Classical IV foundations and refinement

The paper proposes two new nonparametric estimators (kernel-based and orthogonal series) for the regression function g in a model where an instrumental variable is used because the explanatory variable is correlated with the error.

“We suggest two nonparametric approaches, based on kernel methods and orthogonal series to estimating regression functions in the presence of instrumental variables.”◌ not checked against the paper’s text as it now stands

It formalizes the instrumental variable condition E(Ui|Wi)=0 and the requirement of a sufficiently strong relationship between X and W for identification.

“Suppose, however, that for each i we have available another observed data value, Wi, say (an instrumental variable), for which E(Ui|Wi) = 0 (1.2) and there is a “sufficiently strong” relationship between Xi and Wi.”◌ not checked against the paper’s text as it now stands

The paper derives, for the first time in this class of instrumental-variables problems, optimal nonparametric convergence rates and shows they are attained by the proposed estimators.

“For the first time in this class of problems, we derive optimal convergence rates, and show that they are attained by particular estimators.”◌ not checked against the paper’s text as it now stands

It characterizes the identification of g via instrumental variables as an ill-posed inverse problem whose difficulty depends on the eigenvalues of an integral operator determined by the joint density of the endogenous and instrumental variables, and addresses this via ridge-type regularization.

“In the presence of instrumental variables the relation that identifies the regression function also defines an ill-posed inverse problem, the “difficulty” of which depends on eigenvalues of a certain integral operator which is determined by the joint density of endogenous and instrumental variables.”◌ not checked against the paper’s text as it now stands

The paper argues that despite nonstationarity, the same rank condition for identification holds, requiring some form of instrumental variable estimator to achieve consistency.

“The same rank condition for identification holds for stationary and nonstationary data and some sort of instrumental variable estimators will have to be employed to yield consistency.”◌ not checked against the paper’s text as it now stands

The paper shows that classical instrumental variable estimators need modification to yield valid statistical inference under nonstationarity.

“Classical instrumental variable estimators have to be modified to ensure valid inference.”◌ not checked against the paper’s text as it now stands

The paper reviews least squares and instrumental variable estimators, focusing specifically on the two-stage least squares (2SLS) estimator for structural vector autoregressions.

“Section 4 discusses the least squares and instrumental variable estimators, in particular, the two stage least squares estimator (2SLS) for a SVAR.”◌ not checked against the paper’s text as it now stands

The paper notes that for stationary structural VARs, 2SLS and 3SLS using lagged variables as instruments are consistent and asymptotically normal, motivating the need for modified versions in the nonstationary case discussed later.

The paper applies its invariance-based MILE framework to an instrumental variable model with N observations and K instruments.

“Section 4 considers an instrumental variable (IV) model with N observations and K instruments.”◌ not checked against the paper’s text as it now stands

It shows that for the orthogonal group of transformations, the MILE in the IV model coincides with the LIMLK estimator.

“For the orthogonal group of transformations, MILE coincides with the LIMLK estimator.”◌ not checked against the paper’s text as it now stands

The paper uses this framework to establish consistency of the MLE in the IV setup under both strong and many weak instruments asymptotics.

“In particular, we are able to (i) show consistency of the MLE in the IV setup even under MWIV asymptotics from the perspective of likelihood maximization; (ii) derive the asymptotic distribution of the MLE directly from the objective function under SIV and MWIV asymptotics; and (iii) provide an explanation for optimality of MLE within the class of regular invariant estimators.”◌ not checked against the paper’s text as it now stands

The paper derives the exact density of the maximal invariant statistic in the IV model and uses it to prove consistency and asymptotic normality results, recovering the known limiting distribution of LIMLK.

The paper reviews and situates recent statistics-literature IV methods within the older econometric tradition, contrasting them on treatment binariness, heterogeneity, potential outcomes, and reduced-form focus.

“Although this recent statistics literature builds on the earlier econometric literature, there are nevertheless important differences. First, the recent statistics literature primarily focuses on the binary treatment case. Second, the recent literature explicitly allows for treatment effect heterogeneity. Third, the recent instrumental variables literature (starting with […]; […]; […]; […]; and […]) explicitly uses the potential outcome framework used by Neyman for randomized experiments and generalized to observational studies by Rubin ( […], […], […]).”◌ not checked against the paper’s text as it now stands

The paper traces the historical origins of instrumental variables methods to econometricians in the 1920s and identifies antecedents such as Zelen's encouragement designs.

“Instrumental Variables (IV) refers to a set of methods developed in econometrics starting in the 1920s to draw causal inferences in settings where the treatment of interest cannot be credibly viewed as randomly assigned, even after conditioning on additional covariates, that is, settings where the assumption of no unmeasured confounders does not hold.”◌ not checked against the paper’s text as it now stands

The paper argues that the early econometric IV work is useful for improving modern applications and identifying potential instruments, and it reframes this literature using modern potential outcome notation.

The paper defines weak identification for linear IV models with many instruments via necessary and sufficient conditions based on the concentration parameter relative to the square root of the number of instruments.

“We define weak identification as a situation where an analog of the concentration parameter divided by the square root of the number of instruments stays bounded in large samples.”◌ not checked against the paper’s text as it now stands

The paper proposes a jackknifed Anderson-Rubin test statistic that is robust to weak identification with many instrumental variables and heteroscedasticity.

“We propose a jackknifed version of the classical weak identification-robust Anderson-Rubin (AR) test statistic. Large-sample inference based on the jackknifed AR is valid under heteroscedasticity and weak identification.”◌ not checked against the paper’s text as it now stands

The paper develops a pre-test to determine whether instrumental variables are weakly or strongly identified in settings with many instruments, guiding a two-step inference procedure.

“We also develop a pre-test for weak identification that is related to the size property of a Wald test based on the Jackknife Instrumental Variable Estimator. This new pre-test is valid under heteroscedasticity and with many instruments.”

The paper develops wild bootstrap Wald and Anderson-Rubin tests for instrumental variables regressions under a fixed-number-of-clusters asymptotic framework, showing size control under weak identification in some or all clusters.

“We study the wild bootstrap inference for instrumental variable regressions under an alternative asymptotic framework that the number of independent clusters is fixed, the size of each cluster diverges to infinity, and the within cluster dependence is sufficiently weak. We first show that the wild bootstrap Wald test controls size asymptotically up to a small error as long as the parameters of endogenous variables are strongly identified in at least one of the clusters.”◌ not checked against the paper’s text as it now stands

The paper extends the wild restricted efficient (WRE) cluster bootstrap to general k-class IV estimators, proving its validity when IV strength varies across clusters and the number of clusters is fixed.

“Instead, we focus on extending the wild restricted efficient (WRE) cluster bootstrap, which is popular among empirical researchers, for general k-class IV estimators. Therefore, our procedure cannot be formulated as a score bootstrap in the GMM setting.”◌ not checked against the paper’s text as it now stands

The paper establishes the local power properties of the bootstrap Wald test for IV regressions, both with and without cluster-robust variance estimation, showing the CRVE-studentized version is more powerful for distant alternatives.

“Second, we study the local power for the Wald test both with and without CRVE. The power analysis of the Wald test with CRVE is new to the literature and technically involved because with a fixed number of clusters, the CRVE has a random limit.”

The paper formally defines the two conditions a valid instrumental variable must satisfy: relevance and exogeneity.

“A valid instrument Z_i must satisfy two conditions:”◌ not checked against the paper’s text as it now stands

The paper derives the two-stage least squares (2SLS) instrumental variable estimator as a feasible approximation to the infeasible optimal IV estimator.

“β̂_2SLS= ((𝐗'𝐙)(𝐙'𝐙)^-1𝐙'𝐗)^-1𝐗'𝐙(𝐙'𝐙)^-1𝐙'𝐘”◌ not checked against the paper’s text as it now stands

The paper analyzes how weak instruments cause bias and inconsistency in the standard IV/2SLS estimator.

“In the weak instruments case, the 2SLS estimator fails to provide unbiased, reliable estimates.”◌ not checked against the paper’s text as it now stands

The paper explores and derives alternative instrumental variable estimators (JIVE and LIML) intended to have better finite-sample properties than 2SLS under weak instruments.

“Both of these estimators, the Jackknife IV estimator and the LIML estimator, fall under the broader class of k-class estimators. These estimators partially robust ie less sensitive to weak instruments; they are more reliable in comparison to 2SLS estimates.”

The paper studies the first-stage F-statistic as a test for the relevance/underidentification of instrumental variables in a 2SLS model.

“It is commonplace to report the first-stage F-statistic as a test for underidentification in linear single endogenous variable models estimated by two-stage least squares (2SLS).”◌ not checked against the paper’s text as it now stands

It analyzes a grouped-data IV model where mutually exclusive group membership indicators serve as the instruments.

“The […] design is the same as a grouped data one, see […] and the discussion in […], where the instruments are mutually exclusive group membership indicators.”◌ not checked against the paper’s text as it now stands

The paper proposes a new GMM instrumental-variable estimator (GMMf) whose weight matrix is based on first-stage residuals rather than structural residuals, to better utilize instrument information under heteroskedasticity.

“An estimator that corrects for this is a robust GMM estimator, denoted GMMf, with the robust weight matrix not based on the structural residuals, but on the first-stage residuals.”◌ not checked against the paper’s text as it now stands

The paper extends its analysis of the zero conditional mean assumption to show that 2SLS estimation with a linear IV model can fail to have a causal interpretation even when the instrument is uncorrelated with the error.

“even if D is endogenous i.e., Cov[D,ε]≠ 0, estimating λ by 2SLS using an instrument Z that is uncorrelated with ε does not guarantee a causal interpretation.”◌ not checked against the paper’s text as it now stands

The paper derives a specific IV data generating process showing that E[ε_i|Z_i]=0 holds yet the 2SLS coefficient does not equal the true causal effect unless an omitted-variable term vanishes.

“Therefore, E[ε_i|Z_i]=0 is satisfied, but λ, the coefficient of a 2SLS regression of Y on D using Z as an instrument, does not have a causal interpretation unless βδ =0.”◌ not checked against the paper’s text as it now stands

The paper contrasts its result with prior work by showing that a pseudo-parameter problem in 2SLS can arise even without non-linearity or non-additivity in the outcome equation.

“Our derivation shows that the DGP does not need to feature a non-linearity or non-additivity for the 2SLS estimand to lack a causal interpretation despite E[ε_i|Z_i]=0.”

The paper notes that in general dynamic panel models the Nickell bias is typically corrected using instrumental variable methods or analytical formulas, contrasting this with its own SPJ approach.

“For general dynamic panel models, the Nickell bias is usually addressed with the instrumental variable (IV) method […] or analytical formulas […].”◌ not checked against the paper’s text as it now stands

The paper points out a key weakness of the IV method, namely its inefficiency and poor finite-sample performance when the regressor is persistent.

“The IV method is subject to the lack of efficiency and performs poorly with finite samples when the regressor is persistent […].”◌ not checked against the paper’s text as it now stands

The paper argues that the IV method requires case-by-case adjustments and derivations, motivating the use of SPJ instead.

“Therefore, case-by-case adjustments and mathematical derivations are necessary for the IV and analytical methods.”◌ not checked against the paper’s text as it now stands

The paper positions SPJ as an alternative to IV that avoids the weak instrument problem associated with instrumental variable approaches to bias correction.

The paper analyzes the model with a scalar endogenous variable and a K_n-dimensional vector of instrumental variables in a first-stage regression.

“y_i=Y_iβ+u_i, Y_i=π'𝐙_i+v_i, for i=1,…,n, where y_i is a scalar outcome, Y_i is a scalar endogenous variable, 𝐙_i is a K_n× 1 vector of instrument variables.”◌ not checked against the paper’s text as it now stands

It shows that with many instruments, the appropriate measure of instrument strength shifts from the concentration parameter to the re-scaled concentration parameter, redefining when instruments are considered weak.

“When K_n →∞, a more appropriate measure of the strength of instruments is μ_n^2/√(K_n) that leverages the effect of many instruments.”◌ not checked against the paper’s text as it now stands

The paper proves that the first-stage F test used to detect weak instruments becomes asymptotically normal rather than noncentral Chi-squared and exhibits size distortions when the number of instruments is large.

“We show that the more appropriate distribution of the F statistic shifts to the normal distribution, instead of the conventional noncentral Chi-squared distribution. The inadequacy of the noncentral Chi-squared distribution provides poor finite sample approximations to the F statistic with many instruments, leading to size distortion of the classical F test.”◌ not checked against the paper’s text as it now stands

The paper states that Wright's Appendix B is generally credited with introducing instrumental variables estimation.

“In particular, it is generally credited with introducing instrumental variables (IV) estimation.”◌ not checked against the paper’s text as it now stands

The paper formally derives the IV estimator formulas for the demand and supply elasticities from Wright's structural model using method of moments.

“Solving this system of equations, we recover the IV formulas α_1 = 𝔼 ( Y Z^s)/𝔼 ( P Z^s) and β_1 = 𝔼 ( Y Z^d)/𝔼 ( P Z^d),”◌ not checked against the paper’s text as it now stands

The paper identifies the instrument relevance condition required for Wright's IV formulas to be valid.

“which identify the demand and supply elasticities α_1 and β_1, provided that the denominators are not equal to zero; that is, the supply and demand shifters have a nontrivial effect on the equilibrium price. This is the instrument relevance condition in econometrics.”◌ not checked against the paper’s text as it now stands

The paper explains how well-chosen instrumental variables can serve as supply or demand shifters to identify the other curve's elasticity, provided they are uncorrelated with the other curve's shocks.