13 Estimation for Instrumental Variables

Learning Objectives

By the end of this chapter, students should be able to:

Construct the Wald estimator as the sample analog of the Wald estimand and interpret its numerator and denominator as the reduced form and first stage. Extend this to the IV regression estimator with covariates as the direct sample analog of the residualized-covariance ratio \(\mathrm{Cov}(\tilde Z,\,Y)/\mathrm{Cov}(\tilde Z,\,T)\), where \(\tilde Z\) is the linear-projection residual of \(Z\) on the intercept and \(X\), and recognize the Wald estimator as a special case.
Distinguish the structural form of a linear SEM from its reduced form; derive the reduced form by solving out endogenous variables; interpret the reduced-form coefficient \(\phi = \beta\pi\) as the product of the structural parameter and the first-stage strength; define the reduced form regression estimator as OLS applied to the reduced form equation; and interpret \(\hat\phi_{\mathrm{RF}}\) as the intent-to-treat (ITT) effect when the instrument is binary and randomly assigned.
Derive the scalar IV regression estimator from the valid sample moment equations, derive 2SLS through projection, and prove their equivalence in the single-instrument case.
Interpret 2SLS as a method-of-moments estimator based on the IV orthogonality condition, connect it to the estimating-equation framework of Chapter 10, and extend to GMM for overidentified models.
State the asymptotic distribution of the GMM estimator, specialize the sandwich variance formula to the exactly identified and efficient GMM cases, and construct cluster-robust standard errors.
Explain the weak-instrument problem, state precisely in what sense 2SLS is “biased toward OLS,” distinguish the conventional homoskedastic first-stage \(F\)-statistic from its robust and effective counterparts, and construct and interpret an Anderson–Rubin confidence set obtained by test inversion.

Learning Objectives — Advanced Enrichment (Section 13.8)

Students who complete the advanced section should additionally be able to describe generalized empirical likelihood as a one-step alternative to efficient GMM, identify empirical likelihood, exponential tilting, and the continuous updating estimator as representative members of the GEL family, and state their first-order equivalence with efficient GMM under standard regularity and strong-identification conditions.

13.1 From Identification to Estimation

Chapter 7 established what IV identifies and under what assumptions. This chapter shows how to estimate that target from finite data using sample analogs of the same orthogonality restrictions.

Chapter 7 showed that when an instrument \(Z\) satisfies relevance, exogeneity, and exclusion, the residualized-covariance ratio \(\mathrm{Cov}(\tilde Z, Y)/\mathrm{Cov}(\tilde Z, T)\) — where \(\tilde Z = Z - L(Z\mid 1, X)\) is the linear-projection residual of \(Z\) on the intercept and \(X\) — is a well-defined function of observable quantities, but it does not identify any specific causal parameter without a fourth structural assumption. This chapter works within the linear constant-effect model \[Y = \alpha + \beta T + \gamma^\top X + \varepsilon, \tag{13.1}\] \[T = a_T + \pi Z + \delta^\top X + \eta, \tag{13.2}\] where \(a_T\) is the first-stage intercept and \(X \in \mathbb{R}^p\) is the unit-level vector of non-constant covariates (the corresponding stacked sample object \(\mathbf{X} \in \mathbb{R}^{n \times p}\) is the covariate matrix).

The fourth structural assumption is the constant-effect restriction: for binary \(T\), \(Y_i(1) - Y_i(0) = \beta\) for all \(i\); for continuous \(T\), \(Y_i(t) - Y_i(t') = \beta(t - t')\) for all \(i\) and all \(t, t'\) in the treatment domain. The difference form is the primitive assumption; when \(Y_i(\cdot)\) is differentiable on a connected domain it is equivalent to \(\partial Y_i(t)/\partial t = \beta\). Under the three core assumptions plus this restriction, the structural coefficient \(\beta\) is identified by the Wald formula: \[\beta = \frac{\mathrm{Cov}(\tilde Z,\, Y)}{\mathrm{Cov}(\tilde Z,\, T)}.\] Under heterogeneous treatment effects, the interpretation depends on the nature of the treatment. In the binary-instrument, binary-treatment setting, the same ratio identifies the LATE for the complier population, provided the additional monotonicity assumption (\(T_i(1) \geq T_i(0)\) for all \(i\)) serves as the fourth structural assumption instead. With a continuous or multi-valued treatment the corresponding IV estimand is generally a weighted average of marginal or incremental causal effects rather than a single unqualified LATE; that interpretive point is developed in Chapter 7. In the simplest binary-instrument, no-covariate case, this reduces to the Wald estimand: \[\beta = \frac{\E[Y \mid Z{=}1] - \E[Y \mid Z{=}0]}{\E[T \mid Z{=}1] - \E[T \mid Z{=}0]}.\]

Identification establishes that \(\beta\) is a well-defined functional of the observable distribution \(P(Y, T, Z, X)\). This chapter addresses the estimation problem: how to recover \(\beta\) from a finite sample. The answer is not a single formula but a family of estimators — the Wald estimator, two-stage least squares (2SLS), and the generalized method of moments (GMM) — each of which is a sample analog of the same underlying orthogonality restriction, viewed from a different angle. This chapter develops estimators for the IV estimands identified in Chapter 7; it does not weaken or replace the substantive assumptions needed for identification.

This chapter assumes that a valid IV design has already been justified. Our goal here is not to discover instruments, but to estimate an already-identified IV estimand and quantify its uncertainty.

The chapter has a core sequence and one optional extension.

Core material (Section 13.2–Section 13.7). These sections develop the Wald estimator, IV regression, two-stage least squares, moment-condition estimation, GMM, and robust variance estimation. Section 13.2 introduces the Wald estimator and derives the scalar IV regression estimator with covariates from the valid sample IV moment equations. Section 13.3 extends this to two-stage least squares for multi-valued and multiple instruments. Section 13.4 proves the equivalence of the IV regression estimator and 2SLS. Section 13.5 interprets both as method-of-moments estimators and extends to GMM for overidentified models. Section 13.6 develops the asymptotic distribution of the GMM estimator, specializes the sandwich variance formula, and covers cluster-robust inference. Section 13.7 studies weak instruments, first-stage diagnostics, and weak-instrument-robust inference.

Advanced enrichment (Section 13.8). This section introduces generalized empirical likelihood as an alternative implementation of the same IV moment restrictions. The detailed convex-duality derivation, the minimum-discrepancy representation, the computational algorithm, and the extended table of special cases are collected in the Chapter 13 Supplement. The section presupposes the core material but can be omitted without loss of continuity.

The figure below displays this architecture. Each arrow is a step taken in this chapter; the identification assumptions at the top are the subject of Chapter 7 and are maintained, not re-examined, below that point.

The estimation pipeline of this chapter. Identification (green, Chapter 7) supplies the moment condition; the middle of the chain supplies increasingly general implementations of that same restriction; the last step (red) asks whether the resulting large-sample inference can be trusted.

13.2 The Wald Estimator and the IV Regression Estimator

The Wald estimator is the canonical just-identified special case. It is the cleanest entry point for understanding IV estimation, but most empirical applications with covariates or multiple instruments are implemented through regression or GMM formulations. We begin with the Wald estimator to establish the sample-analog logic before generalizing.

13.2.1 The Wald Estimator

We begin with the simplest setting: a binary instrument \(Z \in \{0,1\}\), a scalar treatment \(T\), a scalar outcome \(Y\), and no additional covariates \(X\). Chapter 7 established that under relevance, exogeneity, exclusion, and the constant-effect assumption (\(\tau_i = \beta\) for all \(i\)), the parameter \(\beta\) is identified by the Wald estimand \[\beta = \frac{\E[Y \mid Z{=}1] - \E[Y \mid Z{=}0]}{\E[T \mid Z{=}1] - \E[T \mid Z{=}0]}.\] Each expectation in this ratio is a population mean conditional on instrument status. Replacing each with its sample analog yields the Wald estimator.

Definition 13.1 (Wald Estimator) Let \(n_z = \sum_{i=1}^n \mathbf{1}(Z_i = z)\) for \(z \in \{0,1\}\), and define the within-group sample means \[\bar{Y}_z = \frac{1}{n_z}\sum_{i:\,Z_i=z} Y_i, \qquad \bar{T}_z = \frac{1}{n_z}\sum_{i:\,Z_i=z} T_i.\] The Wald estimator is \[\hat\beta_{\mathrm{Wald}} = \frac{\bar{Y}_1 - \bar{Y}_0}{\bar{T}_1 - \bar{T}_0}. \tag{13.3}\]

Under heterogeneous treatment effects, the Wald estimator remains consistent for the IV estimand defined by the same ratio formula, but the causal interpretation of that estimand depends on the structure of \(T\): for binary \(T\) it is the LATE, and for continuous \(T\) it is generally a weighted average of marginal causal effects rather than a single treatment-control contrast.

The Wald estimator has a transparent structural interpretation. The numerator \(\bar{Y}_1 - \bar{Y}_0\) estimates the reduced form: the total effect of the instrument on the outcome. The denominator \(\bar{T}_1 - \bar{T}_0\) estimates the first stage: the effect of the instrument on treatment uptake. Their ratio recovers the effect of treatment on the outcome by attributing all of the instrument’s effect on \(Y\) to the path \(Z \to T \to Y\). Three assumptions do distinct work here. Exogeneity rules out any noncausal \(Z\)–\(Y\) association; exclusion rules out a direct path from \(Z\) to \(Y\), so that the reduced-form effect operates through \(T\); and relevance guarantees a nonzero denominator. The fourth structural restriction then determines what the ratio means — a homogeneous effect, a LATE, or a weighted average causal response.

The three regression objects serve distinct purposes: the first stage answers how strongly the instrument moves treatment; the reduced form answers how the instrument changes the outcome; and the Wald (or 2SLS) ratio converts that intent-to-treat effect into a per-unit-of-treatment causal effect under the IV assumptions. All three should be reported in applied work — not because they are redundant, but because each is informative about a different aspect of the design.

Remark: Consistency by the Continuous Mapping Theorem

The within-group means \(\bar{Y}_z\) and \(\bar{T}_z\) are consistent estimators of \(\E[Y \mid Z{=}z]\) and \(\E[T \mid Z{=}z]\) by the law of large numbers. Consistency of \(\hat\beta_{\mathrm{Wald}}\) for \(\beta\) then follows from the continuous mapping theorem, provided the denominator \(\E[T \mid Z{=}1] - \E[T \mid Z{=}0] \neq 0\) (the relevance condition).

Remark: What the Wald Estimator Targets under Heterogeneous Effects

When treatment effects are heterogeneous, the causal estimand depends on the structure of \(T\).

Binary treatment. For binary \(T\), Chapter 7 showed that under the additional monotonicity assumption (\(T_i(1) \geq T_i(0)\) for all \(i\), ruling out defiers) — the fourth structural assumption in Framework 2 — the Wald estimand identifies the Local Average Treatment Effect (LATE) for the subpopulation of compliers: units whose treatment status is changed by the instrument. The Wald estimator therefore consistently estimates the LATE, not the population average treatment effect.

Continuous treatment. For continuous \(T\) with binary \(Z\), no single per-unit-of-treatment causal contrast exists for the estimator to target. Under monotonicity in the treatment response function, the IV estimand identifies a weighted average of marginal causal effects \(\partial Y_i(t)/\partial t\) across the treatment distribution, with weights determined by the instrument-induced shift in the distribution of \(T\) (Angrist, Graddy, and Imbens, 2000). The numerator-denominator algebra of the Wald estimator carries through, but “the LATE” is no longer the right name for what it estimates.

The estimator’s consistency for its IV estimand is statistical and holds in both cases; the interpretation of that estimand as a causal parameter depends on the structure of \(T\) and the substantive identifying assumptions.

13.2.2 The IV Regression Estimator with Covariates

When the IV identifying restrictions are conditional on observed covariates \(X \in \mathbb{R}^p\), or when covariate adjustment is desired for precision, the unadjusted group-mean Wald ratio is generally not the appropriate sample analog: the binary group-mean difference does not partial out \(X\). (The presence of covariates does not by itself invalidate the unadjusted ratio — under unconditional randomization of \(Z\) it can remain consistent, with adjustment serving only precision — but whenever the exogeneity conditions hold only given \(X\), partialling out is required.) We derive the general IV estimator directly from the estimating-equation principle: an exogenous variable is valid as an instrument precisely because it satisfies an orthogonality condition with the structural error that the endogenous treatment does not.

Notation. Stack observations into vectors \(\mathbf{Y}, \mathbf{T} \in \mathbb{R}^n\), \(\mathbf{Z} \in \mathbb{R}^n\) (scalar instrument), and the covariate matrix \(\mathbf{X} \in \mathbb{R}^{n \times p}\). Let \(\bar{\mathbf{X}} = [\mathbf{1}_n,\,\mathbf{X}] \in \mathbb{R}^{n \times (p+1)}\) denote the design matrix obtained by adjoining a column of ones to \(\mathbf{X}\), and define the annihilator matrix \[M_X = I_n - \bar{\mathbf{X}}\bigl(\bar{\mathbf{X}}^\top\bar{\mathbf{X}}\bigr)^{-1}\bar{\mathbf{X}}^\top, \tag{13.4}\] the orthogonal projection onto the orthogonal complement of \(\mathrm{col}(\bar{\mathbf{X}})\). For any vector \(\mathbf{W} \in \mathbb{R}^n\), \(M_X\mathbf{W}\) is the vector of OLS residuals from regressing \(\mathbf{W}\) on the intercept and the columns of \(\mathbf{X}\); the \(i\)-th entry of \(M_X\mathbf{W}\) is the within-\(X\) residual \(\tilde{W}_i\). The matrix \(M_X\) is symmetric and idempotent. Throughout this chapter, “partialling out \(X\)” always means partialling out the intercept together with the columns of \(X\).

Derivation from the sample IV equations. The structural model imposes \[Y_i = \alpha + \beta T_i + \gamma^\top X_i + \varepsilon_i, \qquad \E\!\left[\begin{pmatrix} 1 \\ X_i \\ Z_i \end{pmatrix}\varepsilon_i\right] = 0, \qquad \E[T_i\,\varepsilon_i] \neq 0. \tag{13.5}\] The components of the moment vector encode exogeneity of the intercept (trivially), exogeneity of \(X\), and the joint exogeneity–exclusion restriction on \(Z\); the inequality \(\E[T\varepsilon] \neq 0\) is the endogeneity that motivates IV in the first place and is not a mean-independence statement that one is willing to assume.

To translate this moment condition into estimation, define the structural residual as a function of the parameters, \[\mathbf{e}(\beta, \bar\gamma) = \mathbf{Y} - \mathbf{T}\beta - \bar{\mathbf{X}}\bar\gamma, \qquad \bar\gamma = (\alpha, \gamma^\top)^\top,\] and consider the three families of sample equations that could be imposed on it: \[\mathbf{T}^\top \mathbf{e}(\beta,\bar\gamma) = \mathbf{0}, \tag{S-T}\] \[\bar{\mathbf{X}}^\top \mathbf{e}(\beta,\bar\gamma) = \mathbf{0}, \tag{S-X}\] \[\mathbf{Z}^\top \mathbf{e}(\beta,\bar\gamma) = \mathbf{0}. \tag{S-Z}\] Equations (S-X) and (S-Z) are the sample analogs of the population-valid moment conditions in Equation 13.5: they are exactly the exogeneity restrictions on the intercept, on \(X\), and on \(Z\). Equation (S-T), by contrast, is the equation OLS would impose on the endogenous regressor, and its population analog \(\E[T_i\varepsilon_i] = 0\) is false by assumption.

Note that these are not the normal equations of any single least-squares problem. In particular, they are not obtained by regressing \(\mathbf{Y}\) on \((\mathbf{T},\bar{\mathbf{X}},\mathbf{Z})\): that regression would give the residual a coefficient on \(\mathbf{Z}\), and constraining that coefficient to zero does not produce (S-Z). The argument here is a moment-equation argument, not a normal-equation argument. IV estimation retains the valid exogenous-regressor equations and replaces the invalid treatment equation (S-T) by the valid instrument equation (S-Z).

We therefore solve the \((p+2)\)-dimensional system (S-X), (S-Z). From (S-X): \[\hat{\bar\gamma} = (\bar{\mathbf{X}}^\top\bar{\mathbf{X}})^{-1}(\bar{\mathbf{X}}^\top\mathbf{Y} - \bar{\mathbf{X}}^\top\mathbf{T}\beta).\] Substituting into (S-Z) and simplifying using \(M_X\) gives \(\mathbf{Z}^\top M_X \mathbf{T}\,\beta = \mathbf{Z}^\top M_X \mathbf{Y}\). Provided \(\mathbf{Z}^\top M_X \mathbf{T} \neq 0\) (the relevance condition after partialling out the intercept and \(X\)), this has a unique solution.

Definition 13.2 (IV Regression Estimator) The IV regression estimator in the linear IV model with scalar instrument \(Z\) and covariates \(X\) is \[\hat\beta_{\mathrm{IV}} = \frac{\mathbf{Z}^\top M_X \mathbf{Y}}{\mathbf{Z}^\top M_X \mathbf{T}} = \frac{\widehat{\mathrm{Cov}}(\tilde{Z},\, Y)}{\widehat{\mathrm{Cov}}(\tilde{Z},\, T)}, \tag{13.6}\] where \(\tilde{Z}_i\), \(\tilde{Y}_i\), \(\tilde{T}_i\) are the within-\(X\) residuals (entries of \(M_X\mathbf{Z}\), \(M_X\mathbf{Y}\), \(M_X\mathbf{T}\)).

The estimator forces the residual \(\hat{\mathbf{e}} = \mathbf{Y} - \mathbf{T}\hat\beta_{\mathrm{IV}} - \bar{\mathbf{X}}\hat{\bar\gamma}\) to satisfy both (S-X) and (S-Z); it is the unique solution obtained by substituting the instrument equation for the corrupted treatment equation.

Remark: Wald Estimator as a Special Case

When \(Z \in \{0,1\}\) is binary and there are no covariates beyond the intercept (so \(\bar{\mathbf{X}} = \mathbf{1}_n\)), \(M_X = I_n - n^{-1}\mathbf{1}\mathbf{1}^\top\) is the centering matrix and \(M_X\mathbf{W}\) is the mean-centered \(\mathbf{W}\). One can verify that \(\mathbf{Z}^\top M_X\mathbf{Y} \propto (\bar{Y}_1 - \bar{Y}_0)\) and \(\mathbf{Z}^\top M_X\mathbf{T} \propto (\bar{T}_1 - \bar{T}_0)\), so \(\hat\beta_{\mathrm{IV}} = \hat\beta_{\mathrm{Wald}}\).

13.2.3 Structural Form, Reduced Form, and the Reduced Form Regression

The terminology reduced form only makes sense in contrast to something that has not been reduced. This subsection introduces that contrast, derives the reduced form of the linear SEM, and then presents the reduced form regression as the procedure for estimating the relevant coefficient.

The structural form. The system Equation 13.1–Equation 13.2 is called the structural form of the model. It represents the causal mechanism directly: each equation describes how one variable is determined by others, including endogenous variables on the right-hand side. In particular, the outcome equation \(Y = \alpha + \beta T + \gamma^\top X + \varepsilon\) contains the endogenous regressor \(T\) on its right-hand side. The structural coefficient \(\beta\) has a causal interpretation (it is the effect of a unit change in \(T\) on \(Y\), holding \(X\) and \(\varepsilon\) fixed), but \(T\) is correlated with \(\varepsilon\), so OLS applied to the structural form is inconsistent.

The reduced form. The reduced form is obtained by solving the structural system so that each endogenous variable is expressed purely as a function of the exogenous variables \(Z\) and \(X\), with no endogenous regressors on the right-hand side. Substituting the first-stage equation Equation 13.2 into the outcome equation Equation 13.1 gives \[Y = \underbrace{(\alpha + \beta a_T)}_{\alpha_{\mathrm{rf}}} + \underbrace{\beta\pi}_{\phi}\, Z + \underbrace{(\beta\delta + \gamma)^\top}_{\gamma_{\mathrm{rf}}^\top}\, X + \underbrace{(\beta\eta + \varepsilon)}_{\nu}. \tag{13.7}\] Writing this compactly, \[Y = \alpha_{\mathrm{rf}} + \phi\, Z + \gamma_{\mathrm{rf}}^\top X + \nu, \tag{13.8}\] where \(\alpha_{\mathrm{rf}} = \alpha + \beta a_T\), \(\phi = \beta\pi\), \(\gamma_{\mathrm{rf}} = \beta\delta + \gamma\), and \(\nu = \beta\eta + \varepsilon\). The reduced-form equation for \(T\) is simply the first-stage equation Equation 13.2 itself, which is already in reduced form because \(Z\) and \(X\) are exogenous.

Two features of the reduced form are immediate from Equation 13.7. First, both right-hand-side variables (\(Z\) and \(X\)) are exogenous, so OLS applied to Equation 13.8 is consistent for \(\phi\). Second, \(\phi = \beta\pi\) is a product of the structural parameter of interest \(\beta\) and the first-stage strength \(\pi\). Neither \(\beta\) nor \(\pi\) is separately identified from the reduced form alone; identifying \(\beta\) requires also using the first-stage equation — which is the essence of IV.

Because the exclusion restriction rules out any direct path \(Z \to Y\), the full effect of the instrument on the outcome flows through the treatment channel \(Z \to T \to Y\), so \(\phi = \beta\pi\) exactly — and not \(\phi = \beta\pi + \kappa\) for some additional direct-effect coefficient \(\kappa \neq 0\) that would be present if \(Z\) entered the outcome equation directly.

The reduced form regression. The reduced form regression is simply OLS applied to the reduced form equation Equation 13.8. Its purpose is to estimate the reduced-form coefficient \(\phi\).

Definition 13.3 (Reduced Form Regression Estimator) The reduced form regression estimator is the OLS estimator of \(\phi\) in the reduced form equation Equation 13.8: \[\hat\phi_{\mathrm{RF}} = \frac{\widehat{\mathrm{Cov}}(\tilde{Z},\, Y)}{\widehat{\mathrm{Var}}(\tilde{Z})}, \tag{13.9}\] where \(\tilde{Z}\) is the residual from projecting \(Z\) on \((1, X^\top)^\top\). It is a consistent estimator of \(\phi = \beta\pi\) under the reduced-form exogeneity condition \(\E[\nu \mid Z, X] = 0\). Since \(\nu = \beta\eta + \varepsilon\), this condition follows from two conditions: \(\E[\varepsilon \mid Z, X] = 0\), instrument exogeneity for the structural outcome disturbance; and \(\E[\eta \mid Z, X] = 0\), the first-stage conditional-mean restriction under which \(\pi\) and \(\delta\) are identified by OLS. Exclusion is a distinct structural restriction and is already encoded by the absence of \(Z\) from the outcome equation Equation 13.1; it entered when the reduced form Equation 13.7 was derived, not here. All three conditions are assumed throughout this chapter.

Intent-to-treat interpretation. When \(Z \in \{0,1\}\) is binary and randomly assigned (as in a lottery or randomized encouragement design), \(\hat\phi_{\mathrm{RF}}\) estimates the intent-to-treat (ITT) effect: the average effect on \(Y\) of being assigned \(Z = 1\) rather than \(Z = 0\), regardless of whether treatment is actually taken up. For a continuous or multi-valued instrument, \(\phi\) is instead the coefficient in the specified linear reduced-form projection, and carries an ITT reading only when a particular intervention contrast on \(Z\) is separately defined. The ITT is a valid causal estimand in its own right — it requires only exogeneity of \(Z\), not the exclusion restriction — and it is often of direct policy interest when the instrument corresponds to an implementable intervention.

Relationship to the first stage and the IV estimator. The first-stage regression estimator is \[\hat\pi_{\mathrm{FS}} = \frac{\widehat{\mathrm{Cov}}(\tilde{Z},\, T)}{\widehat{\mathrm{Var}}(\tilde{Z})}.\] Since \(\phi = \beta\pi\), we have \(\beta = \phi/\pi\), and the sample analog gives \[\hat\beta_{\mathrm{IV}} = \frac{\hat\phi_{\mathrm{RF}}}{\hat\pi_{\mathrm{FS}}}, \tag{13.10}\] which reproduces the IV regression estimator Equation 13.6. The reduced form delivers the instrument’s total effect on the outcome; the first stage scales it by the instrument’s effect on treatment; the ratio removes the scaling and recovers the structural parameter. Equation Equation 13.10 also shows that the IV estimator is undefined when \(\hat\pi_{\mathrm{FS}} = 0\) — the sample-level manifestation of the relevance requirement.

The ratio form Equation 13.10 is useful because it exposes the logic of IV, not because all IV estimation should be carried out by separately estimating and then dividing two coefficients. In practice, 2SLS (Section 13.3) solves the estimating equations jointly and produces valid standard errors automatically; forming \(\hat\phi_{\mathrm{RF}}/\hat\pi_{\mathrm{FS}}\) by hand and dividing their separate standard errors is incorrect.

Remark: Reporting All Three Quantities

In applied work it is standard practice to report the first stage, reduced form, and IV (or 2SLS) estimates side by side, as in the KIPP table of Chapter 7. The reduced form and first stage each have a transparent OLS interpretation and can be assessed independently before the ratio is formed. Reporting all three allows the reader to verify instrument strength (first stage), the ITT effect (reduced form), and the per-unit-of-treatment causal effect (IV/2SLS) simultaneously. The IV estimate carries no identifying content beyond what the reduced form and first stage together contain.

13.3 Two-Stage Least Squares

The Wald estimator is limited to binary instruments and no covariates. Two-stage least squares (2SLS) extends the identification-to-estimation step to settings with continuous or multi-valued instruments, multiple instruments, and observed covariates. The key idea is that 2SLS replaces the endogenous regressor by its projection onto the instrument space, thereby retaining only the variation in treatment that is explained by the exogenous instruments and covariates. Although it is defined through a two-stage procedure below, 2SLS is a single estimator with a closed-form expression; the stages are a transparent derivation device, not a description of two separate estimation problems.

13.3.1 Setup

Return to the structural model Equation 13.1–Equation 13.2, where \(T\) is a scalar endogenous variable, \(X \in \mathbb{R}^p\) is a vector of exogenous covariates, and \(Z \in \mathbb{R}^q\) is a vector of instruments with \(q \geq 1\). The IV assumptions require:

Relevance: the first-stage coefficient \(\pi \neq 0\). Provided the residualized second-moment matrix \(\E[\tilde Z\tilde Z^\top]\) is nonsingular, this is equivalent to \(\mathrm{Cov}(\tilde Z, T) \neq 0\), where \(\tilde Z = Z - L(Z\mid 1, X)\) is the linear-projection residual of \(Z\) on the intercept and \(X\).
Exclusion: \(Z\) does not enter the structural outcome equation Equation 13.1 directly.
Instrument exogeneity: \(\E[\varepsilon \mid Z, X] = 0\).
First-stage conditional-mean specification: \(\E[\eta \mid Z, X] = 0\).

Exclusion is a restriction on which variables appear in the structural equation; exogeneity is a conditional-mean restriction on the structural disturbance. They are logically separate and are listed separately here. The last two conditions are stronger than what GMM requires: the estimation theory of Section 13.5–Section 13.6 uses only the unconditional moments \(\E[W\varepsilon] = 0\). The conditional-mean form is maintained here because it is what justifies interpreting the first-stage and reduced-form OLS fits as conditional expectations rather than as mere linear projections. Endogeneity means \(\mathrm{Cov}(T, \varepsilon) \neq 0\), arising because \(\varepsilon\) and \(\eta\) share an unobserved common driver.

13.3.2 Stage 1: The First-Stage Regression

Regress \(T\) on \(Z\) and \(X\) by OLS using the first-stage model \(T = a_T + \pi^\top Z + \delta^\top X + \eta\), and denote the OLS fitted values \(\hat{T}_i = \hat{a}_T + \hat\pi^\top Z_i + \hat\delta^\top X_i\). In matrix form, \(\hat{\mathbf{T}} = P_W \mathbf{T}\) where \(P_W\) is the projection onto the column space of the instrument design matrix \(\mathbf{W}\) with rows \(W_i = (1,\,X_i^\top,\,Z_i^\top)^\top\). The fitted value \(\hat{T}_i\) is the linear projection of \(T_i\) onto the space spanned by the intercept, \(Z_i\), and \(X_i\). Because OLS residuals are orthogonal to the regressors, \(\hat{T}_i\) is by construction uncorrelated (in the sample) with the first-stage residual \(\hat\eta_i = T_i - \hat{T}_i\). The fitted values \(\hat{T}_i\) are not used because they predict treatment well in a generic regression sense; they are used because they isolate the component of treatment variation spanned by \((1, Z, X)\) — the variation that is exogenous under IV validity.

13.3.3 Stage 2: The Second-Stage Regression

Regress \(Y\) on \(\hat{T}\) and \(X\) by OLS: \(Y = \alpha + \beta \hat{T} + \gamma^\top X + \text{error}\).

Definition 13.4 (2SLS Estimator) The two-stage least squares estimator \(\hat\beta_{\mathrm{2SLS}}\) is the OLS coefficient on \(\hat{T}\) in the second-stage regression.

Equivalently, 2SLS projects the endogenous regressors onto the linear span of the instruments and exogenous covariates, and then uses only that projected component in the structural regression. The component of \(T\) orthogonal to \((Z, X)\) — the residual \(\hat\eta_i = T_i - \hat{T}_i\), which is correlated with \(\varepsilon_i\) when \(T\) is endogenous — is dropped before the causal coefficient is estimated. The fitted values \(\hat T_i\) are not literally randomized; they are the variation in \(T\) that is linearly explained by \((Z, X)\), and their exogeneity with respect to \(\varepsilon\) is inherited from the maintained IV assumptions (instrument exogeneity and exclusion), not guaranteed by the projection algebra alone.

To obtain a closed form, let \(\tilde{Y}_i\), \(\tilde{T}_i\), and \(\tilde{Z}_i\) denote the residuals from projecting \(Y_i\), \(T_i\), and \(Z_i\), respectively, onto \(X_i\) by OLS (i.e., the within-\(X\)-residuals). In the single-instrument case (\(q = 1\)), the 2SLS estimator simplifies to \[\hat\beta_{\mathrm{2SLS}} = \frac{\sum_{i=1}^n \tilde{Z}_i\, Y_i}{\sum_{i=1}^n \tilde{Z}_i\, T_i} = \frac{\widehat{\mathrm{Cov}}(\tilde{Z},\, Y)}{\widehat{\mathrm{Cov}}(\tilde{Z},\, T)}. \tag{13.11}\] This is structurally parallel to the Wald ratio: a ratio of the instrument-by-outcome sample covariance to the instrument-by-treatment sample covariance, after partialling out the observed covariates \(X\).

Standard Errors from the Second-Stage OLS Are Incorrect

A common error is to report the standard errors from the second-stage OLS regression directly. Those standard errors use \(\hat{T}\) rather than \(T\) and residuals that do not equal the structural errors \(\varepsilon_i\), so the resulting standard errors are invalid. Correct inference requires the sandwich variance formula derived in Section 13.6, or software that implements 2SLS natively.

13.4 Equivalence of the IV Regression Estimator and 2SLS

The IV regression estimator Equation 13.6 and the 2SLS estimator look like different procedures: one computes a ratio of partialled-out covariances directly, while the other constructs first-stage fitted values and runs a second regression. In the single-instrument case they are in fact numerically identical for any covariate vector \(X\). This theorem is not just algebraic housekeeping: it shows that the ratio-of-covariances view and the staged-regression view are two computational faces of the same IV estimator. Knowing both is useful — the ratio form exposes the logic of IV estimation while the staged-regression form generalizes to multiple instruments and connects naturally to GMM.

Theorem 13.1 (IV Regression–2SLS Equivalence) In the linear IV model with a scalar instrument \(Z\) and covariates \(X \in \mathbb{R}^p\), the IV regression estimator and the 2SLS estimator coincide: \(\hat\beta_{\mathrm{2SLS}} = \hat\beta_{\mathrm{IV}}\).

The intuition is compact. In Section 13.2 we derived \(\hat\beta_{\mathrm{IV}}\) by replacing the invalid (S-T) with the valid instrument equation and solving the system (S-X), (S-Z), obtaining \(\hat\beta_{\mathrm{IV}} = (\mathbf{Z}^\top M_X \mathbf{T})^{-1}\mathbf{Z}^\top M_X \mathbf{Y}\). The 2SLS procedure reaches the same formula by a projection argument.

Proof. Retain the matrix notation of Section 13.2: \(\bar{\mathbf{X}} = [\mathbf{1}_n,\,\mathbf{X}]\), \(M_X = I_n - \bar{\mathbf{X}}(\bar{\mathbf{X}}^\top\bar{\mathbf{X}})^{-1}\bar{\mathbf{X}}^\top\), scalar instrument \(\mathbf{Z} \in \mathbb{R}^n\).

First-stage projection. The first-stage fitted value is \(\hat{\mathbf{T}} = P_{[\bar{\mathbf{X}},\mathbf{Z}]}\mathbf{T}\), where \(P_{[\bar{\mathbf{X}},\mathbf{Z}]}\) projects onto \(\mathrm{col}(\bar{\mathbf{X}}, \mathbf{Z})\). By the Frisch–Waugh–Lovell theorem applied to Stage 1: \[M_X\hat{\mathbf{T}} = M_X\mathbf{Z}(\mathbf{Z}^\top M_X\mathbf{Z})^{-1}\mathbf{Z}^\top M_X\mathbf{T}.\]

2SLS second-stage closed form. Applying FWL to the second-stage regression of \(\mathbf{Y}\) on \((\hat{\mathbf{T}}, \bar{\mathbf{X}})\): \[\hat\beta_{\mathrm{2SLS}} = \frac{\hat{\mathbf{T}}^\top M_X \mathbf{Y}}{\hat{\mathbf{T}}^\top M_X \hat{\mathbf{T}}} = \frac{\mathbf{T}^\top M_X\mathbf{Z}(\mathbf{Z}^\top M_X\mathbf{Z})^{-1}\mathbf{Z}^\top M_X\mathbf{Y}}{\mathbf{T}^\top M_X\mathbf{Z}(\mathbf{Z}^\top M_X\mathbf{Z})^{-1}\mathbf{Z}^\top M_X\mathbf{T}}.\] Under exact identification (\(q = 1\)), the scalars \(\mathbf{T}^\top M_X\mathbf{Z}\) and \((\mathbf{Z}^\top M_X\mathbf{Z})^{-1}\) cancel from numerator and denominator, leaving \[\hat\beta_{\mathrm{2SLS}} = \frac{\mathbf{Z}^\top M_X\mathbf{Y}}{\mathbf{Z}^\top M_X\mathbf{T}} = \hat\beta_{\mathrm{IV}}. \qquad\square\]

In the scalar-instrument linear model, Wald, IV regression, and 2SLS are therefore not competing methods; they are different representations of the same sample analog of the identification formula.

Remark: What the Proof Reveals

The proof makes the logic of IV transparent at the level of estimating equations. OLS imposes residual orthogonality with every regressor. When \(T\) is endogenous, the treatment equation \(\mathbf{T}^\top\mathbf{e}(\theta) = 0\) is invalid at the structural parameter, since \(\E(T_i\varepsilon_i) \neq 0\). IV retains the valid equations for the exogenous regressors and replaces the invalid treatment equation by the instrument equation \(\mathbf{Z}^\top\mathbf{e}(\theta) = 0\), which is valid because \(Z\) is exogenous. The result is a method-of-moments construction, not an OLS normal-equation system: the retained equations happen to coincide with OLS equations for the exogenous block, but the system as a whole is not the first-order condition of any least-squares problem. The 2SLS derivation reaches the same estimator from a different direction, by projection.

Remark: Wald Estimator as a Special Case of the Equivalence

When \(Z \in \{0,1\}\) is binary and \(X\) is absent, the IV regression estimator reduces to the Wald estimator (Definition 13.1), as shown in Section 13.2. Theorem 13.1 therefore implies \(\hat\beta_{\mathrm{2SLS}} = \hat\beta_{\mathrm{Wald}}\) in this case as well.

Remark: Generalization to Multiple Instruments

With \(q > 1\) instruments, the IV regression formula no longer applies directly because the system \(\sum_i \tilde{Z}_i Y_i / \sum_i \tilde{Z}_i T_i\) is not well defined for vector \(\tilde{Z}_i\). 2SLS resolves this by using the first-stage projection of \(T\) onto the full instrument vector, producing a single scalar \(\hat{T}_i\). The resulting estimator is efficient in the class of linear IV estimators under homoskedasticity and reduces to the IV regression estimator when \(q = 1\).

13.5 The Moment-Condition View and GMM

The proof of Theorem 13.1 already contains the key idea: IV estimation amounts to solving the normal equations of the structural model with the invalid \(T\)-block replaced by the valid instrument equation \(\mathbf{Z}^\top\hat{\mathbf{e}} = 0\). This replacement is not an algebraic trick but a direct expression of the moment condition implied by IV validity: \(\E[(1, X^\top, Z^\top)^\top\,\varepsilon] = 0\), which follows from the IV exogeneity statement \(\E[\varepsilon\mid Z, X] = 0\) together with the (trivial) exogeneity of the intercept and of \(X\). The advantage of the moment-condition view is that it separates the identifying restrictions from the specific algebra of 2SLS, making clear which part of the problem is about causal assumptions and which part is about estimation efficiency. Recognizing 2SLS as a method-of-moments estimator for this condition connects it to the general estimating-equation framework of Chapter 10 and motivates the generalized method of moments for overidentified models.

13.5.1 2SLS as a Method-of-Moments Estimator

Stack the constant, the covariates, and the instrument into the \((p+1+q)\)-dimensional vector \(W = (1,\,X^\top,\,Z^\top)^\top\), which contains every variable that should be exogenous under IV validity. The IV moment condition is \[\E\bigl[W\,\bigl(Y - \alpha - \beta T - \gamma^\top X\bigr)\bigr] = 0.\] Setting \(\theta = (\alpha, \beta, \gamma^\top)^\top \in \mathbb{R}^{p+2}\) and defining the moment function \(U(O;\,\theta) = W\,(Y - \alpha - \beta T - \gamma^\top X)\), the identifying condition is \(\E\{U(O;\,\theta_0)\} = 0\), which is exactly an estimating equation in the sense of Chapter 10. The number of moment conditions is \(p+1+q\) and the number of parameters is \(p+2\); the model is exactly identified when \(q = 1\) and overidentified when \(q > 1\).

Theorem 13.2 (2SLS as a Method-of-Moments Estimator) In the exactly identified linear IV model (\(q = 1\)), suppose the sample rank condition holds: \(n^{-1}\sum_i W_i D_i^\top\) is nonsingular. (Equality of the number of equations and the number of parameters is only the order condition; it does not by itself guarantee a unique solution.) Then the 2SLS estimator \(\hat\theta_{\mathrm{2SLS}} = (\hat\alpha,\,\hat\beta_{\mathrm{2SLS}},\,\hat\gamma^\top)^\top\) is the unique solution to the sample moment system \[\frac{1}{n}\sum_{i=1}^n W_i\,\bigl(Y_i - \hat\alpha - \hat\beta_{\mathrm{2SLS}}\,T_i - \hat\gamma^\top X_i\bigr) = 0, \qquad W_i = (1,\,X_i^\top,\,Z_i)^\top \in \mathbb{R}^{p+2}.\] The first row of this \((p+2)\)-dimensional system is the intercept-orthogonality condition (residuals sum to zero); the middle \(p\) rows are the \(X\)-orthogonality conditions; the last row is the instrument-orthogonality condition.

This moment-condition view clarifies why the first-stage fitted values \(\hat{T}_i\) are not merely a computational device. They encode the orthogonality structure implied by instrument validity: replacing \(T_i\) with \(\hat{T}_i\) ensures the second-stage regression uses only the exogenous variation in treatment. IV is therefore not a separate estimation philosophy from the rest of Part III; it is a special case of the general estimating-equation framework of Chapter 10, with moment functions induced by instrument validity.

13.5.2 Overidentification and GMM

When \(q > 1\) instruments are available, the model is overidentified: there are more moment conditions than parameters. The system \(\mathbb{P}_n U(O;\,\theta) = 0\) is generically overdetermined and has no exact solution.

When the model is overidentified, different linear combinations of the sample moments contain the same target parameter but with different noise levels. GMM chooses the combination that solves the moment equations as well as possible, with a weighting scheme that can exploit the heteroskedastic structure of the moment scores for greater efficiency. The weighting matrix \(\hat\Omega_n\) determines which combination is used: identity weighting treats all moment conditions equally, while efficient weighting downweights noisy directions in the moment vector and accounts for correlation among the moment conditions through the full inverse covariance matrix \(\Sigma^{-1}\). Only when \(\Sigma\) is diagonal does this reduce to weighting individual moments by their inverse variances.

The generalized method of moments (GMM) formalizes this by minimizing a weighted quadratic form in the sample moments: \[\hat\theta_{\mathrm{GMM}} = \arg\min_{\theta}\; \bigl[\mathbb{P}_n U(O;\,\theta)\bigr]^\top\,\hat\Omega_n\,\bigl[\mathbb{P}_n U(O;\,\theta)\bigr],\] where \(\hat\Omega_n\) is a positive-definite weighting matrix. Different choices of \(\hat\Omega_n\) yield different estimators within the IV class:

The choice \(\hat\Omega_n = (n^{-1}\sum_i W_i W_i^\top)^{-1}\) — the inverse of the second-moment matrix of the instrument vector \(W_i = (1, X_i^\top, Z_i^\top)^\top\) — yields the 2SLS estimator.
The choice \(\hat\Omega_n = \bigl[\mathbb{P}_n U(O;\,\hat\theta)\,U(O;\,\hat\theta)^\top\bigr]^{-1}\) — the inverse of the estimated moment covariance — yields the efficient GMM estimator, which achieves the semiparametric efficiency bound for the IV moment-condition model.

These comparisons have content only when \(m > k\). In an exactly identified model every positive-definite weighting matrix yields the same estimator and the same asymptotic variance, so 2SLS and efficient GMM coincide whatever the conditional variance structure. In an overidentified model, homoskedasticity is sufficient for 2SLS to be efficient; under general heteroskedasticity, overidentified 2SLS need not be efficient, and efficient GMM is weakly (often strictly) more efficient.

Remark: Overidentification as a Testable Restriction

An overidentified model imposes more moment conditions than are needed for point identification. If all instruments are valid, the sample moments should be jointly near zero. The Sargan–Hansen \(J\)-statistic (Sargan 1958; Hansen 1982) tests this restriction: under the null that all moment conditions hold, \(J \xrightarrow{d} \chi^2_{q-1}\). A rejection indicates that the full set of maintained moment conditions is incompatible with the data. This may reflect an exclusion failure, an exogeneity failure, or other model misspecification; the test does not identify which assumption failed or which instrument is responsible. Overidentification therefore allows us to check whether the full set of moment restrictions is jointly compatible with the data, but a failure to reject is consistent with all instruments being valid, or with violations that happen to cancel in the joint test.

Example 13.1 (GMM with Two Instruments) We illustrate the GMM procedure in the simplest overidentified setting: one endogenous regressor, two instruments, no additional covariates.

Setup. Consider the structural model \(Y = \beta T + \varepsilon\) (intercept suppressed after demeaning). Two instruments \(Z_1\) and \(Z_2\) are available, giving the moment conditions \(\E[Z_1\varepsilon] = 0\) and \(\E[Z_2\varepsilon] = 0\). The model is overidentified (\(q = 2\), one free parameter), so the two moment conditions cannot generally be satisfied simultaneously in finite samples.

Sample moments. Suppose the following sample covariances are observed: \[\widehat{\mathrm{Cov}}(Z_1,\,Y) = 0.40, \quad \widehat{\mathrm{Cov}}(Z_1,\,T) = 0.50, \quad \widehat{\mathrm{Cov}}(Z_2,\,Y) = 0.60, \quad \widehat{\mathrm{Cov}}(Z_2,\,T) = 0.80.\] The just-identified IV estimate using each instrument alone is \[\hat\beta^{(1)} = \frac{0.40}{0.50} = 0.80, \qquad \hat\beta^{(2)} = \frac{0.60}{0.80} = 0.75.\] The two instruments give different point estimates. Under the linear constant-effect model maintained in this example, both moment conditions target the same parameter \(\beta\), so the discrepancy between \(\hat\beta^{(1)}\) and \(\hat\beta^{(2)}\) reflects sampling variation alone. (Under treatment-effect heterogeneity, instruments that move different complier subpopulations can additionally target different local estimands, in which case overidentified combinations weight several distinct LATEs, and the resulting combination need not be a convex average of the instrument-specific local estimands: depending on the first-stage structure and the estimator, some weights can be negative.) The GMM objective combines both moment conditions.

GMM moment function. With \(a = (\widehat{\mathrm{Cov}}(Z_1,T),\;\widehat{\mathrm{Cov}}(Z_2,T))^\top = (0.50,\;0.80)^\top\) and \(c = (\widehat{\mathrm{Cov}}(Z_1,Y),\;\widehat{\mathrm{Cov}}(Z_2,Y))^\top = (0.40,\;0.60)^\top\), the sample moment vector is \[\hat{U}(\beta) = c - a\beta = \begin{pmatrix} 0.40 - 0.50\beta \\ 0.60 - 0.80\beta \end{pmatrix}.\]

GMM with identity weighting (\(\Omega = I_2\)). The GMM criterion is \(Q(\beta) = \hat{U}(\beta)^\top \hat{U}(\beta)\). The first-order condition \(\partial Q/\partial\beta = 0\) gives \[a^\top(c - a\beta) = 0 \;\;\Longrightarrow\;\; \hat\beta_{\mathrm{GMM}} = \frac{a^\top c}{a^\top a} = \frac{(0.50)(0.40)+(0.80)(0.60)}{(0.50)^2+(0.80)^2} = \frac{0.68}{0.89} \approx 0.764.\] Under identity weighting this is a first-stage-sensitivity-weighted average of the two just-identified estimates, with weights proportional to \(a_j^2\): it lies between \(0.75\) and \(0.80\), with more weight on \(Z_2\) because it has the larger first-stage covariance (\(0.80 > 0.50\)). These weights are not in general inverse-variance (precision) weights; identity weighting ignores \(\Sigma\) entirely.

Effect of the weighting matrix. With a diagonal weight \(\Omega = \mathrm{diag}(w_1, w_2)\), the GMM estimator becomes \[\hat\beta_{\mathrm{GMM}}(\Omega) = \frac{w_1(0.50)(0.40) + w_2(0.80)(0.60)}{w_1(0.50)^2 + w_2(0.80)^2}.\] Setting \(w_1 = 4\) and \(w_2 = 1\) (upweighting \(Z_1\)) gives \[\hat\beta_{\mathrm{GMM}} = \frac{4(0.20)+1(0.48)}{4(0.25)+1(0.64)} = \frac{1.28}{1.64} \approx 0.780,\] closer to \(\hat\beta^{(1)} = 0.80\) as expected. As \(w_1/w_2 \to \infty\), the estimate converges to \(\hat\beta^{(1)}\); as \(w_1/w_2 \to 0\), it converges to \(\hat\beta^{(2)}\). The efficient weighting \(\Omega^\ast = \hat\Sigma^{-1}\) (Section 13.6) minimizes the asymptotic variance over all weightings. If \(\hat\Sigma\) is diagonal, this amounts to choosing \(w_j = \hat\Sigma_{jj}^{-1}\), so noisier moments receive less weight. In general \(\hat\Sigma^{-1}\) is not diagonal: efficient GMM also uses the off-diagonal covariances among the moment conditions, and so weights directions in moment space rather than each instrument separately — the diagonal family considered in this example does not contain the optimum unless the moments are uncorrelated.

Sargan–Hansen \(J\)-test. At \(\hat\beta \approx 0.764\) under identity weighting, the residual sample moments are \[\hat{U}_1 = 0.40 - (0.50)(0.764) = 0.018, \qquad \hat{U}_2 = 0.60 - (0.80)(0.764) = -0.011.\] Neither is exactly zero, reflecting the overidentification. The \(J\)-statistic is \(J = n\,\hat{U}(\hat\beta)^\top\hat\Sigma^{-1}\hat{U}(\hat\beta)\), and has a \(\chi^2_1\) null distribution (one overidentifying restriction) when evaluated at the efficiently weighted estimator using the corresponding \(\hat\Sigma\). For this example we adopt the convention \(\hat\Sigma = I_2\) with \(n = 200\), so that identity weighting is the assumed optimal weighting; then \[J = 200\bigl\{(0.018)^2 + (-0.011)^2\bigr\} \approx 0.09,\] far below the 5% critical value \(\chi^2_{1,0.95} = 3.84\).

The maintained moment restrictions are therefore not rejected, and that is the whole of the conclusion. Failure to reject does not establish that both instruments satisfy exclusion. A rejection could arise from exogeneity failure, exclusion failure, failure of the common constant-effect restriction, functional-form misspecification, or any other failure of the maintained moments, and the test cannot say which.

13.6 Asymptotic Theory of the GMM Estimator

Section 13.5 placed IV estimation within the GMM framework. This section derives the asymptotic distribution of the GMM estimator for the linear IV model and then specializes to 2SLS and to the efficient GMM estimator.

13.6.1 Setup and Notation

Stack the structural regressors and instruments into vectors: \[D_i = \begin{pmatrix} 1 \\ T_i \\ X_i \end{pmatrix} \in \mathbb{R}^k, \qquad W_i = \begin{pmatrix} 1 \\ X_i \\ Z_i \end{pmatrix} \in \mathbb{R}^m,\] where \(k = p + 2\) and \(m = p + 1 + q\). The structural model is \(Y_i = D_i^\top\theta_0 + \varepsilon_i\) with \(\theta_0 = (\alpha_0, \beta_0, \gamma_0^\top)^\top\), and the IV moment condition is \[\E\bigl[U(O_i;\,\theta_0)\bigr] = 0, \qquad U(O_i;\,\theta) = W_i\,(Y_i - D_i^\top\theta).\] The order condition requires \(m \geq k\) (at least as many instruments as parameters), i.e., \(q \geq 1\). For identification, the rank condition requires \(A = \E[W_i D_i^\top] \in \mathbb{R}^{m \times k}\) to have full column rank \(k\).

Define the moment variance matrix \[\Sigma = \E\bigl[U(O_i;\,\theta_0)\,U(O_i;\,\theta_0)^\top\bigr] = \E[\varepsilon_i^2\, W_i W_i^\top] \;\in \mathbb{R}^{m \times m}.\] Under homoskedasticity \(\E[\varepsilon_i^2 \mid W_i] = \sigma^2\), this simplifies to \(\Sigma = \sigma^2\,\E[W_i W_i^\top]\).

13.6.2 Asymptotic Distribution of the GMM Estimator

The large-sample logic is the same as for any method-of-moments estimator: sample orthogonality converges to population orthogonality by the law of large numbers, and the estimator fluctuates around the truth according to how sensitive the moment equations are to the parameter and how noisy the sample moments are. The sensitivity is captured by the matrix \(A = -\E[\partial U/\partial\theta^\top]\), which measures how sharply the moments respond to a change in \(\theta\); the noise is captured by \(\Sigma = \E[U(O;\theta_0)U(O;\theta_0)^\top]\), the variance of the moment scores at the truth. The asymptotic variance is a sandwich that combines both.

Recall that the GMM estimator minimizes \([\mathbb{P}_n U(O;\theta)]^\top \hat\Omega_n [\mathbb{P}_n U(O;\theta)]\) for some positive-definite weighting matrix \(\hat\Omega_n \xrightarrow{p} \Omega\).

Theorem 13.3 (Asymptotic Distribution of the GMM Estimator) Suppose the standard GMM regularity conditions hold: the IV moment condition is satisfied at a unique population minimizer \(\theta_0\) lying in the interior of a compact parameter space of fixed finite dimension; \(O_i = (Y_i, T_i, X_i, Z_i)\) are i.i.d. with finite fourth moments; the number of moment conditions \(m\) is fixed; \(\hat\Omega_n \xrightarrow{p} \Omega\) positive definite; \(\hat\theta_{\mathrm{GMM}} \xrightarrow{p} \theta_0\); \(A\) has full column rank; and \(\Sigma\) is positive definite. Then \[\sqrt{n}\,\bigl(\hat\theta_{\mathrm{GMM}} - \theta_0\bigr) \;\xrightarrow{d}\; N\bigl(0,\; V_{\mathrm{GMM}}(\Omega)\bigr),\] where \[V_{\mathrm{GMM}}(\Omega) = (A^\top\Omega A)^{-1}\,A^\top\Omega\,\Sigma\,\Omega A\,(A^\top\Omega A)^{-1}. \tag{13.12}\]

Intuition: the Sandwich Structure

The asymptotic variance has the same logic as every sandwich formula in semiparametric estimation: one part measures how responsive the moments are to the parameter (\(A\)), and another measures how variable the moments themselves are (\(\Sigma\)). Estimation is more precise when the moments react strongly to the parameter (large \(A\)) and fluctuate little across samples (small \(\Sigma\)). The weighting matrix \(\Omega\) scales the two layers but does not alter their fundamental roles.

The formula Equation 13.12 is the sandwich variance for GMM, and is an instance of the general estimating-equation theory from Chapter 10: the sensitivity matrix \(A\) plays the role of \(-\E[\partial U/\partial\theta^\top]\) and the moment variance \(\Sigma\) plays the role of \(\E[UU^\top]\).

13.6.3 Two Important Special Cases

Exactly identified case (\(q = 1\), \(m = k\)). When there is a single instrument, \(A\) is square and invertible by the rank condition. The GMM objective has an exact solution \(\mathbb{P}_n U(O;\hat\theta) = 0\) regardless of \(\Omega\), so all weighting matrices yield the same estimator. The sandwich variance simplifies to \[V_{\mathrm{IV}} = A^{-1}\,\Sigma\,(A^\top)^{-1}, \tag{13.13}\] matching the general moment-estimator formula from Chapter 10. In the scalar no-covariate case (\(p = 0\)), \(W_i = (1, Z_i)^\top\) and \(D_i = (1, T_i)^\top\), so \(A = \E[W_i D_i^\top]\) is a \(2 \times 2\) matrix. Partitioning the inverse and reading off the block corresponding to \(\beta\) yields the scalar variance \[V_\beta = \frac{\E[\varepsilon^2 \tilde{Z}^2]}{\bigl(\E[\tilde{Z}\,T]\bigr)^2}, \qquad \tilde{Z} = Z - \E[Z].\] Centering the instrument is essential when an intercept is included. The uncentered expression \(\E[\varepsilon^2 Z^2]/(\E[ZT])^2\) is valid under the convention \(\E[Z] = 0\); if in addition \(\E[T] = 0\), then \(\E[ZT] = \mathrm{Cov}(Z,T)\). Mean-centering \(Y\) and \(T\) alone does not remove the need to center \(Z\): the numerator \(\E[\varepsilon^2 Z^2]\) still differs from \(\E[\varepsilon^2\tilde Z^2]\) whenever \(\E[Z] \neq 0\). Under homoskedasticity \(\E[\varepsilon^2 \mid Z] = \sigma^2\) and the first-stage relation \(T = a_T + \pi Z + \eta\) with \(\E[\eta \mid Z] = 0\), the variance reduces to \[V_\beta = \frac{\sigma^2}{\pi^2\,\mathrm{Var}(Z)}.\]

Efficient GMM. Among all choices of \(\Omega\), the asymptotic variance \(V_{\mathrm{GMM}}(\Omega)\) is minimized (in the positive semidefinite sense) by \(\Omega^\ast = \Sigma^{-1}\). Substituting into Equation 13.12: \[V_{\mathrm{eff}} = \bigl(A^\top\Sigma^{-1}A\bigr)^{-1}. \tag{13.14}\] This is the efficient GMM variance and is the semiparametric efficiency lower bound for IV estimation within the linear IV moment-restriction model defined by \(\E[W\varepsilon] = 0\); it is the best achievable variance among all estimators that use only these moment conditions, but it does not in general coincide with efficiency bounds from richer semiparametric models that exploit additional structure. For any other weighting matrix \(\Omega\), \(V_{\mathrm{GMM}}(\Omega) - V_{\mathrm{eff}} \succeq 0\).

Remark: When 2SLS Is Efficient

Begin with the exactly identified case \(m = k\). There \(A\) is square and invertible, every positive-definite \(\Omega\) yields the same estimator, and \[A^{-1}\Sigma (A^\top)^{-1} = (A^\top\Sigma^{-1}A)^{-1},\] so 2SLS is efficient GMM regardless of the conditional variance structure — heteroskedasticity included. The efficiency question only has content when \(m > k\).

In the overidentified case, the 2SLS estimator corresponds to the weighting matrix \(\Omega_{\mathrm{2SLS}} = \E[W_i W_i^\top]^{-1}\). Under homoskedasticity, \(\Sigma = \sigma^2\E[WW^\top]\), so \(\Sigma^{-1} = \sigma^{-2}\E[WW^\top]^{-1} \propto \Omega_{\mathrm{2SLS}}\). Since scalar multiples of \(\Omega\) do not change the minimizer, 2SLS achieves \(V_{\mathrm{eff}}\) under homoskedasticity. Under heteroskedasticity, \(\Sigma\) is not in general a scalar multiple of \(\E[WW^\top]\), and the optimal weighting \(\hat\Omega_n = (\mathbb{P}_n \hat\varepsilon_i^2 W_i W_i^\top)^{-1}\) is weakly more efficient than \(\Omega_{\mathrm{2SLS}}\), with strict improvement except in the special heteroskedastic designs where \(\Sigma\) happens to remain proportional to \(\E[WW^\top]\) — one simple sufficient condition for no gain, though equality can also occur in special designs where the 2SLS and efficient-GMM influence matrices coincide on the identifying directions. Whenever homoskedasticity is plausible, efficient GMM brings no gain over 2SLS; under heteroskedasticity, efficient GMM is generically strictly more efficient, and the gain can be substantial when the conditional variance of \(\varepsilon\) varies sharply with \(W\). This tells students when they can stop at 2SLS without sacrificing asymptotic efficiency.

13.6.4 Consistent Variance Estimation

Let \(\hat\varepsilon_i = Y_i - D_i^\top\hat\theta\) denote the structural residuals evaluated at the GMM estimates. Consistent estimators of \(A\) and \(\Sigma\) are \[\hat{A} = \frac{1}{n}\sum_{i=1}^n W_i D_i^\top, \qquad \hat\Sigma = \frac{1}{n}\sum_{i=1}^n \hat\varepsilon_i^2\, W_i W_i^\top.\] Substituting into Equation 13.12 gives the heteroskedasticity-robust sandwich variance estimator: \[\hat{V}_{\mathrm{GMM}} = (\hat{A}^\top\hat\Omega_n\hat{A})^{-1}\,\hat{A}^\top\hat\Omega_n\,\hat\Sigma\,\hat\Omega_n\hat{A}\,(\hat{A}^\top\hat\Omega_n\hat{A})^{-1}. \tag{13.15}\] In the exactly identified case, \(\hat{A}\) is square and Equation 13.15 reduces to \(\hat{A}^{-1}\hat\Sigma(\hat{A}^\top)^{-1}\), independent of \(\hat\Omega_n\). For the efficient GMM estimator, the optimal weighting is \(\hat\Omega_n = \hat\Sigma^{-1}\), and the variance estimator simplifies to \((\hat{A}^\top\hat\Sigma^{-1}\hat{A})^{-1}\).

Do Not Forget the Factor \(1/n\)

Theorem 13.3 is stated for \(\sqrt{n}(\hat\theta - \theta_0)\), so \(\hat{V}_{\mathrm{GMM}}\) in Equation 13.15 estimates the covariance matrix of the root-\(n\) scaled estimator, not of \(\hat\theta\) itself. Standard errors for the coefficients are therefore \[\widehat{\mathrm{Var}}(\hat\theta) = \frac{\hat{V}_{\mathrm{GMM}}}{n}, \qquad \widehat{\mathrm{se}}(\hat\theta_j) = \sqrt{\frac{\hat{V}_{\mathrm{GMM},jj}}{n}}.\] Reporting \(\sqrt{\hat{V}_{\mathrm{GMM},jj}}\) directly inflates every standard error by a factor of \(\sqrt{n}\).

In applications, the default variance estimator should rarely be the i.i.d. formula. Heteroskedasticity-robust, and when appropriate cluster-robust, standard errors are the practical baseline for IV work. For independent clusters \(c = 1, \ldots, G\), define the cluster score \(\hat{S}_c = \sum_{i \in c} \hat\varepsilon_i W_i\), and estimate the moment covariance by \[\hat\Sigma_{\mathrm{cl}} = \frac{1}{n}\sum_{c=1}^{G} \hat{S}_c \hat{S}_c^\top. \tag{13.16}\] Substituting \(\hat\Sigma_{\mathrm{cl}}\) for \(\hat\Sigma\) in Equation 13.15 and dividing the result by \(n\) gives \(\widehat{\mathrm{Var}}(\hat\theta)\) as above. Note that \(\hat\Sigma_{\mathrm{cl}}\) reduces to the heteroskedasticity-robust \(\hat\Sigma\) when every cluster is a singleton. The asymptotics here are in the number of clusters, not the number of observations: they require independence across clusters, \(G \to \infty\), no single cluster dominating the sample, and finite cluster-level second moments.

Cluster-robust standard errors are required whenever observations within a group share unmodeled common shocks — as in the KIPP application of Chapter 7, revisited in Section 13.7.4, where each student contributes several test observations and the errors are clustered at the student level.

Remark: Two-Step Efficient GMM

Because the optimal weighting \(\hat\Omega_n = \hat\Sigma^{-1}\) requires preliminary residuals \(\hat\varepsilon_i\), efficient GMM is typically implemented in two steps: obtain a consistent first-step estimator (e.g., 2SLS) to form \(\hat\varepsilon_i\), construct \(\hat\Sigma\), and then re-minimize the GMM criterion with \(\hat\Omega_n = \hat\Sigma^{-1}\). The resulting estimator achieves \(V_{\mathrm{eff}}\) asymptotically.

13.7 Weak Instruments and Inferential Fragility

Every result in Section 13.6 was obtained under strong identification.

Where the Standard Asymptotic Theory Stops

Theorem 13.3 assumes that the population Jacobian \(A = \E(W D^\top)\) has full rank and stays bounded away from singularity along the sequence of models being approximated. The weak-instrument sequences studied in this section violate exactly that regularity condition. Consequently the normal approximation of Theorem 13.3 and the Wald confidence interval built from the sandwich variance estimator Equation 13.15 are not uniformly valid: their coverage can be arbitrarily poor over the region of the parameter space in which the first stage is close to zero, no matter how large \(n\) is.

The GMM and 2SLS estimators are well behaved when the instrument strongly predicts treatment. When the first-stage relationship is weak, IV estimation becomes fragile.

It is worth being precise about what kind of fragility this is. A small fixed first-stage coefficient is not an asymptotic impossibility: if \(\pi \neq 0\) is held fixed, increasing \(n\) eventually restores consistency and asymptotic normality. Weak-instrument asymptotics instead study sequences such as \(\pi_n = c/\sqrt{n}\), along which the concentration parameter stays bounded and the conventional normal approximation does not improve with sample size. That device is a modeling choice designed to approximate the finite-sample behavior of a design in which \(\pi\) is small relative to its sampling variability, and it is the right approximation to have in mind whenever the first stage is weak relative to \(n\). A weak instrument is not merely an efficiency problem: it distorts the sampling distribution of IV estimators, pulls their center toward OLS, and undermines conventional confidence intervals. These are not asymptotic curiosities but practical hazards that arise routinely in empirical work, and they are among the most important practical concerns in IV design.

13.7.1 The Weak-Instrument Problem

The 2SLS closed form Equation 13.11 divides by the sample covariance \(\widehat{\mathrm{Cov}}(\tilde{Z}, T)\). When the population first-stage coefficient \(\pi\) is near zero, this denominator is near zero in the population and highly variable in finite samples. Dividing a noisy numerator by a noisy, near-zero denominator is the source of every symptom listed below (Bound et al. 1995).

Center pulled toward OLS. Under weak-instrument asymptotics the center of the 2SLS sampling distribution is pulled toward the OLS probability limit.
Non-Gaussian sampling distribution. The distribution of \(\hat\beta_{\mathrm{2SLS}}\) can be markedly skewed, bimodal, or heavy-tailed, so the \(N(0,\,V_{\mathrm{2SLS}})\) approximation is unreliable.
Size distortion. Wald-type confidence intervals derived from the asymptotic normal approximation can severely undercover the true parameter, and the corresponding \(t\)-tests can reject far more often than their nominal level.

The first of these three statements requires care, because the phrase “finite-sample bias toward OLS” is often used more loosely than the underlying mathematics permits.

In What Sense Is 2SLS “Biased Toward OLS”?

Under standard weak-instrument asymptotics, the center of the 2SLS distribution is pulled toward the OLS probability limit in a local-to-zero, Nagar-expansion, or median-bias sense. In just-identified ratio models, however, the ordinary finite-sample moments of \(\hat\beta_{\mathrm{2SLS}}\) may fail to exist, so “bias toward OLS” should not automatically be read as a statement about \(\E(\hat\beta_{\mathrm{2SLS}} - \beta)\). A fixed nonzero population first stage still yields consistency as \(n \to \infty\); weak instruments instead make the finite-sample approximation and conventional Wald inference unreliable.

The distinction matters because it identifies the correct remedy. If the problem were a bias in the usual sense, one would look for a bias correction. The problem is instead that the approximating distribution is wrong, which is why the remedy in Section 13.7.3 is a different test, not a different point estimator. The figure below conveys the contrast schematically.

Figure 13.1: Schematic comparison of the sampling behavior of \(\hat\beta_{\mathrm{2SLS}}\) under strong and weak first stages. With a strong first stage the distribution is approximately normal and concentrated at \(\beta\), and the conventional Wald interval is reliable. With a weak first stage the distribution is displaced toward the OLS probability limit \(\beta_{\mathrm{OLS}}\), skewed, and heavy-tailed; the conventional Wald interval is too short and centered in the wrong place, whereas the Anderson–Rubin set remains valid at the cost of being wide. The curves are conceptual and are not the exact density of any particular weak-instrument model.

13.7.2 First-Stage Diagnostics

The most widely used diagnostic for weak instruments is an \(F\)-statistic from the first-stage regression, testing the joint significance of the excluded instruments \(Z\) after partialling out \(X\). There is, however, no single “first-stage \(F\)”: at least three distinct statistics circulate under that name, each with its own critical values.

The conventional (homoskedastic) first-stage \(F\). This is the classical \(F\)-statistic for \(H_0\colon \pi = 0\) computed under homoskedastic, serially uncorrelated errors. Staiger and Stock (1997) argued informally for the rule of thumb \(F \geq 10\), and Stock and Yogo (2005) supplied formal critical values, tabulating thresholds at which either the 2SLS bias relative to OLS bias or the size of the nominal 5% Wald test exceeds a stated tolerance. These critical values depend on the number of instruments and on which of the two criteria is used; \(10\) is a rounded stand-in for one cell of that table, not a universal constant.
Robust and cluster-robust first-stage diagnostics. When the errors are heteroskedastic or correlated within clusters, the conventional \(F\) no longer has the distribution assumed by the Stock–Yogo tabulation. The robust analog is a Wald statistic for \(\pi = 0\) built from a robust or cluster-robust variance estimator; with multiple endogenous regressors, the Kleibergen–Paap rk Wald \(F\)-statistic plays this role.
The effective \(F\)-statistic. Montiel Olea and Pflueger (2013) proposed an effective \(F\)-statistic designed specifically for the case of heteroskedasticity, serial correlation, or clustering, with its own critical values indexed by a chosen tolerance for relative bias. It is formulated for a single endogenous regressor and permits multiple instruments; with several endogenous regressors, further diagnostics are required.

Do Not Compare a Robust \(F\) to the Threshold 10

The number \(10\) belongs to the conventional homoskedastic first-stage \(F\). A heteroskedasticity-robust, cluster-robust, or effective \(F\)-statistic should be compared with the critical value derived for that statistic, not with the conventional rule of thumb. Two further cautions apply to all three statistics. First, \(10\) is a coarse threshold even in its home setting: it may be far too lenient with multiple endogenous regressors or many instruments, and Lee et al. (2022) show that in just-identified models a much larger first-stage \(F\) is needed before a conventional \(t\)-test attains its nominal size. Second, and more fundamentally, every first-stage diagnostic is a relevance diagnostic. A large \(F\) says nothing about exogeneity or exclusion: a strong but invalid instrument passes the first-stage test with no difficulty.

Because the diagnostics are noisy and the thresholds are statistic-specific, a natural response is to stop conditioning inference on a preliminary strength test and instead use a procedure whose validity does not depend on first-stage strength at all. That is the subject of the next subsection.

13.7.3 Weak-Instrument-Robust Inference

The Anderson–Rubin (AR) test (Anderson and Rubin 1949) inverts the question. Rather than asking whether the data are consistent with a particular estimate, it asks whether a hypothesized value \(\beta_0\) is consistent with the IV moment condition. Substituting \(\beta_0\) into the structural model gives \[Y - \beta_0 T = \alpha + \gamma^\top X + \bigl\{\varepsilon + (\beta - \beta_0) T\bigr\}.\] If \(\beta_0 = \beta\), the composite error reduces to \(\varepsilon\) and is uncorrelated with \(Z\) by instrument validity. The AR test therefore regresses \(Y - \beta_0 T\) on \(Z\) and \(X\) and tests the null that the coefficient on \(Z\) is zero.

The decisive feature of this construction is that the tested restriction is linear in the observables once \(\beta_0\) is fixed: the null hypothesis has been converted into a statement about a reduced-form coefficient, and no division by an estimated first stage occurs anywhere. Under the classical homoskedastic Gaussian linear model the resulting \(F\)-statistic has an exact finite-sample \(F\) distribution regardless of instrument strength; without normality the same statistic has an asymptotic \(\chi^2_q/q\) distribution under homoskedasticity, and heteroskedasticity- and cluster-robust versions remain asymptotically valid more generally. In all cases the size of the test does not depend on \(\pi\).

Definition 13.5 (Anderson–Rubin Confidence Set) For a nominal level \(1 - a\), the Anderson–Rubin confidence set is obtained by test inversion: \[\mathcal{C}_{1-a} = \bigl\{\beta_0 : \text{the AR test of } H_0\colon\beta = \beta_0 \text{ is not rejected at level } a \bigr\}.\] In practice \(\mathcal{C}_{1-a}\) is computed by evaluating the AR statistic on a grid of candidate values \(\beta_0\) and retaining those that are not rejected; in the single-instrument case it can also be obtained in closed form by solving a quadratic inequality in \(\beta_0\).

Because the AR test has correct size at every \(\beta_0\) whatever the first-stage strength, \(\mathcal{C}_{1-a}\) has correct coverage whether or not the instrument is weak. What varies with instrument strength is not the validity of the set but its shape.

Reading an Unusual Anderson–Rubin Set

An inverted AR confidence set may be

wide or even unbounded (possibly the whole real line), when the instrument is too weak to rule out large values of \(\beta\);
disconnected, a union of intervals, when the quadratic inequality defining the set opens downward;
empty — but only in an overidentified model, when the maintained moment restrictions are rejected at every candidate value \(\beta_0\).

The last case cannot arise when the model is just identified with \(\hat\pi \neq 0\): at \(\beta_0 = \hat\phi/\hat\pi\) the estimated reduced-form coefficient on \(Z\) is exactly zero, so the AR statistic is zero and that value is never rejected. A just-identified AR set therefore always contains the point estimate and is never empty.

None of these outcomes is a numerical failure. They are substantive information. An unbounded set reports honestly that the data identify \(\beta\) weakly or not at all — precisely the information a spuriously short Wald interval conceals. An empty set is an overidentification rejection: no single value of \(\beta\) reconciles all the instruments, so the maintained model, not just the candidate parameter value, is in doubt.

What Weak-Instrument Robustness Does Not Deliver

The AR test is robust to weak instruments under maintained instrument validity. Its derivation uses \(\E[Z\varepsilon] = 0\) and the exclusion restriction at the very first step. If exogeneity or exclusion fails, the AR test is testing the wrong null, and no amount of identification-robustness repairs it. Identification strength and identification validity are separate questions, and only the first is addressed here.

Robustness to weak identification is also not automatic robustness to arbitrary heteroskedasticity or clustering. Those require the appropriate robust or cluster-robust covariance estimator for the reduced-form coefficient, together with the matching asymptotic reference distribution; the exact \(F\) result above holds only in the classical homoskedastic Gaussian model.

The conditional likelihood ratio (CLR) test of Moreira (2003) extends the AR idea more efficiently to the multiple-instrument case. With a single endogenous regressor and a single instrument the practical distinction between AR and CLR is small; with several instruments, CLR can be substantially more powerful, because AR spends degrees of freedom testing all \(q\) reduced-form restrictions jointly rather than concentrating power on the direction of interest.

Remark: Two Distinct IV Concerns

The weak-instrument problem is a statistical concern: it questions the quality of estimation and inference given a valid identification strategy. It is distinct from the estimand concern raised in Chapter 7. With a binary instrument and binary treatment under monotonicity, a valid instrument generally identifies a LATE rather than the population ATE; with a continuous or multi-valued treatment, it generally identifies an instrument-specific weighted average causal response. Instrument strength does not convert either local or instrument-specific estimand into a population ATE. Any complete IV analysis must address both concerns.

13.7.4 Applied Example: End-to-End Estimation for the KIPP Lottery

Section 7.11 used the KIPP Lynn admission lottery of Angrist et al. (2012) to illustrate what the IV assumptions identify. We now carry the same design through the estimation machinery of this chapter. Recall the structure: \(Z_i\) is the lottery offer, \(s_{igt}\) is years of KIPP attendance, \(y_{igt}\) is the MCAS score, and the model is just identified with one excluded instrument. The KIPP table of Chapter 7 reports, for math scores, a first stage of \(\hat\pi = 1.221\) \((0.068)\), a reduced form of \(\hat\phi = 0.430\) \((0.067)\), and a 2SLS estimate of \(\hat\beta = 0.352\) \((0.053)\), all with standard errors clustered at the student level and \(N = 833\) student-by-test observations.

Step 1: the point estimate. The model is just identified, so 2SLS is the Wald ratio of Section 13.2: \[\hat\beta_{\mathrm{2SLS}} = \frac{\hat\phi}{\hat\pi} = \frac{0.430}{1.221} = 0.352.\] Each additional year at KIPP raises math scores by about \(0.35\) standard deviations. Because the endogenous treatment is years of attendance rather than a binary attendance indicator, this is not a binary-complier LATE. Under the maintained monotone-dose interpretation it is a lottery-specific weighted average causal response per additional year of KIPP attendance, over the attendance margins the offer actually shifted — the interpretation given in Chapter 7.

Step 2: the robust standard error. Because students contribute several test observations, the moment function is correlated within student, and the cluster-robust form of the sandwich estimator Equation 13.15 described in Section 13.6 is the appropriate variance estimator; the reported \(0.053\) is of this form. Equivalently, by the delta method applied to the ratio \(\hat\phi/\hat\pi\), \[\widehat{\mathrm{se}}(\hat\beta) = \frac{1}{|\hat\pi|}\Bigl\{\widehat{\mathrm{se}}(\hat\phi)^2 - 2\hat\beta\,\widehat{\mathrm{Cov}}(\hat\phi, \hat\pi) + \hat\beta^{2}\,\widehat{\mathrm{se}}(\hat\pi)^2\Bigr\}^{1/2}.\] Matching this expression to the reported standard errors implies a correlation of about \(0.27\) between \(\hat\phi\) and \(\hat\pi\). That covariance is not itself reported; it is backed out from rounded published estimates and standard errors, so every quantity below that uses it is a reconstruction from summary statistics rather than a direct analysis of the microdata. The conventional 95% Wald interval is \(0.352 \pm 1.96 \times 0.053 = (0.248,\, 0.456)\).

Step 3: the strength diagnostic. The reported first-stage standard error is cluster-robust, so the statistic this computes is the cluster-robust first-stage Wald \(F\): the \(t\)-ratio is \(1.221/0.068 = 17.96\), giving \(F = 17.96^2 \approx 322\). Because the model is just identified with a single endogenous regressor, this statistic also coincides with the effective \(F\) of Section 13.7.2, and it is far above the relevant effective-\(F\) critical values, so weak identification is not a practical concern in this application. Note what has and has not been computed: the conventional homoskedastic first-stage \(F\) is a different statistic and has not been calculated here. This is the expected outcome for a well-run admission lottery: the offer is a direct administrative determinant of enrollment.

Step 4: the identification-robust interval. With a single instrument, the AR statistic for \(H_0\colon\beta = \beta_0\) is the squared \(t\)-ratio of the coefficient on \(Z\) in the regression of \(y_{igt} - \beta_0 s_{igt}\) on the covariates and \(Z\). That coefficient is estimated by \(\hat\phi - \beta_0\hat\pi\), so \[\mathrm{AR}(\beta_0) = \frac{(\hat\phi - \beta_0\hat\pi)^2}{\widehat{\mathrm{se}}(\hat\phi)^2 - 2\beta_0\widehat{\mathrm{Cov}}(\hat\phi, \hat\pi) + \beta_0^2\,\widehat{\mathrm{se}}(\hat\pi)^2},\] and the 95% AR set collects the \(\beta_0\) with \(\mathrm{AR}(\beta_0) \le 1.96^2\). This inequality is quadratic in \(\beta_0\); solving it with the covariance implied by the rounded reported estimates gives the approximate reconstructed set \[\mathcal{C}_{0.95} \approx (0.249,\; 0.458),\] essentially indistinguishable from the Wald interval \((0.248,\, 0.456)\). That agreement is the point: when the first stage is strong, the identification-robust set and the conventional interval nearly coincide, and the extra work buys reassurance rather than a different answer.

Step 5: what a weak first stage would have done. The coefficient on \(\beta_0^2\) in the quadratic inequality is \(\hat\pi^2 - 1.96^2\,\widehat{\mathrm{se}}(\hat\pi)^2\), which is positive precisely when the first-stage \(F\) exceeds \(1.96^2 = 3.84\). Suppose counterfactually that the lottery had moved attendance by only \(\hat\pi = 0.10\) years with the same standard error \(0.068\) (so \(F \approx 2.2\)), with a reduced form of \(\hat\phi = 0.035\) \((0.067)\) chosen to leave the point estimate at \(0.35\). The conventional Wald interval would then be \((-0.918,\, 1.618)\): wide, but bounded and apparently informative. The AR set, by contrast, would be the entire real line. Both procedures use the same data; only one of them reports honestly that those data exclude nothing.

Step 6: interpretation. Four distinct claims have been kept separate throughout, and a complete report should keep them separate too.

Instrument validity — exogeneity holds by randomization of the lottery; exclusion is an assumption, defended institutionally in Section 7.11 and not testable here, since the model is just identified.
Instrument strength — established empirically by \(F \approx 322\), which licenses the conventional normal approximation.
The estimand — with heterogeneous effects, \(0.352\) is a lottery-specific weighted average causal response per additional year of attendance, over the dose margins shifted by the offer (Chapter 7). It is neither a binary-complier LATE nor the population ATE.
Sampling uncertainty — quantified by the cluster-robust interval \((0.248,\, 0.456)\), confirmed by the AR set \((0.249,\, 0.458)\).

Chapter 7 supplied the first and third of these; this chapter supplies the second and fourth.

Advanced Enrichment: Section 13.8

The core development of the chapter is complete. The remaining section presents one extension beyond it. It presupposes familiarity with Section 13.2–Section 13.7 but can be omitted without loss of continuity, and nothing later in the book depends on it. Section 13.8 introduces generalized empirical likelihood (GEL) as a one-step alternative to efficient GMM for the same moment restrictions. Its convex-duality derivation, minimum-discrepancy representation, computational algorithm, extended special-cases table, and additional problems are collected in the Chapter 13 Supplement.

13.8 Generalized Empirical Likelihood

13.8.1 GEL as a One-Step Moment Estimator

Section 13.6 showed that efficient GMM attains the bound \((A^\top\Sigma^{-1}A)^{-1}\) but requires two steps: a preliminary consistent estimator is used to form \(\hat\Sigma\), which is then plugged in as the weighting matrix for a second optimization. That structure raises a practical question of how many iterations to run, and the estimated weighting matrix contributes a source of finite-sample bias (Newey and Smith 2004).

Generalized empirical likelihood (GEL) is a one-step alternative that uses exactly the same moment restrictions. The contrast with GMM is structural rather than substantive:

GMM minimizes a quadratic form in the sample moments, with a separately estimated weighting matrix; GEL profiles a concave criterion over a multiplier attached to the same moments, and the optimal weighting emerges from that inner problem rather than being supplied in advance.

Retain the notation of Section 13.6: let \(U_i(\theta) = U(O_i;\theta) = W_i(Y_i - D_i^\top\theta)\) denote the \(m\)-dimensional moment function, and write \(\bar{U}(\theta) = n^{-1}\sum_{i=1}^n U_i(\theta)\).

GEL introduces a strictly convex function \(G\colon \mathcal{V} \to \mathbb{R}\), where \(\mathcal{V}\) is an open interval containing zero. The GEL estimator solves the saddle-point problem \[\hat\theta_{\mathrm{GEL}} = \arg\min_{\theta}\;\sup_{\lambda \in \Lambda_n(\theta)}\;\frac{1}{n}\sum_{i=1}^n \bigl[-G\!\left(\lambda^\top U_i(\theta)\right)\bigr], \tag{13.17}\] where \(\lambda \in \mathbb{R}^m\) is an auxiliary dual variable and \(\Lambda_n(\theta) = \{\lambda : \lambda^\top U_i(\theta) \in \mathcal{V},\;i = 1,\ldots,n\}\) is the feasible set for \(\lambda\) given \(\theta\). No preliminary weighting matrix appears anywhere in Equation 13.17: the inner supremum over \(\lambda\) performs the role of weighting implicitly. The sign and normalization convention of Equation 13.17 is used throughout the chapter and the supplement.

We normalize \(G\) to satisfy \[G(0) = 0, \qquad g(0) = 1, \qquad g'(0) = 1, \tag{13.18}\] writing \(g = G'\). For the regular generators considered here — those with nonzero first derivative and positive finite second derivative at zero, as all named members have — the conditions \(g(0) = g'(0) = 1\) can be imposed by an affine rescaling of the criterion together with a corresponding rescaling of \(\lambda\), without changing the estimator of \(\theta\); they ensure that the GEL first-order conditions reduce to the efficient GMM conditions at the population level. The condition \(G(0) = 0\) is a normalization of the additive constant of \(G\); it likewise leaves the estimator unchanged, but it pins down the correspondence between \(G\) and its convex conjugate in the duality developed in the supplement.

Remark: The Dual Variable as Observation Weights

At the solution, the quantities \(\hat\omega_i = g(\hat\lambda^\top U_i(\hat\theta))\) act as weights attached to individual observations, and normalizing them to sum to one produces a re-weighted weight vector — an empirical distribution proper when the weights are positive, as they are for EL and ET — under which the sample moment condition holds exactly. This is the sense in which GEL re-weights observations where GMM re-weights moments. The precise statement — including the distinction between the raw weights \(\hat\omega_i\) and the normalized probabilities \(\hat\pi_i\), which is easy to get wrong — is developed in the Chapter 13 Supplement: the exact profile identities in its section on profile weights, and the normalization question alongside them.

13.8.2 Representative Members: EL, ET, and CUE

Different choices of \(G\) give different members of the family. Three are standard.

Definition 13.6 (Representative Members of the GEL Family)

Empirical likelihood (EL): \(G(v) = -\log(1 - v)\), \(\mathcal{V} = (-\infty, 1)\).
Exponential tilting (ET): \(G(v) = e^v - 1\), \(\mathcal{V} = \mathbb{R}\).
Continuous updating estimator (CUE): \(G(v) = v + v^2/2\), \(\mathcal{V} = \mathbb{R}\).

Each satisfies the normalization Equation 13.18.

The third case connects the GEL family directly back to GMM.

Theorem 13.4 (CUE as a GEL Estimator (Newey and Smith 2004)) If \(G(v)\) is quadratic (i.e. \(G(v) = v + v^2/2\) up to rescaling), then \(\hat\theta_{\mathrm{GEL}} = \hat\theta_{\mathrm{CUE}}\), where the continuous updating estimator solves \[\hat\theta_{\mathrm{CUE}} = \arg\min_\theta\;\bar{U}(\theta)^\top\Bigl[n^{-1}{\textstyle\sum_i} U_i(\theta)U_i(\theta)^\top\Bigr]^{-1}\bar{U}(\theta).\]

Unlike two-step efficient GMM, CUE uses a weighting matrix that is continuously updated as \(\theta\) changes, so no preliminary estimator is needed. It has the same first-order asymptotics as efficient GMM but in practice requires a global nonlinear optimizer, because the criterion is no longer quadratic in \(\theta\). Among the three named members, CUE is the most transparent computationally and EL is often preferred for its bias properties; ET, whose weights \(e^{\lambda^\top U_i}\) are positive and defined for every real argument, avoids the domain restriction that constrains EL. Additional members, the full table of generating functions and their conjugates, and the two-loop algorithm used in practice are given in the Chapter 13 Supplement.

13.8.3 First-Order Equivalence with Efficient GMM

The central asymptotic result is that the choice within the family, and the choice between the family and efficient GMM, is asymptotically immaterial.

Theorem 13.5 (First-Order Asymptotic Equivalence (Newey and Smith 2004)) Suppose the moment condition \(\E[U(O;\theta_0)] = 0\) holds with \(\theta_0\) in the interior of a compact parameter space of fixed finite dimension, \(O_i\) are i.i.d. with sufficient moments, the rank condition holds so that identification is strong in the sense required at the start of Section 13.7, and \(G\) satisfies the regularity conditions of Newey and Smith (2004). Then for any GEL estimator, \[\sqrt{n}\,\bigl(\hat\theta_{\mathrm{GEL}} - \theta_0\bigr) \;\xrightarrow{d}\; N\!\left(0,\; (A^\top\Sigma^{-1}A)^{-1}\right),\] the same limiting distribution as the efficient GMM estimator.

The theorem is stated under the same strong-identification premise flagged in Section 13.7; it is a statement about which estimators agree in large samples, not a statement about which approximations are trustworthy in a given data set. Each GEL member also supplies an overidentification statistic — a profile-divergence analog of the Sargan–Hansen \(J\)-statistic of Section 13.5 — with the same \(\chi^2_{m-k}\) null limit; its exact form and its relation to \(J\) are derived in the Chapter 13 Supplement.

13.8.4 What GEL Does and Does Not Improve

Differences among asymptotically equivalent estimators appear at higher order.

Remark: A Qualified Higher-Order Statement

Newey and Smith (2004) show that, to higher asymptotic order, GEL estimators remove the bias component that arises in two-step GMM from estimating the Jacobian \(\E[\partial U_i(\theta_0)/\partial\theta^\top]\), and that EL additionally removes the component arising from estimating the weighting matrix \(\Sigma\). These are genuine improvements in models with several overidentifying restrictions, the number of moment conditions being held fixed, where such bias components are a real concern. They are also particular higher-order components, established under additional smoothness and moment assumptions, and they are not statements about many-instrument asymptotics in the modern sense of \(m\) growing with \(n\). No universal finite-sample ranking of GEL over two-step GMM follows, and none should be asserted: which estimator performs better in a given design remains an empirical question.

The Boundary of What GEL Solves

GEL changes how a given moment model is estimated. It does not change the model. In particular, GEL does not solve

invalid instruments — if \(\E[W\varepsilon] \neq 0\), every member of the family is inconsistent, exactly as GMM is;
weak identification — Theorem 13.5 assumes strong identification, and the fragility described in Section 13.7 applies to GEL as well;
incorrect structural moment conditions — a misspecified \(U_i(\theta)\) is misspecified under any estimation criterion;
poor support or finite-sample feasibility problems. For the positive-weight members EL, ET, and Hellinger, feasibility requires the origin to lie in the appropriate convex hull of the sample moments \(\{U_i(\theta)\}\), which can fail for small \(n\) or many moments. CUE permits signed weights and imposes no convex-hull requirement; it instead requires nonsingularity of the sample second-moment matrix entering its quadratic inner problem.

GEL belongs to the same maintained model as GMM, and inherits every identification assumption of that model.

Remark: Where the Rest of the GEL Theory Lives

The Chapter 13 Supplement develops the material omitted here: the Fenchel-conjugate derivation and the minimum-discrepancy primal problem, the raw-versus-normalized weight distinction and the exact profile identities, the full table of generators, conjugates, and implied divergences, the two-loop computational algorithm, higher-order comparisons, and additional problems.

13.9 Chapter Summary

This chapter developed the main estimation tools for the linear instrumental-variable model. The organizing principle is the same as in Chapter 10: identification provides a target, and the estimator is a sample analog of the identification formula. Chapter 7 established what the IV assumptions identify; this chapter showed how the resulting moment restrictions are estimated, how the associated uncertainty is quantified, and why weak identification requires different inferential tools. The chapter’s architecture is the single chain \[\text{identification} \;\longrightarrow\; \text{sample moments} \;\longrightarrow\; \text{2SLS/GMM} \;\longrightarrow\; \text{robust inference},\] and the key results along it are the following.

Wald estimator, reduced form, and first stage. The Wald estimator is the direct sample analog of the Wald estimand for a binary instrument with no covariates: the ratio of the reduced-form difference-in-means to the first-stage difference-in-means. The first stage answers how strongly the instrument moves treatment; the reduced form answers how the instrument changes the outcome; the ratio converts the intent-to-treat effect into a per-unit-of-treatment causal effect under the IV assumptions.
IV regression estimator and 2SLS. The IV regression estimator extends the Wald estimator to the general single-instrument setting with covariates \(X\), as the ratio of within-\(X\) instrument-outcome to instrument-treatment sample covariances, derived from the valid sample moment equations. Two-stage least squares extends this further to multiple instruments: the first stage projects \(T\) onto \((Z, X)\) and the second stage regresses \(Y\) on the fitted values and \(X\).
Equivalence and moment conditions. The IV regression estimator and 2SLS are numerically identical in the single-instrument case for any \(X\) (Theorem 13.1), with the Wald estimator as the further special case \(Z \in \{0,1\}\), \(X\) absent. Both are method-of-moments estimators based on the IV orthogonality condition \(\E[W\varepsilon] = 0\), an instance of the general estimating-equation framework of Chapter 10.
GMM and efficient weighting. In an exactly identified model all positive-definite weighting matrices give the same estimator and the same asymptotic variance, so 2SLS and efficient GMM coincide whatever the conditional variance structure. In overidentified models, efficient GMM generalizes 2SLS and achieves the semiparametric efficiency bound within the maintained IV moment-restriction model; homoskedasticity is sufficient for 2SLS to attain that bound, while under general heteroskedasticity overidentified 2SLS need not, and optimally weighted GMM is weakly (often strictly) more efficient.
Robust variance estimation under strong identification. Provided the population Jacobian \(A = \E(WD^\top)\) has full rank and is bounded away from singularity, the asymptotic distribution of the GMM estimator is normal with sandwich variance \(V_{\mathrm{GMM}} = (A^\top\Omega A)^{-1} A^\top\Omega\Sigma\Omega A (A^\top\Omega A)^{-1}\), where \(A\) is the sensitivity matrix and \(\Sigma\) is the moment variance. In applications, the default should be heteroskedasticity-robust standard errors; cluster-robust standard errors are required when observations within groups share common shocks.
Weak instruments and identification-robust inference. Strong identification is a premise, not a guarantee. When the first stage is weak, the sampling distribution of \(\hat\beta_{\mathrm{2SLS}}\) is displaced toward the OLS probability limit and is neither normal nor well approximated by one, and conventional Wald intervals undercover. “Bias toward OLS” is a statement about the center of a limiting approximation, not necessarily about \(\E(\hat\beta_{\mathrm{2SLS}} - \beta)\), whose finite-sample moments may not exist. First-stage diagnostics come in conventional, robust, and effective versions with different critical values, and all of them speak only to relevance, never to validity. Anderson–Rubin confidence sets, obtained by inverting a test of \(H_0\colon\beta = \beta_0\), have correct coverage regardless of first-stage strength; they may be wide, unbounded, or disconnected, and — in an overidentified model — empty. Each of those shapes is informative rather than a malfunction.
GEL as an optional extension. Generalized empirical likelihood provides alternative one-step implementations of the same moment restrictions — empirical likelihood, exponential tilting, and the continuous updating estimator among them — and shares the first-order limiting distribution of efficient GMM under standard regularity and strong-identification conditions (Theorem 13.5). Higher-order bias improvements are available for particular components under additional conditions; no universal finite-sample ranking follows, and GEL changes only how a moment model is estimated, never whether the model is correct.

Remark: Omitted Topics and Further Reading

Two topics adjacent to this chapter are deliberately left out. The control function approach rewrites the endogeneity problem by augmenting the outcome regression with the first-stage residual; in the linear model it reproduces the 2SLS coefficient exactly, so it adds a representation rather than an estimator, and its useful extensions require a triangular structural model with a scalar monotone first-stage disturbance — a different set of assumptions from those maintained here (Imbens and Newey 2009). Many-instrument and many-weak-instrument asymptotics, in which the number of moment conditions grows with \(n\), likewise change the approximating sequence rather than the estimator. Both belong to a separate treatment of nonlinear and high-dimensional IV models.

More generally, this chapter stays within a low-dimensional linear model with a fixed number of moment conditions. The general principles of orthogonal scores, sample splitting, and cross-fitting — which allow nuisance components to be estimated by machine learning at slower than \(\sqrt{n}\) rates — are developed in Chapter 12, with the ATE as the principal example. Their IV-specific implementations, including orthogonalized first-stage and covariate-adjustment scores, lie outside the scope of these notes.

13.10 Exercises

1. The Wald estimator as a ratio-of-moments estimator. Consider the binary-instrument, no-covariate setting of Section 13.2. Let \(\mu_Y(z) = \E[Y \mid Z = z]\) and \(\mu_T(z) = \E[T \mid Z = z]\) for \(z \in \{0,1\}\), and write \(\Delta_Y = \mu_Y(1) - \mu_Y(0)\), \(\Delta_T = \mu_T(1) - \mu_T(0)\).

Augment \(\beta\) with an intercept parameter \(\alpha\) and express the structural model \(Y = \alpha + \beta T + \varepsilon\) as the solution to the two-dimensional moment condition \[\E\!\left[\begin{pmatrix} 1 \\ Z \end{pmatrix}(Y - \alpha - \beta T)\right] = 0,\] so that \(U(O;\,\alpha,\beta) = (1, Z)^\top (Y - \alpha - \beta T) \in \mathbb{R}^2\). Verify that the system has two moment conditions and two parameters \((\alpha, \beta)\), so the order condition for exact identification is satisfied. State the accompanying rank condition — here \(\Delta_T \neq 0\) — and explain why counting equations alone does not guarantee a unique solution. Then solve the system and recover \(\beta = \Delta_Y / \Delta_T\) together with an expression for \(\alpha\).
Using the fact that \(\bar{Y}_z \xrightarrow{p} \mu_Y(z)\) and \(\bar{T}_z \xrightarrow{p} \mu_T(z)\), prove that \(\hat\beta_{\mathrm{Wald}} \xrightarrow{p} \beta\) via the continuous mapping theorem. State the condition on \(\Delta_T\) that is required.
Apply the delta method to the vector \((\hat\Delta_Y, \hat\Delta_T)^\top\) to show that \[\sqrt{n}\,(\hat\beta_{\mathrm{Wald}} - \beta) \;\xrightarrow{d}\; N\!\left(0,\;\frac{1}{\Delta_T^2}\sum_{z \in \{0,1\}}\frac{\mathrm{Var}(Y_i - \beta T_i \mid Z_i = z)}{p_z}\right),\] where \(p_z = \Pr(Z = z)\). Confirm that this matches the IV variance formula Equation 13.13 in the scalar no-covariate case (\(p = 0\)).

2. Matrix form of 2SLS and why the second-stage standard errors are wrong. Stack observations into matrices: let \(\mathbf{Y} \in \mathbb{R}^n\), \(\mathbf{D} \in \mathbb{R}^{n \times k}\) be the full regressor matrix (including the intercept, \(T\), and \(X\)), and \(\mathbf{W} \in \mathbb{R}^{n \times m}\) be the full instrument matrix (including the intercept, \(X\), and \(Z\)). Let \(P_{\mathbf{W}} = \mathbf{W}(\mathbf{W}^\top\mathbf{W})^{-1}\mathbf{W}^\top\) be the orthogonal projection onto the column space of \(\mathbf{W}\).

Show that the 2SLS estimator can be written as \(\hat\theta_{\mathrm{2SLS}} = (\mathbf{D}^\top P_{\mathbf{W}}\mathbf{D})^{-1}\mathbf{D}^\top P_{\mathbf{W}}\mathbf{Y}\). (Hint: the first stage replaces \(\mathbf{D}\) by its projection \(\hat{\mathbf{D}} = P_{\mathbf{W}}\mathbf{D}\); then apply OLS to the second-stage regression of \(\mathbf{Y}\) on \(\hat{\mathbf{D}}\), using idempotency of \(P_{\mathbf{W}}\).)
In the single-instrument, no-covariate case (\(k = m = 2\), intercept plus \(T\) or \(Z\)), verify from the matrix formula above that \(\hat\beta_{\mathrm{2SLS}} = \hat\beta_{\mathrm{IV}}\), reproducing Theorem 13.1 without the Frisch–Waugh–Lovell argument.
The second-stage OLS regression uses the fitted regressor matrix \(\hat{\mathbf{D}}\) in place of \(\mathbf{D}\). Let \(\hat{\boldsymbol{\varepsilon}}_{\mathrm{2nd}} = \mathbf{Y} - \hat{\mathbf{D}}\hat\theta_{\mathrm{2SLS}}\) denote the second-stage residuals actually used by OLS. Show that \(\hat{\boldsymbol{\varepsilon}}_{\mathrm{2nd}} \neq \mathbf{Y} - \mathbf{D}\hat\theta_{\mathrm{2SLS}}\) in general. Explain why this discrepancy makes the second-stage OLS standard errors invalid, and identify the correct residuals that should enter the sandwich variance estimator Equation 13.15.

3. Efficiency of GMM and the Sargan–Hansen \(J\)-statistic.

Prove that \(V_{\mathrm{GMM}}(\Omega) \succeq V_{\mathrm{eff}}\) for every positive-definite \(\Omega\), where \(V_{\mathrm{eff}} = (A^\top\Sigma^{-1}A)^{-1}\) is the efficient GMM variance Equation 13.14. (Hint: factor \(V_{\mathrm{GMM}}(\Omega) - V_{\mathrm{eff}}\) as a product of the form \(C^\top \Sigma^{-1} C\) for a suitable matrix \(C\) by completing the square; recall that \(B^\top\Sigma^{-1}B \succeq 0\) for any matrix \(B\) and positive-definite \(\Sigma\).)
Under homoskedasticity, show that 2SLS is the efficient GMM estimator by verifying that the 2SLS weighting matrix \(\Omega_{\mathrm{2SLS}} = \E[W_i W_i^\top]^{-1}\) is a scalar multiple of \(\Sigma^{-1}\). Conclude that no efficiency gain over 2SLS is possible in the homoskedastic linear IV model.
Return to Example 13.1 with two instruments and no covariates. The GMM estimator under identity weighting gives \(\hat\beta \approx 0.764\) and residual sample moments \(\hat{U}_1 = 0.018\), \(\hat{U}_2 = -0.011\). Treating \(\hat\Sigma = I_2\) and \(n = 200\), compute the Sargan–Hansen \(J\)-statistic \(J = n\,\hat{U}(\hat\beta)^\top\hat\Sigma^{-1}\hat{U}(\hat\beta)\) and determine, using the \(\chi^2_1\) critical value at the 5% level, whether the overidentifying restriction is rejected. Interpret the result in terms of the joint maintained moment restrictions, and explain why failure to reject does not establish that either instrument is valid.

4. Constructing a cluster-robust sandwich by hand. Consider a just-identified IV model with \(W_i = (1, Z_i)^\top\) and \(D_i = (1, T_i)^\top\), so \(m = k = 2\). Suppose \(n = 6\) observations fall into \(G = 3\) clusters, \(c_1 = \{1,2\}\), \(c_2 = \{3,4\}\), \(c_3 = \{5,6\}\), and that the estimated moment contributions \(\hat\varepsilon_i W_i \in \mathbb{R}^2\) are

\(\hat\varepsilon_1 W_1 = (0.6,\ 1.2)^\top\)	\(\hat\varepsilon_2 W_2 = (0.4,\ 0.8)^\top\)	\(\hat\varepsilon_3 W_3 = (-0.5,\ 0.3)^\top\)
\(\hat\varepsilon_4 W_4 = (-0.3,\ -0.1)^\top\)	\(\hat\varepsilon_5 W_5 = (0.2,\ -0.7)^\top\)	\(\hat\varepsilon_6 W_6 = (-0.4,\ -1.5)^\top\)

Compute the three cluster scores \(\hat{S}_c = \sum_{i \in c}\hat\varepsilon_i W_i\), and then form \(\hat\Sigma_{\mathrm{cl}}\) using Equation 13.16.
Compute the heteroskedasticity-robust \(\hat\Sigma = n^{-1}\sum_i \hat\varepsilon_i^2 W_i W_i^\top\) for the same data, and compare the two. Which entries grow, and what feature of the within-cluster scores explains the difference? Construct a rearrangement of the same six vectors into three clusters for which both diagonal elements of \(\hat\Sigma_{\mathrm{cl}}\) are smaller than the corresponding diagonal elements of \(\hat\Sigma\), and explain why clustering is not automatically conservative.
Suppose \(\hat{A} = n^{-1}\sum_i W_i D_i^\top\) equals \(\bigl(\begin{smallmatrix} 1 & 2 \\ 0.5 & 3 \end{smallmatrix}\bigr)\). Form the exactly identified sandwich \(\hat{V} = \hat{A}^{-1}\hat\Sigma_{\mathrm{cl}}(\hat{A}^\top)^{-1}\) and report the cluster-robust standard error of \(\hat\beta\). Do not forget the factor \(1/n\).
State the conditions under which the cluster-robust approximation is justified, and explain why \(G = 3\) would be inadequate in practice even though the arithmetic goes through.

5. The continuous updating estimator as a GEL member. For the continuous updating estimator (CUE), the GEL generating function is \(G(v) = v + v^2/2\) on \(\mathcal{V} = \mathbb{R}\).

Verify that \(G\) satisfies the GEL normalization Equation 13.18.
Solve the inner supremum over \(\lambda\) in Equation 13.17 explicitly at fixed \(\theta\): show that \[\hat\lambda = -\Bigl[n^{-1}{\textstyle\sum_i} U_i(\theta)U_i(\theta)^\top\Bigr]^{-1}\bar{U}(\theta),\] and substitute back to confirm that the profile objective equals \(\tfrac{1}{2}\bar{U}(\theta)^\top[n^{-1}\sum_i U_i(\theta)U_i(\theta)^\top]^{-1}\bar{U}(\theta)\), establishing Theorem 13.4.
Explain why the factor \(\tfrac{1}{2}\) in part (b) has no effect on \(\hat\theta_{\mathrm{CUE}}\), and why the objective in Theorem 13.4, unlike the two-step efficient GMM criterion, is not quadratic in \(\theta\) even in the linear IV model.

6. Weak-instrument-robust inference by Anderson–Rubin. Consider the linear IV model \(Y_i = \alpha + \beta T_i + \varepsilon_i\) with a scalar instrument \(Z_i\). Maintain the joint moment restriction \[\E\!\left[\begin{pmatrix} 1 \\ Z_i \end{pmatrix} \varepsilon_i\right] = 0,\] that is, \(\E[\varepsilon_i] = 0\) together with \(\E[(Z_i - \E Z_i)\varepsilon_i] = 0\), but do not assume that the first-stage coefficient is bounded away from zero.

For a proposed value \(\beta_0\), define \(R_i(\beta_0) = Y_i - \beta_0 T_i\). Show that under \(H_0\colon \beta = \beta_0\), \(\E[Z_i\{R_i(\beta_0) - \alpha\}] = 0\).
Explain why testing \(H_0\) can therefore be implemented by testing the coefficient on \(Z_i\) in a regression of \(R_i(\beta_0)\) on an intercept and \(Z_i\), and identify where the intercept moment \(\E[\varepsilon_i] = 0\) is used. (With only \(\E[Z_i\varepsilon_i] = 0\) and no intercept restriction, the population slope in this regression is \(\mathrm{Cov}(Z_i, \varepsilon_i)/\mathrm{Var}(Z_i)\), which need not vanish.) State how the procedure should be modified when additional exogenous covariates \(X_i\) are included, using the residualized instrument \(\tilde{Z}_i = Z_i - L(Z_i \mid 1, X_i)\).
Explain why the null-size validity of this test does not require the first-stage coefficient to be bounded away from zero. Contrast this with the Wald interval built from Theorem 13.3, and identify exactly which step of that construction fails when \(\pi \approx 0\).
Describe how inversion of the test over candidate values \(\beta_0\) produces an Anderson–Rubin confidence set (Definition 13.5).
Fix a significance level \(a \in (0,1)\) (the symbol \(\alpha\) is in use as the structural intercept). Write the level-\(a\) AR set as the solution of a quadratic inequality \(c_2\beta_0^2 + c_1\beta_0 + c_0 \le 0\) in the just-identified case, using the reduced-form and first-stage coefficient estimates \((\hat\phi, \hat\pi)\) and their joint covariance matrix. Show that the sign of \(c_2\) is determined by whether the first-stage Wald statistic \(\hat\pi^2 / \widehat{\mathrm{Var}}(\hat\pi)\) exceeds the critical value, and use this to enumerate the possible shapes of the set: a bounded interval, an unbounded or disconnected set, a half-line in the degenerate linear case, or the whole real line.
Show that in this just-identified setting the AR set can never be empty when \(\hat\pi \neq 0\), by evaluating the AR statistic at \(\beta_0 = \hat\phi/\hat\pi\). Then explain why an AR set can be empty in an overidentified model, and what an empty set says about the maintained moment restrictions.
Reproduce the two numerical calculations of Section 13.7.4: the AR set for the reported KIPP first stage \(\hat\pi = 1.221\), and the AR set for the counterfactual weak first stage \(\hat\pi = 0.10\). Confirm that the first is essentially the Wald interval and the second is the whole real line, and explain in one or two sentences what the contrast shows about relying on a Wald interval when the first stage is uncertain.
Explain why weak-instrument robustness does not protect the procedure against violation of instrument exogeneity or exclusion. In particular, explain what the AR test is testing when the exclusion restriction fails, and why an empty AR set is evidence against the maintained model rather than against any particular value of \(\beta\).

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