Comparing IV With Structural Models: What Simple IV Can and Cannot Identify

Abstract

This paper compares the economic questions addressed by instrumental variables estimators with those addressed by structural approaches. We discuss Marschak’s Maxim: estimators should be selected on the basis of their ability to answer well-posed economic problems with minimal assumptions. A key identifying assumption that allows structural methods to be more informative than IV can be tested with data and does not have to be imposed.

1 Introduction

The primary question regarding the choice of an empirical approach to analyzing economic data should be What economic question does the analyst seek to answer? Explicit economic models make it easier to formulate and answer economic questions. Advocates of atheoretical approaches to analyzing economic data appeal to randomization as an ideal and invoke IV (or matching or regression discontinuity designs) as a surrogate for randomization. However, even perfectly executed randomizations do not answer all questions of economic interest. There are important examples where structural models produce more information about preferences than experiments.¹

A valid instrument is not guaranteed to identify parameters of economic interest when responses to choices vary among individuals, and these variations influence choices taken.² Different valid instruments answer different questions.³ The sign of the IV estimator can be different from that of the true causal effect.⁴

No one trained in economics can doubt the value of credible explicit economic models in interpreting economic data. They are designed to answer a variety of well-posed economic questions and to be invariant to classes of policy interventions.⁵ The problem with this approach is that after 60 years of experience with fitting structural models on a variety of data sources, empirical economists have come to appreciate the practical difficulty in identifying, and precisely estimating, the full array of structural parameters that answer the large variety of policy questions contemplated by the Cowles Commission economists — the fathers of structural estimation.⁶

Proponents of IV are less ambitious in the range of questions they seek to answer. The method often gains precision by asking narrower questions. The problem that plagues the IV approach is that the questions it answers are usually defined as probability limits of estimators and not by well-formulated economic problems. Unspecified “effects” replace clearly defined economic parameters as the objects of empirical interest.

As noted by Marschak (1953), there is a middle ground. Marschak’s Maxim emphasizes that one should solve well-posed economic problems with minimal assumptions. Marschak noted that for many problems of policy analysis, it is not necessary to identify fully specified structural models with parameters that are invariant to classes of policy modifications — the goal of structural analysis as conceived by the Cowles pioneers and successor generations of structural economists.⁷ All that is required to conduct many policy analyses or to answer many well-posed economic questions are combinations of the structural parameters that are often much easier to identify than the individual parameters themselves.⁸

Heckman and Vytlacil (1999, 2005) bridge the structural and IV literatures. They develop an economically interpretable parameter — the marginal treatment effect (MTE) — which is a marginal willingness to pay for the benefit of treatment for persons at well-defined margins of choice. It is invariant to a class of policy modifications. The MTE can be identified by local variation in instruments. They show how different instrumental variables weight the MTE differently. These weights need not be positive for all values of the argument of the MTE.⁹ For classes of well-posed economic questions, it is possible, under the conditions given in Heckman and Vytlacil (2005), to fashion functions of instruments that answer well-posed economic questions. Alternatively, it is possible to use an estimated MTE and construct weights to estimate economically interpretable parameters. This approach is unusual in the standard IV literature, which traditionally defines the parameter of interest to be an “effect” identified by an instrument.

Many economists follow Imbens and Angrist (1994) and interpret IV as identifying a weighted average of the gains to persons induced to change their choice (or state) by a change in the instrument. Imbens and Angrist work with a two choice model. Heckman, Urzua, and Vytlacil (2006, 2008) and Heckman and Vytlacil (2007b) extend this analysis to an ordered choice model and to general unordered choice models.¹⁰ This paper develops the unordered case further and gives a precise characterization of the generalization of the MTE that is appropriate for this case.

The original Imbens-Angrist intuition applies, but in general unordered choice models, agents attracted into a state by a change in an instrument come from many origin states, so there are many margins of choice. Structural models can identify the gains arising from these separate margins. This is a difficult task for IV without invoking structural assumptions. Structural models can also identify the fraction of persons induced into a state coming from each origin state. IV alone cannot.

For some economic questions, these are unimportant distinctions. For others, they are crucial. For specificity, consider an analysis of the GED program. The GED is a test by which high school dropouts can exam certify to be the equivalents of ordinary high school graduates. Heckman, LaFontaine, and Rodríguez (2008) show that the presence of a GED program induces some persons to drop out of high school. It also induces some persons who would remain dropouts to exam certify. Within this context we ask: What are the wage benefits for those induced to take the GED from the dropout state? For those induced to drop out of high school? What proportion of persons induced to take a GED come from each of the other states?

IV cannot answer these questions except under structural assumptions. It can identify the mean gross gain to the GED for those induced to take it, compared to the next best alternative. This is a weighted average of the effects from each possible origin state that the structural approach can separately identify. In the IV approach, when there are multiple origin states, the weights on the individual effects cannot be estimated without using structural methods. As shown in Heckman and Vytlacil (2007b), IV needs to be supplemented with explicit choice theory to answer many interesting questions, including questions of economic welfare regarding introduction of policies as well as distributional questions such as the percentage of persons harmed by a policy.

This paper demonstrates these points. We first establish a precise framework for discussing IV, and relate it to economic models.

2 The Choice Model and Assumptions

Following Heckman, Urzua, and Vytlacil (2006, 2008) and Heckman and Vytlacil (2007b), consider the following model with multiple choices and associated multiple outcome states. Let 𝒥 denote the agent’s choice set, where 𝒥 contains a finite number of elements. For example, 𝒥 enumerates possible schooling states (e.g., GED, high school dropout, high school graduate). The value to the agent of choosing j ∈ 𝒥 is

$$R_j(Z_j)={\vartheta }_j(Z_j)-V_j,$$ (1)

where Z_j are the agent’s observed characteristics that affect the utility from choosing j, and V_j is the unobserved shock to the agent’s utility from choice j. We sometimes write R_j for R_j(Z_j) to simplify notation. Let Z denote the random vector containing all unique elements of {Z_j}_j∈𝒥. We write R_j(Z) for R_j(Z_j), leaving implicit the condition that R_j(·) only depends on the elements of Z that are contained in Z_j. Let D_j be a variable indicating whether the agent would choose j if confronted with choice set 𝒥:¹¹

$$D_j=\{\begin{array}{cc}1\hfill & \text{if}{R}_j\ge R_k\forall k\in 𝒥\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}$$

Array the D_j into a vector D. Let Y be the outcome that would be observed if the agent faced choice set 𝒥, defined as

$$Y={\displaystyle \sum _{j\in 𝒥}{D}_j{Y}_j,}$$

where Y_j is a potential outcome observed only if option j is chosen. Y_j is determined by

$$Y_j={\mu }_j(X_j,U_j),$$

where X_j is a vector of the agent’s observed characteristics and U_j is an unobserved random vector. Let X denote the random vector containing all unique elements of {X_j}_j∈𝒥. (Z, X, D, Y) is assumed to be observed by the analyst.¹²

Define R_𝒥 as the maximum obtainable value given choice set 𝒥:

$$\begin{array}{cc}{R}_{𝒥}\hfill & ={\text{max}}_{j\in 𝒥}\left\{R_j\right\}\hfill \\ \hfill & ={\displaystyle {\sum }_{j\in 𝒥}{D}_j{R}_j.}\hfill \end{array}$$ (2)

This is the traditional representation of the decision process that if choice j is optimal, choice j is better than the “next best” option:

$$D_j=1⟺R_j\ge R_{𝒥\backslash j}.$$

Heckman, Urzua, and Vytlacil (2006, 2008) and Heckman and Vytlacil (2007b) show that this simple, well-known, representation is the key intuition for understanding how instrumental variables estimate the effect of a given choice versus the “next best” alternative. IV is a weighted average of the effects for people induced into a choice from different margins. Analogous to the definition of R_𝒥, we define R_𝒥(z) to be the maximum obtainable value given choice set 𝒥 when instruments are fixed at Z = z,

$$R_{𝒥}(z)=\underset{j\in 𝒥}{\text{max}}\{R_j(z)\}.$$

Following the analysis in Heckman, Urzua, and Vytlacil (2006, 2008) and Heckman and Vytlacil (2007b), we assume:

• (A-1)The distribution of ({V_j}_j∈𝒥) is continuous.¹³
• (A-2){(V_j, U_j)}_j∈𝒥 is independent of Z conditional on X.
• (A-3)E | Y_j| < ∞ for all j ∈ 𝒥.
• (A-4)Pr(D_j = 1 | X) > 0 for all j ∈ 𝒥.

In addition, we assume an exclusion restriction that requires some additional notation.¹⁴ Let Z^[−l] denote all elements of Z except for the lth component. We assume (A-5) For each j ∈ 𝒥, their exists at least one element of Z, say Z^[l], such that the distribution of ϑ_j(Z_j) conditional on (X, Z^[−l]) is continuous.

With these assumptions, one can generalize the analysis of Heckman and Vytlacil (1999, 2001, 2005) to the unordered case. Assumptions (A-1) and (A-2) imply that R_j ≠ R_k (with probability 1) for j ≠ k, so that argmax_j∈𝒥{R_j} is unique (with probability 1). Assumption (A-2) assures the existence of an instrument. Assumption (A-3) is required for mean treatment parameters to be well defined. It also allows one to integrate to the limit and to produce well-defined means. Assumption (A-4) requires that at least some individuals participate in each choice for all X. Assumption (A-5) imposes the requirement that one be able to independently vary the index for the given value function. It imposes a type of exclusion restriction, that for any j ∈ 𝒥, Z contains an element such that (i) it is contained in Z_j; (ii) it is not contained in any Z_k for k ≠ j, and (iii) ϑ_j(·) is a nontrivial function of that element conditional on all other regressors.¹⁵

In a series of papers, Heckman and Vytlacil (1999, 2001, 2005, 2007b), develop the method of local instrumental variables (LIV) to estimate the marginal treatment effect (MTE) for the case of binary choices. We now define and interpret the MTE and LIV in the case of general unordered choices.

3 Interpreting Local Instrumental Variables in the Unordered Case

We define local instrumental variables (LIV) using a variable that shifts people toward (or against) choice j by operating only on R_j(Z_j). LIV identifies an average marginal return to j vs. the next best alternative across persons.¹⁶ However, without further assumptions, LIV will not decompose the average marginal return into its component parts corresponding to the effects for persons induced into j from each of the possible origin states.

To see this, consider a three outcome case, 𝒥 = {1, 2, 3}. For concreteness, we pursue the education example previously stated and let 1 be GED, 2 be high school dropout, and 3 be high school graduate. Our results are more general but the three outcome case is easy to exposit.

In this section, we assume that Z₁, Z₂, Z₃ are disjoint sets of regressors so Z = (Z₁, Z₂, Z₃) but they are not necessarily statistically independent. We can easily relax this assumption but making it simplifies the notation. We condition on X and keep it implicit throughout the analysis of this paper.¹⁷ In this notation,

$$\begin{array}{cc}E(Y|Z)\hfill & =E\left[{\displaystyle \sum _{j=1}^3{Y}_j{D}_j}|Z\right]\hfill \\ \hfill & =E(Y_1{D}_1|Z)+E(Y_2{D}_2|Z)+E(Y_3{D}_3|Z).\hfill \end{array}$$ (3)

E(Y|Z) and its components can be estimated from data on (Y, Z). IV is based on (3). From (2), choices are generated by the following inequalities:

$$\begin{array}{c}{D}_1=\mathbf{1}(R_1\ge R_2,R_1\ge R_3)\hfill \\ D_2=\mathbf{1}(R_2\ge R_1,R_2\ge R_3)\hfill \\ D_3=\mathbf{1}(R_3\ge R_1,R_3\ge R_2).\hfill \end{array}$$

We define the marginal change in Y with respect to Z₁. IV methods are based on such types of variation. The local instrumental variable estimator using Z₁ as an instrument is the sample analogue of

$$\frac{\frac{\partial E(Y|Z)}{\partial Z_1}}{\frac{\partial \text{Pr}(D_1=1|Z)}{\partial Z_1}}{|}_{Z=z}=\text{LIV}(z),$$

where LIV is a function of z. In the case of three choices, there are two margins from which persons can be attracted into or out of choice 1 by Z₁.¹⁸

From local variations in Z₁, one can recover the following combinations of parameters from the data on Y₁D₁:

$$\begin{array}{c}\frac{\partial E(Y_1{D}_1|Z=z)}{\partial Z_1}\hfill \\ =\frac{\partial }{\partial Z_1}{\displaystyle \int {\displaystyle {\int }_{-\infty }^{{\vartheta }_1(Z_1)-{\vartheta }_2(Z_2)}{\displaystyle {\int }_{-\infty }^{{\vartheta }_1(Z_1)-{\vartheta }_3(Z_3)}{y}_1{f}_{Y_1,V_1-V_2,V_1-V_3}(y_1,v_1-v_2,v_1-v_3)d(v_1-v_3)d(v_1-v_2)d{y}_1{|}_{Z=z}}}}\hfill \\ =\frac{\partial {\vartheta }_1(Z_1)}{\partial Z_1}{|}_{Z_1=z_1}\left[\begin{array}{c}{\displaystyle \int y_1{\displaystyle {\int }_{-\infty }^{{\vartheta }_1(z_1)-{\vartheta }_3(z_3)}{f}_{Y_1,V_1-V_2,V_1-V_3}(y_1,{\vartheta }_1(z_1)-{\vartheta }_2(z_2),v_1-v_3)d(v_1-v_3)d{y}_1}}\hfill \\ +{\displaystyle \int y_1{\displaystyle {\int }_{-\infty }^{{\vartheta }_1(z_1)-{\vartheta }_2(z_2)}{f}_{Y_1,V_1-V_2,V_1-V_3}(y_1,v_1-v_2,{\vartheta }_1(z_1)-{\vartheta }_3(z_3))d(v_1-v_2)d{y}_1}}\hfill \end{array}\right].\hfill \end{array}$$ (4)

By similar reasoning, we can recover the following combination of parameters from the data on Y₂D₂:

$$\begin{array}{c}\frac{\partial E(Y_2{D}_2|Z=z)}{\partial Z_1}\hfill \\ =\frac{\partial }{\partial Z_1}{\displaystyle \int y_2{\displaystyle {\int }_{-\infty }^{{\vartheta }_2(Z_2)-{\vartheta }_1(Z_1)}{\displaystyle {\int }_{-\infty }^{{\vartheta }_2(Z_2)-{\vartheta }_3(Z_3)}{f}_{Y_2,V_2-V_1,V_2-V_3}(y_2,v_2-v_1,v_2-v_3)d(v_2-v_3)d(v_2-v_1)d{y}_2{|}_{Z=z}}}}\hfill \\ =\frac{-\partial {\vartheta }_1(Z_1)}{\partial Z_1}{|}_{Z_1=z_1}\left[{\displaystyle \int y_2{\displaystyle {\int }_{-\infty }^{{\vartheta }_2(z_2)-{\vartheta }_3(z_3)}{f}_{Y_2,V_2-V_1,V_2-V_3}(y_2,{\vartheta }_2(z_2)-{\vartheta }_3(z_1),v_2-v_3)d(v_2-v_3)d{y}_2}}\right].\hfill \end{array}$$ (5)

From data on Y₃D₃, we obtain the following combination of parameters:

$$\begin{array}{c}\frac{\partial E(Y_3{D}_3|Z_1=z)}{\partial Z_1}\hfill \\ =\frac{-\partial {\vartheta }_1(Z_1)}{\partial Z_1}{|}_{Z_1=z_1}{\displaystyle \int y_3{\displaystyle {\int }_{-\infty }^{{\vartheta }_3(z_3)-{\vartheta }_2(z_2)}{f}_{Y_3,V_3-V_1,V_3-V_2}(y_3,{\vartheta }_3(z_3)-{\vartheta }_1(z_1),v_3-v_2)d(v_3-v_2)d{y}_3}.}\hfill \end{array}$$ (6)

Agents induced into 1 come from 2 and 3. There are two margins:

$$(R_1=R_2)\text{ and }(R_1\ge R_3)(\text{margin}\text{of}\text{indifference}\text{between}1\text{and}2),$$

and

$$(R_1=R_3)\text{ and }(R_1\ge R_2)(\text{margin}\text{of}\text{indifference}\text{between}1\text{and}3).$$

Unaided, IV does not enable analysts to identify the returns at each of the different margins. Instead, it identifies a weighted average of returns. It does not identify the density of persons at the various margins, i.e., the proportion of people induced into (or out of) 1 from each possible alternative state by a change in the instrument.

Collecting terms and rewriting in more easily interpretable components, which generalize the MTE developed for a two choice model to a multiple choice unordered model:¹⁹

$$\begin{array}{c}\frac{\left(\frac{\partial E(Y|Z)}{\partial Z_1}\right)}{\left(\frac{\partial {\vartheta }_1}{\partial Z_1}\right)}{|}_{Z=z}=\hfill \\ \left[\begin{array}{c}\stackrel{\begin{array}{c}\text{Generalization}\text{of}\text{MTE}\text{for}\text{persons}\text{indifferent}\\ \text{between}1\text{and}2,\text{where}\text{choice}3\text{is}\text{dominated}\end{array}}{\overbrace{[E(Y_1-Y_2|R_1(z_1)=R_2(z_2),R_1(z_1)\ge R_3(z_3))]}}\text{Pr}(R_1(z_1)=R_2(z_2),R_1(z_1)\ge R_3(z_3))\\ +\underset{\begin{array}{c}\text{Generalization}\text{of}\text{MTE}\text{for}\text{persons}\text{indifferent}\\ \text{between}1\text{and}3,\text{where}\text{choice}2\text{is}\text{dominated}\end{array}}{\underbrace{[E(Y_1-Y_3|R_1(z_1)=R_3(z_3),R_1(z_1)\ge R_2(z_2))]}}\text{Pr}(R_1(z_1)=R_3(z_3),R_1(z_1)\ge R_2(z_2))\end{array}\right]\hfill \end{array}.$$

This is a weighted return to alternative 1 for persons coming from two separate margins: alternative 1 versus alternative 2, and alternative 1 versus alternative 3, i.e., the return to people induced into 1 from their next best choice. The weights are the proportion of people induced into 1 from each margin. This combination of parameters can be identified from IV. The components of the sum cannot be identified by IV without further assumptions. Note that it is possible that a group at one margin gains while a group at another margin loses. IV only estimates a net effect, which might be zero.

Notice that from representation (2.1) and the assumption that the Z_j (j∈𝒥) are distinct, pairwise monotonicity, an extension of the monotonicity assumption invoked by Imbens and Angrist (1994) for the binary choice case, is satisfied.²⁰ In the context of a model with multiple choices, pairwise monotonicity means the same pattern of flow between any two states is experienced by everyone. Thus, as Z_j increases, there is a flow from i to j but not from j to i (or vice versa). From (1), changing Z₁ induces all persons to move in the same direction (i.e. from 1 to 2 or 2 to 1 but not both, and from 1 to 3 or 3 to 1 but not both). Pairwise monotonicity does not rule out the possibility that a change in an instrument causes people to move in the direction from j to i but to move away from the direction from k to i for j ≠ k, and j, k ≠ i.

By the chain rule, the derivative of Pr (D₁ = 1 | Z) is:

$$\frac{\partial \text{Pr}(D_1=1|Z=z)}{\partial Z_1}=\frac{\partial {\vartheta }_1}{\partial Z_1}{|}_{Z_1=z_1}\left[\begin{array}{c}\text{Pr}(R_1(z_1)=R_2(z_2),R_1(z_1)\ge R_3(z_3))\hfill \\ +\text{Pr}(R_1(z_1)=R_3(z_3),R_1(z_1)\ge R_2(z_2))\hfill \end{array}\right].$$

We can define LIV in terms of the preceding ingredients as

$$\text{LIV}(z)=\frac{\left(\frac{\partial E(Y|Z)}{\partial Z_1}\right)}{\left(\frac{\partial \text{Pr}(D_1=1|Z)}{\partial Z_1}\right)}{|}_{Z=z}=\left[\begin{array}{c}E(Y_1-Y_2|R_1(z_1)=R_2(z_2),R_1(z_1)\ge R_3(z_3)){\omega }_{12}\hfill \\ +E(Y_1-Y_3|R_1(z_1)=R_3(z_3),R_1(z_1)\ge R_2(z_2)){\omega }_{13}\hfill \end{array}\right].$$ (7)

The combination of terms can be identified by LIV from the data on (Y, D, Z).

The IV weights are:

$${\omega }_{12}=\frac{\text{Pr}(R_1(z_1)=R_2(z_2),R_1(z_1)\ge R_3(z_3))}{\left[\begin{array}{c}\text{Pr}(R_1(z_1)=R_2(z_2),R_1(z_1)\ge R_3(z_3))\hfill \\ +\text{Pr}(R_1(z_1)=R_3(z_3),R_1(z_1)\ge R_2(z_2))\hfill \end{array}\right]}$$ (8) (9)

and

$${\omega }_{13}=\frac{\text{Pr}(R_1(z_1)=R_3(z_3),R_1(z_1)\ge R_2(z_2))}{\left[\begin{array}{c}\text{Pr}(R_1(z_1)=R_2(z_2),R_1(z_1)\ge R_3(z_3))\hfill \\ +\text{Pr}(R_1(z_1)=R_3(z_3),R_1(z_1)\ge R_2(z_2))\hfill \end{array}\right]}.$$ (10)

The weights can be identified from a structural discrete choice analysis.²¹ They cannot be identified by an unaided instrumental variable analysis. Thus it is not possible to identify the component parts of (5) by LIV alone, i.e., one cannot separately identify the generalized MTEs:

$$E(Y_1-Y_2|R_1(z_1)=R_2(z_2),R_1(z_1)\ge R_3(z_3))$$

and

$$E(Y_1-Y_3|R_1(z_1)=R_3(z_3),R_1(z_1)\ge R_2(z_2)),$$

unless one invokes “identification at infinity” arguments or alternative arguments using local variation in regressors developed by Fox and Gandhi (2008).²² Carneiro, Heckman, and Vytlacil (2009) and Heckman and Vytlacil (2007b) develop classes of economically interpretable parameters that do not require “identification at infinity” and that can be identified using estimated marginal treatment effects.

Using a structural model, one can estimate the components of (7) and determine the flow into (or out of) state 1 from all sources. We illustrate this point in Section 5. First we consider what standard IV estimates.

4 What does standard IV estimate?

To see what standard IV estimates, consider the following linear-in-schooling model of earnings that receives much attention in the literature in labor economics.²³ Let Y denote log earnings and write S as years of schooling. The model writes

$$Y=\alpha +\beta S+U$$ (11)

where

$$S={\displaystyle \sum _{j=1}^{3}j{D}_j,}$$ (12)

and Y is defined as in Section 2. It is interpreted in this section as an approximation to the general model presented in Section 2. S is assumed to be correlated with U, and β is a random variable that may be statistically dependent on S. The model of Section 2 does not, in general, imply (11). Indeed, there is much empirical evidence against model (11).²⁴ An analysis of what IV estimates when linearity in S is imposed as an approximation, even though it may be inappropriate, is an interesting exercise because linearity is so often invoked.

Suppose Z₁ is a valid instrument. We now interpret what

$${\mathrm{\Delta }}_{Z_1}^{\text{IV}}=\frac{\text{Cov}(Z_1,Y)}{\text{Cov}(Z_1,S)}$$ (13)

estimates. We do this by decomposing ${\mathrm{\Delta }}_{Z_1}^{\text{IV}}$ into components analogous to the decomposition produced by Heckman, Urzua, and Vytlacil (2006, 2008) and Heckman and Vytlacil (2007b). The Appendix presents the derivation of the following decomposition of IV into our pairwise generalization of MTE for the unordered case:

$$\frac{\begin{array}{c}{\mathrm{\Delta }}_{Z_1}^{\text{IV}}=\frac{\text{Cov}(Z_1,Y)}{\text{Cov}(Z_1,S)}=\hfill \\ \left(\begin{array}{c}{\displaystyle {\int }_{-\infty }^{\infty }}{\displaystyle {\int }_{-\infty }^{\infty }}\stackrel{\text{Generalized}\text{MTE}(2\to 1)\text{not}\text{identified}\text{from}\text{LIV}}{\overbrace{E(Y_1-Y_2|V_2-V_1=v_2-v_1,{\vartheta }_2(z_2)-{\vartheta }_3(z_3)\ge V_2-V_3)}}\hfill \\ \times \underset{\begin{array}{c}\hfill \text{weight}\text{identified}\text{from}\text{discrete}\hfill \\ \hfill \text{choice}\text{analysis}\hfill \end{array}}{\underbrace{{\eta }_{{\vartheta }_2(Z_2)-{\vartheta }_3(Z_3),V_2-V_1}({\vartheta }_2(z_2)-{\vartheta }_3(z_3),v_2-v_1)}}d(v_2-v_1)d({\vartheta }_2(z_2)-{\vartheta }_3(z_3))\hfill \end{array}\right)\hfill \\ +\left(\begin{array}{c}{\displaystyle {\int }_{-\infty }^{\infty }}{\displaystyle {\int }_{-\infty }^{\infty }}\stackrel{\text{Generalized}\text{MTE}(3\to 1)\text{not}\text{identified}\text{from}\text{LIV}}{\overbrace{E(Y_1-Y_3|V_3-V_1=v_3-v_1,{\vartheta }_3(z_3)-{\vartheta }_2(z_2)\ge V_3-V_2)}}\hfill \\ \times \underset{\begin{array}{c}\hfill \text{weight}\text{identified}\text{from}\text{discrete}\hfill \\ \hfill \text{choice}\text{analysis}\hfill \end{array}}{\underbrace{{\eta }_{{\vartheta }_3(Z_3)-{\vartheta }_2(Z_2),V_3-V_1}({\vartheta }_3(z_3)-{\vartheta }_2(z_2),v_3-v_1)}}d(v_3-v_1)d({\vartheta }_3(z_3)-{\vartheta }_2(z_2))\hfill \end{array}\right)\hfill \end{array}}{\begin{array}{c}{\displaystyle {\int }_{-\infty }^{\infty }}{\displaystyle {\int }_{-\infty }^{\infty }}\underset{\begin{array}{c}\hfill \text{weight}\text{identified}\text{from}\text{discrete}\hfill \\ \hfill \text{choice}\text{analysis}\hfill \end{array}}{\underbrace{[-{\eta }_{{\vartheta }_2(Z_2)-{\vartheta }_3(Z_3),V_2-V_1}({\vartheta }_2(z_2)-{\vartheta }_3(z_3),v_2-v_1)]}}d({\vartheta }_2(z_2)-{\vartheta }_3(z_3))d(v_2-v_1)\hfill \\ +2{\displaystyle {\int }_{-\infty }^{\infty }}{\displaystyle {\int }_{-\infty }^{\infty }}\underset{\begin{array}{c}\hfill \text{weight}\text{identified}\text{from}\text{discrete}\hfill \\ \hfill \text{choice}\text{analysis}\hfill \end{array}}{\underbrace{[-{\eta }_{{\vartheta }_3(Z_3)-{\vartheta }_2(Z_2),V_3-V_1}({\vartheta }_3(z_3)-{\vartheta }_2(z_2),v_3-v_1)]}}d(v_3-v_1)d({\vartheta }_3(z_3)-{\vartheta }_2(z_2)).\hfill \end{array}}$$ (14)

IV identifies a weighted average of gains to state 1 compared to the next best alternative which may be 2 or 3. The two terms of the decomposition are defined as generalized MTEs and are weighted averages of the gain of moving from state 2 to state 1 for persons on the margin of indifference between 1 and 2 and for whom 2 is a better choice than 3 (the first term) and the gain of moving from 3 to 1 for persons on the margin of indifference between 1 and 3 and for whom 3 is a better choice than 2 (the second term).²⁵

In the Appendix, we derive the weights on the generalized MTEs and show that they do not sum to 1 even when normalized by the denominator. The mathematical reason for this result is simple. The weights in the numerator do not sum to the weights in the denominator. The second term in the denominator receives twice as much weight as the corresponding term in the numerator. This is a consequence of the definition of S (12), which plays no role in the numerator term. Thus, IV applied to the general model produces an arbitrarily weighted sum of generalized MTEs with weights that do not sum to 1, and which, in general, places more weight on the first generalized MTE term than on the second term, compared to the weights placed on the corresponding terms in the denominator.²⁶ Using IV alone, we cannot decompose (14) into its component parts, even though the weights can be identified from discrete choice analysis.²⁷,²⁸

4.1 The Mincer Model

The Mincer (1974) model is a specialization of the general model discussed in Section 2 of this paper that justifies the precise functional form of equation (11).²⁹ For this case, the weights in (14) in the numerator and denominator are the same. The Mincer model is formulated in terms of log earnings for Y₁, Y₂, and Y₃:

$$\begin{array}{c}{Y}_2=\text{ln}(1+g)+Y_1,\hfill \\ Y_3=\text{ln}(1+g)+Y_2=2\text{ln}(1+g)+Y_1,\hfill \end{array}$$

where g is a growth factor for income that varies in the population. Earnings at each schooling level depend on two parameters: (g, Y₁). In this case, letting α = ln(1 + g),

$$\begin{array}{cc}{\mathrm{\Delta }}_{Z_1}^{\text{IV}}\hfill & =\frac{\text{Cov}(Z_1,Y)}{\text{Cov}(Z_1,S)}\hfill \\ \hfill & =[\left(\begin{array}{c}{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }E(\alpha |V_2-V_1=v_2-v_1,{\vartheta }_2(z_2)-{\vartheta }_3(z_3)\ge V_2-V_3)}}\hfill \\ \begin{array}{c}\times {\eta }_{{\vartheta }_2(Z_2)-{\vartheta }_3(Z_3),V_2-V_1}({\vartheta }_2(z_2)-{\vartheta }_3(z_3),v_2-v_1)d(v_2-v_1)d({\vartheta }_2(z_2)-{\vartheta }_3(z_3))\hfill \end{array}\hfill \end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{c}{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }E(\alpha |V_3-V_1=v_3-v_1,}{\vartheta }_3(z_3)-{\vartheta }_2(z_2)\ge V_3-V_2)}\hfill \\ \times 2{\eta }_{{\vartheta }_3(Z_3)-{\vartheta }_2(Z_2),V_3-V_1}({\vartheta }_3(z_3)-{\vartheta }_2(z_2),v_3-v_1)d(v_3-v_1)d({\vartheta }_3(z_3)-{\vartheta }_2(z_2))\hfill \end{array}\right)]\hfill \\ \hfill & \times {\left[\begin{array}{c}{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }{\eta }_{{\vartheta }_2(Z_2)-{\vartheta }_3(z_3),V_2-V_1}({\vartheta }_2(z_2)-{\vartheta }_3(z_3),v_2-v_1)d(v_2-v_1)d({\vartheta }_2(z_2)-{\vartheta }_3(z_3))}}\hfill \\ +2{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }{\eta }_{{\vartheta }_3(Z_3)-{\vartheta }_2(Z_2),V_3-V_1}({\vartheta }_3(z_3)-{\vartheta }_2(z_2),v_3-v_1)d(v_3-v_1)d({\vartheta }_3(z_3)-{\vartheta }_2(z_2))}}\hfill \end{array}\right]}^{-1}.\hfill \end{array}$$ (16)

In this case, the weights now sum to 1. The weights for the numerator term now are the same as the weights for the denominator term. But again, unaided IV does not identify the component parts of the term bundled in IV — the mean gains at each margin.³⁰,³¹

5 An Example

It is instructive to summarize our analysis with an example. Consider a 3 choice model with associated outcomes. This corresponds to the GED, high school dropout and high school graduate example that we have used throughout the paper. Under conditions presented in Heckman and Vytlacil (2007a, Appendix B), the structural model is nonparametrically identified. A key assumption in their proof is the “identification at infinity” assumption previously discussed.³² This assumes the ability to vary (Z₁, Z₂, Z₃) freely and the existence of limit sets such that fixing any two of (Z₁, Z₂, Z₃), one makes the R_j associated with Z_j arbitrarily small.³³

Heckman and Vytlacil (2007b) show that if one augments the IV assumptions with the same identification at infinity assumptions used in structural models, one can use IV in the limit to identify the components of (3.5). In the limit sets, one can identify

$$E(Y_1-Y_2|R_1(z_1)=R_2(z_2))$$ (17)

and

$$E(Y_1-Y_3|R_1(z_1)=R_3(z_3))$$ (18)

by setting Z₃ and Z₂ respectively to limit set values. Essentially one can use the limit sets to make a three choice model into a two choice model, and the standard results for the two choice model apply.³⁴ Under these assumptions, and additional mild regularity assumptions, using structural methods, one can identify the distributions of (Y₁, Y₂) and (Y₁, Y₃) so that one can identify distributions of treatment effects, Y₂ − Y₁ and Y₃ − Y₁, in addition to the mean parameters identified by IV.³⁵ One can also identify the proportion of people induced into 1 from each alternative state using variation in the instrument.

Consider the model with the parameters presented in Table 1. This is a discrete choice model with associated outcome variables. The Z_j, j = 1, …, 3, are assumed to be scalar and mutually independent. They are normally distributed so they satisfy large support (“identification at infinity”) conditions. Table 2 shows how a change in Z₁, which increases it by .75 standard deviations, shifts people across categories. This corresponds to making GED attainment easier.³⁶ The estimates reported in Table 2 can be obtained from a structural discrete choice model. The percentage initially in 1 (GED) increases from 33.17% to 38.8%. The percentage in 2 (dropout) decreases from 29.11% to 25.91%. The percentage in 3 (graduating high school) declines from 37.72% to 35.29%.

Table 1.

Potential Outcomes, Choice Model and Parameterizations

$\begin{array}{l}\begin{array}{ccc}\text{Outcomes}& & \text{Choice Model}\\ Y_j={\alpha }_j+U_j\text{ with }j\in 𝒥=\{1,2,3\}& & D_j=\{\begin{array}{cc}1& \text{if}{R}_j\ge R_k\forall j\in 𝒥\\ 0& \text{otherwise}\end{array}\\ Y={\displaystyle {\sum }_{j\in 𝒥}{Y}_j{D}_j}& & R_j={\gamma }_j{Z}_j-V_j\text{ with }j\in 𝒥\\ & \begin{array}{c}\text{Parameterization}\end{array}& \\ & (U_1,U_2,U_3,V_1,V_2,V_3)\sim N(\mathbf{0},{\mathbf{\Sigma }}_{\mathit{\text{UV}}}),(Z_1,Z_2,Z_3)\sim N({\mu }_Z,{\mathbf{\Sigma }}_Z)& \end{array}\\ {\mathbf{\Sigma }}_{\mathit{\text{UV}}}=\left[\begin{array}{cccccc}0.64& 0.16& 0.16& 0.024& -0.32& 0.016\\ 0.16& 1& 0.20& 0.020& -0.30& 0.010\\ 0.16& 0.20& 1& 0.020& -0.40& 0.040\\ 0.024& 0.020& 0.020& 1& 0.6& 0100\\ -0.32& -0.30& -0.40& 0.6& 1& 0.2\\ 0.016& 0.01& 0.040& 0100& 0.2& 1\end{array}\right],{\mu }_Z=(1.0,0.5,1.5)\text{and}{\mathbf{\Sigma }}_Z=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\\ \begin{array}{ccc}& \begin{array}{c}\left[\begin{array}{ccc}{\alpha }_1& {\alpha }_2& {\alpha }_3\end{array}\right]=[\begin{array}{ccc}0.3& 0.1& 0.7\end{array}],\left[\begin{array}{ccc}{\gamma }_1& {\gamma }_2& {\gamma }_3\end{array}\right]=[\begin{array}{ccc}0.2& 0.3& 0.1]\end{array}\end{array}& \end{array}\end{array}$

Table 2.

Transition Matrix Obtained from the Change in the Instrument Z₁

The Instrument Increases by 0.75 Standard Deviation

		New Value of Instrument (Z͂ = Z1 + 0.75)
		D1 = 1	D2 = 1	D3 = 1	Total
Original Value of Instrument (Z1)	D1 = 1	33.17%	0%	0%	33.17%
	D2 = 1	3.20%	25.91%	0%	29.11%
	D3 = 1	2.43%	0%	35.29%	37.72%
	Total	38.80%	25.91%	35.29%	100%

The IV estimate is −.032. (See the base of Table 3) This is the only number produced by an IV analysis using Z₁ as an instrument that changes within the specified range. The structural analysis in Table 3 shows that the net effect produced by the change in Z₁ is composed of 2 terms. It arises from a gain of .199 for the switchers 2 → 1 (dropout to GED) and a loss of .336 (3 → 1) (graduate to GED).

Table 3.

Marginal Gains Identified from the Change in the Instrument Z₁

The Instrument Increases by 0.75 Standard Deviation

	Gains to Switchers	Fraction of Population Switching
From 2 to 1	0.199	3.20%
From 3 to 1	−0.336	2.43%
Overall (IV estimate)	−0.032	5.63%

IV Estimate:

$$E[Y|{\tilde{Z}}_1]-E[Y|Z_1]=\frac{3.20}{3.20+2.43}\times 0.199-\frac{2.43}{3.20+2.43}\times 0.336=-0.032$$

Figure 1 shows what can be identified from the structural model. It plots the distributions of gains for persons going from 2 to 1 and from 3 to 1 as well as the overall distribution of gains to the switchers. Persons switching from 3 to 1 are harmed in gross terms by the policy that changes Z₁, while those who switch from 2 to 1 gain in gross terms. In utility terms, (R_j), people are better off.³⁷ In terms of gross gains, about 56.8% of the people who switch from 2 to 1 are better off while 39.3% of the people who switch from 3 to 1 are better off. Overall, 49.2% are better off in gross terms even though the IV estimate is slightly negative. If one seeks to understand the distributional effects of the policy associated with a change Z₁, the structural analysis is clearly much more revealing. The IV estimate, which is a mean gross gain aggregating over origin states, does not capture the rich information about choices afforded by a structural analysis. However, it does identify the average gain to the program compared to the next-best alternatives. If that is the object of interest, linear IV is the right tool to use.

Figure 1.

Distribution of Gains in Outcomes Induced by the Change in the Instrument Z₁

6 Summary and Discussion

The choice between using IV or a more structural approach for a particular problem should be made on the basis of Marschak’s Maxim: use minimal assumptions to answer well-posed economic questions. Most IV studies do not clearly formulate the economic question being answered by the IV analysis. The probability limit of the IV estimator is defined to be the object of interest. In the binary outcome case, even if Z is a valid instrument, if Z is a vector, and analysts use only one component of the vector as an instrument, and do not condition on the other components of Z, the weights on the MTE can be negative over certain ranges. The practice of not conditioning on the other instruments is common in the literature.³⁸ In this case, IV can estimate the wrong sign for the true causal effect.³⁹ Recent analyses show how to improve on this practice and to design functions of standard instrumental variables that answer classes of well-posed economic questions.⁴⁰

We have discussed a model with three or more choices where there is no particular order among the choices. Such examples arise routinely in applied economics. In this case, under conditions specified in this paper, IV estimates a weighted average of the mean gross gain to persons induced into a choice state by a change in the instrument (policy) compared to their next best alternative.⁴¹ It averages the returns to a destination state over all origin states. It does not produce the distribution of gains overall or by each origin state. Again, as in the binary choice case, for vector Z, using one component of Z as an instrument, and not conditioning on the other components can produce negative weights so that the sign of an IV can be opposite to that of the true causal effect which can be identified by a structural analysis.

Structural methods provide a more complete description of the effect of the instrument or the policy associated with the instrument. They identify mean returns as well as distributions of returns for agents coming to a destination state from each margin. They also identify the proportion of people induced into a state from each origin state.

Structural methods come at a cost. Unless distributional assumptions for unobservables are invoked, structural methods often require some form of an “identification at infinity” assumption.⁴² However, in the general case in which responses to treatment are heterogeneous, IV requires the same assumption if one seeks to identify average treatment effects.⁴³ An identification at infinity assumption can be checked in any sample so it does not require imposing a priori beliefs onto the data. Heckman, Stixrud, and Urzua (2006) present an example of how to test an identification at infinity assumption. See also the discussion in Abbring and Heckman (2007).⁴⁴ Recent advances by Fox and Gandhi (2008) show that in structural models generated by finite mixtures, identification at infinity is not required to establish model identification, although continuity properties on functions and some of the regressors that constitute their arguments are required.⁴⁵ Carneiro, Heckman, and Vytlacil (2009) develop a rich class of economically interesting parameters that do not require identification at infinity and can be identified from local variation in instruments. An “identification at infinity” assumption is not intrinsic to the structural approach.

Many proponents of IV point to the strong distributional and functional form assumptions required to implement structural methods. They ignore recent progress in econometrics that identifies and empirically implements robust semiparametric and nonparametric approaches to structural analysis.⁴⁶ Recent developments respond to arguments against the use of explicit econometric models made by a generation of applied economists that emerged in the 1980s. Those arguments are more properly directed against 1980s versions of structural models that were based on linearity and normality. Structural econometricians in the 21^st century have listened to the critics and have perfected their tools to response to the criticism.

The appeal to standard IV as a preferred estimator is sometimes made on the basis of “simplicity and robustness”. Standard IV is certainly simple to compute although problems with weak instruments can make it empirically unstable.⁴⁷ Since, in the general case, different instruments identify different parameters, IV is not robust to the choice of instrument.⁴⁸ Since the sign of an IV can be different from the true causal effect, IV may even produce a misleading guide to policy or inference, so it is not robust.

The meaning of “simplicity” is highly subjective. How simple is the economic interpretation of IV? Certainly decomposition (14) is not simple. The fact that simple IV can estimate wrong signs for true causal effects should give pause to those who claim that it is “robust”. The weak instrument literature cautions us against uncritical claims about the sturdiness of IV estimators.

The ability of different statistical estimators to answer questions of economic interest, or to show why they cannot be answered, should drive the choice of empirical techniques for analyzing data. Consider a worst case for structural estimation. Suppose that application of recently developed procedures for testing for structural identification reveal that a structural model is not identified or is only partially identified. Does this conclusion suggest that IV is a better choice for an estimator? That disguising identification problems by a statistical procedure is preferable to an honest discussion of the limits of the data? For underidentified structural models, it is possible to conduct sensitivity analyses guided by economic theory to explore the consequences of ignorance about features of the model. With IV, unaided by structural analysis, this type of exercise is not possible. Problems of identification and interpretation are swept under the rug and replaced by “an effect” identified by IV that is often very difficult to interpret as an answer to an interesting economic question.

A Derivation of the Standard IV Estimator

We first study the numerator of ${\mathrm{\Delta }}_{Z_1}^{\text{IV}}$ in the text. Recall that we keep the conditioning on X implicit. Using Z͂₁ = Z₁ − Z͂₁,

$$\text{Cov}(Y,Z_1)=E\left({\tilde{Z}}_1(Y_1{D}_1+Y_2{D}_2+Y_3{D}_3)\right).$$

Using D₁ = 1 − D₂ − D₃, we obtain

$$\begin{array}{cc}\text{Cov}(Y,Z_1)\hfill & =E\left({\tilde{Z}}_1(Y_1+(Y_2-Y_1)D_2+(Y_3-Y_1)D_3)\right)\hfill \\ \hfill & =E\left({\tilde{Z}}_1{Y}_1\right)+E\left({\tilde{Z}}_1(Y_2-Y_1)D_2\right)+E\left({\tilde{Z}}_1(Y_3-Y_1)D_3\right),\hfill \end{array}$$

where E (Z͂₁Y₁) = 0. It is natural to decompose this expression using choice “1” as the base, because Z₁ only shifts R₁(Z₁). The final two terms can be written as

$$\begin{array}{c}\text{Cov}(Y,Z_1)\hfill \\ =E\left({\tilde{Z}}_1(Y_2-Y_1)\mathbf{1}(R_2(Z_2)\ge R_1(Z_1),R_2(Z_2)\ge R_3(Z_3))\right)+E\left({\tilde{Z}}_1(Y_3-Y_1)\mathbf{1}(R_3(Z_3)\ge R_1(Z_1),R_3(Z_3)\ge R_2(Z_2))\right)\hfill \\ =E\left[{\tilde{Z}}_1(Y_2-Y_1)\mathbf{1}\left(({\vartheta }_2(Z_2)-{\vartheta }_1(Z_1)\ge V_2-V_1),({\vartheta }_2(Z_2)-{\vartheta }_3(Z_3)\ge V_2-V_3)\right)\right]\hfill \\ +E\left[{\tilde{Z}}_1(Y_3-Y_1)\mathbf{1}\left(({\vartheta }_3(Z_3)-{\vartheta }_1(Z_1)\ge V_3-V_1),({\vartheta }_3(Z_3)-{\vartheta }_2(Z_2)\ge V_3-V_2)\right)\right]\hfill \\ ={\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }{\tilde{z}}_1(y_2-y_1)}}}}\hfill \\ \times \left({\displaystyle {\int }_{-\infty }^{{\vartheta }_2(z_2)-{\vartheta }_1(z_1)}{\displaystyle {\int }_{-\infty }^{{\vartheta }_2(z_2)-{\vartheta }_3(z_3)}{f}_{Y_2-Y_1,V_2-V_1,V_2-V_3}(y_2-y_1,v_2-v_1,v_2-v_3)d(v_2-v_3)d(v_2-v_1)d(y_2-v_1)}}\right)\hfill \\ \times f_{{\tilde{Z}}_1,{\vartheta }_2(Z_2)-{\vartheta }_1(Z_1),{\vartheta }_2(Z_2)-{\vartheta }_3(Z_3)}({\tilde{z}}_1,{\vartheta }_2(z_2)-{\vartheta }_1(z_1),{\vartheta }_2(z_2)-{\vartheta }_3(z_3))d({\vartheta }_2(z_2)-{\vartheta }_3(z_3))d({\vartheta }_2(z_2)-{\vartheta }_1(z_1))d{\tilde{z}}_1\hfill \\ +{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }{\tilde{z}}_1(y_3-y_1)}}}}\hfill \\ \begin{array}{c}\times \left({\displaystyle {\int }_{-\infty }^{{\vartheta }_3(z_3)-{\vartheta }_1(z_1)}{\displaystyle {\int }_{-\infty }^{{\vartheta }_3(z_3)-{\vartheta }_2(z_2)}{f}_{Y_3-Y_1,V_3-V_1,V_3-V_2}}(y_3-y_1,v_3-v_1,v_3-v_2)d(v_3-v_2)d(v_3-v_1)d(y_3-y_1)}\right)\hfill \\ \times f_{{\tilde{Z}}_1,{\vartheta }_3(Z_3)-{\vartheta }_1(Z_1),{\vartheta }_3(Z_3)-{\vartheta }_2(Z_2)}({\tilde{z}}_1,{\vartheta }_3(z_3)-{\vartheta }_1(z_1),{\vartheta }_3(z_3)-{\vartheta }_2(z_2))d({\vartheta }_3(z_3)-{\vartheta }_2(z_2))d({\vartheta }_3,(z_3)-{\vartheta }_1(z_1))d{\tilde{z}}_1.\hfill \end{array}\hfill \end{array}$$

By Fubini’s Theorem, we can simplify the expressions and obtain for the first term:

$$\begin{array}{c}{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }E(Y_2-Y_1|V_2-V_1=v_2-v_1,{\vartheta }_2(z_2)-{\vartheta }_3(z_3)\ge V_2-V_3)}}\hfill \\ \times \{{\displaystyle {\int }_{-\infty }^{\infty }{\tilde{z}}_1}[\left({\displaystyle {\int }_{-\infty }^{{\vartheta }_2(z_2)-{\vartheta }_3(z_3)}{h}_{V_2-V_1,V_2-V_3}(v_2-v_1,v_2-v_3)d(v_2-v_3)}\right)\hfill \\ \begin{array}{c}\times \left({\displaystyle {\int }_{v_2-v_1}^{\infty }{f}_{{\tilde{Z}}_1,{\vartheta }_2(Z_2)-{\vartheta }_1(Z_1),{\vartheta }_2(Z_2)-{\vartheta }_3(Z_3)}}({\tilde{z}}_1,{\vartheta }_2(z_2)-{\vartheta }_1(z_1),{\vartheta }_2(z_2)-{\vartheta }_3(z_3))d({\vartheta }_2(z_2)-{\vartheta }_1(z_1))\right)]d{\tilde{z}}_1\}\\ \hfill \times d({\vartheta }_2(z_2)-{\vartheta }_3(z_3))d(v_2-v_1).\hfill \end{array}\hfill \end{array}$$ (19)

h_{V2 − V1,V2 − V3}(·) is the joint density of V₂ − V₁, V₂ − V₃. Define the weighting term in braces in (19) as η_{ϑ2(Z2) − ϑ3(Z3),V2 − V1}(ϑ₂ (z₂) − ϑ₃ (z₃), v₂ − v₁). It is necessary to fix both ϑ₂ (z₂) − ϑ₃ (z₃) and v₂ − v₁ in forming the weight. This weight can be estimated from a structural discrete choice analysis and the joint distribution of (Z, D₁, D₂, D₃). The terms multiplying the weight are marginal treatment effects generalized to the unordered case. (A.1) cannot be decomposed using IV. An alternative representation of the term in braces, η_{ϑ2(Z2) − ϑ3(Z3),V2 − V1}(ϑ₂ (z₂) − ϑ₃ (z₃), v₂ − v₁) is

$$\begin{array}{c}{\eta }_{{\vartheta }_2(Z_2)-{\vartheta }_3(Z_3),V_2-V_1}({\vartheta }_2(z_2)-{\vartheta }_3(z_3),v_2-v_1)=\hfill \\ E(Z_1-E(Z_1)|{\vartheta }_2(Z_2)-{\vartheta }_3(Z_3)={\vartheta }_2(z_2)-{\vartheta }_3(z_3),{\vartheta }_2(Z_2)-{\vartheta }_1(Z_1)\ge v_2-v_1)\hfill \\ \hfill \times \text{Pr}({\vartheta }_2(Z_2)-{\vartheta }_3(Z_3)={\vartheta }_2(z_2)-{\vartheta }_3(z_3),{\vartheta }_2(Z_2)-{\vartheta }_1(Z_1)\ge v_2-v_1).\hfill \end{array}$$

An analysis parallel to the preceding one shows that the second term can be written as

$$\begin{array}{c}{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }E(Y_3-Y_1|V_3-V_1=v_3-v_1,{\vartheta }_3(z_3)-{\vartheta }_2(z_2)\ge V_3-V_2)}}\hfill \\ \times \{{\displaystyle {\int }_{-\infty }^{\infty }{\tilde{z}}_1}[\left({\displaystyle {\int }_{-\infty }^{{\vartheta }_3(z_3)-{\vartheta }_2(z_2)}{h}_{V_3-V_1,V_3-V_2}(v_3-v_1,v_3-v_2)d(v_3-v_2)}\right)\hfill \\ \times \left({\displaystyle {\int }_{v_3-v_1}^{\infty }{f}_{{\tilde{Z}}_1,{\vartheta }_3(Z_3)-{\vartheta }_1(Z_1),{\vartheta }_3(Z_3)-{\vartheta }_2(Z_2)}}({\tilde{z}}_1,{\vartheta }_3(z_3)-{\vartheta }_1(z_1),{\vartheta }_3(z_3)-{\vartheta }_2(z_2))d({\vartheta }_3(z_3)-{\vartheta }_1(z_1))\right)]d{\tilde{z}}_1\}\hfill \\ \hfill \times d({\vartheta }_3(z_3)-{\vartheta }_2(z_2))d(v_3-v_1).\hfill \end{array}$$ (20)

Define the term in braces in (20) as the weight η_{ϑ3(Z3) − ϑ2(Z2),V3 − V1}(ϑ₃ (z₃) − ϑ₂ (z₂), v₃ − v₁).

To obtain the denominator for the IV, recall that $S={\displaystyle {\sum }_{j=1}^{3}j{D}_j}$. Substitute D₁ = 1 − D₂ − D₃,

$$\begin{array}{ll}{\displaystyle \sum _{j=1}^{3}j{D}_j}\hfill & =(1-D_2-D_3)+2D_2+3D_3\hfill \\ \hfill & =1+D_2+2D_3.\hfill \end{array}$$

Then

$$\begin{array}{cc}\text{Cov}(S,{\tilde{Z}}_1)\hfill & =E\left({\tilde{Z}}_1{D}_2\right)+2E\left({\tilde{Z}}_1{D}_3\right)\hfill \\ \hfill & =E\left({\tilde{Z}}_1(\mathbf{1}(R_2\ge R_1,R_2\ge R_3))\right)\hfill \\ \hfill & +2E\left({\tilde{Z}}_1(\mathbf{1}(R_3\ge R_1,R_3\ge R_2))\right).\hfill \end{array}$$ (21)

Using reasoning similar to that invoked for the analysis of the numerator terms, we obtain expressions for the terms corresponding to the two terms of (19) and (20). We obtain for the first term of (21)

$$\begin{array}{c}{\displaystyle {\int }_{-\infty }^{\infty }{\tilde{z}}_1}[{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{{\vartheta }_2(z_2)-{\vartheta }_1(z_1)}{f}_{{\tilde{Z}}_1,{\vartheta }_2(Z_2)-{\vartheta }_1(Z_1),{\vartheta }_2(Z_2)-{\vartheta }_3(Z_3)}}({\tilde{z}}_1,{\vartheta }_2(z_2)-{\vartheta }_1(z_1),{\vartheta }_2(z_2)-{\vartheta }_3(z_3))}}\hfill \\ \times \left({\displaystyle {\int }_{-\infty }^{{\vartheta }_2(z_2)-{\vartheta }_3(z_3)}{h}_{V_2-V_1,V_2-V_3}(v_2-v_1,v_2-v_3)d(v_2-v_3)}\right)d(v_2-v_1)\hfill \\ \times d({\vartheta }_2(z_2)-{\vartheta }_3(z_3))d({\vartheta }_2(z_2)-{\vartheta }_1(z_1))]d{\tilde{z}}_1.\hfill \end{array}$$ (22)

By Fubini’s Theorem, we obtain:

$$\begin{array}{c}{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }{\tilde{z}}_1}[{\displaystyle {\int }_{-\infty }^{\infty }\left({\displaystyle {\int }_{-\infty }^{{\vartheta }_2(z_2)-{\vartheta }_3(z_3)}{h}_{V_2-V_1,V_2-V_3}}(v_2-v_1,v_2-v_3)d(v_2-v_3)\right)}}\hfill \\ \times ({\displaystyle {\int }_{v_2-v_1}^{\infty }{f}_{{\tilde{Z}}_1,{\vartheta }_2(Z_2)-{\vartheta }_1(Z_1),{\vartheta }_2(Z_2)-{\vartheta }_3(Z_3)}}({\tilde{z}}_1,{\vartheta }_2(z_2)-{\vartheta }_1(z_1),{\vartheta }_2(z_2)-{\vartheta }_3(z_3))\hfill \\ \hfill \times d({\vartheta }_2(z_2)-{\vartheta }_1(z_1)))d({\vartheta }_2(z_2)-{\vartheta }_3(z_3))]d(v_2-v_1)d{\tilde{z}}_1\hfill \\ ={\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }{\eta }_{{\vartheta }_2(Z_2)-{\vartheta }_3(Z_3),V_2-V_1}}({\vartheta }_2(z_2)-{\vartheta }_3(z_3),v_2-v_1)d(v_2-v_1)d({\vartheta }_2(z_2)-{\vartheta }_3(z_3)).}\hfill \end{array}$$ (23)

By parallel logic, we obtain for the second term in 21:

$$\begin{array}{c}2{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }{\tilde{z}}_1}[{\displaystyle {\int }_{-\infty }^{\infty }\left({\displaystyle {\int }_{-\infty }^{{\vartheta }_3(z_3)-{\vartheta }_2(z_2)}{h}_{V_3-V_1,V_3-V_2}}(v_3-v_1,v_3-v_2)d(v_3-v_2)\right)}}\hfill \\ \times \left({\displaystyle {\int }_{v_3-v_1}^{\infty }{f}_{{\tilde{Z}}_1,{\vartheta }_3(Z_3)-{\vartheta }_1(Z_1),{\vartheta }_3(Z_3)-{\vartheta }_2(Z_2)}}({\tilde{z}}_1,{\vartheta }_3(z_3)-{\vartheta }_1(z_1),{\vartheta }_3(z_3)-{\vartheta }_2(z_2))d({\vartheta }_3(z_3)-{\vartheta }_1(z_1))\right)\hfill \\ \times d({\vartheta }_3(z_3)-{\vartheta }_2(z_2))]d(v_3-v_1)d{\tilde{z}}_1\hfill \\ =2{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }{\eta }_{{\vartheta }_3(Z_3)-{\vartheta }_2(Z_2),V_3-V_1}({\vartheta }_3(z_3)-{\vartheta }_2(z_2),v_3-v_1)d(v_3-v_1)d({\vartheta }_3(z_3)-{\vartheta }_2(z_2)).}}\hfill \end{array}$$

These terms can be identified from a structural analysis using the joint distribution of (Z, D₁, D₂, D₃). Collecting results, we obtain decomposition (14) in the text if we multiply both the numerator and denominator by −1.