12 08

w o r k i n g

p a p e r

F E D E R A L R E S E R V E B A N K O F C L E V E L A N D

12 08

Local Average Neighborhood Effects from Moving to Opportunity

Dionissi Aliprantis and Francisca G.-C. Richter

Working papers of the Federal Reserve Bank of Cleveland are preliminary materials circulated to stimulate discussion and critical comment on research in progress. They may not have been subject to the formal editorial review accorded offi cial Federal Reserve Bank of Cleveland publications. The views stated herein are those of the authors and are not necessarily those of the Federal Reserve Bank of Cleveland or of the Board of Governors of the Federal Reserve System.

Working papers are available on the Cleveland Fed’s website at:

www.clevelandfed.org/research ^.

Working Paper 12-08 March 2012

Local Average Neighborhood Effects from Moving to Opportunity Dionissi Aliprantis and Francisca G.-C. Richter

This paper estimates Local Average Treatment Effects (LATEs) of neighborhood quality from the Moving to Opportunity (MTO) housing mobility experiment in a generalized model with multiple treatment levels. We propose a new approach to identifying parameters that exploits the identifi cation of unobservables in the multi-level model. The variation in neighborhood quality induced by MTO only allows us to identify LATEs of moving from the fi rst to the second decile of the national distribution of quality, but in other applications the approach may allow for the estimation of Marginal Treatment Effects. Estimated LATEs on employ- ment, labor force participation rates, earnings, income, welfare receipt, and body mass index are consistent with standard theories of neighborhood externalities.

Keywords: Marginal Treatment Effect, Local Average Treatment Effect,

Essential Heterogeneity, Strong Ignorability, Average Causal Response, Moving to Opportunity.

JEL Codes: C31, C36, C50, D04, I38, R23.

Dionissi Aliprantis is at the Federal Reserve Bank of Cleveland (dionissi.alipran- tis@clev.frb.org). Francisca G.-C. Richter is also at the Bank (francisca.g.richter@

clev.frb.org). The authors thank Joel Elvery, Fabian Lange, Jon James, and Ben

Craig for helpful comments. They also thank Emily Burgen and Nelson Oliver for

valuable research assistance.

1 Introduction

Since neighborhood externalities are widely believed to be a major determinant of outcomes in the United States, empirical evidence on the existence and magnitude of such neighborhood effects has been of great interest to social scientists. However, neighborhood characteristics are examples of treatments that are infeasible to randomly allocate to individuals. Researchers have not agreed on how best to apply non-experimental methods to investigate neighborhood effects, with many researchers broadly distrustful of non-experimental tools.

The results from the Moving to Opportunity (MTO) housing mobility program are a prominent example of the difficulties confronted when studying neighborhood effects. In MTO rental vouchers were randomly assigned to public housing residents of poor neighborhoods that could be used to re- locate to low-poverty neighborhoods. The lack of large effects from the program were disappointing (Kling et al. (2007), Sanbonmatsu et al. (2006)), especially given the strong positive effects from the program serving as its motivation, Gautreaux (Polikoff (2006), Rubinowitz and Rosenbaum (2000)). There has been controversy regarding the most appropriate techniques to learn about neighborhood effects from MTO (Ludwig et al. (2001), Clampet-Lundquist and Massey (2008), Sampson (2008)).

This paper generalizes previous analyses of effects from neighborhood quality using MTO voucher assignment as an instrumental variable. Specifically, we take as our starting point the analysis in Kling et al. (2007) and relax two key identifying assumptions (See Aliprantis (2012) for a full discussion.). We first relax the assumption that neighborhood quality is selected by households under strong ignorability by assuming selection occurs in a model with essential het- erogeneity. We also create and use in our analysis a linear index of neighborhood quality that uses several neighborhood characteristics in addition to the neighborhood poverty rate.

In order to identify neighborhood effects under these relaxed assumptions we specify and esti- mate a joint model of potential outcomes and selection into a multi-valued treatment. Our model adapts the assumptions from Heckman et al. (2006) to the specific application of MTO. We iden- tify individuals’ unobservable component in an ordered choice model, something not possible in the case of a binary treatment. We use these estimates and the exogenous variation induced by MTO on an underlying latent index to identify those individuals contributing to specific Local Average Treatment Effects (LATEs). In the context of our model these LATEs are the components of the Average Causal Response (ACR) from Angrist and Imbens (1995). Although the variation from MTO does not precisely estimate Marginal Treatment Effects (MTEs), we illustrate why we believe this approach also has potential for identifying specific MTEs in other applications with discrete instruments affecting all choice margins simultaneously.

The estimation results from the ordered choice model indicate that MTO had strongly het- erogeneous effects across sites. Those sites with residents living in the lowest quality neighbor- hoods experienced the greatest impact on neighborhood quality from the offer of an MTO voucher.

Ordered choice estimates also show that the Section 8 voucher was more effective at improving

neighborhood quality at very low levels, but that the experimental voucher became more effective

before the fifth percentile of the national distribution of neighborhood quality.

The variation in neighborhood quality induced by MTO is only able to precisely identify LATE parameters pertaining to moves from the first to the second decile of the national distribution of neighborhood quality. This move is exogenously induced by the MTO experimental voucher for about 13 percent of a restricted subsample of MTO volunteers. The LATE estimates for this subpopulation are consistent with the theory that improving neighborhood characteristics improves outcomes. There was a large increase in household income for those who moved from the first to second decile of neighborhood quality, although this increase was not a result of increases in the earnings of the respondent. The LATE on employment was quite large, and the fact that this estimate was larger than the increase in labor force participation rates is suggestive that employment improved due to changes in the local labor market rather than changes in household behavior alone.

Adult respondents and their children were also less likely to currently be on welfare if the program induced them from the first to second deciles of quality. And the LATE on body mass index also indicates there were improvements in health from exogenously increasing neighborhood quality.

The remainder of the paper is organized as follows: Section 2 develops the ordered choice model, defines the the treatment effect parameters of interest, and discusses assumptions necessary to identify these parameters. Section 3 describes the MTO experiment and Section 4 discusses the data used in estimation. Sections 5 and 6 present the estimation results, and Section 7 concludes.

2 Model

The model we use is based on the Marginal Treatment Effect (MTE) framework developed in Heckman et al. (2006) and Heckman and Vytlacil (2005), henceforth referred to as HUV and HV, respectively, and originally introduced by Bj¨orklund and Moffitt (1987). In a multi-valued treat- ment setting, identification of level-specific MTEs over the whole range of unobserved heterogeneity would typically be achieved using continuous instruments for each transition. Experimental voucher assignment from MTO is not such a set of instruments. Thus while we are able to relax some of the identifying assumptions in Kling et al. (2007), the three-level instrument from MTO only allows us to identify level-specific Local Average Treatment Effects (LATEs) for those transitions that are induced by the instrument.

2.1 Ordered Choice Model

We begin by modeling the way agents select into different levels of a multi-valued treatment.

Consider a model in which there is a treatment D with J levels (ie, D ∈ {1, . . . , J }), a binary instrument Z (ie, Z ∈ {0, 1}), and individuals have observable characteristics X. We assume that for each treatment level j ≤ J − 1 there is a latent index D

^∗_j

representing the net marginal benefit of moving to treatment level j + 1 from level j. Agents select level k

^∗

that maximizes total utility u(D):

k

^∗

= argmax

k∈{1,...,J}

u(k) (1)

where

u(k) =

 



u(1) if k = 1;

u(1) + $

_k−1

j=1

D

_j^∗

if k ≥ 2.

We specify the J − 1 latent indeces to be functions of observable characteristics (Z, X) and unob- servable characteristics V with the following structure:

A1: D

^∗_ij

= µ(X

i

) + γ

_j

Z

_i

− V

_i

− C

_j

for j = 1, . . . , J − 1,

A2: C

_j

< C

_j+1

for j = 1, . . . , J − 1 with C

₀

= −∞ and C

_J

= ∞,

A3: C

_j

− γ

_j

< C

_j+1

− γ

_j+1

for j = 1, . . . , J − 1 with C

₀

− γ

₀

= −∞ and C

_J

+ γ

_J

= ∞.

Here µ(X) is the gain from moving up one level and is independent of the level. C

_j

is a transition-specific cost that increases at each level. V is a continuously distributed random variable representing the unobserved cost for individual i from moving up one level. And γ

_j

can be inter- preted as a reduction in the transition specific cost brought about by varying the instrument that preserves the cost ranking among levels (A3). Note that the initial level u(1) is important only as a way of indexing the model.

Let D

₀^∗

> max{0, D

₁^∗

} and define D

_J^∗

≡ −∞. Then given Z, the net marginal benefit D

^∗_j

is decreasing in j, so for k

^∗

∈ {1, . . . , J }

D = k

^∗

⇐⇒ D

_k^∗∗

≤ 0 < D

^∗_k∗−1

. (2) The model described in Equation 1, together with assumptions A1-A3, allow us to write condition 2 as an ordered choice model in terms of the unobservable characteristics that determine selection into treatment, V :

D = j ⇐⇒ µ(X) + γ

j

Z − C

j

< V ≤ µ(X) + γ

_j−1

Z − C

_j−1

. (3) Denoting the CDF of V conditional on X by F

_V|X

(·), we can define π

^Z_j

(X) ≡ P r(D > j|Z, X) = F

_V|X

(µ(X) + γ

_j

Z − C

_j

) and U

_D

≡ F

_V|X

(V |X). Defining π

^Z₀

≡ 1 for the sake of exposition, we can alternatively write the model in terms of the conditional quantiles of V , U

_D

:

D(Z, X) =

%

J

j=1

j × 1 &

π

_j^Z

(X) < U

_D

≤ π

_j−1^Z

(X) ' .

Figure 1 shows the ordered choice model in A1-A3 graphically. The top panel reflects marginal

utilities D

^∗_j

and the bottom panel illustrates the level of utility. D = k is selected by the agent

with the given realization of (Z, X, V ) since D

^∗_k−1

> 0 and D

^∗_k

≤ 0.

2.2 Marginal and Local Average Treatment Effects

We are interested in learning about features of the joint distribution of some outcome Y given treatment levels D ∈ {1, . . . , J}, observable characteristics X, and the unobservable com- ponent of selection into treatment, U

_D

. In this analysis we will focus on the conditional mean E[Y (D, X, U

_D

)|U

D

= u

_D

]. We assume potential outcomes are a function of treatment level, ob- servable characteristics, and level-specific unobservable characteristics:

Y

_j

(X, U

j

) = Y (D = j, X, U

_j

) for j = 1, . . . , J. (4) As in HUV we add the following assumptions to A1-A3:

A4 (U

_ij

, V

_i

) ⊥ ⊥ Z

_i

|X

i

for all j = 1, . . . , J

A5 γ

_j

≥ 0 for all j = 1, . . . , J − 1 and γ

_j

> 0 for at least one j ∈ {1, . . . , J − 1}

A6 V

_i

is continuously distributed A7 E[ |Y

j

| ] < ∞ for all j = 1, . . . , J

A8 0 < P r(D

i

= j|X

i

) < 1 for all j = 1, . . . , J for all X ⊂ suppX

Given a realization of (X, U

D

) = (x, u

D

), the Marginal Treatment Effect (MTE) for the transition j to j + 1 is defined as the average effect of moving individuals with (x, u

_D

) from level j to j + 1:

,

^{M T E}_j,j+1

(x, u

D

) ≡ E[Y

_j+1

− Y

j

|X = x, U

D

= u

D

] for j = 1, . . . , J − 1. (5) Given a realization of X = x, the corresponding Local Average Treatment Effect (LATE) for individuals with U

_D

∈ (a, b) is defined as the average MTE over that interval:

,

^{LAT E}_j,j+1

(X, a, b) ≡ (

b

a

,

^{M T E}_j,j+1

(X, u

D

)du

D

b − a for j = 1, . . . , J − 1. (6) 2.3 Identification of MTE Parameters

One way of identifying MTE parameters in this model is to augment assumptions A1-A8, as done by HUV, with an assumption about the ordered choice model. HUV assumes there exist instrumental variables W

_j

for each j = 1, . . . , J −1 such that the distribution of C

_j

(W

j

), conditional on Z, X, and {C

h

: h -= j}, is nondegenerate and continuous. We label this assumption, which does not hold in our model, A9. Under A9 each margin of choice can be varied exogenously and independently of all others.

When evaluated at u

_D

= π

_j^Z

(X), ,

^{M T E}_j,j+1

(X, u

_D

= π

^Z_j

(X)) represents the gross gain of mov-

ing from j to j + 1 for individuals that are indifferent between levels j and j + 1. HUV show

that index sufficiency holds in this model so that E[Y |Z, X] is equivalent to E[Y | π

^Z

(X)], where

π ≡ [π

^Z₁

(X), π

^Z₂

(X), . . . , π

^Z_J−1

(X)]. HUV further show that E[Y | π

^Z

] is differentiable due to As- sumptions A4, A6, and A7. The j to j + 1 MTE can be interpreted as the change in mean outcome due to externally increasing π

_j^Z

while leaving all other π

_k^Z

’s fixed for k -= j:

,

^{M T E}_j,j+1

(X, u

D

= π

^Z_j

) = ∂E[Y |X, π

^Z

]

∂π

_j^Z

. (7)

Identification of MTE parameters is achieved in HUV using the exogenous variation in π

^Z_j

(X) induced by the W

_j

to estimate the right hand side of Equation 7.

It is difficult to imagine a set of instruments {W

j

} each of which exogenously varies one margin of choice while leaving all other margins of choice unaffected in the context of residential choice.

Even if they existed, it is doubtful these instruments could be manipulated to identify the MTE function over the entire support of the distribution of (X, U

D

). It would be more likely that each level-specific instrument W

_j

would vary the choice margin over some interval, but not over the entire unit interval. A discussion of related issues can be found in Carneiro et al. (2011) for a binary context.

The discrete instrument of voucher assignment in MTO does not continuously vary π

_j^Z

(X) for any j nor does it provide an independent source of variation for each of the transitions. Violation of assumption A9 leads us to adopt an alternative approach to identifying parameters that is based on the identification of U

_D

in the ordered choice model and is therefore not possible in the binary model. Remembering that D = j iff π

_j^Z

< U

D

≤ π

^Z_j−1

and assuming π

⁰_j

(x) < π

¹_j

(x) < π

⁰_j−1

(x), an implication from the ordered choice model is that when Z = 0 an individual with X = x and u

_D

∈ [π

_j⁰

(x), π

¹_j

(x)] selects into D = j, but exogenously changing Z to 1 induces such individuals into D = j + 1. Thus for u

D

∈ [π

_j⁰

(x), π

_j¹

(x)],

E[Y |x, u

D

, Z = 1] − E[Y |x, u

D

, Z = 0] = E[Y (j + 1)|x, u

D

] − E[Y (j)|x, u

D

] (8)

= ,

^{M T E}_j,j+1

(x, u

D

).

Given estimates π ) and u )

_D

, we can construct an empirical estimate of , )

^{M T E}_j,j+1

(x, u

_D

) using the left hand side of Equation 8. Specifically, when ) π

_j⁰

(x) < π )

¹_j

(x) < π )

⁰_j−1

(x), for ) u

D

∈ [ ) π

_j⁰

(x), π )

_j¹

(x)]

we can estimate the MTE parameter as

, )

^{M T E}_j,j+1

(x, u

_D

) = E[Y |x, ) u

_D

, Z = 1] − E[Y |x, ) u

_D

, Z = 0]. (9) 2.4 Identification of LATE Parameters

As the building blocks of LATEs and other parameters, MTEs are the ultimate object of interest

in our model. Although we can identify the MTE function over a subset of its support as discussed

above, there are still reasons for estimating LATE parameters. One reason is that LATE parameters

will be more precisely estimated and will be less dependent on functional form and distributional

assumptions than MTE parameters. Another reason for estimating LATEs is that discussing their

identification will also help to clarify the identification of MTEs in our model.

Turning to the estimation of LATEs, we start by deriving an expression for E[Y |x, u

D

, Z = 1] − E[Y |x, u

D

, Z = 0] general enough to allow for any ordering relationship between π

⁰

(X) and π

¹

(X). Assumption A5 requires that π

¹_j

(X) ≥ π

_j⁰

(X) for all j and π

_j¹

(X) > π

_j⁰

(X) for some j, since the instrument monotonically increases individuals’ latent index. And from A4 the variation in average outcomes induced by the instrument can be attributed to changing π

^Z=0

to π

^Z=1

(from this point forward we keep conditioning on X implicit for the sake of exposition):

E[Y |Z = 1] − E[Y |Z = 0] =

%

J

j=1

*

₁

0

1{π

¹_j

≤ u

_D

< π

_j−1¹

}E[Y (D = j, u

_D

)]du

D

(10)

−

%

J

j=1

*

1

0

1{π

_j⁰

≤ u

_D

< π

_j−1⁰

}E[Y (D = j, u

_D

)]du

D

=

%

J

j=1

+*

π¹_j−1 π¹_j

E[Y (D = j, u

_D

)]du

D

(11)

−

*

π¹_j−1

π¹_j

%

J

k=1

,

1{π

_k⁰

≤ u

_D

< π

_k−1⁰

}E[Y (D = k, u

_D

)]

- du

_D

. .

Given equation 6 and since

E[Y (D = j, u

_D

)] − E[Y (D = j − m, u

_D

)] =

j−m+1

%

k=j

/

E[Y (D = k, u

_D

)] − E[Y (D = k − 1, u

_D

)]

0

we can rewrite Equation 10 as:

E[Y |Z = 1] − E[Y |Z = 0] =

J−1

%

j=1

*

π¹_j

π⁰_j

,

^{M T E}_j,j+1

(u

D

)du

D

=

J−1

%

j=1

/

,

^{LAT E}_j,j+1

(π

⁰_j

, π

¹_j

) 1

π

¹_j

− π

⁰_j

2 0

(12)

Given estimates π ) and u )

_D

from the ordered choice model we can construct empirical estimates of LATEs. When ) π

_j⁰

< ) π

¹_j

< π )

⁰_j−1

the LATE is estimated as:

, )

^{LAT E}_j,j+1

(π

_j⁰

, π

_j¹

) = E[Y | u )

_D

∈ [ ) π

_j⁰

, π )

¹_j

], Z = 1] − E[Y | u )

_D

∈ [ ) π

_j⁰

, ) π

_j¹

], Z = 0]. (13) The precise LATEs identified will be determined by the exogenous variation in the choice probabil- ities π

^Z

induced by the instrument. Two examples with D ∈ {1, 2, 3} are illustrated in Figures 2 and 3, and these examples are discussed in the Appendix to clarify issues related to identification.

A related discussion can also be found in Section VI A of HUV.

2.5 Discussion of Identifying Assumptions

We believe Assumptions A1-A8 represent the weakest assumptions under which we can identify interesting parameters in a model with a multi-valued treatment. We now consider alternative sets of identifying assumptions and discuss why we believe these assumptions are either too strong or too weak relative to A1-A8.

2.5.1 Strengthening A1-A8

One alternative approach would be to strengthen assumptions A1-A8. One particular assump- tion would allow us to estimate average treatment effects over the support of the distribution of a continuous treatment D using the generalized propensity score as developed in Imai and van Dyk (2004) or Hirano and Imbens (2005). However, while relatively standard, the Strong Ignorability (SI) assumption necessary for identification in Imai and van Dyk (2004) is restrictive relative to the framework we have adopted from HV and HUV.

¹

When f denotes the distribution of potential outcomes, SI can be written as:

f{Y (D = j)|X} = f {Y (D = j)|u

D

∈ [a, b], X} for all j ∈ {1, . . . , J }. (14) From our specification of potential outcomes in Equation 4, this is the same as:

f {u

j

|X} = f {u

j

|u

D

∈ [a, b], X} for all j ∈ {1, . . . , J }, (15) or

u

j

⊥ ⊥ u

D

|X for all j ∈ {1, . . . , J }. (16) When calculating E[Y (D = j|U

D

)] as in Figures 2 and 3, the expectation is taken over the distri- bution of U

j

conditional on X and U

D

. Thus SI requires that conditional on X, the distribution of the Y

_j

, and therefore their expected values as well, would have to be the same for all values of U

_D

as shown in Figure 4. This assumption is most likely to hold when observable characteristics in X are able to explain most of the variability in choice, so deciding whether to adopt the SI assumption will depend on the particular application and data available.

Figure 5 shows a binary example to help illustrate the differences between the heterogeneity in treatment effects allowed under our assumptions A1-A8 and under SI. Let θ = µ(X) be an index of observable characteristics. The average effect of treatment varies across θ as shown in the top panel of the Figure. In the center panel we can see a cross section of potential outcomes conditional on θ = θ

^∗

. Since E[β |θ] = E[Y (1) − Y (0)|θ] is the same for all u

D

conditional on θ = θ

^∗

, any variation in treatment identifies E[β|θ]. We could use variation in treatment induced by an instrument, but we could also simply compare those individuals in the population or control group with u

_D

< π

₁⁰

and those with u

_D

> π

₁⁰

to estimate the treatment effect. That is, when matching under SI there is no need for an instrument.

1

See Imbens (2004) for a discussion of this assumption in models with a binary treatment.

In contrast to the center panel, the bottom panel shows a possible example of MTEs that depend on u

_D

even conditional on θ. This is defined in HUV as a model with Essential Heterogeneity (EH).

In this case we need an instrument to generate variation in treatment status, and the variation gen- erated by the instrument determines what part of the distribution of Y (2)−Y (1)|θ

^∗

we can identify.

Since E[Y (2) − Y (1)|θ

^∗

] = (

₁

0

,

^{M T E}_1,2

(θ

^∗

, u

_D

)du

D

, in the example in the bottom panel we cannot identify E[Y (2) − Y (1)|θ

^∗

], but rather only (

π₁¹

π⁰₁

,

^{M T E}_1,2

(θ

^∗

, u

_D

)du

_D

. That the treatment effects we can identify are determined by the response of individuals to the instrument re-emphasizes that the neighborhood effects identified by MTO, or any other housing mobility experiment, depend on how people endogenously respond to the experiment. That is, under EH it is not possible to clearly interpret the neighborhood effects we observe through MTO without first understanding how the experiment impacted selection into treatment (Aliprantis (2012), Clampet-Lundquist and Massey (2008), Sampson (2008)).

2.5.2 Weakening A1-A8

Another alternative would be to weaken A1-A8, which would allow us to estimate the Average Causal Response (ACR) parameter introduced in Angrist and Imbens (1995). Under Assumptions A1-A8 the ACR is:

,

^ACR

(Z = 0, Z = 1) = E[Y |Z = 1] − E[Y |Z = 0]

E[D|Z = 1] − E[D|Z = 0] =

$

_J−1

j=1

[ (

^π1_j

π⁰_j

,

^{M T E}_j,j+1

(u

D

)du

D

]

$

_J−1

j=1

(j + 1) × (π

¹_j

− π

⁰_j

)

By relaxing Assumptions A2 and A3 we could specify a set of identifying assumptions equivalent to the assumptions in Angrist and Imbens (1995) (See Vytlacil (2006) for a proof.), and we would still be allowing for EH. We do not estimate ACRs under these weaker assumptions, however, because they yield only one summary parameter that is quite difficult to interpret. By imposing the structure of the choice model in Section 2.1 we are able to allow for EH while at the same time decomposing the ACR into its contributing LATEs. These components of the ACR are considerably more interesting than the single ACR parameter by itself. Given our current econometric tools we believe assumptions A1-A8 strike the right balance of structure for interpretation.

3 Moving To Opportunity (MTO)

Moving To Opportunity (MTO) was inspired by the promising results of the Gautreaux pro-

gram. Following a class-action lawsuit led by Dorothy Gautreaux, in 1976 the Supreme Court

ordered the Department of Housing and Urban Development (HUD) and the Chicago Housing Au-

thority (CHA) to remedy the extreme racial segregation experienced by public-housing residents in

Chicago. One of the resulting programs gave families awarded Section 8 public housing vouchers

the ability to use them beyond the territory of CHA, giving families the option to be relocated

either to suburbs that were less than 30 percent black or to black neighborhoods in the city that

were forecast to undergo “revitalization” (Polikoff (2006)).

The initial relocation process of the Gautreaux program created a quasi-experiment, and its results indicated housing mobility could be an effective policy. Relative to city movers, suburban movers from Gautreaux were more likely to be employed (Mendenhall et al. (2006)), and the chil- dren of suburban movers attended better schools, were more likely to complete high school, attend college, be employed, and had higher wages than city movers (Rosenbaum (1995)).

²

MTO was designed to replicate these beneficial effects, offering housing vouchers to eligible households between September 1994 and July 1998 in Baltimore, Boston, Chicago, Los Angeles, and New York (Goering (2003)). Households were eligible to participate in MTO if they were low-income, had at least one child under 18, were residing in either public housing or Section 8 project-based housing located in a census tract with a poverty rate of at least 40%, were current in their rent payment, and all families members were on the current lease and were without criminal records (Orr et al. (2003)).

Families were drawn from the MTO waiting list through a random lottery. After being drawn, families were randomly allocated into one of three treatment groups. The experimental group was offered Section 8 housing vouchers, but were restricted to using them in census tracts with 1990 poverty rates of less than 10 percent. However, after one year had passed, families in the experimental group were then unrestricted in where they used their Section 8 vouchers. Families in this group were also provided with counseling and education through a local non-profit. Families in the Section-8 only comparison group were provided with no counseling, and were offered Section 8 housing vouchers without any restriction on their place of use. And families in the control group received project-based assistance.

³

4 Data

The first source of data we use in our analysis is the MTO Interim Evaluation. The MTO Interim Evaluation contains variables listing the census tracts in which households lived at both the baseline and in 2002, the time the interim evaluation was conducted. These census tracts are used to merge the MTO sample with decennial census data from the National Historical Geographic Information System (NHGIS, Minnesota Population Center (2004)), which provide measures of neighborhood characteristics.

4.1 Variables

A variable measuring neighborhood quality is created as a linear combination of several neigh- borhood characteristics. Neighborhood characteristics measured by NHGIS variables are first trans-

2

It has also been found that suburban movers have much lower male youth mortality rates Votruba and Kling (2009) and tend to stay in high-income suburban neighborhoods many years after their initial placement (DeLuca and Rosenbaum (2003), Keels et al. (2005)).

3

Section 8 vouchers pay part of a tenant’s private market rent. Project-based assistance gives the option of a

reduced-rent unit tied to a specific structure.

formed into percentiles of the national distribution from the 2000 census. Principal components analysis is then used to determine which single vector combines the most information about the national distribution of the poverty rate, the percent with high school degrees, the percent with BAs, the percent of single-headed households, the male Employed-to-Population Ratio (EPR), and the female unemployment rate. Table 1 shows that the resulting univariate index explains 63 per- cent of the variance of these neighborhood characteristics, and that no additional eigenvector would explain more than 13 percent of the variance of these variables. Table 2 displays the coefficients relating each of these variables to the index vector. Note that the magnitudes of the coefficients for most variables are similar to the magnitude of the coefficient for poverty.

In addition to the quality as measured in 2000 of the neighborhood of residence at the baseline and in 2002, another neighborhood characteristic recorded at the baseline of the MTO data used in the analysis is whether the respondent had family living in their neighborhood of residence. The household characteristic variables used in the analysis were measured at the MTO baseline and include whether the adult respondent had ever applied for a Section 8 voucher, whether a member of the household was a victim of a crime in the previous 6 months, whether the respondent had a car that ran, and whether there were teenage children in the household.

Outcome variables from the MTO Interim Evaluation include the self-reported total household income (all sources summed), the individual earnings in 2001 of the sample adult, the labor market status of the adult at the time of the interim survey (ie, Whether they were employed, unemployed, or not in the labor force at all.), and the respondent’s body mass index (BMI). Weights are used in constructing all estimates.

⁴

4.2 Sample and Descriptive Statistics

The focus of this analysis is adults in the MTO Interim Evaluation sample. We choose not to include children in our analysis for two reasons. First, careful analysis of program effects from MTO on children’s outcomes has already been conducted (Sanbonmatsu et al. (2006), Kling et al.

(2007)). Second, as shown in Figure 6 and investigated in depth in DeLuca and Rosenblatt (2010), MTO did not induce large changes in school quality.

⁵

Both intuition and previous findings in the literature suggest schools may be the most relevant “neighborhood” for children (Oreopoulos (2003)).

We further restrict our sample in three ways to ensure that we are focusing on a relatively homogeneous population, facilitating the estimation of the ordered choice model. We first exclude Boston from the analysis because it is clearly an outlier relative to the other MTO sites. In Figure 7a we see that unlike all of the other sites, the baseline neighborhood quality in Boston was

4

Weights are used for two reasons. First, random assignment ratios varied both from site to site and over different time periods of sample recruitment. Randomization ratio weights are used to create samples representing the same number of people across groups within each site-period. This ensures neighborhood effects are not conflated with time trends. Second, sampling weights must be used to account for the sub-sampling procedures used during the interim evaluation data collection.

5

The advent of school choice may be important for these results: Thirty percent of MTO control group children

in Chicago and Los Angeles were attending magnet schools (Sanbonmatsu et al. (2006), p 684).

not confined to the first percentiles of the national distribution of neighborhood quality. Table 3 quantifies these differences precisely. We can see that the control group in Boston looks more like the experimental group in every other site. Observations from Boston comprised a little less than a quarter of the full sample.

⁶

Second, we drop all households living at baseline in a neighborhood above the tenth percentile of the national distribution of quality. Returning to Table 3, we can see that the median baseline neighborhood quality for MTO participants was below the first percentile of the national distri- bution. For Chicago, Los Angeles, and New York City, nearly all participants lived at baseline in neighborhoods below the 10th percentile of the national distribution. In Baltimore, however, at baseline about 20 percent of participants lived in much higher quality neighborhoods, driven mainly by the male EPR and the share of adults holding a BA in their neighborhoods. These individuals are also dropped from our sample.

Finally, we drop individuals who moved to neighborhoods in 2002 above the median of the national distribution of quality in 2000. We can see from Figure 8c that this is a small subsample to discard.

The resulting sample used in the analysis has more than 2,500 adults, approximately 65 percent of the original adult sample. Approximately 1,800 are in either the experimental or control groups.

Figure 8 shows the distribution of the sample both at the baseline and in 2002. 75 percent of the sample lives in 2002 in a neighborhood below the 10th percentile of the 2000 national distribution of neighborhood quality. Figure 8a confirms that MTO induced small shares of participants into high quality neighborhoods (Aliprantis (2012)). Neighborhoods in New York City are among the ones with the lowest quality in the sample at baseline. Figure 7f shows that in 2002, neighborhood quality for the New York City control group remained almost unchanged, while at other sites quality experienced small improvements.

5 Ordered Choice Model Estimates

5.1 Specification

In the ordered choice model from Section 2.1 the marginal benefit of choosing to move from treatment level j to j + 1 is

D

_j^∗

= µ(X) + Zγ

_j

− C

_j

− V.

We specify

µ(X) = β

₁

X

₁

+ · · · + β

_m

X

_m

(17)

γ

_j

= γ

₀

+ γ

_j

(18)

C

_j

= δ

₀

+ δ

_j

(19)

6

Section 5 will also present estimation results from a model including data from Boston that further confirms the

singular nature of the site.

where both γ

₀

and δ

₀

are site-specific, and we assume V ∼ N (0, 1). X is composed of variables measuring baseline neighborhood quality, previous use of a Section 8 voucher, whether a household member was the victim of a certain crimes in the past six months, whether the respondent had a car, whether there were teens in the house, and whether the respondent had family members in the neighborhood. The instrument Z is an indicator variable for assignment to the experimental voucher group in the MTO experiment: Z

_i

= 1{i is in experimental group}.

⁷

An implication of the ordered choice model that was stated in Equation 3 is:

D = j ⇐⇒ µ(X) + Zγ

_j

− C

_j

< V ≤ µ(X) + Zγ

_j−1

− C

_j−1

. (20) So if treatment levels are defined such that D = j for q

^∗

∈ [q

_j

, q

_j

], then the probability of observing D = j is

P r(D = j) = F

_V

[µ(X) + Zγ(q

_j

) − C(q

_j

)] − F

_V

[µ(X) + Zγ(q

_j

) − C(q

_j

)], allowing us to write the log-likelihood of the data as

LL =

%

N

i=1

%

J

j=1

1{D

i

= j} ln(P r(D

i

= j)). (21)

This ordered probit model is estimated with the gologit2 procedure in Stata.

5.2 The Identification of Unobservables

We now discuss how we use the parameter estimates from the discrete ordered choice model θ )

_J

≡ ( β, ) ) γ, δ) to recover estimates of each individual’s unobservable ) V . If we take the limit of the ordered choice model as J → ∞, the benefit of moving to level j + 1 from j, D

_j^∗

, becomes the marginal benefit of moving to a neighborhood with higher quality. We denote the marginal benefit of moving to a higher quality neighborhood when in a neighborhood of quality q as

D

^∗

(q) = µ(X) + Zγ(q) − C(q) − V

where the cost and program effect components of marginal benefit are now allowed to be continuous functions of quality:

µ(X) = β

₁

X

₁

+ · · · + β

_m

X

_m

(22)

γ (q) = f (q) (23)

C(q) = g(q). (24)

7

We will actually define binary instruments for both the experimental and Section 8 groups, (Z

^exp, Z^S8

). Although

we will use information from both voucher groups and the control group to estimate the ordered choice model, for

the sake of tractability both the ensuing discussion of the ordered choice model and the estimation of LATEs will

focus on comparing the experimental voucher with the control treatment.

Given estimates of γ(q

_j

) = γ

j

from the ordered choice model, we linearly interpolate γ (q) = γ

₀

+ γ

_j

+ (q − q

_j

)( γ

_j+1

− γ

_j

q

_j+1

− q

_j

) for q ∈ (q

_j

, q

_j+1

) = (q

_j+1

, q

_j+1

)

and an analogous function g for cost. We set {q

_j

}

^J−1_j=1

to the cutoffs separating the deciles of the sample distribution of neighborhood quality.

If we assume D

^∗

is strictly decreasing and households maximize their utility at q

^∗

, then q

^∗

satisfies the following first order condition equivalent to conditions in Equations 2 and 3 in the multilevel case:

D

^∗

(q

^∗

) = 0. (25)

Given the estimated parameters and functions θ ≡ ( β, ) ) γ(q), C(q)) constructed from the discrete ) choice model parameters θ )

_J

, we use the FOC in Equation 25 to identify unobservables as:

⁸

V ) = µ(X) + ) Z ) γ (q

^∗

) − C(q )

^∗

).

For each individual we also estimate )

π

_q⁰

j

= Φ[ β )

₁

X

₁

+ · · · + β )

_m

X

_m

− C(q )

_j

)] (26) )

π

_q¹

j

= Φ[ β )

₁

X

₁

+ · · · + β )

_m

X

_m

− C(q )

_j

) + ) γ(q

_j

)]. (27) 5.3 Estimation Results

As a first check on model fit, Figure 9 shows the unconditional average π

_q^Z

from the data together with fractional polynomial fitting of ) π

_q^Z

estimates as functions of µ(X). We can see that ) the model fits the data well.

As measured by the estimated µ(X), the distributions of observable characteristics conditional ) on voucher status confirm that voucher assignment was successfully randomized (Figure 10a). The distribution of unobservables follows a standard normal distribution very closely (Figure 10b), and the distributions of unobservables conditional on experimental group status look similar (Figure 10c).

Figure 11a shows the joint distribution of ( µ(X), ) u )

_D

) along with fractional polynomial fittings of ) π

_q^Z

for several quantiles q as functions of ) µ(X). Looking at this joint distribution more closely in Figure 11b, we can see that from one perspective MTO had only modest effects on neighborhood quality, while from another it had relatively large effects.

One perspective looks at Figures 11a and 11b and sees that even for the experimental group, the vast majority of MTO volunteers remained in a neighborhood in the first decile of the national distribution of quality. Even while remembering that our sample does not include those who moved

8

This is a notable departure from HUV. In the binary case, HUV define the support over which MTEs can be

identified as the intersection of the sets of predicted values of the propensity score obtained from the treated and

untreated samples. With linear interpolation we obtain estimates of

UD

throughout the entire interval (0, 1).

to neighborhoods above the national median, we still see that in our sample (as in the entire MTO population) only a small share of volunteers were induced across the 30th percentile of neighborhood quality.

Another perspective first notes that at baseline the MTO volunteers were representative of the one percent living in the left tail of the distribution of neighborhood quality (Figure 7a). Although MTO volunteers tended to live in better neighborhoods by the time of the interim evaluation, it was still the case that less than 15 percent of the control group in our sample lived in neighborhoods of quality above the 10th percentile of the national distribution (Figure 8d). Given such an initial distribution, the MTO voucher did induce many volunteers into higher quality neighborhoods.

The cutoffs for moving across the 20th and 30th percentiles of the national distribution for the experimental group ( π )

¹₂₀

(X) and ) π

¹₃₀

(X)) look similar to the cutoff for the 10th percentile for the control group ( ) π

₁₀⁰

(X)).

Perhaps the most interesting result from the estimation of the ordered choice model is the strong heterogeneity in the effects of the program as displayed in Figure 12. Those sites with the lowest marginal benefit of increasing neighborhood quality in the absence of the program (Los Angeles and New York City, see Figure 12a.) were also the sites where the MTO experimental voucher had the largest effect on the marginal benefit of moving (Figure 12d). And not only were effects heterogeneous by site, but also by voucher type, which is important for the design of future housing mobility programs (Galiani et al. (2012)). The standard Section 8 housing voucher was more effective at reducing moving costs than the experimental voucher at very low levels of neighborhood quality, but this relationship reverses somewhere before the fifth percentile of the national distribution of neighborhood quality.

Figure 12c shows the observed component of the marginal benefit function for the estimated model that includes Boston. We can see again that Boston is a clear outlier relative to the other MTO sites, warranting its removal from the analysis. Table 4 reports the estimated coefficients in µ(X).

6 Neighborhood Effects Estimates

6.1 LATEs

When estimating effects we redefine treatment levels in terms of deciles of the national distri- bution (ie, D = 1 if q

^∗

∈ [0, 10), . . . , D = 5 if q

^∗

∈ [40, 50).). Referring back to Figure 11b, it is not surprising that either single-level LATEs or combinations of several level LATEs are imprecisely estimated as one moves to higher quality. The only single-level LATE we are able to precisely estimate is ,

^{LAT E}_1,2

(max{ π )

¹₂₀

, ) π

⁰₁₀

}, ) π

₁₀¹

).

Table 5 shows estimates for several outcomes of ,

^{LAT E}_1,2

that are averaged over observables X together with estimates of the interval (max{ ) π

₂₀¹

, ) π

₁₀⁰

}, π )

₁₀¹

) to which the LATE pertains. Standard errors for each of these estimates are computed using 400 bootstrap replications.

⁹

9

Weights were used in constructing point estimates, but were not used in constructing standard errors. Weighted

A first point we can see in Table 5 is that the identified parameter ,

^{LAT E}_1,2

is the average MTE over only 12.5 percent of our subsample of MTO volunteers, with this 12.5 percent being located among those individuals with low unobservables U

_D

. Although we are only able to identify a few parameters among many of interest, the estimated LATEs for this subpopulation conform to the theory that improving neighborhood characteristics improves outcomes.

There was a large increase in household income for those who moved from the first to second decile of neighborhood quality, although this increase was not a result of increases in the earnings of the respondent. The LATE on employment was quite large, and the fact this estimate was larger than the increase in labor force participation rates is suggestive that employment improved due to changes in the local labor market rather than changes in household behavior alone. Adult respondents and their children were also less likely to currently be on welfare if the program induced them from the first to second deciles of neighborhood quality. And the LATE on body mass index also indicates there were improvements in health from exogenously increasing neighborhood quality.

6.2 MTEs

Figure 13 shows estimates of E[Y (D = j)|U

D

] when Y (D) is assumed to be linear in U

_D

using for identification all ) u

D

∈ [ ) π

_j^Z

, π )

_j−1^Z

] for each voucher group. We see the expected pattern of improved outcomes as quality increases. In the case of MTO it is not possible to precisely estimate MTEs using identified unobservables u )

_D

and functional form assumptions as illustrated in Figure 13, and as a result this Figure is merely suggestive of the true parameters. We nevertheless show Figure 13 because we believe this approach may be able to estimate MTEs more precisely within an ordered choice framework in other applications.

7 Conclusion

This paper presented a new approach to identifying MTE and LATE parameters that exploits the identification of unobservables in a multi-level model of treatment. Estimation results from an ordered choice model indicated that the Moving to Opportunity (MTO) housing mobility experi- ment had strongly heterogeneous effects across sites. Those sites with residents living in the lowest quality neighborhoods experienced the greatest impact on neighborhood quality from the offer of an MTO voucher. Ordered choice estimates also showed that the Section 8 voucher was more effective at improving neighborhood quality at very low levels, but that the experimental voucher became more effective before the fifth percentile of the national distribution of neighborhood quality.

The variation in neighborhood quality induced by MTO was only able to precisely identify LATE parameters pertaining to a subset of about 13 percent of a restricted subsample of MTO volunteers.

These LATE estimates for employment, labor force participation rates, earnings, income, welfare receipt, and body mass index conformed to standard theories of neighborhood externalities.

and unweighted point estimates were similar.

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8 Appendix: Two Examples with D ∈ {1, 2, 3}

We illustrate the identification of LATEs in our model for two examples with J = 3 as shown in Figures 2-3. First suppose we tighten Assumption A6 to:

A6* π

¹_j

> π

_j−1⁰

for all j ∈ {1, . . . , J − 1}.

Example I in Figure 2 shows such an ordering when J = 3. Under A6* the right hand side of Equation 12 can be derived quickly since

E[Y |Z = 1] =

%

J

j=1

*

π¹_j−1

π¹_j

E[Y (D = j, u

_D

)]du

D

=

*

1

π¹₁

E[Y (D = 1, u

D

)]du

D

+

J−1

%

j=1

/*

π¹_j π⁰_j

E[Y (D = j + 1, u

D

)]du

D

+

*

π⁰_j

π¹_j+1

E[Y (D = j + 1, u

D

)]du

D

0

and

E[Y |Z = 0] =

%

J

j=0

*

π⁰_j−1

π⁰_j

E[Y (D = j, u

_D

)]du

D

=

*

1

π₁¹

E[Y (D = 1, u

D

)]du

D

+

J−1

%

j=1

/*

π_j¹ π⁰_j

E[Y (D = j, u

_D

)]du

_D

+

*

π_j⁰

π¹_j+1

E[Y (D = j + 1, u

_D

)]du

_D

0

.

Thus the difference in expected outcomes due to changes in the instrument is the sum of integrated MTEs:

E[Y |Z = 1] − E[Y |Z = 0] =

J

%

−1

j=1

3*

π¹_j π⁰_j

E[Y (D = j + 1, u

D

)]du

D

−

*

π¹_j

π_j⁰

E[Y (D = j, u

D

)]du

D

4

=

J

%

−1

j=1

*

π_j¹

π⁰_j

,

^{M T E}_j,j+1

(u

D

)du

D

=

J−1

%

j=1

/

,

^{LAT E}_j,j+1

(π

⁰_j

, π

¹_j

) 1

π

_j¹

− π

⁰_j

2 0 . (28)

Since we can recover π

^Z

= [π

^Z₁

, π

₂^Z

, . . . , π

^Z_J−1

] and we can tell in which [π

⁰_j

, π

_j¹

] interval u

D

lies from

the data, we can estimate these LATEs using Equation 13. The variation in treatment induced by

the instrument identifies:

,

^{LAT E}_1,2

(π

₁⁰

, π

₁¹

) = (

π₁¹

π⁰₁

,

^{M T E}_1,2

(u

D

)du

D

π

₁¹

− π

⁰₁

= E[Y |u

D

∈ [π

⁰₁

, π

₁¹

], Z = 1] − E[Y |u

D

∈ [π

⁰₁

, π

¹₁

], Z = 0]

and

,

^{LAT E}_2,3

(π

⁰₂

, π

₂¹

) = (

π¹₂

π⁰₂

,

^{M T E}_2,3

(u

D

)du

D

π

¹₂

− π

⁰₂

= E[Y |u

D

∈ [π

⁰₂

, π

¹₂

], Z = 1] − E[Y |u

D

∈ [π

₂⁰

, π

¹₂

], Z = 0].

An important issue to remember is that all of the preceding has been implicitly conditioning on observable characteristics X. Figure 14 shows that Example I in Figure 2 is just one cross section taken from an interval of observable characteristics.

Now suppose that we drop A6* and replace it with the less restrictive original Assumption A6.

In this case it is possible that π

¹_j

> π

_j−1⁰

for some j. Let u

_D

∈ [π

¹_m

, π

_m−1¹

] and u

_D

∈ [π

_n⁰

, π

_n−1⁰

] for some m, n ∈ {1, . . . , J − 1} where m > n. Then Equation 12 implies

E[Y (u

D

)|Z = 1] − E[Y (u

D

)|Z = 0] =

m−1

%

j=n

,

^{M T E}_j,j+1

(u

D

).

Thus if a = max{π

¹_m

, π

_n⁰

} and b = min{π

¹_m−1

, π

⁰_n−1

}, we can identify:

E[Y |u

D

∈ [a, b], Z = 1] − E[Y |u

D

∈ [a, b], Z = 0] = (

b

a

$

_m−1

j=n

,

^{M T E}_j,j+1

(u

D

)du

D

b − a (29)

This scenario highlights that the precise LATEs identified will be determined by the exogenous variation in the choice probabilities π

^Z

induced by the instrument. Note that if π

_j¹

> π

_j−1⁰

for some j, the corresponding LATE parameter is still separately identified over the interval

,

max{π

_j⁰

, π

¹_j+1

}, min{π

⁰_j−1

, π

_j¹

} -

.

But if max{π

⁰_j

, π

¹_j+1

} -= π

⁰_j

or min{π

⁰_j−1

, π

_j¹

} -= π

_j¹

, then the level-specific LATE parameters will not be separately identified over the entire interval [π

⁰_j

, π

_j¹

].

Comparing Example I and Example II in Figure 3 helps to illustrate how the ordering of the π

_j^Z

determines identification. Since π

¹₂

> π

₀¹

in Example II, the instrument identifies

,

^{LAT E}_1,2

(π

₂¹

, π

₁¹

) = (

π₁¹

π¹₂

,

^{M T E}_1,2

(u

D

)du

D

π

₁¹

− π

¹₂

= E[Y |u

D

∈ [π

¹₂

, π

₁¹

], Z = 1] − E[Y |u

D

∈ [π

¹₂

, π

¹₁

], Z = 0]

However, over the interval [π

₁⁰

, π

¹₂

] = [π

₁⁰

, min{π

₂¹

, π

₁¹

}] we cannot separately identify each level- specific LATE. Instead the instrument identifies:

,

^{LAT E}_1,3

(π

₁⁰

, π

¹₂

) = ,

^{LAT E}_1,2

(π

⁰₁

, π

₂¹

) + ,

^{LAT E}_2,3

(π

₁⁰

, π

¹₂

)

= E[Y |u

D

∈ [π

₁⁰

, π

¹₂

], Z = 1] − E[Y |u

D

∈ [π

₁⁰

, π

₂¹

], Z = 0]

= (

π¹₂

π₁⁰

5 ,

^{M T E}_1,2

(u

_D

) + ,

^{M T E}_2,3

(u

_D

) 6 du

_D

π

¹₂

− π

⁰₁

.

But over the interval [π

₂⁰

, π

⁰₁

] = [π

₂⁰

, min{π

₁⁰

, π

¹₂

}] the instrument does again separately identify the LATE parameter:

,

^{LAT E}_2,3

(π

⁰₂

, π

₁⁰

) = E[Y |u

D

∈ [π

⁰₂

, π

₁⁰

], Z = 1] − E[Y |u

D

∈ [π

⁰₂

, π

⁰₁

], Z = 0] = (

π₁⁰

π⁰₂

,

^{M T E}_2,3

(u

_D

)du

_D

π

⁰₁

− π

₂⁰

.

Figure 15 shows these LATEs graphically.

Figures

Marginal Utility D

^∗_j

| (Z, X, V )

k-3 k-2 k-1 k k+1 j

-3 -2 -1 0 1 2 3 4 D

_j^∗

u(D) | (Z, X, V ) where k − 2 = 3 and J = k + 2 = 7

0 j 1 2 3 4 5 6 7 u(D = j)

k-2 k-1 k k+1 k+2

Figure 1: The First Stage Ordered Choice Model

Observed E[Y (D)|v, Z] when D ∈ {1, 2, 3}

= E[Y |v, Z = 1]

= E[Y |v, Z = 0]

0 v 5 10 15 20 25 30 35 Y

E[Y

(3)|v]

E[Y

(2)|v]

E[Y

(1)|v]

#^{M T E}2,3 (v)

#^{M T E}1,2 (v)

F

_V

(·) and π

^Z_j

1

0.75

0.50

0.25

v 0 π

₁¹

π

₁⁰

π

₂¹

π

₂⁰

UD=FV(·|X)

Figure 2: Example I: Potential Outcomes and Marginal Treatment Effects Given µ(X)

Example I: Observed E[Y (D)|u

D

, Z ] when D ∈ {1, 2, 3}

= E[Y |u

D

, Z = 1]

= E[Y |u

D

, Z = 0]

π

₁¹

π

₁⁰

π

₂¹

π

⁰₂

U

_D

0 5 10 15 20 25 30 35 Y

E[Y

(3)|u

D

]

E[Y

(2)|u

D

]

E[Y

(1)|u

D

]

#^{M T E}2,3 (uD)

#^{M T E}1,2 (u_D)

Example II: Observed E[Y (D)|u

D

, Z ] when D ∈ {1, 2, 3}

= E[Y |u

D

, Z = 1]

= E[Y |u

D

, Z = 0]

π

₁¹

π

¹₂

π

₁⁰

π

⁰₂

U

D

0 5 10 15 20 25 30 35 Y

E[Y

(3)|u

D

]

E[Y

(2)|u

D

]

E[Y

(1)|u

D

]

#^{M T E}2,3 (uD)

#^{M T E}1,2 (u_D)

Figure 3: Examples I and II: Potential Outcomes and Marginal Treatment Effects Given µ(X)