Nonparametric and semiparametric IV models

The paper applies its general local identification theory to obtain new, primitive identification conditions for nonseparable quantile instrumental variable models and semiparametric single-index IV models.

“We apply these results to obtain new, primitive identification conditions in several important models, including nonseparable quantile instrumental variable (IV) models, single-index IV models, and semiparametric consumption-based asset pricing models.”◌ not checked against the paper’s text as it now stands

It provides a primitive local identification result for nonparametric endogenous quantile IV models where such conditions had previously only been established for discrete regressors.

“One example gives primitive conditions for local identification of the nonparametric endogenous quantile models, where primitive identification conditions had only been given previously for discrete regressors.”◌ not checked against the paper’s text as it now stands

It connects its nonparametric rank condition to the completeness condition used in instrumental variable identification, and treats it as a weak condition when instruments are as numerous as endogenous regressors, analogous to unrestricted reduced form in linear IV models.

“For 𝒩^'=𝒜 this is the completeness condition discussed in Newey and Powell (2003). Andrews (2011) has recently shown that if X and W are continuously distributed, there are at least as many instruments in W as regressors in X, and the conditional distribution of X given W is unrestricted (except for a mild regularity condition), then the completeness condition holds generically, in a sense defined in that paper. In Section 3 we also give a genericity result for a different range of models. For this reason we think of Assumption 1 with 𝒩 ^'=𝒜 as a weak condition when there are as many continuous instruments W as the endogenous regressors X, just as it is in a parametric linear instrumental variables model with unrestricted reduced form.”◌ not checked against the paper’s text as it now stands

The paper notes that its conditional moment restriction framework, central to instrumental variable settings, is distinct from and more general than prior local identification work such as Florens and Sbai (2010), which did not apply to IV-type conditional moment restrictions.

“Florens and Sbai (2010) gave local identification conditions for games but their conditions do not apply to the kind of conditional moment restrictions that arise in instrumental variable settings and are a primary subject of this paper.”◌ not checked against the paper’s text as it now stands

The paper derives sharp bounds on the ACE under the nonparametric IV model using four different sets of independence assumptions on potential outcomes.

“Under any of the assumptions,,,, the set of possible joint distributions P(Y(x_0), Y(x_1)) are characterized by the 8K inequalities:”◌ not checked against the paper’s text as it now stands

The authors show that their bounds reduce to the classic Balke-Pearl bounds when the instrument has only two states.

“In the case where K=2, these bounds reduce to those given by […], who assume (i).”◌ not checked against the paper’s text as it now stands

They clarify that the natural bounds derived under weaker mean-independence IV assumptions are not sharp without additional no-defiers assumptions.

“[…] and […] derived what are called the “natural bounds” on the ACE under the weaker assumption that Z Y(x_0) and Z Y(x_1). As noted by Imbens, without further assumptions these bounds are not sharp. However, the natural bounds are sharp under (i) or (iii), if, in addition, we assume there are no Defiers (an assumption that has testable implications).”◌ not checked against the paper’s text as it now stands

The paper formalizes a graphical representation (SWIG) of the instrumental variable model that encodes the independence assumptions among the instrument and potential outcomes.

The paper explicitly notes that uniform confidence bands in nonparametric instrumental variables (NPIV) models exemplify the broader class of ill-posed inverse problems to which deconvolution belongs.

“See […] for uniform confidence bands in the context of nonparametric instrumental variables (NPIV) models, one of the popular classes of econometric models with ill-posedness.”◌ not checked against the paper’s text as it now stands

The paper explicitly relates its identification strategy to nonseparable IV models, noting similarity to prior IV-based identification approaches for nonseparable models.

“This identification approach is similar to that of […], […], and […], who all identify nonseparable IV models.”◌ not checked against the paper’s text as it now stands

It highlights that these related IV identification papers use the same normalization assumption as the one used in this paper.

“[…] and […] use the same normalization as Assumption I3'.”◌ not checked against the paper’s text as it now stands

The paper adapts a construction from a nonseparable IV identification result, showing that identifying a monotone transformation function analogous to those in IV settings allows point identification of the structural function.

“[…] show that under appropriate assumptions, if for all x and x', we identify the function T_x',x(y) that is strictly increasing in y and satisfies g(x',u) = T_x',x(g(x,u)), then we can identify the structural function g(x,u). We can also construct similar functions and show that g_t is point identified.”◌ not checked against the paper’s text as it now stands

The paper defines the instrumental variable Z as shifting the disutility of working (selection equation) while being independent of the outcome shocks.

“where (μ, ν(Z)) are the means of the outcome and selection, Z is an instrumental variable that shifts the disutility of working but does not affect the offered wage, and (U,V) are centered stochastic shocks independent of Z.”◌ not checked against the paper’s text as it now stands

The paper shows that point identification of the latent offered wage distribution is achieved as long as the instrumental variable Z takes on at least two values, and demonstrates overidentification and testability when Z has more than two values.

“Our contribution is to demonstrate that the distribution of the latent offered wage Y^* is still identified from the observed distribution of wages and employment if the instrumental variable Z takes on at least two values, and to provide tractable estimation and inference methods.”◌ not checked against the paper’s text as it now stands

The paper formally states exclusion restrictions on the instrumental variable Z (non-degeneracy, relevance, outcome exclusion, and selection sorting exclusion) needed for identification.

“[Exclusion Restrictions] There is a binary random variable Z that satisfies: * Non-Degeneracy: 0 < P(D = 1) < 1 and 0 < P(Z=1 | D = 1) <1. * Relevance: P(D = 1 | Z = 0) < P(D = 1 | Z = 1) < 1. * Outcome exclusion: μ(y | z) = μ(y) for all y ∈ℝ and z ∈{0,1}. * Selection Sorting exclusion: ρ(y,0 | z) = ρ(y,0) for all y ∈ℝ and z ∈{0,1}.”

The paper proposes a new IV estimand (OLIVA) that shares OLS's nonparametric interpretation as the optimal linear approximation to the structural regression function, but under endogeneity.

“The main goal of our paper is to fill this gap and to propose an IV estimand that has the same nonparametric interpretation as OLS, but under endogeneity.”◌ not checked against the paper’s text as it now stands

It proves that a necessary condition for regular identification of OLIVA is also sufficient for the existence of an IV estimand in a linear structural regression, using an unknown transformation of the instrument.

“The main contribution of our paper is to show that a necessary condition for regular identification of the OLIVA is also sufficient for existence of an IV estimand in a linear structural regression.”◌ not checked against the paper’s text as it now stands

The paper develops a Two-Step IV (TSIV) estimator that first estimates the unknown instrument via Tikhonov-regularized PSMD and then applies standard linear IV, establishing its asymptotic normality without assuming completeness or identification of the instrument.

“A Two-Step IV (TSIV) estimator based on Tikhonov regularization is proposed, which can be implemented by standard regression routines. We establish the asymptotic normality of the TSIV estimator assuming neither completeness nor identification of the instrument.”

The paper shows that standard instrumental variable methods fail to point-identify the structural function when the instrument has fewer support points than the endogenous variable.

“Models with a discrete endogenous variable are typically underidentified when the instrument takes on too few values.”◌ not checked against the paper’s text as it now stands

It develops a new method that supplements a weak instrument with covariate-based matching points to restore point identification in triangular models.

“This paper presents a new method that matches pairs of covariates and instruments to restore point identification in this scenario in a triangular model.”◌ not checked against the paper’s text as it now stands

The paper demonstrates concretely that the standard IV approach leaves the outcome function underidentified when the instrument's support is too small relative to the endogenous variable's support.

“For a fixed value x_0∈ S(X), g^*(x_0,·) is underidentified by the standard IV approach (for example […]).”◌ not checked against the paper’s text as it now stands

The paper's global identification result is shown to also extend to the standard nonparametric quantile IV approach when the instrument has large support.

The paper focuses on settings where the instrumental variable is only binary, and studies how to extrapolate treatment effects identified via such a binary IV.

“This paper investigates the possibility of extrapolating local treatment effects to different counterfactual settings when instrumental variables are only binary.”◌ not checked against the paper’s text as it now stands

The paper proposes to exploit full statistical independence of the instrument (rather than just mean independence) to achieve tighter identification.

“Our framework is flexible enough to fully incorporate statistical independence (rather than mean independence) of instruments and a large menu of identifying assumptions beyond the shape restrictions on the MTE that have been considered in prior studies.”◌ not checked against the paper’s text as it now stands

The paper formalizes an exclusion restriction and conditional statistical independence assumption for the instrument Z (and W) used throughout the identification analysis.

“For given $(d,w) \{0,1\} $, $(Y(d,w),U) (Z,W)|X$. When there is no $W$ this assumption and all below are understood as $W$ being degenerate. Assumption EX imposes the exclusion restriction and conditional statistical independence for $Z$ (and $W$).”◌ not checked against the paper’s text as it now stands

The paper models h_0 as identified through a conditional moment restriction using instrumental variables W where X and/or Y are endogenous.

“𝔼[Y - h_0(X) |W] = 0 , where Y (a scalar) and/or some elements of X (a vector) are endogenous, W is a vector of instrumental variables, and the conditional distribution of (X,Y) given W is otherwise unspecified.”◌ not checked against the paper’s text as it now stands

The paper develops a data-driven choice of sieve dimension for sieve nonparametric instrumental variables (TSLS) estimators that achieves minimax sup-norm rates.

“Our methods are developed for the popular class of sieve nonparametric IV estimators.[See […], […], […], and […].] That is, h_0 is approximated by a linear combination of several basis functions (e.g., B-splines), with the coefficients estimated by Two Stage Least Squares (TSLS) regression of Y on the basis functions of X, using functions of W as instruments”◌ not checked against the paper’s text as it now stands

The paper shows that standard cross validation methods used for regression are invalid for choosing tuning parameters in instrumental variable models due to endogeneity.

“We briefly explain why the usual approach of cross validation (CV) for regression is not a valid method for choosing J in models with endogeneity.”

The paper investigates how well various artificial neural network architectures perform when used to estimate nonparametric instrumental variables (NPIV) models with moderately high-dimensional covariates.

“We investigate the performance of various ANNs in nonparametric instrumental variables (NPIV) models of moderately high dimensional covariates that are relevant to empirical economics.”◌ not checked against the paper’s text as it now stands

It develops two new efficient estimation and inference procedures for a weighted average derivative of the unknown NPIV function using ANN sieves.

“We present two efficient procedures for estimation and inference on a weighted average derivative (WAD): an orthogonalized plug-in with optimally-weighted sieve minimum distance (OP-OSMD) procedure and a sieve efficient score (ES) procedure.”◌ not checked against the paper’s text as it now stands

The paper conducts extensive Monte Carlo simulations comparing ANN-based NPIV estimators to spline-based NPIV estimators across increasingly complex designs.

“We compare their finite-sample performances in various simulation designs that involve smooth NPIV function of up to 13 continuous covariates, different nonlinearities and covariate correlations.”◌ not checked against the paper’s text as it now stands

The paper introduces the nonparametric instrumental variable (IV) regression model as the leading example of a linear inverse problem on which fairness constraints are imposed.

“A leading example of this setting are nonparametric instrumental regressions […], as mentioned above, but many other models, such as linear and non-linear parametric regressions and additive nonparametric regressions can fit in this general framework […].”◌ not checked against the paper’s text as it now stands

The paper develops a fully worked-out linear IV model example with instruments W satisfying the moment condition E[W'U]=0, deriving the closed-form unconstrained IV estimator and its projection onto the fair subspace.

“Consider the example of a linear model in which φ(X) = Z^'β + S^'γ, with Z,β∈ℝ^p and S,γ∈ℝ^q. We take both Z and S to be potentially endogenous and we have a vector of instruments W ∈ℝ^k, such that k ≥ p+q and E[ W^' U ] = 0.”◌ not checked against the paper’s text as it now stands

The paper formulates the nonparametric IV (NPIV) regression as an ill-posed inverse problem with instrument W and proposes Tikhonov regularization to obtain a well-posed, fair estimator.

“Recall that the nonparametric instrumental regression (NPIV) model amounts to solving an inverse problem defined as follows. Consider W the instrument, the NPIV regression model can be written as E(Y|W=w)= E(φ(Z,S)|W=w)”

The paper shows that its endogeneity-correction estimator can be reinterpreted as a just-identified instrumental variable estimator using residuals from regressing the endogenous regressor on the generated control function as the instrument.

“This representation of the estimator gives rise to the interpretation of a just-identified IV estimator using the residuals from a regression of z on η as instrumental variable vector v.”◌ not checked against the paper’s text as it now stands

The paper proposes combining this internally generated instrument with external instruments to form an over-identified IV or GMM estimator.

“Accordingly, for the full model we may alternatively estimate the coefficients β and γ from an IV regression using (x_i, v_i) as instruments with v̂_i as the i'th element of the residual vector v. This allows us to combine the `internal instrument' v_i with possible external instruments resulting in an over-identified IV (or GMM) estimator.”◌ not checked against the paper’s text as it now stands

The paper notes that existing weak-instrument testing approaches could be adapted to test the strength of this internally constructed instrument.

“Furthermore the approach of […] may be adapted to test against `weak instruments'.”◌ not checked against the paper’s text as it now stands

The paper extends its identification framework to instrumental variable settings where the shock is endogenous and instrumented by Z_t, incorporating covariates.

“We now generalize the setup in […]sec:proxy by allowing X_t to be endogenous and incorporating covariates 𝐖_t. In particular, we retain the nonparametric structural model ( […]eqn:causal), but drop the independence assumption ( […]eqn:indep) and the selection-on-observables assumption ( […]eqn:selection_obs). Let X_t=ξ(Z_t, 𝐖_t, Ṽ_t) denote the first-stage equation, with Ṽ_t corresponding to the unobservable determinants of X_t.”◌ not checked against the paper’s text as it now stands

It shows that a linear reduced-form regression on the instrument identifies a weighted average of derivatives of the marginal treatment response function, generalizing prior instrumental variable identification results.

“[…] shows that the reduced-form regression of Y_t onto Z_t identifies a weighted average of derivatives of the marginal treatment response function.[An analogous application of […]theorem:rr-covariates to a linear “first-stage regression” of X_t onto Z_t and a vector of controls 𝐖_t shows that it identifies the integral of the weights, E[∫ω(x, 𝐕_t)dx], so that the weights in the associated instrumental variables regression integrate to one.]”◌ not checked against the paper’s text as it now stands

The paper derives a stochastic monotonicity condition on the instrument that ensures non-negative weights in the IV estimand, and links it to first-stage monotonicity (a generalized LATE-type condition).