Invalid Instruments Research Articles

Confidence Intervals for Causal Effects with Invalid Instruments by Using Two-Stage Hard Thresholding with Voting

SummaryA major challenge in instrumental variable (IV) analysis is to find instruments that are valid, or have no direct effect on the outcome and are ignorable. Typically one is unsure whether all of the putative IVs are in fact valid. We propose a general inference procedure in the presence of invalid IVs, called two-stage hard thresholding with voting. The procedure uses two hard thresholding steps to select strong instruments and to generate candidate sets of valid IVs. Voting takes the candidate sets and uses majority and plurality rules to determine the true set of valid IVs. In low dimensions with invalid instruments, our proposal correctly selects valid IVs, consistently estimates the causal effect, produces valid confidence intervals for the causal effect and has oracle optimal width, even if the so-called 50% rule or the majority rule is violated. In high dimensions, we establish nearly identical results without oracle optimality. In simulations, our proposal outperforms traditional and recent methods in the invalid IV literature. We also apply our method to reanalyse the causal effect of education on earnings.

Prediction and Identification in Two-Sided Markets

I introduce a reduced form two-sided market model to study prediction and identification in two-sided markets. The model generates the hallmark features of two-sided markets: potentially below cost or even negative prices to one side of the market, and the “see-saw” or “waterbed” effect of a tendency for price movements across sides to be negatively correlated. I show that the standard “one-sided” model of complements is a special case of the two-sided model, and that it generates those same hallmark features of two-sided markets. The model of complements also performs well in predicting price effects even when the data is actually generated by the two-sided market model; the “wrong” model often delivers the right answers and can be used to estimate market power and pass through rates. I show that even a naive one-sided model that ignores any relationship across goods/sides can perform well when prices to one side of the market are censored at zero, a very common outcome in two-sided markets. The main cost to using a model of complements to estimate cross-group effects in a two-sided market is that it invites the use of invalid instruments. I show that these findings are consistent with the empirical regularities and identification strategies in the existing two-sided market and indirect network effects literatures. I conclude by discussing the general difficulty of separately identifying whether price differences across subgroups of users of a platform are driven by pricing of network effects or simple price discrimination based on price elasticity.

Prediction and Identification in Two-Sided Markets

I introduce a reduced form two-sided market model to study prediction and identification in two-sided markets. The model generates the hallmark features of two-sided markets: potentially below cost or even negative prices to one side of the market, and the “see-saw” or “waterbed” effect of a tendency for price movements across sides to be negatively correlated. I show that the standard “one-sided” model of complements is a special case of the two-sided model, and that it generates those same hallmark features of two-sided markets. The model of complements also performs well in predicting price effects even when the data is actually generated by the two-sided market model; the “wrong” model often delivers the right answers and can be used to estimate market power and pass through rates. I show that even a naive one-sided model that ignores any relationship across goods/sides can perform well when prices to one side of the market are censored at zero, a very common outcome in two-sided markets. The main cost to using a model of complements to estimate cross-group effects in a two-sided market is that it invites the use of invalid instruments. I show that these findings are consistent with the empirical regularities and identification strategies in the existing two-sided market and indirect network effects literatures. I conclude by discussing the general difficulty of separately identifying whether price differences across subgroups of users of a platform are driven by pricing of network effects or simple price discrimination based on price elasticity.

On Solving Endogeneity with Invalid Instruments: An Application to Investment Equations

SummaryRegression models relating investment demand with firms’ Tobin's q and cash flow are fraught with measurement errors which, in turn, cause endogeneity bias. We propose an alternative solution to this problem based on modelling the interaction between the endogenous Tobin's q and the error term in the investment equation as a function of lagged values of Tobin's q. We then study the identification conditions and asymptotic properties of the resulting estimator. Our analysis of a panel of US firms reveals a larger effect of Tobin's q on firms’ investment demand than that obtained by using available estimators in the literature. Moreover, the estimates highlight the importance of cash flow. We find mixed evidence on the relationship between investment demand and firms’ cash flow with respect to different measures of financial constraints. Nevertheless, this evidence is more supportive of the view that firms’ cash flows have a weaker correlation to investment demand when financial conditions tighten.

Identification of additive and polynomial models of mismeasured regressors without instruments

We show nonparametric point identification of a measurement error model with covariates that can be interpreted as invalid instruments. Our main contribution is to replace standard exclusion restrictions with the weaker assumption of additivity in the covariates. Measurement errors are ubiquitous and additive models are popular, so our results combining the two should have widespread potential application. We also identify a model that replaces the nonparametric function of the mismeasured regressor with a polynomial in that regressor and other covariates. This allows for rich interactions between the variables, at the expense of introducing a parametric restriction. Our identification proofs are constructive, and so can be used to form estimators. We establish root-n asymptotic normality for one of our estimators.

Testing the IV Zero-Covariance Assumption and Identification with an Invalid Instrument

This paper discusses the empirical content of the exogeneity (zero-covariance) assumption, the key assumption for identification in the linear IV model. Contrary to the general belief, we show that whenever the outcome is bounded, the exogeneity assumption imposes some testable restrictions on the observables, namely the linear instrumental inequalities (LII). The LII can therefore be used to test the instrument validity. The test can be easily implemented using existing inferential methods for testing multiple inequalities. Whenever, the causal effect parameter is random, an additional critical condition that trivially holds in the linear IV model is required for the IV estimand to identify the average treatment effect (ATE). We show that when this critical condition holds while the exogeneity assumption does not, the magnitude of the violations of the LII allows us to derive (informative) bounds on the ATE. We also derive testable implications of this critical condition which when violated implies that the usual IV estimand cannot be interpreted as the ATE or as the local ATE. We illustrate our results by testing the validity of various instruments present in the literature.

Using invalid instruments on purpose: Focused moment selection and averaging for GMM

In finite samples, the use of a slightly endogenous but highly relevant instrument can reduce mean-squared error (MSE). Building on this observation, I propose a novel moment selection procedure for GMM–the Focused Moment Selection Criterion (FMSC)–in which moment conditions are chosen not based on their validity but on the MSE of their associated estimator of a user-specified target parameter. The FMSC mimics the situation faced by an applied researcher who begins with a set of relatively mild “baseline” assumptions and must decide whether to impose any of a collection of stronger but more controversial “suspect” assumptions. When the (correctly specified) baseline moment conditions identify the model, the FMSC provides an asymptotically unbiased estimator of asymptotic MSE, allowing us to select over the suspect moment conditions. I go on to show how the framework used to derive the FMSC can address the problem of inference post-moment selection. Treating post-selection estimators as a special case of moment-averaging, in which estimators based on different moment sets are given data-dependent weights, I propose simulation-based procedures for inference that can be applied to a variety of formal and informal moment-selection and averaging procedures. Both the FMSC and confidence interval procedures perform well in simulations. I conclude with an empirical example examining the effect of instrument selection on the estimated relationship between malaria and income per capita.

A note on testing instrument validity for the identification of LATE

In this paper, we show that the testable implications derived in Huber and Mellace (Rev Econ Stat 97:398, 2015) are the best possible to detect invalid instruments in the presence of heterogeneous treatment effects and endogeneity. We also provide formal proof of the fact that those testable implications are only necessary, but not sufficient, conditions for instrument validity.

Instrumental variable analysis of multiplicative models with potentially invalid instruments.

Instrumental variable (IV) methods have potential to consistently estimate the causal effect of an exposure on an outcome in the presence of unmeasured confounding. However, validity of IV methods relies on strong assumptions, some of which cannot be conclusively verified from observational data. One such assumption is that the effect of the proposed instrument on the outcome is completely mediated by the exposure. We consider the situation where this assumption is violated, but the remaining IV assumptions hold; that is, the proposed IV (1) is associated with the exposure and (2) has no unmeasured causes in common with the outcome. We propose a method to estimate multiplicative structural mean models of binary outcomes in this scenario in the presence of unmeasured confounding. We also extend the method to address multiple scenarios, including mediation analysis. The method adapts the asymptotically efficient G-estimation approach that was previously proposed for additive structural mean models, and it can be carried out using off-the-shelf software for generalized method of moments. Monte Carlo simulation studies show that the method has low bias and accurate coverage. We applied the method to a case study of circulating vitamin D and depressive symptoms using season of blood collection as a (potentially invalid) instrumental variable. Potential applications of the proposed method include randomized intervention studies as well as Mendelian randomization studies with genetic variants that affect multiple phenotypes, possibly including the outcome. Published 2016. This article is a U.S. Government work and is in the public domain in the USA.

BMI as a Modifiable Risk Factor for Type 2 Diabetes: Refining and Understanding Causal Estimates Using Mendelian Randomization.

This study focused on resolving the relationship between BMI and type 2 diabetes. The availability of multiple variants associated with BMI offers a new chance to resolve the true causal effect of BMI on type 2 diabetes; however, the properties of these associations and their validity as genetic instruments need to be considered alongside established and new methods for undertaking Mendelian randomization (MR). We explore the potential for pleiotropic genetic variants to generate bias, revise existing estimates, and illustrate value in new analysis methods. A two-sample MR approach with 96 genetic variants was used with three different analysis methods, two of which (MR-Egger and the weighted median) have been developed specifically to address problems of invalid instrumental variables. We estimate an odds ratio for type 2 diabetes per unit increase in BMI (kg/m(2)) of between 1.19 and 1.38, with the most stable estimate using all instruments and a weighted median approach (1.26 [95% CI 1.17, 1.34]). TCF7L2(rs7903146) was identified as a complex effect or pleiotropic instrument, and removal of this variant resulted in convergence of causal effect estimates from different causal analysis methods. This indicated the potential for pleiotropy to affect estimates and differences in performance of alternative analytical methods. In a real type 2 diabetes-focused example, this study demonstrates the potential impact of invalid instruments on causal effect estimates and the potential for new approaches to mitigate the bias caused.

Plasma urate and coronary heart disease: fingerprint match, but no smoking gun

Plasma urate and coronary heart disease: fingerprint match, but no smoking gun

Forecasting light-duty vehicle demand using alternative-specific constants for endogeneity correction versus calibration

We investigate parameter recovery and forecast accuracy implications of incorporating alternative-specific constants (ASCs) in the utility functions of vehicle choice models. We compare two methods of incorporating ASCs: (1) a maximum likelihood estimator that computes ASCs post-hoc as calibration constants (MLE-C) and (2) a generalized method of moments estimator that uses instrumental variables (GMM-IV) to correct for price endogeneity. In a synthetic study we observe significant coefficient bias with MLE-C when the price-ASC correlation (endogeneity) is large. GMM-IV successfully mitigates this bias given valid instruments but exacerbates the bias given invalid instruments. Despite greater coefficient bias, MLE-C yields better forecasts than GMM-IV with valid instruments in most of the cases examined, including most cases where the price-ASC correlation present in the estimation data is absent in the prediction data. In a market study of U.S. midsize sedan sales from 2002 – 2006 the GMM-IV model predicts the 1-year-forward market better, but the MLE-C model predicts the 5-year-forward market better. Including an ASC in predictions by any of the methods proposed improves share forecasts, and assuming that the ASC of each new vehicle matches that of its closest competitor vehicle yields the best long term forecasts. We find evidence that the instruments most frequently used in the automotive demand literature may be invalid.

Instrumental Variables Estimation With Some Invalid Instruments and its Application to Mendelian Randomization

Instrumental variables have been widely used for estimating the causal effect between exposure and outcome. Conventional estimation methods require complete knowledge about all the instruments’ validity; a valid instrument must not have a direct effect on the outcome and not be related to unmeasured confounders. Often, this is impractical as highlighted by Mendelian randomization studies where genetic markers are used as instruments and complete knowledge about instruments’ validity is equivalent to complete knowledge about the involved genes’ functions. In this article, we propose a method for estimation of causal effects when this complete knowledge is absent. It is shown that causal effects are identified and can be estimated as long as less than 50% of instruments are invalid, without knowing which of the instruments are invalid. We also introduce conditions for identification when the 50% threshold is violated. A fast penalized ℓ1 estimation method, called sisVIVE, is introduced for estimating the causal effect without knowing which instruments are valid, with theoretical guarantees on its performance. The proposed method is demonstrated on simulated data and a real Mendelian randomization study concerning the effect of body mass index(BMI) on health-related quality of life (HRQL) index. An R package sisVIVE is available on CRAN. Supplementary materials for this article are available online.

Identification and Inference With Many Invalid Instruments

We study estimation and inference in settings where the interest is in the effect of a potentially endogenous regressor on some outcome. To address the endogeneity, we exploit the presence of additional variables. Like conventional instrumental variables, these variables are correlated with the endogenous regressor. However, unlike conventional instrumental variables, they also have direct effects on the outcome, and thus are “invalid” instruments. Our novel identifying assumption is that the direct effects of these invalid instruments are uncorrelated with the effects of the instruments on the endogenous regressor. We show that in this case the limited-information-maximum-likelihood (liml) estimator is no longer consistent, but that a modification of the bias-corrected two-stage-least-square (tsls) estimator is consistent. We also show that conventional tests for over-identifying restrictions, adapted to the many instruments setting, can be used to test for the presence of these direct effects. We recommend that empirical researchers carry out such tests and compare estimates based on liml and the modified version of bias-corrected tsls. We illustrate in the context of two applications that such practice can be illuminating, and that our novel identifying assumption has substantive empirical content.

Using Invalid Instruments on Purpose: Focused Moment Selection and Averaging for GMM, Second Version

Infinite samples, the use of a slightly endogenous but highly relevant instrument can reduce mean-squared error (MSE). Building on this observation, I propose a moment selection criterion for GMM in which moment conditions are chosen based on the MSE of their associated estimators rather than their validity: the focused moment selection criterion (FMSC). I then show how the framework used to derive the FMSC can address the problem of inference post-moment selection. Treating post-selection estimators as a special case of moment-averaging, in which estimators based on different moment sets are given data-dependent weights, I propose a simulation-based procedure to construct valid confidence intervals for a variety of formal and informal moment-selection and averaging procedures. Both the FMSC and confidence interval procedure perform well in simulations. I conclude with an empirical example examining the effect of instrument selection on the estimated relationship between malaria transmission and income.

A novel method to evaluate Young׳s modulus of ceramics at high temperature up to 2100 °C

Although it is important to understand Young׳s modulus and creep behavior of ceramics at ultra-high temperature from 1500°C to 2300°C, measuring these properties accurately at high temperature above 1600°C is not very easy due to the difficulty of the deformation measurement or invalid testing instruments at such high temperature. This work presented a solution to this problem via a combination of the split ring test and relative method (viz. relative split ring method). The real deformation of the split ring at ultra-high temperature could be easily determined from the difference of cross-beam displacements between the split ring and a rigid disk under the same loads. Experiments of graphite and C/SiC/ZrB2 composite at ultra-high temperature demonstrated the validity and convenience of the relative split ring method, which would provide a simple and efficient approach to evaluate Young׳s modulus of ceramic materials at ultra-high temperature.

Mendelian randomization with invalid instruments: effect estimation and bias detection through Egger regression.

Background: The number of Mendelian randomization analyses including large numbers of genetic variants is rapidly increasing. This is due to the proliferation of genome-wide association studies, and the desire to obtain more precise estimates of causal effects. However, some genetic variants may not be valid instrumental variables, in particular due to them having more than one proximal phenotypic correlate (pleiotropy).Methods: We view Mendelian randomization with multiple instruments as a meta-analysis, and show that bias caused by pleiotropy can be regarded as analogous to small study bias. Causal estimates using each instrument can be displayed visually by a funnel plot to assess potential asymmetry. Egger regression, a tool to detect small study bias in meta-analysis, can be adapted to test for bias from pleiotropy, and the slope coefficient from Egger regression provides an estimate of the causal effect. Under the assumption that the association of each genetic variant with the exposure is independent of the pleiotropic effect of the variant (not via the exposure), Egger’s test gives a valid test of the null causal hypothesis and a consistent causal effect estimate even when all the genetic variants are invalid instrumental variables.Results: We illustrate the use of this approach by re-analysing two published Mendelian randomization studies of the causal effect of height on lung function, and the causal effect of blood pressure on coronary artery disease risk. The conservative nature of this approach is illustrated with these examples.Conclusions: An adaption of Egger regression (which we call MR-Egger) can detect some violations of the standard instrumental variable assumptions, and provide an effect estimate which is not subject to these violations. The approach provides a sensitivity analysis for the robustness of the findings from a Mendelian randomization investigation.

A Note on Testing the LATE Assumptions

In this paper we show that the testable implications derived in Huber and Mellace (2013) are the best possible to detect invalid instruments, in the presence of heterogeneous treatment effects and endogeneity. We also provide a formal proof of the fact that those testable implications are only necessary but not sufficient conditions for instrument validity.

Choosing a Set of Instruments among Many and Possibly Invalid Instruments

We consider the instrument selection problem in instrumental variable regression model when there is a large set of instruments with potential invalidity. Existing methods for instrument selection are based on a priori assumptions of an instrument's validity. Moreover, existing approaches based on consistent moment selection or first-order asymptotics can lead to a large finite sample bias with many instruments. We derive higher-order mean square error (MSE) approximation of a two-stage least squares (2SLS) estimator allowing locally invalid instruments. Based on the approximation to the higher-order MSE, we propose an invalidity-robust instrument selection criterion (IRC) that captures two sources of finite sample bias at the same time: bias from using many instruments and bias from invalid instruments. We show that our criterion is an asymptotically unbiased estimator of the higher-order MSE approximation. In various simulation experiments, our criterion performs well in terms of lower bias and better coverage properties of the estimator compare with other existing selection methods. We also apply our criterion to the empirical application in Card (1995).

Using Invalid Instruments on Purpose: Focused Moment Selection and Averaging for GMM, Second Version

In finite samples, the use of a slightly endogenous but highly relevant instrument can reduce mean-squared error (MSE). Building on this observation, I propose a moment selection criterion for GMM in which moment conditions are chosen based on the MSE of their associated estimators rather than their validity: the focused moment selection criterion (FMSC). I then show how the framework used to derive the FMSC can address the problem of inference post-moment selection. Treating post-selection estimators as a special case of moment-averaging, in which estimators based on different moment sets are given data-dependent weights, I propose a simulation-based procedure to construct valid confidence intervals for a variety of formal and informal moment-selection and averaging procedures. Both the FMSC and confidence interval procedure perform well in simulations. I conclude with an empirical example examining the effect of instrument selection on the estimated relationship between malaria transmission and income.