j k m l ] (2) \beta= \frac{\sum_{j \in S_1}\sum_{k \in S_2}\omega_{jk}\left[\sum_{l \in S_1}\sum_{m \in S_2}\Delta_{jklm}P_{jklm}-\Delta_{jkml}P_{jkml}\right]+B_1+B_2}{\sum_{j \in S_1}\sum_{k \in S_2}\omega_{jk}\left[\sum_{l \in S_1}\sum_{m \in S_2}P_{jklm}-P_{jkml}\right]} \tag*{(2)} β = ∑ j ∈ S 1 ∑ k ∈ S 2 ω j k [ ∑ l ∈ S 1 ∑ m ∈ S 2 P j k l m − P j k m l ] ∑ j ∈ S 1 ∑ k ∈ S 2 ω j k [ ∑ l ∈ S 1 ∑ m ∈ S 2 Δ j k l m P j k l m − Δ j k m l P j k m l ] + B 1 + B 2 (2) with B c = ∑ j ∈ S 1 ∑ k ∈ S 2 ω j k ∑ l ∈ S c ∑ m ≠ l ∈ S c Δ j k l m P j k l m B_c=\sum_{j \in S_1}\sum_{k \in S_2}\omega_{jk}\sum_{l \in S_c}\sum_{m\ne l \in S_c}\Delta_{jklm}P_{jklm} B c = ∑ j ∈ S 1 ∑ k ∈ S 2 ω j k ∑ l ∈ S c ∑ m = l ∈ S c Δ j k l m P j k l m Δ j k l m = E [ Y ( l ) − Y ( m ) ∣ D ( j ) = l , D ( k ) = m ] \Delta_{jklm}=\mathbb{E}[Y(l)-Y(m)\mid D(j)=l,D(k)=m] Δ j k l m = E [ Y ( l ) − Y ( m ) ∣ D ( j ) = l , D ( k ) = m ] P j k l m = P [ D ( j ) = l , D ( k ) = m ] P_{jklm}=\mathbb{P}[D(j)=l,D(k)=m] P j k l m = P [ D ( j ) = l , D ( k ) = m ] ω j k = P [ Z = j ∣ Z ∈ S 1 ] P [ Z = k ∣ Z ∈ S 2 ] \omega_{jk}=\mathbb{P}[Z=j\mid Z\in S_1]\mathbb{P}[Z=k\mid Z\in S_2] ω j k = P [ Z = j ∣ Z ∈ S 1 ] P [ Z = k ∣ Z ∈ S 2 ]
The proof is deferred to Section A.1. The decomposition in Lemma 1 shows that the IV estimand in (1) is not necessarily a positively weighted average of the treatment effects corresponding to the response types ( D ( j ) ∈ S 1 , D ( k ) ∈ S 2 ) (D(j) \in S_1, D(k) \in S_2) ( D ( j ) ∈ S 1 , D ( k ) ∈ S 2 ) j ∈ S 1 j \in S_1 j ∈ S 1 k ∈ S 2 k \in S_2 k ∈ S 2 P j k l m P_{jklm} P j k l m ( D ( j ) ∈ S 2 , D ( k ) ∈ S 1 ) (D(j) \in S_2, D(k) \in S_1) ( D ( j ) ∈ S 2 , D ( k ) ∈ S 1 ) P j k m l P_{jkml} P j k m l ( D ( j ) ∈ S 1 , D ( k ) ∈ S 1 ) (D(j) \in S_1, D(k) \in S_1) ( D ( j ) ∈ S 1 , D ( k ) ∈ S 1 ) ( D ( j ) ∈ S 2 , D ( k ) ∈ S 2 ) (D(j) \in S_2, D(k) \in S_2) ( D ( j ) ∈ S 2 , D ( k ) ∈ S 2 ) B 1 B_1 B 1 B 2 B_2 B 2 S 1 = { 1 , 2 } S_1=\{1,2\} S 1 = { 1 , 2 } S 2 = { 3 , 4 } S_2=\{3,4\} S 2 = { 3 , 4 } j = 1 j=1 j = 1 k = 3 k=3 k = 3 ( D ( 1 ) = 3 , D ( 3 ) = 1 ) (D(1)=3,D(3)=1) ( D ( 1 ) = 3 , D ( 3 ) = 1 ) ( D ( 1 ) = 1 , D ( 3 ) = 2 ) (D(1)=1,D(3)=2) ( D ( 1 ) = 1 , D ( 3 ) = 2 ) B 1 B_1 B 1 ( D ( 1 ) = 4 , D ( 3 ) = 3 ) (D(1)=4,D(3)=3) ( D ( 1 ) = 4 , D ( 3 ) = 3 ) B 2 B_2 B 2
Lemma 1 also clarifies how IV assumptions invoked at the cluster level affect the interpretation of β \beta β D ~ 1 ( j ) ≥ D ~ 1 ( k ) \widetilde{D}_1(j) \ge\widetilde{D}_1(k) D 1 ( j ) ≥ D 1 ( k ) j ∈ S 1 j \in S_1 j ∈ S 1 k ∈ S 2 k \in S_2 k ∈ S 2 ( D ( j ) ∈ S 2 , D ( k ) ∈ S 1 ) (D(j) \in S_2,D(k) \in S_1) ( D ( j ) ∈ S 2 , D ( k ) ∈ S 1 ) B 1 B_1 B 1 B 2 B_2 B 2 Y Y Y D ~ 1 \widetilde{D}_1 D 1 Δ j k l m = 0 \Delta_{jklm}=0 Δ j k l m = 0 ( l , m ) ∈ S 1 (l,m) \in S_1 ( l , m ) ∈ S 1 ( l , m ) ∈ S 2 (l,m) \in S_2 ( l , m ) ∈ S 2 B 1 = B 2 = 0 B_1=B_2=0 B 1 = B 2 = 0
Instead of invoking monotonicity directly at the cluster level, we apply the monotonicity assumption in Imbens and Angrist [32 ] at the treatment level within the treatment cluster setting.
Assumption 3 (Monotonicity). D l ( j ) ≥ D l ( k ) D_l(j) \ge D_l(k) D l ( j ) ≥ D l ( k ) j , l ∈ S a j,l \in S_a j , l ∈ S a k ∈ S b k \in S_b k ∈ S b a = 1 a=1 a = 1 b = 2 b=2 b = 2
This assumption rules out all individuals for whom Z ∈ S 1 Z \in S_1 Z ∈ S 1 S 1 S_1 S 1 ( D ( j ) ∈ S 2 , D ( k ) ∈ S 1 ) (D(j) \in S_2,D(k) \in S_1) ( D ( j ) ∈ S 2 , D ( k ) ∈ S 1 ) ( D ( j ) ∈ S 1 , D ( k ) ∈ S 1 ) (D(j) \in S_1,D(k) \in S_1) ( D ( j ) ∈ S 1 , D ( k ) ∈ S 1 ) D ( j ) ≠ D ( k ) D(j) \ne D(k) D ( j ) = D ( k ) j ∈ S 1 j \in S_1 j ∈ S 1 k ∈ S 2 k \in S_2 k ∈ S 2 ( D ( j ) = k , D ( k ) = j ) (D(j)=k,D(k)=j) ( D ( j ) = k , D ( k ) = j ) Z = j Z=j Z = j j j j
The plausibility of Assumption 3 depends on how treatments are grouped into clusters and how individuals respond to particular encouragements. Suppose that S 1 = { 1 , 2 } S_1=\{1,2\} S 1 = { 1 , 2 } S 2 = { 3 , 4 } S_2=\{3,4\} S 2 = { 3 , 4 } ( D ( 1 ) = 2 , D ( 3 ) = 4 ) (D(1) = 2, D(3) = 4) ( D ( 1 ) = 2 , D ( 3 ) = 4 )
The assumption instead rules out a STEM offer inducing enrollment in a different STEM field from the one chosen following a non-STEM offer. For example, it rules out ( D ( 1 ) = 1 , D ( 3 ) = 2 ) (D(1) = 1, D(3) = 2) ( D ( 1 ) = 1 , D ( 3 ) = 2 ) ( D ( 1 ) = 1 , D ( 3 ) = 1 ) (D(1) = 1, D(3) = 1) ( D ( 1 ) = 1 , D ( 3 ) = 1 ) ( D ( 1 ) = 2 , D ( 3 ) = 2 ) (D(1) = 2, D(3) = 2) ( D ( 1 ) = 2 , D ( 3 ) = 2 )
Table 1 shows that after invoking Assumption 3, the negative terms and B 1 B_1 B 1 ( D ( j ) ∈ S 2 , D ( k ) ∈ S 2 ) (D(j) \in S_2, D(k) \in S_2) ( D ( j ) ∈ S 2 , D ( k ) ∈ S 2 ) D ( j ) ≠ D ( k ) D(j) \ne D(k) D ( j ) = D ( k ) B 2 B_2 B 2
Assumption 4 (Irrelevance). If D l ( j ) = D l ( k ) = 0 D_l(j) = D_l(k) = 0 D l ( j ) = D l ( k ) = 0 l ∈ S a l \in S_a l ∈ S a D n ( j ) = D n ( k ) D_n(j) = D_n(k) D n ( j ) = D n ( k ) n ∉ S a n \notin S_a n ∈ / S a j ∈ S a j \in S_a j ∈ S a k ∈ S b k \in S_b k ∈ S b a = 1 a = 1 a = 1 b = 2 b = 2 b = 2
This assumption rules out all individuals for whom D ( j ) ≠ D ( k ) D(j) \ne D(k) D ( j ) = D ( k ) D ( j ) D(j) D ( j ) D ( k ) D(k) D ( k ) S 1 S_1 S 1 ( D ( j ) ∈ S 2 , D ( k ) ∈ S 2 ) (D(j) \in S_2, D(k) \in S_2) ( D ( j ) ∈ S 2 , D ( k ) ∈ S 2 ) D ( j ) ≠ D ( k ) D(j) \ne D(k) D ( j ) = D ( k ) j ∈ S 1 j \in S_1 j ∈ S 1 k ∈ S 2 k \in S_2 k ∈ S 2 [36 ] . For example, with S 1 = { 1 , 2 } S_1 = \{1, 2\} S 1 = { 1 , 2 } S 2 = { 3 , 4 } S_2 = \{3, 4\} S 2 = { 3 , 4 } j = 1 j = 1 j = 1 k = 3 k = 3 k = 3 ( D ( 1 ) = 2 , D ( 3 ) = 4 ) (D(1) = 2, D(3) = 4) ( D ( 1 ) = 2 , D ( 3 ) = 4 )
Table 1 shows that, under Assumptions 3 and 4, the only response types with D ( j ) ≠ D ( k ) D(j) \ne D(k) D ( j ) = D ( k ) ( D ( j ) ∈ S 1 , D ( k ) ∈ S 2 ) (D(j) \in S_1, D(k) \in S_2) ( D ( j ) ∈ S 1 , D ( k ) ∈ S 2 ) j ∈ S 1 j \in S_1 j ∈ S 1 k ∈ S 2 k \in S_2 k ∈ S 2 Z = j Z = j Z = j Z = k Z = k Z = k j j j k k k l l l m m m j j j l l l k k k m m m β \beta β
Theorem 1 (Causal interpretation IV estimand). Under Assumptions 1-4, it holds for β \beta β (1) that
β = ∑ j ∈ S 1 ∑ k ∈ S 2 ∑ l ∈ S 1 ∑ m ∈ S 2 λ j k l m Δ j k l m , (3) \beta= \sum_{j\in S_1}\sum_{k\in S_2}\sum_{l\in S_1}\sum_{m\in S_2} \lambda_{jklm}\Delta_{jklm}, \tag*{(3)} β = j ∈ S 1 ∑ k ∈ S 2 ∑ l ∈ S 1 ∑ m ∈ S 2 ∑ λ j k l m Δ j k l m , (3) with the LATEs Δ j k l m \Delta_{jklm} Δ j k l m λ j k l m \lambda_{jklm} λ j k l m
Δ j k l m = E [ Y ( l ) − Y ( m ) ∣ D ( j ) = l , D ( k ) = m ] , (4) \Delta_{jklm} = \mathbb{E}[Y(l)-Y(m)\mid D(j)=l,D(k)=m], \tag*{(4)} Δ j k l m = E [ Y ( l ) − Y ( m ) ∣ D ( j ) = l , D ( k ) = m ] , (4) λ j k l m = ω j k P [ D ( j ) = l , D ( k ) = m ] ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ m ∈ S b ω j k P [ D ( j ) = l , D ( k ) = m ] , (5) \lambda_{jklm} =
\frac{\omega_{jk}\mathbb{P}[D(j)=l,D(k)=m]}
{\sum_{j\in S_a}\sum_{k\in S_b}\sum_{l\in S_a}\sum_{m\in S_b}\omega_{jk}\mathbb{P}[D(j)=l,D(k)=m]}, \tag*{(5)} λ j k l m = ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ m ∈ S b ω j k P [ D ( j ) = l , D ( k ) = m ] ω j k P [ D ( j ) = l , D ( k ) = m ] , (5) ω j k = P [ Z = j ∣ Z ∈ S a ] P [ Z = k ∣ Z ∈ S b ] , (6) \omega_{jk}=\mathbb{P}[Z=j\mid Z\in S_a]\mathbb{P}[Z=k\mid Z\in S_b], \tag*{(6)} ω j k = P [ Z = j ∣ Z ∈ S a ] P [ Z = k ∣ Z ∈ S b ] , (6) where a = 1 a=1 a = 1 b = 2 b=2 b = 2
Since ∑ j ∈ S 1 ∑ k ∈ S 2 ∑ l ∈ S 1 ∑ m ∈ S 2 λ j k l m = 1 \sum_{j\in S_1}\sum_{k\in S_2}\sum_{l\in S_1}\sum_{m\in S_2}\lambda_{jklm}=1 ∑ j ∈ S 1 ∑ k ∈ S 2 ∑ l ∈ S 1 ∑ m ∈ S 2 λ j k l m = 1 0 ≤ λ j k l m ≤ 1 0\leq\lambda_{jklm}\leq1 0 ≤ λ j k l m ≤ 1 β \beta β Δ j k l m \Delta_{jklm} Δ j k l m S 1 S_1 S 1 S 2 S_2 S 2 S 1 S_1 S 1 S 2 S_2 S 2
The LATE Δ j k l m \Delta_{jklm} Δ j k l m l ∈ S 1 l\in S_1 l ∈ S 1 m ∈ S 2 m\in S_2 m ∈ S 2 ( D ( j ) = l , D ( k ) = m ) (D(j)=l,D(k)=m) ( D ( j ) = l , D ( k ) = m ) Δ j k l m \Delta_{jklm} Δ j k l m λ j k l m \lambda_{jklm} λ j k l m Δ j k l m \Delta_{jklm} Δ j k l m β \beta β λ j k l m \lambda_{jklm} λ j k l m ( D ( j ) = l , D ( k ) = m ) (D(j)=l,D(k)=m) ( D ( j ) = l , D ( k ) = m ) j , l ∈ S 1 j,l\in S_1 j , l ∈ S 1 k , m ∈ S 2 k,m\in S_2 k , m ∈ S 2
Theorem 1 delivers a transparent expression for the IV estimand in terms of LATEs using that treatment and instrument have the same number of values and the same labels. This restriction can be weakened if instrument values can be partitioned into sets T c T_c T c S c S_c S c ∣ T c ∣ = ∣ S c ∣ \lvert T_c\rvert=\lvert S_c\rvert ∣ T c ∣ = ∣ S c ∣ j ∈ S c j \in S_c j ∈ S c z ∈ T c z \in T_c z ∈ T c [9 ] .
The TSLS estimand with multiple treatment clusters
Suppose we have c = 1 , … , C c = 1, \ldots, C c = 1 , … , C S c S_c S c S = { S c } c = 1 C S = \{S_c\}_{c=1}^{C} S = { S c } c = 1 C C C C Z ~ b = ( 1 , Z ~ 1 , … , Z ~ b − 1 , Z ~ b + 1 , … , Z ~ C ) ′ \widetilde{Z}_b = (1, \widetilde{Z}_1, \ldots, \widetilde{Z}_{b-1}, \widetilde{Z}_{b+1}, \ldots, \widetilde{Z}_C)' Z b = ( 1 , Z 1 , … , Z b − 1 , Z b + 1 , … , Z C ) ′ D ~ b = ( 1 , D ~ 1 , … , D ~ b − 1 , D ~ b + 1 , … , D ~ C ) ′ \widetilde{D}_b = (1, \widetilde{D}_1, \ldots, \widetilde{D}_{b-1}, \widetilde{D}_{b+1}, \ldots, \widetilde{D}_C)' D b = ( 1 , D 1 , … , D b − 1 , D b + 1 , … , D C ) ′ Z ~ c = ∑ n ∈ S c Z n \widetilde{Z}_c = \sum_{n \in S_c} Z_n Z c = ∑ n ∈ S c Z n D ~ c = ∑ n ∈ S c D n \widetilde{D}_c = \sum_{n \in S_c} D_n D c = ∑ n ∈ S c D n
β b = E [ Z ~ b D ~ b ′ ] − 1 E [ Z ~ b Y ] , (7) \beta_b = \mathbb{E}[\widetilde{Z}_b\widetilde{D}_b']^{-1}\mathbb{E}[\widetilde{Z}_bY], \tag*{(7)} β b = E [ Z b D b ′ ] − 1 E [ Z b Y ] , (7) where β b = ( β 0 b , β 1 b , … , β b − 1 , b , β b + 1 , b , … , β C b ) \beta_b = (\beta_{0b}, \beta_{1b}, \ldots, \beta_{b-1,b}, \beta_{b+1,b}, \ldots, \beta_{Cb}) β b = ( β 0 b , β 1 b , … , β b − 1 , b , β b + 1 , b , … , β C b ) β 12 \beta_{12} β 12 β \beta β C = 2 C = 2 C = 2 E [ D ~ b Z ~ b ′ ] \mathbb{E}[\widetilde{D}_b\widetilde{Z}_b'] E [ D b Z b ′ ]
We assume (7) is considered if the effect of taking treatment from cluster S c S_c S c S b S_b S b c = 1 , … , C c = 1, \ldots, C c = 1 , … , C β b \beta_b β b b = 1 , … , C b = 1, \ldots, C b = 1 , … , C S b S_b S b
Table 2 describes all response types with D ( j ) ≠ D ( k ) D(j) \ne D(k) D ( j ) = D ( k ) Table 1 with a = 1 a = 1 a = 1 b = 2 b = 2 b = 2 S a S_a S a S b S_b S b S a S_a S a S b S_b S b j ∈ S a j \in S_a j ∈ S a k ∈ S b k \in S_b k ∈ S b ( D ( j ) ∉ { S a , S b } , D ( k ) ∈ S a ) (D(j) \notin\{S_a, S_b\}, D(k) \in S_a) ( D ( j ) ∈ / { S a , S b } , D ( k ) ∈ S a ) Z ∈ S a Z \in S_a Z ∈ S a S a S_a S a ( D ( j ) ∈ S b , D ( k ) ∉ { S a , S b } ) (D(j) \in S_b, D(k) \notin\{S_a, S_b\}) ( D ( j ) ∈ S b , D ( k ) ∈ / { S a , S b }) ( D ( j ) ∉ { S a , S b } , D ( k ) ∈ S b ) (D(j) \notin\{S_a, S_b\}, D(k) \in S_b) ( D ( j ) ∈ / { S a , S b } , D ( k ) ∈ S b ) ( D ( j ) ∉ { S a , S b } , D ( k ) ∉ { S a , S b } ) (D(j) \notin\{S_a, S_b\}, D(k) \notin\{S_a, S_b\}) ( D ( j ) ∈ / { S a , S b } , D ( k ) ∈ / { S a , S b }) Notes: For a ≠ b a \ne b a = b j ∈ S a j \in S_a j ∈ S a k ∈ S b k \in S_b k ∈ S b D ( j ) ≠ D ( k ) D(j) \ne D(k) D ( j ) = D ( k ) S b S_b S b S a S_a S a a ≠ b a \ne b a = b ( S a , S b ) (S_a,S_b) ( S a , S b )
Table 2. Response types with multiple treatment clusters
for whom D ( j ) ≠ D ( k ) D(j) \ne D(k) D ( j ) = D ( k ) D ( j ) D(j) D ( j ) D ( k ) D(k) D ( k ) S a S_a S a ( D ( j ) ∈ S a , D ( k ) ∉ { S a , S b } ) (D(j) \in S_a, D(k) \notin\{S_a,S_b\}) ( D ( j ) ∈ S a , D ( k ) ∈ / { S a , S b })
To prevent these response types, we introduce the following additional assumption.
Assumption 5 (Control group). If there is an l ∈ S a l \in S_a l ∈ S a D l ( j ) = 1 D_l(j) = 1 D l ( j ) = 1 D n ( k ) = 0 D_n(k) = 0 D n ( k ) = 0 j ∈ S a j \in S_a j ∈ S a k ∈ S b k \in S_b k ∈ S b n ∉ { S a , S b } n \notin\{S_a,S_b\} n ∈ / { S a , S b }
This assumption states that obtaining a treatment in S a S_a S a S a S_a S a S a S_a S a S b S_b S b S b S_b S b ( D ( j ) ∈ S a , D ( k ) ∉ { S a , S b } ) (D(j) \in S_a, D(k) \notin\{S_a,S_b\}) ( D ( j ) ∈ S a , D ( k ) ∈ / { S a , S b }) j ∈ S a j \in S_a j ∈ S a k ∈ S b k \in S_b k ∈ S b
Assumption 5 may hold in settings with one or more control groups among the J J J S b S_b S b
control. For instance, Duflo et al. [23 ] discuss randomized control trials with multiple treatment arms in development economics, where the control group typically receives no intervention. Behaghel et al. [7 ] analyze a job market activation program with multiple treatment arms where the control group receives standard services. In the absence of a control group, the literature follows Kirkebøen et al. [36 ] and Dahl et al. [18 ] : They assume that individuals have a preference ranking over treatments, which is revealed before Z Z Z r m r_m r m m m m r 2 r_2 r 2 S b S_b S b
Note that Assumption 5 allows for D ( j ) = D ( k ) D(j)=D(k) D ( j ) = D ( k ) D ( k ) = k D(k)=k D ( k ) = k [36 ] . Bhuller and Sigstad [9 ] also discuss that the latter assumption can be relaxed to allow for D ( j ) = D ( k ) D(j)=D(k) D ( j ) = D ( k )
Under Assumptions 3 to 5, the only response types with D ( j ) ≠ D ( k ) D(j)\ne D(k) D ( j ) = D ( k ) ( D ( j ) ∈ S a , D ( k ) ∈ S b ) (D(j)\in S_a,D(k)\in S_b) ( D ( j ) ∈ S a , D ( k ) ∈ S b ) j ∈ S a j\in S_a j ∈ S a k ∈ S b k\in S_b k ∈ S b Z = j Z=j Z = j Z = k Z=k Z = k j j j k k k β a b \beta_{ab} β ab (7) has a similar causal interpretation as β \beta β (1) , and boils down to the same expression with C = 2 C=2 C = 2
Theorem 2 (Causal interpretation TSLS estimand). Under Assumptions 1-5, it holds for β b = ( β 0 b , β 1 b , … , β b − 1 , b , β b + 1 , b , … , β C b ) ′ \boldsymbol{\beta}_b=(\beta_{0b},\beta_{1b},\ldots,\beta_{b-1,b},\beta_{b+1,b},\ldots,\beta_{Cb})' β b = ( β 0 b , β 1 b , … , β b − 1 , b , β b + 1 , b , … , β C b ) ′ (7) that
β a b = ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ m ∈ S b λ j k l m Δ j k l m , (8) \beta_{ab}=\sum_{j\in S_a}\sum_{k\in S_b}\sum_{l\in S_a}\sum_{m\in S_b}\lambda_{jklm}\Delta_{jklm}, \tag*{(8)} β ab = j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ m ∈ S b ∑ λ j k l m Δ j k l m , (8) with a = 1 , … , b − 1 , b + 1 , … , C a=1,\ldots,b-1,b+1,\ldots,C a = 1 , … , b − 1 , b + 1 , … , C λ j k l m \lambda_{jklm} λ j k l m Δ j k l m \Delta_{jklm} Δ j k l m (4) –(6) .
The proof is deferred to Section A.2. From hereon, we present results for an arbitrary number of clusters C C C C = 2 C=2 C = 2
Theorems 1 and 2 have direct implications for IV estimation with treatment clusters. Applying IV at the cluster level without specifying assumptions at the unclustered treatment level leaves violations unnoticed (Imbens and Rubin, [34 ] ), and complicates the discussion of the validity of these assumptions. Our results allow researchers to precisely consider the validity of the required assumptions. Furthermore, they show that the parameter targeted by IV estimation at the cluster level has a well-defined causal interpretation as a weighted average of LATEs under appropriate assumptions, clarifying which LATEs contribute and the weight they receive. In Appendix B, we provide three strategies for obtaining further insights into the components of this weighted average.
Testable conditions
The estimand in (7) only requires data at the treatment cluster level. If data at the unclustered treatment level is available, necessary conditions of the identifying assumptions can be tested. It follows from Theorem 2 that in the absence of certain response types, the estimand in (7) has a causal interpretation. Table 2 summarizes which types are excluded by the underlying assumptions: Only types described by D ( j ) ∈ S a D(j) \in S_a D ( j ) ∈ S a j ∈ S a j \in S_a j ∈ S a D ( k ) ∈ S b D(k) \in S_b D ( k ) ∈ S b k ∈ S b k \in S_b k ∈ S b
Proposition 1 (Necessary conditions causal interpretation).
Let A A A Y Y Y
ϕ l j k A = E [ I Y ∈ A D l ∣ Z = j ] − E [ I Y ∈ A D l ∣ Z = k ] = P [ Y ( l ) ∈ A , D ( j ) = l , D ( k ) ≠ l ] − P [ Y ( l ) ∈ A , D ( j ) ≠ l , D ( k ) = l ] , (9) \phi_{ljk}^{A} = \mathbb{E}[\mathbb{I}_{Y \in A}D_l \mid Z = j] - \mathbb{E}[\mathbb{I}_{Y \in A}D_l \mid Z = k]
= \mathbb{P}[Y(l) \in A, D(j) = l, D(k) \ne l] - \mathbb{P}[Y(l) \in A, D(j) \ne l, D(k) = l], \tag*{(9)} ϕ l j k A = E [ I Y ∈ A D l ∣ Z = j ] − E [ I Y ∈ A D l ∣ Z = k ] = P [ Y ( l ) ∈ A , D ( j ) = l , D ( k ) = l ] − P [ Y ( l ) ∈ A , D ( j ) = l , D ( k ) = l ] , (9) with the necessary conditions for the causal interpretation of β a b \beta_{ab} β ab
ϕ l j k A ≥ 0 \phi_{ljk}^{A} \ge0 ϕ l j k A ≥ 0 l , j ∈ S a l,j \in S_a l , j ∈ S a k ∈ S b k \in S_b k ∈ S b
ϕ l j k A ≤ 0 \phi_{ljk}^{A} \le0 ϕ l j k A ≤ 0 l , k ∈ S b l,k \in S_b l , k ∈ S b j ∈ S a j \in S_a j ∈ S a
ϕ l j k A = 0 \phi_{ljk}^{A} = 0 ϕ l j k A = 0 l ∉ { S a , S b } l \notin\{S_a,S_b\} l ∈ / { S a , S b } j ∈ S a j \in S_a j ∈ S a k ∈ S b k \in S_b k ∈ S b
Suppose that S 1 = { 1 , 2 } S_1 = \{1,2\} S 1 = { 1 , 2 } S 2 = { 3 , 4 } S_2 = \{3,4\} S 2 = { 3 , 4 } j = 1 j = 1 j = 1 k = 3 k = 3 k = 3 ϕ 1 , 1 , 3 A ≥ 0 \phi_{1,1,3}^{A} \ge0 ϕ 1 , 1 , 3 A ≥ 0 ϕ 2 , 1 , 3 A ≥ 0 \phi_{2,1,3}^{A} \ge0 ϕ 2 , 1 , 3 A ≥ 0
Depending on the support of Y Y Y [44 ] show that such conditions can be written as conditional moment inequalities with Y Y Y Z Z Z [16 ] . Bai et al. [6 ] follow the same strategy for testing IV assumptions with multiple unordered treatments without clusters.
The Stata package clrtest by Chernozhukov et al. [15 ] enables joint testing of the conditional moment inequalities within each estimation sample. Mourifié and Wan [44 ] discuss the implementation of the multiple testing procedure of Holm [31 ] when the clrtest is used to test the conditions across a large number of samples. Statistical power may be further improved by testing the conditions across covariate-specific subsamples using machine learning ([24 ] ).
Failure to reject the necessary conditions in Proposition 1 is not sufficient to conclude that Assumptions 1 and 3 to 5 hold. For instance, under Assumption 1, it may be the case that P [ Y ( l ) ∈ A , D ( j ) = l , D ( k ) ≠ l ] > P [ Y ( l ) ∈ A , D ( j ) ≠ l , D ( k ) = l ] > 0 \mathbb{P}[Y(l) \in A, D(j) = l, D(k) \ne l] > \mathbb{P}[Y(l) \in A, D(j) \ne l, D(k) = l] > 0 P [ Y ( l ) ∈ A , D ( j ) = l , D ( k ) = l ] > P [ Y ( l ) ∈ A , D ( j ) = l , D ( k ) = l ] > 0 ϕ l j k A ≥ 0 \phi_{ljk}^{A} \ge0 ϕ l j k A ≥ 0 β a b \beta_{ab} β ab
Researchers may use rejections of specific necessary conditions to find clusters that do meet the assumptions, for instance by sequentially testing the necessary conditions. However, this approach introduces a pre-testing problem when inferring the treatment effect estimates, and may yield classifications for which the assumptions cannot be rejected, rather than classifications for which the assumptions actually hold. Researchers may therefore prefer to use the falsification tests to check the validity of a classification motivated by the empirical context or economic theory.
Fuzzy RDD with treatment clusters
This section provides an expression for the TSLS estimand with treatment clusters in a fuzzy regression discontinuity design. Fuzzy RDDs are commonly used in settings with multiple unordered treatments and revealed preference rankings over treatments. This is also the setting in § 6 .
In a fuzzy RDD, the instrument Z Z Z
Z = { r 1 S ≥ 0 , r 2 S < 0 , (10) Z =
\begin{cases}
r_1 & S \ge0, \\
r_2 & S < 0,
\end{cases} \tag*{(10)} Z = { r 1 r 2 S ≥ 0 , S < 0 , (10) where r m r_m r m m m m S S S S = 0 S = 0 S = 0
The fuzzy RDD replaces Assumption 1(a) by the local continuity assumption ([22 , 27 ] ):
Assumption 6 (Continuity). Y ( d , z ) Y(d,z) Y ( d , z ) D ( z ) D(z) D ( z ) S S S S = 0 S = 0 S = 0 d , z d,z d , z
This assumption implies that individuals just below and above the cutoff have similar potential outcomes. We control for smooth variation in the running variable with a polynomial function of the running variable f h ( S ) f_h(S) f h ( S ) f h ( 0 ) = 0 f_h(0) = 0 f h ( 0 ) = 0 h h h D D D Y Y Y [33 ] ).
We extend the TSLS estimand in (7) to the RDD. Define the p p p X h = ( F ′ , B h ′ ) X_h = (\mathbf{F}^{\prime}, \mathbf{B}_h^{\prime}) X h = ( F ′ , B h ′ ) F \mathbf{F} F f j k f_{jk} f j k ( r 1 , r 2 ) = ( j , k ) (r_1,r_2) = (j,k) ( r 1 , r 2 ) = ( j , k ) j ≠ k ∈ S b j \ne k \in S_b j = k ∈ S b B h ′ \mathbf{B}'_h B h ′ f h ( S ) f_h(S) f h ( S )
( δ b h ′ , β b h ′ ) ′ = E h [ Z ~ b , h D ~ b , h ′ ] − 1 E h [ Z ~ b , h Y ] , (11) (\delta_b^{h\prime},\beta_b^{h\prime})'=\mathbb{E}_h[\widetilde{Z}_{b,h}\widetilde{D}_{b,h}^{\prime}]^{-1}\mathbb{E}_h[\widetilde{Z}_{b,h}Y], \tag*{(11)} ( δ b h ′ , β b h ′ ) ′ = E h [ Z b , h D b , h ′ ] − 1 E h [ Z b , h Y ] , (11) where Z ~ b , h = ( X h ′ , Z ~ b ) ′ \widetilde{Z}_{b,h}=(X'_h,\widetilde{Z}_b)' Z b , h = ( X h ′ , Z b ) ′ D ~ b , h = ( X h ′ , D ~ b ) ′ \widetilde{D}_{b,h}=(X'_h,\widetilde{D}_b)' D b , h = ( X h ′ , D b ) ′ E h [ A ] = E [ A ∣ ∣ S ∣ ≤ h , r 2 ∈ S b ] \mathbb{E}_h[A]=\mathbb{E}[A\mid|S|\le h,r_2\in S_b] E h [ A ] = E [ A ∣ ∣ S ∣ ≤ h , r 2 ∈ S b ]
Y = ∑ k ∈ S b ∑ l ≠ k ν l k h f l k + ∑ a ≠ b β a b h D ~ a + f h , b ( S ) + ε h , b , (12) Y=\sum_{k\in S_b}\sum_{l\ne k}\nu^h_{lk}f_{lk}+\sum_{a\ne b}\beta^h_{ab}\widetilde{D}_a+f_{h,b}(S)+\varepsilon_{h,b}, \tag*{(12)} Y = k ∈ S b ∑ l = k ∑ ν l k h f l k + a = b ∑ β ab h D a + f h , b ( S ) + ε h , b , (12) D ~ a = ∑ k ∈ S b ∑ l ≠ k η a l k h f l k + ∑ c ≠ b π a c h Z ~ c + f h , a b ( S ) + u h , a b , for all a ≠ b . (13) \widetilde{D}_a=\sum_{k\in S_b}\sum_{l\ne k}\eta^h_{alk}f_{lk}+\sum_{c\ne b}\pi^h_{ac}\widetilde{Z}_c+f_{h,ab}(S)+u_{h,ab},\qquad\text{for all }a\ne b. \tag*{(13)} D a = k ∈ S b ∑ l = k ∑ η a l k h f l k + c = b ∑ π a c h Z c + f h , ab ( S ) + u h , ab , for all a = b . (13) Additional controls can be taken into account with saturated specifications that control for covariates nonparametrically ([11 ] ), or by double/debiased machine learning methods ([14 ] ).
Theorem 3 shows that the TSLS estimand in the fuzzy RDD has a causal interpretation under the local continuity assumption.
Theorem 3 (Causal interpretation fuzzy RDD estimand).
Under Assumptions 1(b) and 2-6, and f h ( S ) f_h(S) f h ( S ) β b h = ( β 0 b h , β 1 b h , … , β b − 1 , b h , β b + 1 , b h , … , β C b h ) ′ \beta_b^h=(\beta_{0b}^h,\beta_{1b}^h,\ldots,\beta_{b-1,b}^h,\beta_{b+1,b}^h,\ldots,\beta_{Cb}^h)' β b h = ( β 0 b h , β 1 b h , … , β b − 1 , b h , β b + 1 , b h , … , β C b h ) ′
lim h ↓ 0 β a b h = ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ m ∈ S b λ j k l m Δ j k l m , (14) \lim_{h\downarrow0}\beta_{ab}^h=\sum_{j\in S_a}\sum_{k\in S_b}\sum_{l\in S_a}\sum_{m\in S_b}\lambda_{jklm}\Delta_{jklm}, \tag*{(14)} h ↓ 0 lim β ab h = j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ m ∈ S b ∑ λ j k l m Δ j k l m , (14) with a = 1 , … , b − 1 , b + 1 , … , C a=1,\ldots,b-1,b+1,\ldots,C a = 1 , … , b − 1 , b + 1 , … , C Δ j k l m \Delta_{jklm} Δ j k l m λ j k l m \lambda_{jklm} λ j k l m
Δ j k l m = E [ Y ( l ) − Y ( m ) ∣ D ( j ) = l , D ( k ) = m , f j k = 1 , S = 0 ] , \Delta_{jklm}=\mathbb{E}[Y(l)-Y(m)\mid D(j)=l,D(k)=m,f_{jk}=1,S=0], Δ j k l m = E [ Y ( l ) − Y ( m ) ∣ D ( j ) = l , D ( k ) = m , f j k = 1 , S = 0 ] , λ j k l m = ω j k P [ D ( j ) = l , D ( k ) = m ∣ f j k = 1 , S = 0 ] ∑ j ∈ S a ∑ k ∈ S b ω j k ∑ l ∈ S a ∑ m ∈ S b P [ D ( j ) = l , D ( k ) = m ∣ f j k = 1 , S = 0 ] , \lambda_{jklm}=\frac{\omega_{jk}\mathbb{P}[D(j)=l,D(k)=m\mid f_{jk}=1,S=0]}{\sum_{j\in S_a}\sum_{k\in S_b}\omega_{jk}\sum_{l\in S_a}\sum_{m\in S_b}\mathbb{P}[D(j)=l,D(k)=m\mid f_{jk}=1,S=0]}, λ j k l m = ∑ j ∈ S a ∑ k ∈ S b ω j k ∑ l ∈ S a ∑ m ∈ S b P [ D ( j ) = l , D ( k ) = m ∣ f j k = 1 , S = 0 ] ω j k P [ D ( j ) = l , D ( k ) = m ∣ f j k = 1 , S = 0 ] , ω j k = lim h ↓ 0 P [ f j k = 1 ∣ r 2 ∈ S b , ∣ S ∣ ≤ h ] . \omega_{jk}=\lim_{h\downarrow0}\mathbb{P}[f_{jk}=1\mid r_2\in S_b,|S|\le h]. ω j k = h ↓ 0 lim P [ f j k = 1 ∣ r 2 ∈ S b , ∣ S ∣ ≤ h ] . The proof is deferred to Appendix § 8 .§ 3 . The fuzzy RDD estimand identifies a weighted average of LATEs, with weights 0 ≤ λ j k l m ≤ 1 0 \le\lambda_{jklm} \le1 0 ≤ λ j k l m ≤ 1 ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ m ∈ S b λ j k l m = 1 \sum_{j \in S_a}\sum_{k \in S_b}\sum_{l \in S_a}\sum_{m \in S_b} \lambda_{jklm} = 1 ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ m ∈ S b λ j k l m = 1
Similar to Theorem 2, the weights represent a normalized proportion of response types corresponding to each LATE: the individuals near the cutoff with r 1 = j r_1 = j r 1 = j r 2 = k r_2 = k r 2 = k ( D ( j ) ∈ S a , D ( k ) ∈ S b ) (D(j) \in S_a, D(k) \in S_b) ( D ( j ) ∈ S a , D ( k ) ∈ S b ) j ∈ S a j \in S_a j ∈ S a k ∈ S b k \in S_b k ∈ S b
Application with fields of study
This section illustrates the empirical relevance of our methods in a setting with fields of study as multiple unordered treatments. We consider a natural experiment in the Portuguese public higher education system to study the effect of fields of study on the achieved number of study credit points.
Context and empirical specification
In the final year of high school, prospective students in Portugal apply to public higher education via a centralized admission process. Only 17 percent of the students use a different enrollment system to enroll into private higher education [48 ] . Applicants apply to a field and institution simultaneously, which we will refer to as a course (e.g. , Law at the University of Lisbon). Applicants submit a ranking of up to six courses based on their preferences. Each course has a quota of students it can admit and ranks its applicants by a score based on high school and national exam grades. Because courses weight these components differently, application scores are applicant-course specific. Using these rankings and quotas, a deferred acceptance mechanism similar to the college-proposing Gale-Shapley algorithm [26 ] matches each applicant to at most one course [21 ] . Appendix § 10 .§ 1 describes this mechanism in more detail.
Course cutoffs, instruments, and next-best fields of study
The admission mechanism generates ex-ante unknown application-score cutoffs for each course. Each cutoff represents the lowest score that qualifies an applicant to receive an offer from that course. We construct the preferred and next-best fields from the course cutoffs, application scores, and submitted rankings. First, we retain applicants who rank courses from at least two different fields. Second, we construct preference blocks by assigning the same rank to consecutive courses in the submitted ranking that belong to the same field. Third, we consider the applicant’s matched block and the blocks immediately above and below it. We retain the block below only if the applicant scores above the cutoff for at least one course in that block, ensuring that it represents a feasible alternative. We drop applicants with no available adjacent block. If both adjacent blocks are available, we retain the block above and the matched block.
For these applicants, the higher-ranked retained block defines the preferred field r 1 r_1 r 1 r 2 r_2 r 2
Let Q 1 Q_1 Q 1 s q s_q s q c q c_q c q q q q S = max q ∈ Q 1 { s q − c q } S = \max_{q\in Q_1}\{s_q-c_q\} S = max q ∈ Q 1 { s q − c q } S ≥ 0 S \ge0 S ≥ 0 S < 0 S < 0 S < 0 j j j Z j Z_j Z j j j j (10) applies.
Table 3 illustrates the construction of the preferred and next-best fields and the running variable. Panel A shows an applicant who ranks two courses in field j j j k k k j j j S = 0.1 S = 0.1 S = 0.1 k k k j j j k k k j j j
Table 3. Illustration of the preferred and next-best fields and the fuzzy RDD. Notes: Panel A provides an illustrative example of an applicant’s course ranking and the corresponding outcome of the admission mechanism. Panel B illustrates how this ranking and outcome can be used to construct the preferred and next-best fields for a fuzzy RDD.
Panel B summarizes and compares this applicant to another applicant with the same preferred and next-best fields who falls 0.1 points below the relevant cutoff for field j j j k k k S = 0.1 S = 0.1 S = 0.1 S = − 0.1 S = -0.1 S = − 0.1 j j j k k k
Our preferred and next-best fields may differ from students’ true preferences for three reasons. First, they represent local alternatives around a particular cutoff and need not be the two highest-ranked fields in the submitted ranking. Second, applicants can rank at most six courses, which may induce them to include safe options near the end of their rankings. These first two concerns appear limited in our setting because, for 88% of the sample, r 1 r_1 r 1 r 2 r_2 r 2 [49 ] , potentially introducing error into r 2 r_2 r 2
Bertanha et al. [8 ] discuss how strategic misreporting of preferences complicates identification of causal parameters that depend on true preferences. They also show that strategic reporting may violate the RD continuity assumption. They develop a partial identification approach robust to such behavior in a sharp RDD but leave identification in a fuzzy RDD with “multiple compliance types” to future work. Although we cannot rule out strategic reporting, the balancing and bunching tests reported below do not indicate violations of the RD continuity assumption.
Data and classifications of fields of study
We have access to an administrative data set that includes four cohorts in the Portuguese public higher education system. From the application data we observe each applicant’s submitted course ranking, application scores, and matched course, on top of all course cutoffs. From the student outcome data we observe a student’s enrolled course, whether a student is active in the first, second, or third year of that course, and the achieved number of study credits in each year for that course. Appendix § 10 .§ 2 provides further details on the data and sample construction and presents some descriptive statistics.
Study credits are measured using the European Credit Transfer and Accumulation System (ECTS), which facilitates the accumulation and transfer of learning achievements across European universities. ECTS credits represent learning outcomes and the estimated workload required to achieve them. One ECTS credit typically represents 25 to 30 hours of intended workload, while 60 ECTS correspond to one academic year, or 1,500 to 1,800 hours of workload. Four and eight ECTS typically correspond to a medium-sized and large course, respectively. These standards apply across institutions and fields, making ECTS comparable in terms of intended workload, but not in terms of content, difficulty, or quality.
We interpret achieved ECTS as a measure of study progress rather than a measure of skills or welfare: achieving more credits represents the completion of more intended workload and greater progress toward degree completion, but does not necessarily imply more skill development, enjoyment, or future earnings. Universities in several European countries eventually dismiss students who obtain insufficient credits [50 ] . In those settings, differences in academic progress could affect continued enrollment. However, dropout in Portugal is low, so this is not the focus of this application.
The Portuguese education authorities organize courses into 22 fields of study following the 2-digit International Standard Classification of Education (ISCED) from UNESCO Institute for Statistics (2012). Column (1) of Table 4 shows the 22 fields of study, which we define as our unclustered treatments. Column (2) classifies the 22 fields into two clusters according to the STEM versus non-STEM distinction from OECD (2022). Column (3) constructs eight clusters using the ISCED 1-digit classification: fields that share the same first digit in their 2-digit classification are grouped together in a broader field.
Table 4. STEM and ISCED classification of the fields of study
Empirical specification
We estimate the fuzzy RD model as specified in (12)–(13) using the sample analogue of (11) . For treatment cluster S b S_b S b S b S_b S b Table 4 reports these estimation samples. For the STEM and ISCED classifications, we pool the samples over all next-best fields within the same cluster.
Notes: The first three columns show the 22 fields of study, the crosswalk to STEM versus non-STEM from OECD [46 ] , and the ISCED clusters from UNESCO Institute for Statistics [51 ] . The fourth column reports the number of students in our sample with the field in Column (1) as next-best field k k k (9) , with k k k
The outcome variable Y Y Y D ~ a = ∑ j ∈ S a D j \widetilde{D}_a = \sum_{j \in S_a} D_j D a = ∑ j ∈ S a D j S a S_a S a Z ~ c = ∑ l ∈ S c Z l \widetilde{Z}_c = \sum_{l \in S_c} Z_l Z c = ∑ l ∈ S c Z l S c S_c S c f j k f_{jk} f j k j j j k k k
The function f h ( S ) f_h(S) f h ( S ) f h ( S ) = γ S f_h(S) = \gamma S f h ( S ) = γ S [31 ] correction to account for multiple hypothesis tests within each estimation sample.
Because the treatment variable is defined as second-year course enrollment, our estimation sample is restricted to students with observed second-year enrollment. The local continuity assumption then requires that the course offer does not affect selection into this sample near the cutoff. Non-enrollment and dropout are relatively low in Portugal. Approximately 10% of applicants do not enroll in public higher education in their first year ([20 ] ), while approximately 2.5% of those who enroll drop out before the second year ([25 ] ). The latter is consistent with Portugal’s fixed admission quotas, which give little incentive to induce dropout after admission. These rates suggest limited attrition, while balancing and bunching tests reveal no significant irregularities around the cutoff. Together, these findings are reassuring, but do not entirely rule out offer-induced selection near the cutoff.
With our treatment definition, the exclusion restriction also requires that course offers affect second-year credits only through the second-year field of enrollment. Two potential violations merit discussion. First, offers may affect the institution without changing the field, in which case differences in institutional characteristics could directly affect credits. Such effects are likely to be limited as Portuguese public institutions offering the same fields have similar curricula and academic requirements. Second, offers may change first-year field without changing the second-year field. If these earlier experiences directly affect second-year credits, exclusion is violated. A formal analysis of these multiple treatment stages is beyond the scope of this paper, but Angrist et al. [4 ] discuss identification in these settings with a single instrumental variable.
Identifying assumptions with treatment clusters
The set of identifying assumptions required for a causal interpretation of the TSLS estimand with treatment clusters depends on the number of clusters. In addition to the standard IV assumptions, the STEM classification with two treatment clusters requires Assumptions 3 and 4. We discuss the plausibility of these two assumptions in turn.
First, consider a student whose next-best field is in the non-STEM cluster, such as Personal services, and whose preferred field is in the STEM cluster, such as Physical sciences. A score above the cutoff implies an offer to enroll in Physical sciences rather than Personal services. Assumption 3 rules out students for whom a Physical sciences offer causes them to leave the STEM cluster. This seems plausible, as it is unlikely that receiving an offer from a STEM field reduces the probability of enrolling in any STEM field. The assumption also rules out a STEM offer inducing enrollment in a different STEM field from the one chosen following a non-STEM offer, such as a student who would enroll in Engineering following a Personal services offer but in Physical sciences following a Physical sciences offer. Second, Assumption 4 rules out students who enroll in different non-STEM fields depending on whether they receive a Physical sciences or Personal services offer. If receiving an offer from a STEM field does not make a student enroll into any STEM field, it is indeed unlikely that the offer induces enrollment into any non-STEM field instead.
Since the ISCED classification has more than two treatment clusters, identification requires Assumptions 3, 4, and 5. We discuss these assumptions in the context of the same example. Consider the next-best field of study Personal services in the cluster Services and the preferred field Physical sciences in the cluster Science. Similar to the two treatment cluster case, Assumption 3 ensures that receiving a Physical sciences offer does not make it less likely that a student enrolls into any Science field, and Assumption 4 prevents students who enroll in different non-Science fields depending on whether they receive a Physical sciences or Personal services offer. Assumption 5 additionally requires that a student enrolling into any Science field after receiving an offer from Physical sciences, does not enroll into any non-Services field when receiving an offer from Personal services. This is likely if the Services cluster represents a control group, as is the case for the sample of students with a next-best field in the Services cluster.
The validity of Assumptions 3 to 5 depends on whether clusters effectively group similar fields of study, such that students may update preferences and face low switching costs between fields within, but not across, treatment clusters. Wiswall and Zafar [52 ] find evidence that student preferences are similar for majors within broad academic groups, suggesting that students initially enrolling in Physical sciences (Personal services) may later adjust their preferences towards related fields in Science (Services). Altonji et al. [1 ] discuss that major-switching costs depend on a student’s education history. DGES [19 ] describes that course changes in Portugal require an official request that includes potential accreditation of previous coursework. Since related fields share curricula, accreditation is often easier within clusters.
Results
Two treatment clusters: STEM and non-STEM
We find that STEM enrollment negatively impacts the number of achieved study credits in the second year. Table 5 shows that based on our estimates, STEM enrollment decreases ECTS on average by 3.15 credits for students preferring STEM over non-STEM in their course ranking. This decrease represents approximately 88 fewer hours of intended workload under STEM compared with non-STEM enrollment. The estimated effect is smaller for students preferring non-STEM over STEM: non-STEM enrollment increases ECTS on average by 1.33 credits for these students. The magnitude of this effect represents approximately 37 additional hours of intended workload. In case students are aware of the costs of STEM enrollment, their preference for and enrollment into STEM may be explained by other benefits, such as skill development or future monetary payoffs.
Table 5. Treatment effect and treatment take-up estimates with STEM clusters. Treatment cluster S b S_b S b β a b h \beta^{h}_{ab} β ab h π a c b h \pi^{h}_{acb} π a c b h S b S_b S b β a b h \beta^{h}_{ab} β ab h π a c b h \pi^{h}_{acb} π a c b h ∗ ^{*} ∗ ∗ ∗ ^{**} ∗∗ ∗ ∗ ∗ ^{***} ∗∗∗
The estimates can be interpreted as weighted averages of LATEs between fields of study across the two treatment clusters. The STEM and non-STEM clusters include respectively 8 and 14 fields of study. The weighted averages include the LATEs corresponding to all possible pairs of fields across these two clusters, and for each of those field pairs all possible pairs of instruments between the two clusters. It follows that the treatment effect of STEM versus non-STEM (and vice versa) includes 8 × 14 × 8 × 14 = 12,544 8 \times14 \times8 \times14 = 12{,}544 8 × 14 × 8 × 14 = 12 , 544
The large and precisely estimated treatment take-up effects provide strong evidence that Assumption 2 holds for the STEM classification. Table 5 shows that, for students preferring a field in STEM over non-STEM, scoring above the cutoff and thereby receiving an offer from a STEM field increases the probability of enrollment in a STEM field by 88 percentage points. Conversely, for students preferring non-STEM over STEM, receiving a non-STEM offer increases the probability of non-STEM enrollment by 82 percentage points.
Since we have access to the data at the field of study level, we can also test the necessary conditions in Proposition 1 for a causal interpretation of the treatment cluster effects. We jointly test the (in)equalities in Proposition 1 within each sample defined by a preferred and next-best field using the clrtest package in Stata. Appendix § 10 .§ 4 provides further implementation details. Column (5) of Table 4 summarizes the testing results. After applying a Holm correction, we find no statistically significant violations of the necessary conditions for a causal interpretation of the estimated treatment cluster effects.
This result may be explained by most students having response types that can be described by ( D ( j ) = j , D ( k ) = k ) (D(j) = j, D(k) = k) ( D ( j ) = j , D ( k ) = k ) j ∈ S a j \in S_a j ∈ S a k ∈ S b k \in S_b k ∈ S b π l j k h \pi^h_{ljk} π l j k h π a c b h \pi^h_{acb} π a c b h π l j k h \pi^h_{ljk} π l j k h π j j k h \pi^h_{jjk} π j j k h π l j k h \pi^h_{ljk} π l j k h l ≠ j , k l \ne j,k l = j , k
Multiple treatment clusters: Eight ISCED clusters
Table 6 shows the treatment cluster effect estimates across the eight ISCED clusters. After the Holm correction within each estimation sample, four estimates are statistically significant at the 5% level, compared with twelve before the correction. Their average absolute values are 4.13 and 3.44 ECTS, respectively. These effect sizes imply that some clustered fields can increase or decrease academic progress by the equivalent of approximately 100 Notes: This table reports estimates of β a b \beta_{ab} β ab (12) , using the ISCED classification. The columns correspond to treatment clusters S b S_b S b S a S_a S a S b S_b S b
Table 6. Treatment effect estimates with ISCED clusters
hours of intended workload, or a medium-sized course.
For the ISCED classification, we also find strong evidence that Assumption 2 holds.
Table 8 in Appendix § 10 .§ 5 reports the treatment take-up estimates for enrollment in the cluster containing the preferred field. Consistent with the evidence above that response types are predominantly characterized by ( D ( j ) = j , D ( k ) = k ) (D(j) = j, D(k) = k) ( D ( j ) = j , D ( k ) = k ) Table 4 shows that we reject the necessary conditions for the fields Health and Personal services. These rejections do not affect the conditions for the STEM classification. For instance, the rejected conditions in the sample of next-best Personal services correspond to the preferred field Humanities, and both Personal services and Humanities belong to the non-STEM cluster. Appendix § 10 .§ 6 provides additional empirical results for both the STEM and ISCED classifications, also using the data at the field of study level.
Table 8. Treatment take-up estimates with ISCED clusters.
Conclusion
Many modern policies involve multiple unordered treatments, for which many pairwise treatment comparisons are possible. A common approach is to aggregate these treatments into clusters and apply instrumental variable (IV) methods at the treatment cluster level.
This paper shows that IV methods with treatment clusters identify weighted averages of pairwise effects between treatments across clusters. The required assumptions are formulated at the unclustered treatment level and hence explicit about possible violations. We discuss how practitioners can justify these assumptions in the presence of treatment clusters and show that the assumptions are less strict compared to the assumptions required for identifying the pairwise effects between unclustered treatments. We also derive necessary conditions for these assumptions, and propose falsification tests for settings in which data at the unclustered treatment level is available.
We illustrate the empirical relevance of our methods by analyzing the causal effects of clusters of fields of study using a natural experiment in the Portuguese public higher education system. We consider the STEM versus non-STEM classification and eight clusters defined according to the International Standard Classification of Education. We find meaningful effects of enrollment in different clusters on study progress. Using data at the field of study of study level, we test the necessary conditions for causal interpretation of the treatment cluster effects.
Proofs theoretical results
Proof Lemma 1
The denominator of the IV estimand in (1) can be written as
Δ E [ D ~ 1 ∣ Z ~ 1 ] = ∑ n ∈ S 1 E [ D n ∣ Z ~ = 1 ] − ∑ n ∈ S 1 E [ D n ∣ Z ~ = 2 ] = ∑ j ∈ S 1 ∑ n ∈ S 1 E [ D n ∣ Z = j ] P [ Z = j ∣ Z ∈ S 1 ] − ∑ k ∈ S 2 ∑ n ∈ S 1 E [ D n ∣ Z = k ] P [ Z = k ∣ Z ∈ S 2 ] = ∑ j ∈ S 1 ∑ n ∈ S 1 E [ D n ( j ) ] P [ Z = j ∣ Z ∈ S 1 ] − ∑ k ∈ S 2 ∑ n ∈ S 1 E [ D n ( k ) ] P [ Z = k ∣ Z ∈ S 2 ] = ∑ j ∈ S 1 ∑ k ∈ S 2 ∑ n ∈ S 1 E [ D n ( j ) − D n ( k ) ] ω j k = ∑ j ∈ S 1 ∑ k ∈ S 2 ∑ n ∈ S 1 ( P [ D n ( j ) − D n ( k ) = 1 ] − P [ D n ( j ) − D n ( k ) = − 1 ] ) ω j k = ∑ j ∈ S 1 ∑ k ∈ S 2 ∑ l ∈ S 1 ∑ m ∈ S 2 ( P [ D ( j ) = l , D ( k ) = m ] − P [ D ( j ) = m , D ( k ) = l ] ) ω j k , (15) \begin{aligned}
\Delta\mathbb{E}[\widetilde{D}_1\mid\widetilde{Z}_1]
&=\sum_{n\in S_1}\mathbb{E}[D_n\mid\widetilde{Z}=1]-\sum_{n\in S_1}\mathbb{E}[D_n\mid\widetilde{Z}=2]\\
&=\sum_{j\in S_1}\sum_{n\in S_1}\mathbb{E}[D_n\mid Z=j]\mathbb{P}[Z=j\mid Z\in S_1]-\sum_{k\in S_2}\sum_{n\in S_1}\mathbb{E}[D_n\mid Z=k]\mathbb{P}[Z=k\mid Z\in S_2]\\
&=\sum_{j\in S_1}\sum_{n\in S_1}\mathbb{E}[D_n(j)]\mathbb{P}[Z=j\mid Z\in S_1]-\sum_{k\in S_2}\sum_{n\in S_1}\mathbb{E}[D_n(k)]\mathbb{P}[Z=k\mid Z\in S_2]\\
&=\sum_{j\in S_1}\sum_{k\in S_2}\sum_{n\in S_1}\mathbb{E}[D_n(j)-D_n(k)]\omega_{jk}\\
&=\sum_{j\in S_1}\sum_{k\in S_2}\sum_{n\in S_1}\left(\mathbb{P}[D_n(j)-D_n(k)=1]-\mathbb{P}[D_n(j)-D_n(k)=-1]\right)\omega_{jk}\\
&=\sum_{j\in S_1}\sum_{k\in S_2}\sum_{l\in S_1}\sum_{m\in S_2}\left(\mathbb{P}[D(j)=l,D(k)=m]-\mathbb{P}[D(j)=m,D(k)=l]\right)\omega_{jk}, \tag*{(15)}
\end{aligned} Δ E [ D 1 ∣ Z 1 ] = n ∈ S 1 ∑ E [ D n ∣ Z = 1 ] − n ∈ S 1 ∑ E [ D n ∣ Z = 2 ] = j ∈ S 1 ∑ n ∈ S 1 ∑ E [ D n ∣ Z = j ] P [ Z = j ∣ Z ∈ S 1 ] − k ∈ S 2 ∑ n ∈ S 1 ∑ E [ D n ∣ Z = k ] P [ Z = k ∣ Z ∈ S 2 ] = j ∈ S 1 ∑ n ∈ S 1 ∑ E [ D n ( j )] P [ Z = j ∣ Z ∈ S 1 ] − k ∈ S 2 ∑ n ∈ S 1 ∑ E [ D n ( k )] P [ Z = k ∣ Z ∈ S 2 ] = j ∈ S 1 ∑ k ∈ S 2 ∑ n ∈ S 1 ∑ E [ D n ( j ) − D n ( k )] ω j k = j ∈ S 1 ∑ k ∈ S 2 ∑ n ∈ S 1 ∑ ( P [ D n ( j ) − D n ( k ) = 1 ] − P [ D n ( j ) − D n ( k ) = − 1 ] ) ω j k = j ∈ S 1 ∑ k ∈ S 2 ∑ l ∈ S 1 ∑ m ∈ S 2 ∑ ( P [ D ( j ) = l , D ( k ) = m ] − P [ D ( j ) = m , D ( k ) = l ] ) ω j k , (15) where we use D n = ∑ l = 1 J D n ( l ) × Z l D_n=\sum_{l=1}^{J}D_n(l)\times Z_l D n = ∑ l = 1 J D n ( l ) × Z l ∑ j ∈ S a P [ Z = j ∣ Z ∈ S a ] = 1 \sum_{j\in S_a}\mathbb{P}[Z=j\mid Z\in S_a]=1 ∑ j ∈ S a P [ Z = j ∣ Z ∈ S a ] = 1 ∑ k ∈ S b P [ Z = k ∣ Z ∈ S b ] = 1 \sum_{k\in S_b}\mathbb{P}[Z=k\mid Z\in S_b]=1 ∑ k ∈ S b P [ Z = k ∣ Z ∈ S b ] = 1 ω j k = P [ Z = j ∣ Z ∈ S a ] P [ Z = k ∣ Z ∈ S b ] \omega_{jk}=\mathbb{P}[Z=j\mid Z\in S_a]\mathbb{P}[Z=k\mid Z\in S_b] ω j k = P [ Z = j ∣ Z ∈ S a ] P [ Z = k ∣ Z ∈ S b ]
∑ j ∈ S 1 ∑ k ∈ S 2 ∑ l ∈ S 1 ∑ m ∈ S 1 ( P [ D ( j ) = l , D ( k ) = m ] − P [ D ( j ) = m , D ( k ) = l ] ) ω j k = 0. \sum_{j\in S_1}\sum_{k\in S_2}\sum_{l\in S_1}\sum_{m\in S_1}\left(\mathbb{P}[D(j)=l,D(k)=m]-\mathbb{P}[D(j)=m,D(k)=l]\right)\omega_{jk}=0. j ∈ S 1 ∑ k ∈ S 2 ∑ l ∈ S 1 ∑ m ∈ S 1 ∑ ( P [ D ( j ) = l , D ( k ) = m ] − P [ D ( j ) = m , D ( k ) = l ] ) ω j k = 0. The numerator of (1) can be written as
Δ E [ Y ∣ Z ~ 1 ] = ∑ l = 1 J E [ Y ( l ) D l ∣ Z ~ = 1 ] − ∑ l = 1 J E [ Y ( l ) D l ∣ Z ~ = 2 ] = ∑ j ∈ S 1 ∑ l = 1 J E [ Y ( l ) D l ( j ) ∣ Z = j ] P [ Z = j ∣ Z ∈ S 1 ] − ∑ k ∈ S 2 ∑ l = 1 J E [ Y ( l ) D l ( k ) ∣ Z = k ] P [ Z = k ∣ Z ∈ S 2 ] = ∑ j ∈ S 1 ∑ k ∈ S 2 ∑ l = 1 J E [ Y ( l ) ( D l ( j ) − D l ( k ) ) ] ω j k = ∑ j ∈ S 1 ∑ k ∈ S 2 ∑ l = 1 J ∑ m ≠ l E [ Y ( l ) − Y ( m ) ∣ D ( j ) = l , D ( k ) = m ] P [ D ( j ) = l , D ( k ) = m ] ω j k , (16) \begin{aligned}
\Delta\mathbb{E}[Y\mid\widetilde{Z}_1]
&=\sum_{l=1}^{J}\mathbb{E}[Y(l)D_l\mid\widetilde{Z}=1]-\sum_{l=1}^{J}\mathbb{E}[Y(l)D_l\mid\widetilde{Z}=2]\\
&=\sum_{j\in S_1}\sum_{l=1}^{J}\mathbb{E}[Y(l)D_l(j)\mid Z=j]\mathbb{P}[Z=j\mid Z\in S_1]-\sum_{k\in S_2}\sum_{l=1}^{J}\mathbb{E}[Y(l)D_l(k)\mid Z=k]\mathbb{P}[Z=k\mid Z\in S_2]\\
&=\sum_{j\in S_1}\sum_{k\in S_2}\sum_{l=1}^{J}\mathbb{E}[Y(l)(D_l(j)-D_l(k))]\omega_{jk}\\
&=\sum_{j\in S_1}\sum_{k\in S_2}\sum_{l=1}^{J}\sum_{\substack{m\ne l}}\mathbb{E}[Y(l)-Y(m)\mid D(j)=l,D(k)=m]\mathbb{P}[D(j)=l,D(k)=m]\omega_{jk}, \tag*{(16)}
\end{aligned} Δ E [ Y ∣ Z 1 ] = l = 1 ∑ J E [ Y ( l ) D l ∣ Z = 1 ] − l = 1 ∑ J E [ Y ( l ) D l ∣ Z = 2 ] = j ∈ S 1 ∑ l = 1 ∑ J E [ Y ( l ) D l ( j ) ∣ Z = j ] P [ Z = j ∣ Z ∈ S 1 ] − k ∈ S 2 ∑ l = 1 ∑ J E [ Y ( l ) D l ( k ) ∣ Z = k ] P [ Z = k ∣ Z ∈ S 2 ] = j ∈ S 1 ∑ k ∈ S 2 ∑ l = 1 ∑ J E [ Y ( l ) ( D l ( j ) − D l ( k ))] ω j k = j ∈ S 1 ∑ k ∈ S 2 ∑ l = 1 ∑ J m = l ∑ E [ Y ( l ) − Y ( m ) ∣ D ( j ) = l , D ( k ) = m ] P [ D ( j ) = l , D ( k ) = m ] ω j k , (16) where we use Assumption 1(b) to substitute for potential outcomes Y = ∑ l = 1 J Y ( l ) × D l Y=\sum_{l=1}^{J}Y(l)\times D_l Y = ∑ l = 1 J Y ( l ) × D l
Define Δ j k l m = E [ Y ( l ) − Y ( m ) ∣ D ( j ) = l , D ( k ) = m ] \Delta_{jklm} = \mathbb{E}[Y(l) - Y(m) \mid D(j) = l, D(k) = m] Δ j k l m = E [ Y ( l ) − Y ( m ) ∣ D ( j ) = l , D ( k ) = m ] P j k l m = P [ D ( j ) = l , D ( k ) = m ] P_{jklm} = \mathbb{P}[D(j) = l, D(k) = m] P j k l m = P [ D ( j ) = l , D ( k ) = m ]
∑ l = 1 J ∑ m ≠ l Δ j k l m P j k l m = ∑ l ∈ S 1 ∑ m ∈ S 2 Δ j k l m P j k l m + ∑ l ∈ S 2 ∑ m ∈ S 1 Δ j k l m P j k l m + ∑ l ∈ S 1 ∑ m ≠ l ∈ S 1 Δ j k l m P j k l m + ∑ l ∈ S 2 ∑ m ≠ l ∈ S 2 Δ j k l m P j k l m = ∑ l ∈ S 1 ∑ m ∈ S 2 Δ j k l m P j k l m − Δ j k m l P j k m l + ∑ l ∈ S 1 ∑ m ≠ l ∈ S 1 Δ j k l m P j k l m + ∑ l ∈ S 2 ∑ m ≠ l ∈ S 2 Δ j k l m P j k l m . (17) \begin{aligned}
\sum_{l=1}^{J}\sum_{m\ne l}\Delta_{jklm}P_{jklm}
&= \sum_{l\in S_1}\sum_{m\in S_2}\Delta_{jklm}P_{jklm}
+\sum_{l\in S_2}\sum_{m\in S_1}\Delta_{jklm}P_{jklm} \\
&\quad+\sum_{l\in S_1}\sum_{m\ne l\in S_1}\Delta_{jklm}P_{jklm}
+\sum_{l\in S_2}\sum_{m\ne l\in S_2}\Delta_{jklm}P_{jklm} \\
&= \sum_{l\in S_1}\sum_{m\in S_2}\Delta_{jklm}P_{jklm}
-\Delta_{jkml}P_{jkml} \\
&\quad+\sum_{l\in S_1}\sum_{m\ne l\in S_1}\Delta_{jklm}P_{jklm}
+\sum_{l\in S_2}\sum_{m\ne l\in S_2}\Delta_{jklm}P_{jklm}. \tag*{(17)}
\end{aligned} l = 1 ∑ J m = l ∑ Δ j k l m P j k l m = l ∈ S 1 ∑ m ∈ S 2 ∑ Δ j k l m P j k l m + l ∈ S 2 ∑ m ∈ S 1 ∑ Δ j k l m P j k l m + l ∈ S 1 ∑ m = l ∈ S 1 ∑ Δ j k l m P j k l m + l ∈ S 2 ∑ m = l ∈ S 2 ∑ Δ j k l m P j k l m = l ∈ S 1 ∑ m ∈ S 2 ∑ Δ j k l m P j k l m − Δ j k m l P j k m l + l ∈ S 1 ∑ m = l ∈ S 1 ∑ Δ j k l m P j k l m + l ∈ S 2 ∑ m = l ∈ S 2 ∑ Δ j k l m P j k l m . (17) Hence, according to Assumption 2,
β = ∑ j ∈ S 1 ∑ k ∈ S 2 ω j k [ ∑ l ∈ S 1 ∑ m ∈ S 2 Δ j k l m P j k l m − Δ j k m l P j k m l ] + B 1 + B 2 ∑ j ∈ S 1 ∑ k ∈ S 2 ω j k [ ∑ l ∈ S 1 ∑ m ∈ S 2 P j k l m − P j k m l ] , (18) \beta=
\frac{\sum_{j\in S_1}\sum_{k\in S_2}\omega_{jk}\left[\sum_{l\in S_1}\sum_{m\in S_2}\Delta_{jklm}P_{jklm}-\Delta_{jkml}P_{jkml}\right]+B_1+B_2}
{\sum_{j\in S_1}\sum_{k\in S_2}\omega_{jk}\left[\sum_{l\in S_1}\sum_{m\in S_2}P_{jklm}-P_{jkml}\right]}, \tag*{(18)} β = ∑ j ∈ S 1 ∑ k ∈ S 2 ω j k [ ∑ l ∈ S 1 ∑ m ∈ S 2 P j k l m − P j k m l ] ∑ j ∈ S 1 ∑ k ∈ S 2 ω j k [ ∑ l ∈ S 1 ∑ m ∈ S 2 Δ j k l m P j k l m − Δ j k m l P j k m l ] + B 1 + B 2 , (18) with B c = ∑ j ∈ S 1 ∑ k ∈ S 2 ω j k ∑ l ∈ S c ∑ m ≠ l ∈ S c Δ j k l m P j k l m B_c=\sum_{j\in S_1}\sum_{k\in S_2}\omega_{jk}\sum_{l\in S_c}\sum_{m\ne l\in S_c}\Delta_{jklm}P_{jklm} B c = ∑ j ∈ S 1 ∑ k ∈ S 2 ω j k ∑ l ∈ S c ∑ m = l ∈ S c Δ j k l m P j k l m
Proof Theorem 2
Using Assumption 2, extended to the case with more than two treatment clusters, the expression of the IV estimand in (7) can be written as
E [ Z ~ b D ~ b ′ ] β b = E [ Z ~ b Y ] , (19) \mathbb{E}[\widetilde{Z}_b\widetilde{D}_b^{\prime}]\beta_b=\mathbb{E}[\widetilde{Z}_bY], \tag*{(19)} E [ Z b D b ′ ] β b = E [ Z b Y ] , (19) from which follows that the IV parameters β a b \beta_{ab} β ab
β a b = θ a b π a a b − ∑ c ≠ a , b π c a b π a a b β c b , ∀ a ≠ b , (20) \beta_{ab}=\frac{\theta_{ab}}{\pi_{aab}}-\sum_{c\ne a,b}\frac{\pi_{cab}}{\pi_{aab}}\beta_{cb},\quad\forall a\ne b, \tag*{(20)} β ab = π aab θ ab − c = a , b ∑ π aab π c ab β c b , ∀ a = b , (20) with
θ a b = E [ Y ∣ Z ~ = a ] − E [ Y ∣ Z ~ = b ] and π c a b = E [ D ~ c ∣ Z ~ = a ] − E [ D ~ c ∣ Z ~ = b ] . (21) \theta_{ab}=\mathbb{E}[Y\mid\widetilde{Z}=a]-\mathbb{E}[Y\mid\widetilde{Z}=b]\text{ and }\pi_{cab}=\mathbb{E}[\widetilde{D}_c\mid\widetilde{Z}=a]-\mathbb{E}[\widetilde{D}_c\mid\widetilde{Z}=b]. \tag*{(21)} θ ab = E [ Y ∣ Z = a ] − E [ Y ∣ Z = b ] and π c ab = E [ D c ∣ Z = a ] − E [ D c ∣ Z = b ] . (21) Write π c a b \pi_{cab} π c ab c ≠ a , b c\ne a,b c = a , b
π c a b = ∑ n ∈ S c E [ D n ∣ Z ~ = a ] − ∑ n ∈ S c E [ D n ∣ Z ~ = b ] = ∑ j ∈ S a ∑ n ∈ S c E [ D n ∣ Z = j ] P [ Z = j ∣ Z ∈ S a ] − ∑ k ∈ S b ∑ n ∈ S c E [ D n ∣ Z = k ] P [ Z = k ∣ Z ∈ S b ] . (22) \begin{aligned}
\pi_{cab}=\sum_{n\in S_c}\mathbb{E}[D_n\mid\widetilde{Z}=a]-\sum_{n\in S_c}\mathbb{E}[D_n\mid\widetilde{Z}=b] \\
&=\sum_{j\in S_a}\sum_{n\in S_c}\mathbb{E}[D_n\mid Z=j]\mathbb{P}[Z=j\mid Z\in S_a]-\sum_{k\in S_b}\sum_{n\in S_c}\mathbb{E}[D_n\mid Z=k]\mathbb{P}[Z=k\mid Z\in S_b]. \tag*{(22)}
\end{aligned} π c ab = n ∈ S c ∑ E [ D n ∣ Z = a ] − n ∈ S c ∑ E [ D n ∣ Z = b ] = j ∈ S a ∑ n ∈ S c ∑ E [ D n ∣ Z = j ] P [ Z = j ∣ Z ∈ S a ] − k ∈ S b ∑ n ∈ S c ∑ E [ D n ∣ Z = k ] P [ Z = k ∣ Z ∈ S b ] . (22) Substitute for potential outcomes D n = ∑ l = 1 J D n ( l ) × Z l D_n = \sum_{l=1}^{J} D_n(l) \times Z_l D n = ∑ l = 1 J D n ( l ) × Z l
π c a b = ∑ j ∈ S a ∑ n ∈ S c E [ D n ( j ) ] P [ Z = j ∣ Z ∈ S a ] − ∑ k ∈ S b ∑ n ∈ S c E [ D n ( k ) ] P [ Z = k ∣ Z ∈ S b ] . (23) \pi_{cab} = \sum_{j \in S_a} \sum_{n \in S_c} \mathbb{E}[D_n(j)]\mathbb{P}[Z = j \mid Z \in S_a] - \sum_{k \in S_b} \sum_{n \in S_c} \mathbb{E}[D_n(k)]\mathbb{P}[Z = k \mid Z \in S_b]. \tag*{(23)} π c ab = j ∈ S a ∑ n ∈ S c ∑ E [ D n ( j )] P [ Z = j ∣ Z ∈ S a ] − k ∈ S b ∑ n ∈ S c ∑ E [ D n ( k )] P [ Z = k ∣ Z ∈ S b ] . (23) Since ∑ j ∈ S a P [ Z = j ∣ Z ∈ S a ] = 1 \sum_{j \in S_a} \mathbb{P}[Z = j \mid Z \in S_a] = 1 ∑ j ∈ S a P [ Z = j ∣ Z ∈ S a ] = 1 ∑ k ∈ S b P [ Z = k ∣ Z ∈ S b ] = 1 \sum_{k \in S_b} \mathbb{P}[Z = k \mid Z \in S_b] = 1 ∑ k ∈ S b P [ Z = k ∣ Z ∈ S b ] = 1
π c a b = ∑ j ∈ S a ∑ k ∈ S b ∑ n ∈ S c E [ D n ( j ) − D n ( k ) ] ω j k = ∑ j ∈ S a ∑ k ∈ S b ∑ n ∈ S c ( P [ D n ( j ) − D n ( k ) = 1 ] − P [ D n ( j ) − D n ( k ) = − 1 ] ) ω j k , (24) \begin{aligned}
\pi_{cab} &= \sum_{j \in S_a} \sum_{k \in S_b} \sum_{n \in S_c} \mathbb{E}[D_n(j) - D_n(k)]\omega_{jk} \\
&= \sum_{j \in S_a} \sum_{k \in S_b} \sum_{n \in S_c} \left(\mathbb{P}[D_n(j) - D_n(k) = 1] - \mathbb{P}[D_n(j) - D_n(k) = -1]\right)\omega_{jk}, \tag*{(24)}
\end{aligned} π c ab = j ∈ S a ∑ k ∈ S b ∑ n ∈ S c ∑ E [ D n ( j ) − D n ( k )] ω j k = j ∈ S a ∑ k ∈ S b ∑ n ∈ S c ∑ ( P [ D n ( j ) − D n ( k ) = 1 ] − P [ D n ( j ) − D n ( k ) = − 1 ] ) ω j k , (24) where ω j k = P [ Z = j ∣ Z ∈ S a ] P [ Z = k ∣ Z ∈ S b ] \omega_{jk} = \mathbb{P}[Z = j \mid Z \in S_a]\mathbb{P}[Z = k \mid Z \in S_b] ω j k = P [ Z = j ∣ Z ∈ S a ] P [ Z = k ∣ Z ∈ S b ]
According to Assumption 3, P [ D n ( j ) − D n ( k ) = 1 ] = P [ D n ( j ) − D n ( k ) = 1 , D l ( j ) = 0 ] = P [ D n ( j ) − D n ( k ) = 1 , D l ( j ) = 0 , D l ( k ) = 0 ] \mathbb{P}[D_n(j) - D_n(k) = 1] = \mathbb{P}[D_n(j) - D_n(k) = 1, D_l(j) = 0] = \mathbb{P}[D_n(j) - D_n(k) = 1, D_l(j) = 0, D_l(k) = 0] P [ D n ( j ) − D n ( k ) = 1 ] = P [ D n ( j ) − D n ( k ) = 1 , D l ( j ) = 0 ] = P [ D n ( j ) − D n ( k ) = 1 , D l ( j ) = 0 , D l ( k ) = 0 ] l ∈ S a l \in S_a l ∈ S a l ∈ S a l \in S_a l ∈ S a P [ D n ( j ) − D n ( k ) = 1 ] = 0 \mathbb{P}[D_n(j) - D_n(k) = 1] = 0 P [ D n ( j ) − D n ( k ) = 1 ] = 0 n ∉ S a n \notin S_a n ∈ / S a j ∈ S a j \in S_a j ∈ S a k ∈ S b k \in S_b k ∈ S b π c a b \pi_{cab} π c ab
π c a b = − ∑ j ∈ S a ∑ k ∈ S b ∑ n ∈ S c P [ D n ( j ) − D n ( k ) = − 1 ] ω j k . (25) \pi_{cab} = - \sum_{j \in S_a} \sum_{k \in S_b} \sum_{n \in S_c} \mathbb{P}[D_n(j) - D_n(k) = -1]\omega_{jk}. \tag*{(25)} π c ab = − j ∈ S a ∑ k ∈ S b ∑ n ∈ S c ∑ P [ D n ( j ) − D n ( k ) = − 1 ] ω j k . (25) Assumption 3 together with Assumption 4 impose that when D n ( j ) − D n ( k ) = − 1 D_n(j) - D_n(k) = -1 D n ( j ) − D n ( k ) = − 1 D l ( j ) − D l ( k ) = 1 D_l(j) - D_l(k) = 1 D l ( j ) − D l ( k ) = 1 l ∈ S a l \in S_a l ∈ S a π c a b \pi_{cab} π c ab
π c a b = − ∑ j ∈ S a ∑ k ∈ S b ∑ n ∈ S c ∑ l ∈ S a P [ D n ( j ) − D n ( k ) = − 1 , D l ( j ) − D l ( k ) = 1 ] ω j k . (26) \pi_{cab} = - \sum_{j \in S_a} \sum_{k \in S_b} \sum_{n \in S_c} \sum_{l \in S_a} \mathbb{P}[D_n(j) - D_n(k) = -1, D_l(j) - D_l(k) = 1]\omega_{jk}. \tag*{(26)} π c ab = − j ∈ S a ∑ k ∈ S b ∑ n ∈ S c ∑ l ∈ S a ∑ P [ D n ( j ) − D n ( k ) = − 1 , D l ( j ) − D l ( k ) = 1 ] ω j k . (26) Assumption 5 implies that P [ D n ( j ) − D n ( k ) = − 1 , D l ( j ) = 1 ] = P [ D n ( j ) − D n ( k ) = − 1 , D l ( j ) − D l ( k ) = 1 ] = 0 \mathbb{P}[D_n(j) - D_n(k) = -1, D_l(j) = 1] = \mathbb{P}[D_n(j) - D_n(k) = -1, D_l(j) - D_l(k) = 1] = 0 P [ D n ( j ) − D n ( k ) = − 1 , D l ( j ) = 1 ] = P [ D n ( j ) − D n ( k ) = − 1 , D l ( j ) − D l ( k ) = 1 ] = 0 n ∉ { S a , S b } n \notin\{S_a, S_b\} n ∈ / { S a , S b } j ∈ S a j \in S_a j ∈ S a k ∈ S b k \in S_b k ∈ S b
π c a b = 0 , ∀ c ≠ a , b . (27) \pi_{cab} = 0,\quad\forall c \ne a,b. \tag*{(27)} π c ab = 0 , ∀ c = a , b . (27) It follows from (20) and (27) that
β a b = θ a b π a a b , ∀ a ≠ b . (28) \beta_{ab} = \frac{\theta_{ab}}{\pi_{aab}},\quad\forall a \ne b. \tag*{(28)} β ab = π aab θ ab , ∀ a = b . (28) Use the expression in (24) to write
π a a b = ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ( P [ D l ( j ) − D l ( k ) = 1 ] − P [ D l ( j ) − D l ( k ) = − 1 ] ) ω j k . (29) \pi_{aab} = \sum_{j \in S_a} \sum_{k \in S_b} \sum_{l \in S_a} \left(\mathbb{P}[D_l(j) - D_l(k) = 1] - \mathbb{P}[D_l(j) - D_l(k) = -1]\right)\omega_{jk}. \tag*{(29)} π aab = j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ ( P [ D l ( j ) − D l ( k ) = 1 ] − P [ D l ( j ) − D l ( k ) = − 1 ] ) ω j k . (29) Assumption 3 implies that P [ D l ( j ) − D l ( k ) = − 1 ] = 0 \mathbb{P}[D_l(j)-D_l(k)=-1]=0 P [ D l ( j ) − D l ( k ) = − 1 ] = 0 j , l ∈ S a j,l\in S_a j , l ∈ S a k ∈ S b k\in S_b k ∈ S b
π a a b = ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a P [ D l ( j ) − D l ( k ) = 1 ] ω j k . (30) \pi_{aab}=\sum_{j\in S_a}\sum_{k\in S_b}\sum_{l\in S_a}\mathbb{P}[D_l(j)-D_l(k)=1]\,\omega_{jk}. \tag*{(30)} π aab = j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ P [ D l ( j ) − D l ( k ) = 1 ] ω j k . (30) Since Assumption 5 implies that when D l ( j ) − D l ( k ) = 1 D_l(j)-D_l(k)=1 D l ( j ) − D l ( k ) = 1 D m ( j ) − D m ( k ) = − 1 D_m(j)-D_m(k)=-1 D m ( j ) − D m ( k ) = − 1 m m m S a S_a S a S b S_b S b m ∉ S a m\notin S_a m ∈ / S a
π a a b = ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ m ∈ S b P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] ω j k . (31) \pi_{aab}=\sum_{j\in S_a}\sum_{k\in S_b}\sum_{l\in S_a}\sum_{m\in S_b}\mathbb{P}[D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1]\,\omega_{jk}. \tag*{(31)} π aab = j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ m ∈ S b ∑ P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] ω j k . (31) Next, we use Assumption 1(b) to substitute for potential outcomes Y = ∑ l = 1 J Y ( l ) × D l Y=\sum_{l=1}^{J}Y(l)\times D_l Y = ∑ l = 1 J Y ( l ) × D l
θ a b = ∑ l = 1 J E [ Y ( l ) D l ∣ Z ~ = a ] − ∑ l = 1 J E [ Y ( l ) D l ∣ Z ~ = b ] = ∑ j ∈ S a ∑ l = 1 J E [ Y ( l ) D l ( j ) ∣ Z = j ] P [ Z = j ∣ Z ∈ S a ] − ∑ k ∈ S b ∑ l = 1 J E [ Y ( l ) D l ( k ) ∣ Z = k ] P [ Z = k ∣ Z ∈ S b ] , (32) \begin{aligned}
\theta_{ab}
&=\sum_{l=1}^{J}\mathbb{E}[Y(l)D_l\mid\widetilde{Z}=a]-\sum_{l=1}^{J}\mathbb{E}[Y(l)D_l\mid\widetilde{Z}=b]\\
&=\sum_{j\in S_a}\sum_{l=1}^{J}\mathbb{E}[Y(l)D_l(j)\mid Z=j]\mathbb{P}[Z=j\mid Z\in S_a]\\
&\quad-\sum_{k\in S_b}\sum_{l=1}^{J}\mathbb{E}[Y(l)D_l(k)\mid Z=k]\mathbb{P}[Z=k\mid Z\in S_b], \tag*{(32)}
\end{aligned} θ ab = l = 1 ∑ J E [ Y ( l ) D l ∣ Z = a ] − l = 1 ∑ J E [ Y ( l ) D l ∣ Z = b ] = j ∈ S a ∑ l = 1 ∑ J E [ Y ( l ) D l ( j ) ∣ Z = j ] P [ Z = j ∣ Z ∈ S a ] − k ∈ S b ∑ l = 1 ∑ J E [ Y ( l ) D l ( k ) ∣ Z = k ] P [ Z = k ∣ Z ∈ S b ] , (32) then use Assumption 1(a) to arrive at
θ a b = ∑ j ∈ S a ∑ k ∈ S b ∑ l = 1 J E [ Y ( l ) ( D l ( j ) − D l ( k ) ) ] ω j k , (33) \theta_{ab}=\sum_{j\in S_a}\sum_{k\in S_b}\sum_{l=1}^{J}\mathbb{E}[Y(l)(D_l(j)-D_l(k))]\,\omega_{jk}, \tag*{(33)} θ ab = j ∈ S a ∑ k ∈ S b ∑ l = 1 ∑ J E [ Y ( l ) ( D l ( j ) − D l ( k ))] ω j k , (33) where we split the summation across treatments l l l
θ a b = ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a E [ Y ( l ) ( D l ( j ) − D l ( k ) ) ] ω j k + ∑ j ∈ S a ∑ k ∈ S b ∑ m ∈ S b E [ Y ( m ) ( D m ( j ) − D m ( k ) ) ] ω j k + ∑ j ∈ S a ∑ k ∈ S b ∑ n ∉ S a , S b E [ Y ( n ) ( D n ( j ) − D n ( k ) ) ] ω j k . (34) \begin{aligned}
\theta_{ab}
&=\sum_{j\in S_a}\sum_{k\in S_b}\sum_{l\in S_a}\mathbb{E}[Y(l)(D_l(j)-D_l(k))]\,\omega_{jk}\\
&\quad+\sum_{j\in S_a}\sum_{k\in S_b}\sum_{m\in S_b}\mathbb{E}[Y(m)(D_m(j)-D_m(k))]\,\omega_{jk}\\
&\quad+\sum_{j\in S_a}\sum_{k\in S_b}\sum_{n\notin S_a,S_b}\mathbb{E}[Y(n)(D_n(j)-D_n(k))]\,\omega_{jk}. \tag*{(34)}
\end{aligned} θ ab = j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ E [ Y ( l ) ( D l ( j ) − D l ( k ))] ω j k + j ∈ S a ∑ k ∈ S b ∑ m ∈ S b ∑ E [ Y ( m ) ( D m ( j ) − D m ( k ))] ω j k + j ∈ S a ∑ k ∈ S b ∑ n ∈ / S a , S b ∑ E [ Y ( n ) ( D n ( j ) − D n ( k ))] ω j k . (34) For the second term in (34), Assumption 3 imposes that P [ D m ( j ) − D m ( k ) = 1 ] = P [ D m ( j ) − D m ( k ) = 1 , D l ( j ) = 0 ] = P [ D m ( j ) − D m ( k ) = 1 , D l ( j ) = 0 , D l ( k ) = 0 ] \mathbb{P}[D_m(j)-D_m(k)=1]=\mathbb{P}[D_m(j)-D_m(k)=1,D_l(j)=0]=\mathbb{P}[D_m(j)-D_m(k)=1,D_l(j)=0,D_l(k)=0] P [ D m ( j ) − D m ( k ) = 1 ] = P [ D m ( j ) − D m ( k ) = 1 , D l ( j ) = 0 ] = P [ D m ( j ) − D m ( k ) = 1 , D l ( j ) = 0 , D l ( k ) = 0 ] l ∈ S a l\in S_a l ∈ S a P [ D m ( j ) − D m ( k ) = 1 ] = 0 \mathbb{P}[D_m(j)-D_m(k)=1]=0 P [ D m ( j ) − D m ( k ) = 1 ] = 0 m ∈ S b m\in S_b m ∈ S b j ∈ S a j\in S_a j ∈ S a k ∈ S b k\in S_b k ∈ S b D n ( j ) − D n ( k ) = 0 D_n(j)-D_n(k)=0 D n ( j ) − D n ( k ) = 0 n ∉ { S a , S b } n\notin\{S_a,S_b\} n ∈ / { S a , S b } D m ( j ) − D m ( k ) = − 1 D_m(j)-D_m(k)=-1 D m ( j ) − D m ( k ) = − 1 D l ( j ) − D l ( k ) = 1 D_l(j)-D_l(k)=1 D l ( j ) − D l ( k ) = 1 D l ( j ) − D l ( k ) = 1 D_l(j)-D_l(k)=1 D l ( j ) − D l ( k ) = 1 D m ( j ) − D m ( k ) = − 1 D_m(j)-D_m(k)=-1 D m ( j ) − D m ( k ) = − 1 j , l ∈ S a j,l\in S_a j , l ∈ S a k , m ∈ S b k,m\in S_b k , m ∈ S b θ a b \theta_{ab} θ ab
θ a b = ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ m ∈ S b E [ Y ( l ) − Y ( m ) ∣ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] × \theta_{ab}=\sum_{j\in S_a}\sum_{k\in S_b}\sum_{l\in S_a}\sum_{m\in S_b}\mathbb{E}[Y(l)-Y(m)\mid D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1]\times θ ab = j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ m ∈ S b ∑ E [ Y ( l ) − Y ( m ) ∣ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] × P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] ω j k . (35) \mathbb{P}[D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1]\omega_{jk}. \tag*{(35)} P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] ω j k . (35) Combining (31) and (35) with (28) we have
β a b = ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ m ∈ S b λ j k l m E [ Y ( l ) − Y ( m ) ∣ D ( j ) = l , D ( k ) = m ] , (36) \beta_{ab}=\sum_{j\in S_a}\sum_{k\in S_b}\sum_{l\in S_a}\sum_{m\in S_b}\lambda_{jklm}\mathbb{E}[Y(l)-Y(m)\mid D(j)=l,D(k)=m], \tag*{(36)} β ab = j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ m ∈ S b ∑ λ j k l m E [ Y ( l ) − Y ( m ) ∣ D ( j ) = l , D ( k ) = m ] , (36) with
λ j k l m = ω j k P [ D ( j ) = l , D ( k ) = m ] ∑ j ∈ S a ∑ k ∈ S b ω j k ∑ l ∈ S a ∑ m ∈ S b P [ D ( j ) = l , D ( k ) = m ] , (37) \lambda_{jklm}=\frac{\omega_{jk}\mathbb{P}[D(j)=l,D(k)=m]}{\sum_{j\in S_a}\sum_{k\in S_b}\omega_{jk}\sum_{l\in S_a}\sum_{m\in S_b}\mathbb{P}[D(j)=l,D(k)=m]}, \tag*{(37)} λ j k l m = ∑ j ∈ S a ∑ k ∈ S b ω j k ∑ l ∈ S a ∑ m ∈ S b P [ D ( j ) = l , D ( k ) = m ] ω j k P [ D ( j ) = l , D ( k ) = m ] , (37) where we use that ( D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ) (D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1) ( D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ) ( D ( j ) = l , D ( k ) = m ) (D(j)=l,D(k)=m) ( D ( j ) = l , D ( k ) = m ) C = 2 C=2 C = 2
Proof Theorem 3
The estimand in (11) allows for the expression β b h = E h [ Z ¨ b , h D ~ b ] − 1 E h [ Z ¨ b , h Y ] \beta_b^h=\mathbb{E}_h[\ddot{Z}_{b,h}\widetilde{D}_b]^{-1}\mathbb{E}_h[\ddot{Z}_{b,h}Y] β b h = E h [ Z ¨ b , h D b ] − 1 E h [ Z ¨ b , h Y ] Z ¨ b , h = Z ~ b − P h ( Z ~ b ∣ X h ) \ddot{Z}_{b,h}=\widetilde{Z}_b-P_h(\widetilde{Z}_b\mid X_h) Z ¨ b , h = Z b − P h ( Z b ∣ X h ) P h ( Z ~ b ∣ X h ) = E h [ Z ~ b X h ′ ] E h [ X ~ h X h ′ ] − 1 X ~ h P_h(\widetilde{Z}_b\mid X_h)=\mathbb{E}_h[\widetilde{Z}_bX_h']\mathbb{E}_h[\widetilde{X}_hX_h']^{-1}\widetilde{X}_h P h ( Z b ∣ X h ) = E h [ Z b X h ′ ] E h [ X h X h ′ ] − 1 X h E h [ ( Z ~ b − P h ( Z ~ b ∣ X h ) ) ] = 0 \mathbb{E}_h[(\widetilde{Z}_b-P_h(\widetilde{Z}_b\mid X_h))]=0 E h [( Z b − P h ( Z b ∣ X h ))] = 0 a a a c ≠ b c\ne b c = b E h [ Z ¨ b D ~ b ′ ] \mathbb{E}_h[\ddot{Z}_b\widetilde{D}_b'] E h [ Z ¨ b D b ′ ]
E h [ Z ¨ a , h D ~ c ] = ∑ j ∈ S a ∑ k ∈ S b P h [ f j k = 1 ] E h [ Z ¨ a , h D ~ c ∣ f j k = 1 ] = ∑ j ∈ S a ∑ k ∈ S b P h [ f j k = 1 ] E h [ Z ¨ a , h E h [ D ~ c ∣ S , f j k = 1 ] ∣ f j k = 1 ] , (38) \begin{aligned}
\mathbb{E}_h[\ddot{Z}_{a,h}\widetilde{D}_c]
&=\sum_{j\in S_a}\sum_{k\in S_b}\mathbb{P}_h[f_{jk}=1]\mathbb{E}_h[\ddot{Z}_{a,h}\widetilde{D}_c\mid f_{jk}=1]\\
&=\sum_{j\in S_a}\sum_{k\in S_b}\mathbb{P}_h[f_{jk}=1]\mathbb{E}_h[\ddot{Z}_{a,h}\mathbb{E}_h[\widetilde{D}_c\mid S,f_{jk}=1]\mid f_{jk}=1], \tag*{(38)}
\end{aligned} E h [ Z ¨ a , h D c ] = j ∈ S a ∑ k ∈ S b ∑ P h [ f j k = 1 ] E h [ Z ¨ a , h D c ∣ f j k = 1 ] = j ∈ S a ∑ k ∈ S b ∑ P h [ f j k = 1 ] E h [ Z ¨ a , h E h [ D c ∣ S , f j k = 1 ] ∣ f j k = 1 ] , (38) where we use that the polynomial is flexible enough so that we can write the projection residual per f j k f_{jk} f j k Z ¨ a , h = 1 [ S ≥ 0 ] − P h ( 1 [ S ≥ 0 ] ∣ B h , f j k = 1 ) \ddot{Z}_{a,h}=\mathbb{1}[S\ge0]-P_h(\mathbb{1}[S\ge0]\mid B_h,f_{jk}=1) Z ¨ a , h = 1 [ S ≥ 0 ] − P h ( 1 [ S ≥ 0 ] ∣ B h , f j k = 1 ) f j k = 1 f_{jk}=1 f j k = 1 f j k = 1 f_{jk}=1 f j k = 1 S S S Z ¨ a , h \ddot{Z}_{a,h} Z ¨ a , h S S S
Under Assumption 6, we can write
lim h ↓ 0 E h [ D ~ c ∣ S , f j k = 1 ] = lim v ↓ 0 E [ D ~ c ∣ S = v , f j k = 1 ] + 1 [ S ≥ 0 ] Δ j k c , (39) \lim_{h\downarrow0}\mathbb{E}_h[\widetilde{D}_c|S,f_{jk}=1]=\lim_{v\downarrow0}\mathbb{E}[\widetilde{D}_c|S=v,f_{jk}=1]+\mathbb{1}[S\geq0]\Delta^c_{jk}, \tag*{(39)} h ↓ 0 lim E h [ D c ∣ S , f j k = 1 ] = v ↓ 0 lim E [ D c ∣ S = v , f j k = 1 ] + 1 [ S ≥ 0 ] Δ j k c , (39) where we define Δ j k c = lim v ↓ 0 E [ D ~ c ∣ S = v , f j k = 1 ] − lim v ↑ 0 E [ D ~ c ∣ S = v , f j k = 1 ] \Delta^c_{jk}=\lim_{v\downarrow0}\mathbb{E}[\widetilde{D}_c|S=v,f_{jk}=1]-\lim_{v\uparrow0}\mathbb{E}[\widetilde{D}_c|S=v,f_{jk}=1] Δ j k c = lim v ↓ 0 E [ D c ∣ S = v , f j k = 1 ] − lim v ↑ 0 E [ D c ∣ S = v , f j k = 1 ]
lim h ↓ 0 E h [ Z ¨ a , h E h [ D ~ c ∣ S , f j k = 1 ] ∣ f j k = 1 ] = lim v ↓ 0 E [ D ~ c ∣ S = v , f j k = 1 ] lim h ↓ 0 E h [ Z ¨ a , h ∣ f j k = 1 ] + Δ j k c lim h ↓ 0 E h [ Z ¨ a , h 1 [ S ≥ 0 ] ∣ f j k = 1 ] = Δ j k c κ , (40) \begin{aligned}
\lim_{h\downarrow0}\mathbb{E}_h[\ddot{Z}_{a,h}\mathbb{E}_h[\widetilde{D}_c|S,f_{jk}=1]|f_{jk}=1]
&=\lim_{v\downarrow0}\mathbb{E}[\widetilde{D}_c|S=v,f_{jk}=1]\lim_{h\downarrow0}\mathbb{E}_h[\ddot{Z}_{a,h}|f_{jk}=1]\\
&\quad+\Delta^c_{jk}\lim_{h\downarrow0}\mathbb{E}_h[\ddot{Z}_{a,h}\mathbb{1}[S\geq0]|f_{jk}=1]\\
&=\Delta^c_{jk}\kappa, \tag*{(40)}
\end{aligned} h ↓ 0 lim E h [ Z ¨ a , h E h [ D c ∣ S , f j k = 1 ] ∣ f j k = 1 ] = v ↓ 0 lim E [ D c ∣ S = v , f j k = 1 ] h ↓ 0 lim E h [ Z ¨ a , h ∣ f j k = 1 ] + Δ j k c h ↓ 0 lim E h [ Z ¨ a , h 1 [ S ≥ 0 ] ∣ f j k = 1 ] = Δ j k c κ , (40) where we use that E h [ Z ¨ a , h ∣ f j k = 1 ] = 0 \mathbb{E}_h[\ddot{Z}_{a,h}|f_{jk}=1]=0 E h [ Z ¨ a , h ∣ f j k = 1 ] = 0 lim h ↓ 0 E h [ Z ¨ a , h 1 [ S ≥ 0 ] ∣ f j k = 1 ] = κ \lim_{h\downarrow0}\mathbb{E}_h[\ddot{Z}_{a,h}\mathbb{1}[S\geq0]|f_{jk}=1]=\kappa lim h ↓ 0 E h [ Z ¨ a , h 1 [ S ≥ 0 ] ∣ f j k = 1 ] = κ S / h S/h S / h ∣ S ∣ ≤ h |S|\leq h ∣ S ∣ ≤ h ( j , k ) (j,k) ( j , k ) h → 0 h\to0 h → 0 S S S S = 0 S=0 S = 0
It follows that lim h ↓ 0 E h [ Z ¨ a , h D ~ c ] = κ ∑ j ∈ S a ∑ k ∈ S b ω j k Δ j k c \lim_{h\downarrow0}\mathbb{E}_h[\ddot{Z}_{a,h}\widetilde{D}_c]=\kappa\sum_{j\in S_a}\sum_{k\in S_b}\omega_{jk}\Delta^c_{jk} lim h ↓ 0 E h [ Z ¨ a , h D c ] = κ ∑ j ∈ S a ∑ k ∈ S b ω j k Δ j k c ω j k = lim h ↓ 0 P h [ f j k = 1 ] \omega_{jk}=\lim_{h\downarrow0}\mathbb{P}_h[f_{jk}=1] ω j k = lim h ↓ 0 P h [ f j k = 1 ] D ~ c = ∑ n ∈ S c D n \widetilde{D}_c=\sum_{n\in S_c}D_n D c = ∑ n ∈ S c D n
Δ j k c = ∑ n ∈ S c lim v ↓ 0 E [ D n ∣ S = v , f j k = 1 ] − lim v ↑ 0 E [ D n ∣ S = v , f j k = 1 ] = ∑ n ∈ S c lim v ↓ 0 E [ D n ∣ Z = j , S = v , f j k = 1 ] − lim v ↑ 0 E [ D n ∣ Z = k , S = v , f j k = 1 ] = ∑ n ∈ S c E [ D n ( j ) − D n ( k ) ∣ S = 0 , f j k = 1 ] , (41) \begin{aligned}
\Delta^c_{jk}
&=\sum_{n\in S_c}\lim_{v\downarrow0}\mathbb{E}[D_n|S=v,f_{jk}=1]-\lim_{v\uparrow0}\mathbb{E}[D_n|S=v,f_{jk}=1]\\
&=\sum_{n\in S_c}\lim_{v\downarrow0}\mathbb{E}[D_n|Z=j,S=v,f_{jk}=1]-\lim_{v\uparrow0}\mathbb{E}[D_n|Z=k,S=v,f_{jk}=1]\\
&=\sum_{n\in S_c}\mathbb{E}[D_n(j)-D_n(k)|S=0,f_{jk}=1], \tag*{(41)}
\end{aligned} Δ j k c = n ∈ S c ∑ v ↓ 0 lim E [ D n ∣ S = v , f j k = 1 ] − v ↑ 0 lim E [ D n ∣ S = v , f j k = 1 ] = n ∈ S c ∑ v ↓ 0 lim E [ D n ∣ Z = j , S = v , f j k = 1 ] − v ↑ 0 lim E [ D n ∣ Z = k , S = v , f j k = 1 ] = n ∈ S c ∑ E [ D n ( j ) − D n ( k ) ∣ S = 0 , f j k = 1 ] , (41) where we use the definition of the instruments and Assumption 6. Similarly, we have
lim h ↓ 0 E h [ Z ¨ a , h Y ] = κ ∑ j ∈ S a ∑ k ∈ S b ω j k ∑ n = 1 J E [ Y ( n ) ( D n ( j ) − D n ( k ) ) ∣ S = 0 , f j k = 1 ] , (42) \lim_{h\downarrow0}\mathbb{E}_h[\ddot{Z}_{a,h}Y]=\kappa\sum_{j\in S_a}\sum_{k\in S_b}\omega_{jk}\sum_{n=1}^{J}\mathbb{E}[Y(n)(D_n(j)-D_n(k))|S=0,f_{jk}=1], \tag*{(42)} h ↓ 0 lim E h [ Z ¨ a , h Y ] = κ j ∈ S a ∑ k ∈ S b ∑ ω j k n = 1 ∑ J E [ Y ( n ) ( D n ( j ) − D n ( k )) ∣ S = 0 , f j k = 1 ] , (42) and since the κ \kappa κ
Additional theoretical results
Theorem 2 shows that the TSLS estimand identifies a weighted average of LATEs using only data at the clustered treatment level. When data at the unclustered treatment level is available, this Appendix provides three strategies for providing additional insights into the components of the weighted average. First, we show that the weights of the treatment effects are partially identified. Second, we consider two sets of additional assumptions which allow for the identification of both the weights and the LATEs in the weighted averages. We provide testable conditions for the additional assumptions.
Partial identification of the treatment effect weights
The contribution of each component λ j k l m Δ j k l m \lambda_{jklm}\Delta_{jklm} λ j k l m Δ j k l m β a b \beta_{ab} β ab Δ j k l m \Delta_{jklm} Δ j k l m λ j k l m \lambda_{jklm} λ j k l m
In the general case in which both S a S_a S a S b S_b S b S a S_a S a Z ∈ S a Z \in S_a Z ∈ S a S b S_b S b Z ∈ S b Z \in S_b Z ∈ S b
Corollary 1. (Partial identification treatment effect weights). Under Assumptions 1-5, it holds for λ j k l m \lambda_{jklm} λ j k l m (5) with j , l ∈ S a j,l \in S_a j , l ∈ S a k , m ∈ S b k,m \in S_b k , m ∈ S b
max ( 0 , − π m j k − ∑ n ∈ { S a \ l } π n j k ) ω j k ∑ j ∈ S a ∑ k ∈ S b ω j k ∑ l ∈ S a π l j k ≤ λ j k l m ≤ min ( π l j k , − π m j k ) ω j k ∑ j ∈ S a ∑ k ∈ S b ω j k ∑ l ∈ S a π l j k , (43) \max\left(0,-\pi_{mjk}-\sum_{n\in\{S_a\backslash l\}}\pi_{njk}\right)\frac{\omega_{jk}}{\sum_{j\in S_a}\sum_{k\in S_b}\omega_{jk}\sum_{l\in S_a}\pi_{ljk}} \leq\lambda_{jklm} \leq\frac{\min(\pi_{ljk},-\pi_{mjk})\omega_{jk}}{\sum_{j\in S_a}\sum_{k\in S_b}\omega_{jk}\sum_{l\in S_a}\pi_{ljk}}, \tag*{(43)} max 0 , − π mj k − n ∈ { S a \ l } ∑ π nj k ∑ j ∈ S a ∑ k ∈ S b ω j k ∑ l ∈ S a π l j k ω j k ≤ λ j k l m ≤ ∑ j ∈ S a ∑ k ∈ S b ω j k ∑ l ∈ S a π l j k min ( π l j k , − π mj k ) ω j k , (43) where π l j k = E [ D l ∣ Z = j ] − E [ D l ∣ Z = k ] = P [ D ( j ) = l , D ( k ) ≠ l ] − P [ D ( j ) ≠ l , D ( k ) = l ] \pi_{ljk}=\mathbb{E}[D_l\mid Z=j]-\mathbb{E}[D_l\mid Z=k]=\mathbb{P}[D(j)=l,D(k)\ne l]-\mathbb{P}[D(j)\ne l,D(k)=l] π l j k = E [ D l ∣ Z = j ] − E [ D l ∣ Z = k ] = P [ D ( j ) = l , D ( k ) = l ] − P [ D ( j ) = l , D ( k ) = l ] and ω j k \omega_{jk} ω j k as defined in (6) .
The proof is deferred to Appendix § 9 .§ 4 . Corollary 1 provides individual bounds on each weight. These bounds do not jointly characterize the feasible weights, since extreme values for different weights may not be simultaneously attainable.
Corollary 1 shows that the proportion of types ( D ( j ) = l , D ( k ) = m ) (D(j)=l,D(k)=m) ( D ( j ) = l , D ( k ) = m ) ( D ( j ) = l , D ( k ) ∈ S b ) (D(j)=l,D(k)\in S_b) ( D ( j ) = l , D ( k ) ∈ S b ) ( D ( j ) ∈ S a , D ( k ) = m ) (D(j) \in S_a, D(k) = m) ( D ( j ) ∈ S a , D ( k ) = m ) ( D ( j ) ∈ S a , D ( k ) = m ) (D(j) \in S_a, D(k) = m) ( D ( j ) ∈ S a , D ( k ) = m ) ( D ( j ) ∈ { S a ∖ l } , D ( k ) ∈ S b ) (D(j) \in\{S_a \setminus l\}, D(k) \in S_b) ( D ( j ) ∈ { S a ∖ l } , D ( k ) ∈ S b ) π l j k \pi_{ljk} π l j k ω j k \omega_{jk} ω j k
When either S a S_a S a S b S_b S b S a = { j } S_a = \{j\} S a = { j } ( D ( j ) ∈ S a , D ( k ) = m ) (D(j) \in S_a, D(k) = m) ( D ( j ) ∈ S a , D ( k ) = m ) ( D ( j ) = j , D ( k ) = m ) (D(j) = j, D(k) = m) ( D ( j ) = j , D ( k ) = m ) S b = { k } S_b = \{k\} S b = { k } ( D ( j ) = l , D ( k ) ∈ S b ) (D(j) = l, D(k) \in S_b) ( D ( j ) = l , D ( k ) ∈ S b ) ( D ( j ) = l , D ( k ) = k ) (D(j) = l, D(k) = k) ( D ( j ) = l , D ( k ) = k ) S a = { j } S_a = \{j\} S a = { j } S b = { k } S_b = \{k\} S b = { k }
λ j k j m = − π m j k P [ Z = k ∣ Z ∈ S b ] ∑ k ∈ S b P [ Z = k ∣ Z ∈ S b ] π j j k and λ j k l k = π l j k P [ Z = j ∣ Z ∈ S a ] ∑ j ∈ S a P [ Z = j ∣ Z ∈ S a ] ∑ l ∈ S a π l j k . (44) \lambda_{jkjm} = \frac{-\pi_{mjk}\mathbb{P}[Z = k \mid Z \in S_b]}{\sum_{k \in S_b}\mathbb{P}[Z = k \mid Z \in S_b]\pi_{jjk}}
\quad\text{and}\quad
\lambda_{jklk} = \frac{\pi_{ljk}\mathbb{P}[Z = j \mid Z \in S_a]}{\sum_{j \in S_a}\mathbb{P}[Z = j \mid Z \in S_a]\sum_{l \in S_a}\pi_{ljk}} . \tag*{(44)} λ j k j m = ∑ k ∈ S b P [ Z = k ∣ Z ∈ S b ] π j j k − π mj k P [ Z = k ∣ Z ∈ S b ] and λ j k l k = ∑ j ∈ S a P [ Z = j ∣ Z ∈ S a ] ∑ l ∈ S a π l j k π l j k P [ Z = j ∣ Z ∈ S a ] . (44) Full identification with heterogeneity restrictions
In addition to the treatment weights λ j k l m \lambda_{jklm} λ j k l m Δ j k l m \Delta_{jklm} Δ j k l m Δ j k j k \Delta_{jkjk} Δ j k j k j j j k k k
α k = ( α 0 k , α 1 k , … , α k − 1 , k , α k + 1 , k , … , α J k ) ′ = E [ Z k D k − 1 E [ Z k Y ] , (45) \alpha_k = (\alpha_{0k}, \alpha_{1k}, \ldots, \alpha_{k-1,k}, \alpha_{k+1,k}, \ldots, \alpha_{Jk})^\prime= \mathbb{E}[\mathbf{Z}_k\mathbf{D}_k^{-1}\mathbb{E}[\mathbf{Z}_kY], \tag*{(45)} α k = ( α 0 k , α 1 k , … , α k − 1 , k , α k + 1 , k , … , α J k ) ′ = E [ Z k D k − 1 E [ Z k Y ] , (45) which equals β b \beta_b β b S j = { j } S_j = \{j\} S j = { j } j = 1 , … , J j = 1, \ldots, J j = 1 , … , J S b = { k } S_b = \{k\} S b = { k } α j k = Δ j k j k \alpha_{jk} = \Delta_{jkjk} α j k = Δ j k j k
Corollary 2 (Treatment effect heterogeneity restrictions).
Under Assumptions 1-5 and the following two conditions
If j ∈ S a j \in S_a j ∈ S a Δ j k l m = Δ l k l k − Δ m k m k \Delta_{jklm} = \Delta_{lklk} - \Delta_{mkmk} Δ j k l m = Δ l k l k − Δ mk mk l ∈ S a l \in S_a l ∈ S a k , m ∈ S b k, m \in S_b k , m ∈ S b a ≠ b a \ne b a = b
If j ∈ S b j \in S_b j ∈ S b Δ j k l m = Δ l k l k − Δ m k m k \Delta_{jklm} = \Delta_{lklk} - \Delta_{mkmk} Δ j k l m = Δ l k l k − Δ mk mk k ∈ S b k \in S_b k ∈ S b l , m ∈ S l, m \in S l , m ∈ S
it holds that
β a b = ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a Δ l k l k π l j k ω j k ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a π l j k ω j k − ∑ j ∈ S a ∑ k ∈ S b ∑ m ∈ S b Δ m k m k π m j k ω j k ∑ j ∈ S a ∑ k ∈ S b ∑ m ∈ S b π m j k ω j k , (46) \beta_{ab} =
\frac{\sum_{j \in S_a}\sum_{k \in S_b}\sum_{l \in S_a}\Delta_{lklk}\pi_{ljk}\omega_{jk}}
{\sum_{j \in S_a}\sum_{k \in S_b}\sum_{l \in S_a}\pi_{ljk}\omega_{jk}}
-
\frac{\sum_{j \in S_a}\sum_{k \in S_b}\sum_{m \in S_b}\Delta_{mkmk}\pi_{mjk}\omega_{jk}}
{\sum_{j \in S_a}\sum_{k \in S_b}\sum_{m \in S_b}\pi_{mjk}\omega_{jk}}, \tag*{(46)} β ab = ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a π l j k ω j k ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a Δ l k l k π l j k ω j k − ∑ j ∈ S a ∑ k ∈ S b ∑ m ∈ S b π mj k ω j k ∑ j ∈ S a ∑ k ∈ S b ∑ m ∈ S b Δ mk mk π mj k ω j k , (46) α j k = Δ j k j k , with j ∈ S a and k ∈ S b . (47) \alpha_{jk} = \Delta_{jkjk}, \quad\text{with } j \in S_a \text{ and } k \in S_b. \tag*{(47)} α j k = Δ j k j k , with j ∈ S a and k ∈ S b . (47) The proof is deferred to Appendix B.5. Assumptions 3 to 5 allow for response types ( D ( j ) ∈ S a , D ( k ) ∈ S b ) (D(j) \in S_a, D(k) \in S_b) ( D ( j ) ∈ S a , D ( k ) ∈ S b ) j ∈ S a j \in S_a j ∈ S a k ∈ S b k \in S_b k ∈ S b α j k \alpha_{jk} α j k S a S_a S a S b S_b S b a ≠ b a \ne b a = b S b S_b S b α j k \alpha_{jk} α j k β a b \beta_{ab} β ab
The restrictions in Corollary 2 are substantially stronger than those required for Theorem 1. Treatment effect heterogeneity is not allowed to be systematically related to individuals’ response types. A sufficient condition is mean independence of pairwise treatment effects from the potential treatment choices. This allows expected outcome levels to differ across response types, but requires the relative effects of fields of study not to depend on how students’ enrollment responds to field offers. Such restrictions may be more plausible when students have limited information about their field-specific gains, so that sorting on those gains is less likely and choices instead reflect access constraints or geographic considerations.
Full identification under response type restrictions
Alternatively, the LATEs can be identified by ruling out additional response types. Consider Assumptions 3 to 5, with each cluster only including one treatment, such that S j = { j } S_j = \{j\} S j = { j } j = 1 , … , J j = 1, \ldots, J j = 1 , … , J D ( j ) ≠ D ( k ) D(j) \ne D(k) D ( j ) = D ( k ) ( D ( j ) = j , D ( k ) = k ) (D(j) = j, D(k) = k) ( D ( j ) = j , D ( k ) = k ) j ∈ S a j \in S_a j ∈ S a k ∈ S b k \in S_b k ∈ S b
Corollary 3 (Response type restrictions).
Under Assumption 1, and Assumption 2-5 at the unclustered treatment level, it holds that
β a b = ∑ j ∈ S a ∑ k ∈ S b λ j k j k Δ j k j k , λ j k j k = π j j k ω j k ∑ j ∈ S a ∑ k ∈ S b π j j k ω j k , (48) \beta_{ab} = \sum_{j\in S_a}\sum_{k\in S_b}\lambda_{jkjk}\Delta_{jkjk},\qquad\lambda_{jkjk} = \frac{\pi_{jjk}\omega_{jk}}{\sum_{j\in S_a}\sum_{k\in S_b}\pi_{jjk}\omega_{jk}}, \tag*{(48)} β ab = j ∈ S a ∑ k ∈ S b ∑ λ j k j k Δ j k j k , λ j k j k = ∑ j ∈ S a ∑ k ∈ S b π j j k ω j k π j j k ω j k , (48) α j k = Δ j k j k , with j ∈ S a and k ∈ S b . (49) \alpha_{jk} = \Delta_{jkjk},\ \text{with } j\in S_a\ \text{and } k\in S_b. \tag*{(49)} α j k = Δ j k j k , with j ∈ S a and k ∈ S b . (49) The proof is deferred to Section B.6. The assumptions in Corollary 3 imply that ϕ j j k A ≥ 0 \phi^A_{jjk} \ge0 ϕ j j k A ≥ 0 ϕ l j k A = 0 \phi^A_{ljk} = 0 ϕ l j k A = 0 l ≠ j , k l \ne j,k l = j , k ϕ l j k A \phi^A_{ljk} ϕ l j k A
The conditions at the unclustered level are more demanding than their cluster-level counterparts: if they hold at the unclustered level, the corresponding cluster-level conditions also hold, but not vice versa. Similar conditions for identifying LATEs between unclustered treatments are discussed by Behaghel et al. [7 ] and Bhuller and Sigstad [9 ] . In the context of fields of study, Kirkebøen et al. [36 ] invoke Assumptions 2 to 5 at the unclustered level and subsequently aggregate fields into ten clusters in their empirical application. Corollary 3 shows that the resulting treatment cluster estimand identifies a weighted average of LATEs between individual fields.
Proof Corollary 1
From Assumption 1(a) follows that for all l ≠ k l \ne k l = k
π l j k = E [ D l ( j ) − D l ( k ) ] = P [ D l ( j ) − D l ( k ) = 1 ] − P [ D l ( j ) − D l ( k ) = − 1 ] . (50) \pi_{ljk} = \mathbb{E}[D_l(j)-D_l(k)] = \mathbb{P}[D_l(j)-D_l(k)=1] - \mathbb{P}[D_l(j)-D_l(k)=-1]. \tag*{(50)} π l j k = E [ D l ( j ) − D l ( k )] = P [ D l ( j ) − D l ( k ) = 1 ] − P [ D l ( j ) − D l ( k ) = − 1 ] . (50) From Theorem 2 we have that
λ j k l m = ω j k P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] ∑ j ∈ S a ∑ k ∈ S b ω j k ∑ l ∈ S a ∑ m ∈ S b P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] , (51) \lambda_{jklm} = \frac{\omega_{jk}\mathbb{P}\left[D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1\right]}{\sum_{j\in S_a}\sum_{k\in S_b}\omega_{jk}\sum_{l\in S_a}\sum_{m\in S_b}\mathbb{P}\left[D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1\right]}, \tag*{(51)} λ j k l m = ∑ j ∈ S a ∑ k ∈ S b ω j k ∑ l ∈ S a ∑ m ∈ S b P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] ω j k P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] , (51) where ( D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ) (D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1) ( D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ) ( D ( j ) = l , D ( k ) = m ) (D(j)=l,D(k)=m) ( D ( j ) = l , D ( k ) = m )
∑ j ∈ S a ∑ k ∈ S b ω j k ∑ l ∈ S a ∑ m ∈ S b P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] = ∑ j ∈ S a ∑ k ∈ S b ω j k ∑ l ∈ S a π l j k . (52) \sum_{j\in S_a}\sum_{k\in S_b}\omega_{jk}\sum_{l\in S_a}\sum_{m\in S_b}\mathbb{P}\left[D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1\right] = \sum_{j\in S_a}\sum_{k\in S_b}\omega_{jk}\sum_{l\in S_a}\pi_{ljk}. \tag*{(52)} j ∈ S a ∑ k ∈ S b ∑ ω j k l ∈ S a ∑ m ∈ S b ∑ P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] = j ∈ S a ∑ k ∈ S b ∑ ω j k l ∈ S a ∑ π l j k . (52) For the probability in the numerator we have that
max ( 0 , P [ D m ( j ) − D m ( k ) = − 1 ] − ∑ n ∈ { S a ∖ l } P [ D n ( j ) − D n ( k ) = 1 ] ) ≤ P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] ≤ min ( P [ D l ( j ) − D l ( k ) = 1 ] , P [ D m ( j ) − D m ( k ) = − 1 ] ) . (53) \max\left(0,\mathbb{P}[D_m(j)-D_m(k)=-1]-\sum_{n\in\{S_a\setminus l\}}\mathbb{P}[D_n(j)-D_n(k)=1]\right)
\le\mathbb{P}[D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1]
\le\min\left(\mathbb{P}[D_l(j)-D_l(k)=1],\mathbb{P}[D_m(j)-D_m(k)=-1]\right). \tag*{(53)} max 0 , P [ D m ( j ) − D m ( k ) = − 1 ] − n ∈ { S a ∖ l } ∑ P [ D n ( j ) − D n ( k ) = 1 ] ≤ P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] ≤ min ( P [ D l ( j ) − D l ( k ) = 1 ] , P [ D m ( j ) − D m ( k ) = − 1 ] ) . (53) From Assumption 3 it follows that P [ D l ( j ) − D l ( k ) = − 1 ] = 0 \mathbb{P}[D_l(j)-D_l(k)=-1]=0 P [ D l ( j ) − D l ( k ) = − 1 ] = 0 l ∈ S a l\in S_a l ∈ S a P [ D l ( j ) − D l ( k ) = 1 ] = 0 \mathbb{P}[D_l(j)-D_l(k)=1]=0 P [ D l ( j ) − D l ( k ) = 1 ] = 0 l ∈ S b l\in S_b l ∈ S b
max ( 0 , − π m j k − ∑ n ∈ { S a ∖ l } π n j k ) ≤ P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] ≤ min ( π l j k , − π m j k ) . (54) \max\left(0,-\pi_{mjk}-\sum_{n\in\{S_a\setminus l\}}\pi_{njk}\right)
\le\mathbb{P}[D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1]
\le\min\left(\pi_{ljk},-\pi_{mjk}\right). \tag*{(54)} max 0 , − π mj k − n ∈ { S a ∖ l } ∑ π nj k ≤ P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] ≤ min ( π l j k , − π mj k ) . (54) Proof Corollary 2
From Theorem 2 follows that
β a b = ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ m ∈ S b λ j k l m Δ j k l m , (55) \beta_{ab}=\sum_{j\in S_a}\sum_{k\in S_b}\sum_{l\in S_a}\sum_{m\in S_b}\lambda_{jklm}\Delta_{jklm}, \tag*{(55)} β ab = j ∈ S a ∑ k ∈ S b ∑ l ∈ S a ∑ m ∈ S b ∑ λ j k l m Δ j k l m , (55) where we use additional condition 1 that Δ j k l m = Δ l k l k − Δ m k m k \Delta_{jklm}=\Delta_{lklk}-\Delta_{mkmk} Δ j k l m = Δ l k l k − Δ mk mk j , l ∈ S a j,l\in S_a j , l ∈ S a k , m ∈ S b k,m\in S_b k , m ∈ S b a ≠ b a\ne b a = b
β a b = ∑ j ∈ S a ∑ k ∈ S b ( ∑ l ∈ S a Δ l k l k ∑ m ∈ S b λ j k l m − ∑ m ∈ S b Δ m k m k ∑ l ∈ S a λ j k l m ) , (56) \beta_{ab}=\sum_{j\in S_a}\sum_{k\in S_b}\left(\sum_{l\in S_a}\Delta_{lklk}\sum_{m\in S_b}\lambda_{jklm}-\sum_{m\in S_b}\Delta_{mkmk}\sum_{l\in S_a}\lambda_{jklm}\right), \tag*{(56)} β ab = j ∈ S a ∑ k ∈ S b ∑ ( l ∈ S a ∑ Δ l k l k m ∈ S b ∑ λ j k l m − m ∈ S b ∑ Δ mk mk l ∈ S a ∑ λ j k l m ) , (56) in which, using Assumptions 1-5 and (52),
∑ m ∈ S b λ j k l m = ω j k π l j k ∑ j ∈ S a ∑ k ∈ S b ω j k ∑ l ∈ S a π l j k , ∑ l ∈ S a λ j k l m = ω j k π m j k ∑ j ∈ S a ∑ k ∈ S b ω j k ∑ m ∈ S b π m j k . (57) \sum_{m\in S_b}\lambda_{jklm}=\frac{\omega_{jk}\pi_{ljk}}{\sum_{j\in S_a}\sum_{k\in S_b}\omega_{jk}\sum_{l\in S_a}\pi_{ljk}},\qquad
\sum_{l\in S_a}\lambda_{jklm}=\frac{\omega_{jk}\pi_{mjk}}{\sum_{j\in S_a}\sum_{k\in S_b}\omega_{jk}\sum_{m\in S_b}\pi_{mjk}}. \tag*{(57)} m ∈ S b ∑ λ j k l m = ∑ j ∈ S a ∑ k ∈ S b ω j k ∑ l ∈ S a π l j k ω j k π l j k , l ∈ S a ∑ λ j k l m = ∑ j ∈ S a ∑ k ∈ S b ω j k ∑ m ∈ S b π mj k ω j k π mj k . (57) where we use that π l j k = P [ D l ( j ) − D l ( k ) = 1 ] \pi_{ljk}=\mathbb{P}[D_l(j)-D_l(k)=1] π l j k = P [ D l ( j ) − D l ( k ) = 1 ] π m j k = − P [ D m ( j ) − D m ( k ) = − 1 ] \pi_{mjk}=-\mathbb{P}[D_m(j)-D_m(k)=-1] π mj k = − P [ D m ( j ) − D m ( k ) = − 1 ] l ∈ S a l\in S_a l ∈ S a m ∈ S b m\in S_b m ∈ S b
β a b = ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a Δ l k l k π l j k ω j k ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a π l j k ω j k − ∑ j ∈ S a ∑ k ∈ S b ∑ m ∈ S b Δ m k m k π m j k ω j k ∑ j ∈ S a ∑ k ∈ S b ∑ m ∈ S b π m j k ω j k . (58) \beta_{ab}=
\frac{\sum_{j\in S_a}\sum_{k\in S_b}\sum_{l\in S_a}\Delta_{lklk}\pi_{ljk}\omega_{jk}}{\sum_{j\in S_a}\sum_{k\in S_b}\sum_{l\in S_a}\pi_{ljk}\omega_{jk}}
-
\frac{\sum_{j\in S_a}\sum_{k\in S_b}\sum_{m\in S_b}\Delta_{mkmk}\pi_{mjk}\omega_{jk}}{\sum_{j\in S_a}\sum_{k\in S_b}\sum_{m\in S_b}\pi_{mjk}\omega_{jk}}. \tag*{(58)} β ab = ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a π l j k ω j k ∑ j ∈ S a ∑ k ∈ S b ∑ l ∈ S a Δ l k l k π l j k ω j k − ∑ j ∈ S a ∑ k ∈ S b ∑ m ∈ S b π mj k ω j k ∑ j ∈ S a ∑ k ∈ S b ∑ m ∈ S b Δ mk mk π mj k ω j k . (58) Using Assumption 2, extended to the case with more than two treatments at the unclustered level, the expression of the IV estimand in (45) can be written as
E [ Z k D k ′ ] α k = E [ Z k Y ] , (59) \mathbb{E}[Z_kD_k^{\prime}]\alpha_k = \mathbb{E}[Z_kY], \tag*{(59)} E [ Z k D k ′ ] α k = E [ Z k Y ] , (59) from which follows that
θ j k = ∑ l ≠ k π l j k α l k , ∀ j ≠ k , (60) \theta_{jk} = \sum_{l\ne k}\pi_{ljk}\alpha_{lk}, \qquad\forall j\ne k, \tag*{(60)} θ j k = l = k ∑ π l j k α l k , ∀ j = k , (60) with respectively
θ j k = E [ Y ∣ Z = j ] − E [ Y ∣ Z = k ] and π l j k = E [ D l ∣ Z = j ] − E [ D l ∣ Z = k ] . (61) \theta_{jk} = \mathbb{E}[Y\mid Z=j]-\mathbb{E}[Y\mid Z=k]\quad\text{and}\quad\pi_{ljk}=\mathbb{E}[D_l\mid Z=j]-\mathbb{E}[D_l\mid Z=k]. \tag*{(61)} θ j k = E [ Y ∣ Z = j ] − E [ Y ∣ Z = k ] and π l j k = E [ D l ∣ Z = j ] − E [ D l ∣ Z = k ] . (61) Use Assumption 1(b) to substitute for potential outcomes Y = ∑ l = 1 J Y ( l ) × D l Y=\sum_{l=1}^{J}Y(l)\times D_l Y = ∑ l = 1 J Y ( l ) × D l
θ j k = ∑ l = 1 J ( E [ Y ( l ) D l ∣ Z = j ] − E [ Y ( l ) D l ∣ Z = k ] ) = ∑ l = 1 J ( E [ Y ( l ) D l ( j ) ∣ Z = j ] − E [ Y ( l ) D l ( k ) ∣ Z = k ] ) = ∑ l = 1 J E [ Y ( l ) ( D l ( j ) − D l ( k ) ) ] . (62) \begin{aligned}
\theta_{jk}
&= \sum_{l=1}^{J}\left(\mathbb{E}[Y(l)D_l\mid Z=j]-\mathbb{E}[Y(l)D_l\mid Z=k]\right)\\
&= \sum_{l=1}^{J}\left(\mathbb{E}[Y(l)D_l(j)\mid Z=j]-\mathbb{E}[Y(l)D_l(k)\mid Z=k]\right)\\
&= \sum_{l=1}^{J}\mathbb{E}[Y(l)(D_l(j)-D_l(k))]. \tag*{(62)}
\end{aligned} θ j k = l = 1 ∑ J ( E [ Y ( l ) D l ∣ Z = j ] − E [ Y ( l ) D l ∣ Z = k ] ) = l = 1 ∑ J ( E [ Y ( l ) D l ( j ) ∣ Z = j ] − E [ Y ( l ) D l ( k ) ∣ Z = k ] ) = l = 1 ∑ J E [ Y ( l ) ( D l ( j ) − D l ( k ))] . (62) This can be further rewritten to
θ j k = ∑ l = 1 J E [ Y ( l ) ∣ D l ( j ) − D l ( k ) = 1 ] P [ D l ( j ) − D l ( k ) = 1 ] − ∑ m = 1 J E [ Y ( m ) ∣ D m ( j ) − D m ( k ) = − 1 ] P [ D m ( j ) − D m ( k ) = − 1 ] = ∑ l = 1 J ∑ m ≠ l E [ Y ( l ) ∣ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] × P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] − ∑ m = 1 J ∑ l ≠ m E [ Y ( m ) ∣ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] × P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] = ∑ l = 1 J ∑ m ≠ l Δ j k l m P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] . (63) \begin{aligned}
\theta_{jk}
&= \sum_{l=1}^{J}\mathbb{E}[Y(l)\mid D_l(j)-D_l(k)=1]\mathbb{P}[D_l(j)-D_l(k)=1]\\
&\quad-\sum_{m=1}^{J}\mathbb{E}[Y(m)\mid D_m(j)-D_m(k)=-1]\mathbb{P}[D_m(j)-D_m(k)=-1]\\
&= \sum_{l=1}^{J}\sum_{m\ne l}\mathbb{E}[Y(l)\mid D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1]\\
&\qquad\qquad\times\mathbb{P}[D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1]\\
&\quad-\sum_{m=1}^{J}\sum_{l\ne m}\mathbb{E}[Y(m)\mid D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1]\\
&\qquad\qquad\times\mathbb{P}[D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1]\\
&= \sum_{l=1}^{J}\sum_{m\ne l}\Delta_{jklm}\mathbb{P}[D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1]. \tag*{(63)}
\end{aligned} θ j k = l = 1 ∑ J E [ Y ( l ) ∣ D l ( j ) − D l ( k ) = 1 ] P [ D l ( j ) − D l ( k ) = 1 ] − m = 1 ∑ J E [ Y ( m ) ∣ D m ( j ) − D m ( k ) = − 1 ] P [ D m ( j ) − D m ( k ) = − 1 ] = l = 1 ∑ J m = l ∑ E [ Y ( l ) ∣ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] × P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] − m = 1 ∑ J l = m ∑ E [ Y ( m ) ∣ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] × P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] = l = 1 ∑ J m = l ∑ Δ j k l m P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] . (63) Recall that k ∈ S b k \in S_b k ∈ S b j ∈ S a j \in S_a j ∈ S a l ∈ S a l \in S_a l ∈ S a m ∈ S b m \in S_b m ∈ S b Δ j k l m = Δ l k l k − Δ m k m k \Delta_{jklm} = \Delta_{lklk} - \Delta_{mkmk} Δ j k l m = Δ l k l k − Δ mk mk j ∈ S b j \in S_b j ∈ S b Δ j k l m = Δ l k l k − Δ m k m k \Delta_{jklm} = \Delta_{lklk} - \Delta_{mkmk} Δ j k l m = Δ l k l k − Δ mk mk
θ j k = ∑ l = 1 J ∑ m = 1 m ≠ l J ( Δ l k l k − Δ m k m k ) P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] = ∑ l = 1 J Δ l k l k ∑ m = 1 m ≠ l J P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] − ∑ m = 1 J Δ m k m k ∑ l = 1 l ≠ m J P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] = ∑ l = 1 J Δ l k l k P [ D l ( j ) − D l ( k ) = 1 ] − ∑ m = 1 J Δ m k m k P [ D m ( j ) − D m ( k ) = − 1 ] = ∑ l ≠ k Δ l k l k π l j k . (64) \begin{aligned}
\theta_{jk}
&= \sum_{l=1}^{J}\sum_{\substack{m=1\\m\ne l}}^{J}(\Delta_{lklk}-\Delta_{mkmk})\mathbb{P}[D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1] \\
&= \sum_{l=1}^{J}\Delta_{lklk}\sum_{\substack{m=1\\m\ne l}}^{J}\mathbb{P}[D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1] - \\
&\quad\sum_{m=1}^{J}\Delta_{mkmk}\sum_{\substack{l=1\\l\ne m}}^{J}\mathbb{P}[D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1] \\
&= \sum_{l=1}^{J}\Delta_{lklk}\mathbb{P}[D_l(j)-D_l(k)=1]-\sum_{m=1}^{J}\Delta_{mkmk}\mathbb{P}[D_m(j)-D_m(k)=-1] \\
&= \sum_{l\ne k}\Delta_{lklk}\pi_{ljk}. \tag*{(64)}
\end{aligned} θ j k = l = 1 ∑ J m = 1 m = l ∑ J ( Δ l k l k − Δ mk mk ) P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] = l = 1 ∑ J Δ l k l k m = 1 m = l ∑ J P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] − m = 1 ∑ J Δ mk mk l = 1 l = m ∑ J P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] = l = 1 ∑ J Δ l k l k P [ D l ( j ) − D l ( k ) = 1 ] − m = 1 ∑ J Δ mk mk P [ D m ( j ) − D m ( k ) = − 1 ] = l = k ∑ Δ l k l k π l j k . (64) Combining (60) with the result that θ j k = ∑ l ≠ k Δ l k l k π l j k \theta_{jk}=\sum_{l\ne k}\Delta_{lklk}\pi_{ljk} θ j k = ∑ l = k Δ l k l k π l j k j ≠ k j\ne k j = k k ∈ S b k\in S_b k ∈ S b
∑ l ≠ k ( Δ l k l k − α l k ) π l j k = 0 , ∀ j ≠ k , (65) \sum_{l\ne k}(\Delta_{lklk}-\alpha_{lk})\pi_{ljk}=0,\quad\forall j\ne k, \tag*{(65)} l = k ∑ ( Δ l k l k − α l k ) π l j k = 0 , ∀ j = k , (65) which has the unique solution α l k = Δ l k l k \alpha_{lk}=\Delta_{lklk} α l k = Δ l k l k l ≠ k l\ne k l = k Π k \Pi_k Π k π l j k \pi_{ljk} π l j k E [ D k Z k ′ ] \mathbb{E}[D_kZ_k'] E [ D k Z k ′ ] E [ Z k Z k ′ ] \mathbb{E}[Z_kZ_k'] E [ Z k Z k ′ ] E [ D k Z k ′ ] E [ Z k Z k ′ ] − 1 = [ δ k , Π k ] \mathbb{E}[D_kZ_k']\mathbb{E}[Z_kZ_k']^{-1}=[\delta_k,\Pi_k] E [ D k Z k ′ ] E [ Z k Z k ′ ] − 1 = [ δ k , Π k ] δ k = ( δ 1 k , … , δ k − 1 , k , δ k + 1 , k , … , δ J k ) ′ \delta_k=(\delta_{1k},\ldots,\delta_{k-1,k},\delta_{k+1,k},\ldots,\delta_{Jk})' δ k = ( δ 1 k , … , δ k − 1 , k , δ k + 1 , k , … , δ J k ) ′ Π k \Pi_k Π k π l j k \pi_{ljk} π l j k l ≠ k l\ne k l = k j ≠ k j\ne k j = k
Proof Corollary 3
Using Assumption 3 at the individual treatment level we can write P [ D j ( j ) − D j ( k ) = − 1 ] = 0 \mathbb{P}[D_j(j)-D_j(k)=-1]=0 P [ D j ( j ) − D j ( k ) = − 1 ] = 0 l ≠ j l\ne j l = j P [ D l ( j ) − D l ( k ) = 1 ] = P [ D l ( j ) − D l ( k ) = 1 , D j ( j ) = 0 ] = P [ D l ( j ) − D l ( k ) = 1 , D j ( j ) = 0 , D j ( k ) = 0 ] \mathbb{P}[D_l(j)-D_l(k)=1]=\mathbb{P}[D_l(j)-D_l(k)=1,D_j(j)=0]=\mathbb{P}[D_l(j)-D_l(k)=1,D_j(j)=0,D_j(k)=0] P [ D l ( j ) − D l ( k ) = 1 ] = P [ D l ( j ) − D l ( k ) = 1 , D j ( j ) = 0 ] = P [ D l ( j ) − D l ( k ) = 1 , D j ( j ) = 0 , D j ( k ) = 0 ] D m ( k ) = 0 D_m(k)=0 D m ( k ) = 0 D j ( j ) = 1 D_j(j)=1 D j ( j ) = 1 m ≠ j , k m\ne j,k m = j , k P [ D j ( j ) − D j ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] = 0 \mathbb{P}\left[D_j(j)-D_j(k)=1,D_m(j)-D_m(k)=-1\right]=0 P [ D j ( j ) − D j ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] = 0 m ≠ j , k m\ne j,k m = j , k
P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] = 0 \mathbb{P}\left[D_l(j)-D_l(k)=1,D_m(j)-D_m(k)=-1\right]=0 P [ D l ( j ) − D l ( k ) = 1 , D m ( j ) − D m ( k ) = − 1 ] = 0 l ≠ j l\ne j l = j m ≠ k m\ne k m = k λ j k l m = 0 \lambda_{jklm}=0 λ j k l m = 0 l ≠ j l\ne j l = j m ≠ k m\ne k m = k
β a b = ∑ j ∈ S a ∑ k ∈ S b λ j k j k Δ j k j k , λ j k j k = π j j k ω j k ∑ j ∈ S a ∑ k ∈ S b π j j k ω j k , (66) \beta_{ab}=\sum_{j\in S_a}\sum_{k\in S_b}\lambda_{jkjk}\Delta_{jkjk},\qquad\lambda_{jkjk}=\frac{\pi_{jjk}\omega_{jk}}{\sum_{j\in S_a}\sum_{k\in S_b}\pi_{jjk}\omega_{jk}}, \tag*{(66)} β ab = j ∈ S a ∑ k ∈ S b ∑ λ j k j k Δ j k j k , λ j k j k = ∑ j ∈ S a ∑ k ∈ S b π j j k ω j k π j j k ω j k , (66) where we use that from Assumption 1(a) follows that
π j j k = E [ D j ( j ) − D j ( k ) ] = P [ D j ( j ) − D j ( k ) = 1 ] − P [ D j ( j ) − D j ( k ) = − 1 ] = P [ D j ( j ) − D j ( k ) = 1 ] = P [ D j ( j ) − D j ( k ) = 1 , D k ( j ) − D k ( k ) = − 1 ] , (67) \begin{aligned}
\pi_{jjk}=\mathbb{E}[D_j(j)-D_j(k)]=\mathbb{P}[D_j(j)-D_j(k)=1]-\mathbb{P}[D_j(j)-D_j(k)=-1] \\
&=\mathbb{P}[D_j(j)-D_j(k)=1]=\mathbb{P}[D_j(j)-D_j(k)=1,D_k(j)-D_k(k)=-1], \tag*{(67)}
\end{aligned} π j j k = E [ D j ( j ) − D j ( k )] = P [ D j ( j ) − D j ( k ) = 1 ] − P [ D j ( j ) − D j ( k ) = − 1 ] = P [ D j ( j ) − D j ( k ) = 1 ] = P [ D j ( j ) − D j ( k ) = 1 , D k ( j ) − D k ( k ) = − 1 ] , (67) using subsequently Assumptions 3 and 5 at the individual treatment level.
If Assumptions 3-5 hold on the individual treatment level, we have that α j k = β a b = λ j k j k Δ j k j k = Δ j k j k \alpha_{jk}=\beta_{ab}=\lambda_{jkjk}\Delta_{jkjk}=\Delta_{jkjk} α j k = β ab = λ j k j k Δ j k j k = Δ j k j k S j = { j } S_j=\{j\} S j = { j } j j j
Additional details empirical application
Deferred acceptance in Portugal
Each course has a quota (q q q [21 ] ).
The applicants are matched to courses using an iterative process (Article 37 of DGES [21 ] ), similar to the college-proposing Gale–Shapley algorithm (Gale and Shapley [26 ] ):
The algorithm terminates when the allocation is not updated, and matches each course to the set of q q q
One to two weeks before the start of the academic year, each applicant receives the offer from her matched course. Within that week, applicants have to accept the offer, participate in a second cycle, or withdraw from the application process. The slots that have remained or were not accepted after the first cycle are then allocated to the applicants who participate in the second cycle according to the same algorithm. The process concludes with an identical third cycle.
Data
We use data from two different sources. First, the higher education application data from DGES contains for each applicant the submitted course ranking, their application scores, the matched course, and the course cutoffs, from 2008-09 to 2019-20. We consider for each student the first cycle from the earliest observed application in which the student also enrolls in the first year. We refer to students who applied in that cycle within the same year as a cohort.
The second data set has the student outcome data from the Direção-Geral de Estatísticas da Educação e Ciência (DGEEC; Directorate for Education Statistics). For each enrolled student, this data set contains the enrolled course, whether the student is active in the first, second, or third year of that course, and the achieved number of ECTS study credits in that course in the previous year, from 2013-14 to 2019-20. Our main treatment and outcome variable are enrollment and ECTS credits in the second year of a course. In case a student is active in the second year of a course across multiple years, these variables will refer to the first observed enrollment in the second year of a course.
The student outcome data can be merged to the application data if we observe a student’s enrollment in the first year. Hence, the first cohort for which we can combine the data sets is 2013-14. We take 2016-17 as the final cohort, so that we can use course enrollment in the second year up to 2018-19, allowing for a one-year delay. This results in the final data set with four cohorts from 2013-14 unto 2016-17.
To arrive at the final estimation sample, we implement the procedure described in § 6.1.1 to select applicants with a (locally) preferred and next-best field of study, and exclude a small subset of 751 applicants (representing 1.50% of the remaining applicants). The excluded applicants are characterized by a combination of preferred and next-best unclustered fields of study with (1) fewer than 10 observations, (2) perfect compliance, where all applicants above (below) the cutoff are enrolled in the preferred (next-best) field in second year, or (3) exact multicollinearity, where all applicants score either above or below the cutoff. The preferred and next-best field combinations with perfect compliance and exact multicollinearity make the estimation of the IV model in (12) –(13) impossible.
Panel A of Table 7 shows descriptive statistics of the final estimation sample. This sample includes 27.24 percent of the total number of applicants in the four cohorts. The mean of the running variable is positive, and 68% of the students are assigned to their preferred field. The applicants have on average 5.46 courses on their course ranking. In 88% of these rankings, the top two fields equal our measure for the preferred and next-best field. The enrolled field in year two is different from the initially matched field for 8% of the students. The mean of second-year ECTS across all enrolled students is equal to 54.19. However, there is substantial variation in the number of ECTS between fields. For instance, students enrolled in Physical sciences and Architecture and building have a mean second-year ECTS of less than 50, those in Law and Social and behavioural science have a mean of around 53, and students in Health and Journalism and information have a mean
Table 7. Descriptive statistics for estimation sample. Panel A shows descriptive statistics of our estimation sample of 49,272 applicants from 4 cohorts. Panel B provides an overview of the number of applicants per next-best and preferred field combination.
of almost 60.
Panel B of Table 7 provides an overview of the most common preferred fields for each next-best. All 22 next-best fields have a skewed distribution of students across preferred fields. For instance, 77.83% (16.64%) of the students with Computing as next-best prefer Engineering (Business and administration). The remaining 5.53% prefer any of the other 19 fields. Panel B also shows that the number of observations differs across next-best fields. For example, there are 7,432 students with Business and administration as next-best versus 363 students with Agriculture, forestry and fishery.
Robustness checks and validity of the continuity assumption
We assess the robustness of our results against more flexible specifications. In our baseline specification, we use a polynomial of degree one, so that f h ( S ) = γ S f_h(S) = \gamma S f h ( S ) = γ S S S S ∑ c ≠ b Z ~ c \sum_{c \ne b} \widetilde{Z}_c ∑ c = b Z c (13) . Second, we include S 2 S^2 S 2 (12) and (13) . Third, we assess the robustness of our results via the use of symmetric bandwidths equal to 40, 30, and 20. These bandwidths include 94.59%, 87.02%, and 70.94% of the total sample respectively.
Panel A of Figure 1 presents the results of these robustness checks using the unclustered fields of study. Specifically, the left plot compares the treatment take-up estimates from (13) of our baseline specification with those from each robustness check. The estimates lie close to the 45-degree line, indicating virtually identical results across all specifications. Similarly, the right plot illustrates that the treatment effect estimates from (12) are similar across the robustness checks. The largest deviation occurs with a bandwidth of 20, which also substantially reduces statistical power and weakens the significance. Panels B and C of Figure 1 confirm that these conclusions continue to hold when using the STEM and the ISCED classification.
Figure 1. Notes: This figure shows scatterplots of the estimates from five robustness checks against those from the baseline specification. The left plots display treatment take-up estimates π a c b h \pi^{h}_{acb} π a c b h (13) , while the right plots show treatment effect estimates β a b h \beta^{h}_{ab} β ab h (12) . Panel A shows the results for the unclustered fields of study, panel B for the STEM classification, and panel C for the ISCED classification.
Next, we assess the validity of the local continuity assumption (Assumption 6). First, we examine whether the application score to the next-best field is continuous at the cutoff.
Figure 2 plots the maximum score across all courses in the preference block of the next-best field against the running variable. Consistent with continuous potential outcomes, this score does not show a discontinuity around the cutoff. This result is confirmed with a pooled regression on the students from all next-best fields: We regress the application score to the next-best field on the running variable and a single treatment dummy indicating whether students scored above the cutoff, and the estimate on this treatment dummy is equal to -0.31 and has a p p p
Figure 2. Balancing test. Notes: This figure shows a scatterplot of the application score to the next-best field against the running variable with a fourth-order polynomial estimated with a uniform kernel on the full estimation sample (the default of the Stata program rdplot).
Second, we examine the density of the students across all next-best fields near the course cutoffs [42 ] . If applicants have precise control over their application score, we might observe bunching just above the cutoffs and the local continuity assumption may not be valid ([39 ] . However, Figure 3 shows no sign of excess mass just above the cutoffs. The bias corrected discontinuity test introduced in Cattaneo, Jansson, and Ma [13 ] has a test statistic equal to 0.13 with a p p p
Figure 3. Discontinuity in density test. Notes: This figure shows the density of the running variable estimated with a second-order polynomial and a third-order polynomial for the bias-correction estimate, an optimally-chosen bandwidth with the full estimation sample as plot range, a triangular kernel, and 95% confidence intervals with jackknifed standard errors (the default of the Stata program rddensity).
The continuity assumption is also consistent with the setting at hand. The number of applications is different each year, and hence the cutoffs change from year to year and are unknown at the time of application. The cutoffs only become publicly available when the applicants receive their offer. A regression of the cutoffs across years on course dummies has an R R R
Testing the necessary conditions
The (in)equalities ϕ l j k A ⋚ 0 \phi^{A}_{ljk} \lesseqgtr0 ϕ l j k A ⋚ 0 Y Y Y Z Z Z ϕ l j k A ≥ 0 \phi^{A}_{ljk} \ge0 ϕ l j k A ≥ 0 l , j ∈ S a l,j \in S_a l , j ∈ S a k ∈ S b k \in S_b k ∈ S b
E [ 1 Y ∈ A D l ∣ Z = k ] ≤ E [ 1 Y ∈ A D l ∣ Z = j ] (68) \mathbb{E}[\mathbb{1}_{Y\in A}D_l\mid Z=k] \le\mathbb{E}[\mathbb{1}_{Y\in A}D_l\mid Z=j] \tag*{(68)} E [ 1 Y ∈ A D l ∣ Z = k ] ≤ E [ 1 Y ∈ A D l ∣ Z = j ] (68) E [ 1 Y ∈ A D l Z k ] P [ Z = j ] ≤ E [ 1 Y ∈ A D l Z j ] P [ Z = k ] \mathbb{E}[\mathbb{1}_{Y\in A}D_lZ_k]\mathbb{P}[Z=j] \le\mathbb{E}[\mathbb{1}_{Y\in A}D_lZ_j]\mathbb{P}[Z=k] E [ 1 Y ∈ A D l Z k ] P [ Z = j ] ≤ E [ 1 Y ∈ A D l Z j ] P [ Z = k ] E [ c j D l Z k − c k D l Z j ∣ Y = y ] ≤ 0 , \mathbb{E}[c_jD_lZ_k-c_kD_lZ_j\mid Y=y] \le0, E [ c j D l Z k − c k D l Z j ∣ Y = y ] ≤ 0 , where c j = P [ Z = j ] c_j = \mathbb{P}[Z=j] c j = P [ Z = j ] [44 ] . Similarly, the inequality ϕ l j k A ≤ 0 \phi^{A}_{ljk} \le0 ϕ l j k A ≤ 0 l , k ∈ S b l,k \in S_b l , k ∈ S b j ∈ S a j \in S_a j ∈ S a
E [ 1 Y ∈ A D l ∣ Z = j ] ≤ E [ 1 Y ∈ A D l ∣ Z = k ] (69) \mathbb{E}[\mathbb{1}_{Y\in A}D_l\mid Z=j] \le\mathbb{E}[\mathbb{1}_{Y\in A}D_l\mid Z=k] \tag*{(69)} E [ 1 Y ∈ A D l ∣ Z = j ] ≤ E [ 1 Y ∈ A D l ∣ Z = k ] (69) E [ c k D l Z j − c j D l Z k ∣ Y = y ] ≤ 0. \mathbb{E}[c_kD_lZ_j-c_jD_lZ_k\mid Y=y] \le0. E [ c k D l Z j − c j D l Z k ∣ Y = y ] ≤ 0. For the equality ϕ l j k A = 0 \phi^{A}_{ljk}=0 ϕ l j k A = 0 l ∉ { S a , S b } l \notin\{S_a,S_b\} l ∈ / { S a , S b } j ∈ S a j \in S_a j ∈ S a k ∈ S b k \in S_b k ∈ S b
These conditional moment inequalities can be tested using the intersection bounds framework of Chernozhukov et al. [16 ] , implemented in the Stata package clrtest by Chernozhukov et al. [15 ] . The clrtest allows for the joint testing of inequalities within an estimation sample. Hence, within each sample defined by a preferred and next-best field ( j , k ) (j,k) ( j , k ) ϕ l j k A \phi^{A}_{ljk} ϕ l j k A l l l
For example, similar to the discussion of the identifying assumptions in § 6.1.4 , consider testing the STEM classification with Personal services as the next-best field ( k ) (k) ( k ) ( j ) (j) ( j ) j j j k k k
Now consider testing the ISCED classification. Again take the next-best field Personal services ( k ) (k) ( k ) j j j j j j k k k (68) , weakly decrease it in the 4 fields in the Services cluster according to (69) , and should have no effect on the 14 fields in the other clusters according to (68) and (69) .
To account for multiple testing within each next-best field k k k [44 ] and apply the Holm [31 ] correction to multiple applications of clrtest across different samples. The Holm procedure orders the p p p α \alpha α p p p clrtest does not report p p p 10 − 3 10^{-3} 1 0 − 3 10 − 12 10^{-12} 1 0 − 12
The clrtest package allows both nonparametric and parametric estimation of the conditional moment inequalities. We use the parametric option, which approximates the conditional expectations using a linear regression in Y Y Y ( j , k ) (j,k) ( j , k )
The inequalities are evaluated over values of the conditioning variable Y Y Y Y Y Y Y Y Y
We implement the tests without including the running variable. Since the inequalities are intended to hold locally at the application score cutoff, this corresponds to the local randomization approach to RDDs (Cattaneo et al. [12 ] ). In the baseline test we use the full sample, as in the main RDD analysis. As a robustness check we also repeat the tests using symmetric bandwidths of 40, 30, and 20. The results are similar.
Treatment take-up estimates with ISCED classification
We analyze the treatment take-up estimates for the ISCED classification. Estimating (13) yields a matrix of treatment take-up effects for each treatment cluster S b S_b S b π a c b h \pi_{acb}^{h} π a c b h (13) for enrollment D ~ a \widetilde{D}_{a} D a S a S_{a} S a S b S_{b} S b S a S_{a} S a S b S_{b} S b ∗ * ∗ ∗ ∗ ^{**} ∗∗ ∗ ∗ ∗ ^{***} ∗∗∗
combination of clusters S a S_{a} S a S b S_{b} S b Table 8 reports only the effect of receiving an offer from the preferred field on enrollment in a field from cluster S a S_{a} S a
The smallest estimate is 0.59 for students preferring a field in Health and welfare over one in Agriculture. Overall, these results provide strong evidence that Assumption 2 holds for the ISCED classification.
Additional empirical results
This section applies the results derived in Appendix B to the field of study level data to provide additional information about the weights and LATEs underlying the estimated treatment cluster effects. The required treatment take-up estimates π l j k h \pi^{h}_{ljk} π l j k h (13) and treatment effect estimates β j k h \beta^{h}_{jk} β j k h (12) at the unclustered field of study level are provided as supplementary data in the Supporting Information. Because we do not conduct statistical inference on the estimated bounds or weighted averages, we use them only to describe patterns in our sample and draw no conclusions about their population-level counterparts.
Two treatment clusters: STEM and non-STEM
We estimate the bounds on the weights in Corollary 1 by replacing the treatment take-up estimands π l j k \pi_{ljk} π l j k Figure 4 shows a scatterplot of the estimated upper bounds against the estimated lower bounds for the weights λ j k l m \lambda_{jklm} λ j k l m j = l j=l j = l k = m k=m k = m j ≠ l j\ne l j = l k ≠ m k\ne m k = m D ( j ) = j D(j)=j D ( j ) = j D ( k ) = k D(k)=k D ( k ) = k j ∈ S a j\in S_a j ∈ S a k ∈ S b k\in S_b k ∈ S b
Figure 4. Treatment effect weights and weighted LATEs within STEM clusters
We next examine the assumptions that allow for the identification of the LATEs within
Notes: The left panel shows a scatterplot of the estimated upper bounds of the treatment effect weights against the estimated lower bounds, following (43) from Corollary 1, for both treatment clusters S b S_b S b λ j k l m \lambda_{jklm} λ j k l m j = l j=l j = l k = m k=m k = m j ≠ l j\ne l j = l k ≠ m k\ne m k = m (46) in Corollary 2 against the estimated treatment cluster effects, for both treatment clusters S b S_b S b
the weighted averages. Corollaries 2 and 3 show that the LATEs are identified by α j k \alpha_{jk} α j k (45) under treatment effect heterogeneity restrictions or additional response type restrictions, respectively. Under either set of restrictions, the treatment cluster effects between STEM and non-STEM equal a weighted average of identifiable LATEs. We estimate these weighted averages by replacing the treatment take-up and LATE estimands, π l j k \pi_{ljk} π l j k α j k \alpha_{jk} α j k Figure 4 plots the estimated weighted average under treatment effect heterogeneity restrictions from Corollary 2 against the estimated treatment cluster effects with 95% confidence intervals. The left plot of Figure 5 shows these estimates under response type restrictions from Corollary 3. In both figures, the confidence intervals overlap with the 45-degree line. One explanation for this result may be that treatment effect heterogeneity is restricted. However, it may be more likely that the weighted averages of LATEs assign little weight to the response types ( D ( j ) = l , D ( k ) = m ) (D(j)=l,D(k)=m) ( D ( j ) = l , D ( k ) = m ) j ≠ l j\ne l j = l k ≠ m k\ne m k = m
Figure 5. Weighted LATEs within STEM and ISCED clusters and response type restrictions. Notes: The left panel shows a scatterplot of the estimated weighted average of LATEs from (48) in Corollary 3 against the estimated treatment cluster effects, for both treatment clusters S b S_b S b S b S_b S b
The necessary conditions and the weighted average of LATEs suggest that the estimates according to (45) provide a reasonable approximation to the LATEs in the treatment cluster effects. Hence, we can further unpack the treatment cluster effects. For instance, the non-STEM versus STEM treatment cluster effect (1.33 credits) comprises 30% from Health versus Life sciences (-0.03 credits), 22% from Health versus Engineering and engineering trades (1.08 credits), and 10% from Arts versus Architecture and building (5.02 credits).
With an average absolute value of the estimated LATEs of 4, 628 ECTS credits, all ten estimates with Environmental protection as next-best field are considered unreliable and set to zero when constructing the weighted average of LATEs in Corollaries 2 and 3.
Multiple treatment clusters: Eight ISCED clusters
We bound the weights for the treatment effects between the eight clusters. The left plot in Figure 6 shows the estimated weights when the next-best field belongs to the Science cluster, allowing us to distinguish between the seven clusters containing the preferred fields.
Figure 6. Treatment effect weights and weighted LATEs within ISCED clusters
Notes: The left panel shows a scatterplot of the estimated upper bounds of the treatment effect weights against the estimated lower bounds, following (43) from Corollary 1, for each treatment cluster S a S_a S a S b S_b S b λ j k l m \lambda_{jklm} λ j k l m j = l j=l j = l k = m k=m k = m j ≠ l j\ne l j = l k ≠ m k\ne m k = m (46) in Corollary 2 against the estimated treatment cluster effects, for all treatment clusters S b S_b S b
As with the STEM clusters, the estimated bounds are tight and suggest that most weight is assigned to response types ( D ( j ) = j , D ( k ) = k ) (D(j) = j, D(k) = k) ( D ( j ) = j , D ( k ) = k ) j ∈ S a j \in S_a j ∈ S a k ∈ S b k \in S_b k ∈ S b
We examine the additional assumptions to identify the LATEs within the treatment cluster effects. The right plot in Figure 6 plots the estimated weighted average of LATEs against the treatment effects between the ISCED clusters using treatment effect heterogeneity restrictions. The right plot of Figure 5 shows these estimates under the response type restrictions. Similar to the results with the STEM clusters, the 95% confidence intervals of the treatment cluster effects overlap with the 45-degree line. This may indicate negligible bias in the identification of the LATEs due to little treatment effect heterogeneity or little weight to the response types ( D ( j ) = l , D ( k ) = m ) (D(j) = l, D(k) = m) ( D ( j ) = l , D ( k ) = m ) j ≠ l j \ne l j = l k ≠ m k \ne m k = m
In this case, the estimated LATEs corresponding to the 22 fields of study can be used to provide further insights into the treatment cluster effects. For instance, the estimated treatment cluster effect of Social sciences, business and law versus Science (3.49 credits) comprises 22% from Business and administration versus Computing (2.60 credits), 17% from Business and administration versus Mathematics and statistics (6.69 credits), and 16% from Social and behavioural science versus Life sciences (4.87 credits).
