instrumental variables regression with weak instruments this paper develops asymptotic distribution theory for instrumental variable regression when the partial correlation between the instruments and a single included endogenous variable is weak here modeled as local to zero asymptotic representations are provided for various instrumental variable statistics including the two stage least squares tsls and limited information maximum likelihood liml estimators and their t statistics the asymptotic distributions are found to provide good approximations to sampling distributions with just 20 observations per instrument even in large samples tsls can be badly biased but liml is in many cases approximately median unbiased the theory suggests concrete quantitative guidelines for applied work these guidelines help to interpret angrist and krueger s 1991 estimates of the returns to education whereas tsls estimates with many instruments approach the ols estimate of 6 the more reliable liml and tsls estimates with fewer instruments fall between 8 and 10 with a typical confidence interval of 6 14

estimation and inference with weak semi strong and strong identification this paper analyzes the properties of standard estimators tests and confidence sets cs s for parameters that are unidentified or weakly identified in some parts of the parameter space the paper also introduces methods to make the tests and cs s robust to such identification problems the results apply to a class of extremum estimators and corresponding tests and cs s that are based on criterion functions that satisfy certain asymptotic stochastic quadratic expansions and that depend on the parameter that determines the strength of identification this covers a class of models estimated using maximum likelihood ml least squares ls quantile generalized method of moments generalized empirical likelihood minimum distance and semi parametric estimators the consistency lack of consistency and asymptotic distributions of the estimators are established under a full range of drifting sequences of true distributions the asymptotic sizes in a uniform sense of standard and identification robust tests and cs s are established the results are applied to the arma 1 1 time series model estimated by ml and to the nonlinear regression model estimated by ls in companion papers the results are applied to a number of other models

alternative approximations to the distributions of instrumental variable estimators the paper considers the ols the iv and two method of moments estimators mm and mmk of the coefficients of a single equation where the explanatory variables are correlated with the disturbance term the mm and mmk estimators are generalizations of the liml and limlk estimators respectively multivariate first order approximations to the distributions are derived under normality using a parameter sequence where the number of instruments increases as the number of observations increases numerical results show these approximations are more accurate compared to large sample approximations even if the number of instruments is small the moments of the multivariate limit distributions of the mm and mmk estimators can be consistently estimated under a variety of parameter sequences including the large sample sequence the new approximate confidence regions perform well in terms of exact levels compared to traditional ones the iv estimator of the coefficient of a single explanatory endogenous variable is interpreted as a shrinkage estimator which is dominated in practical cases by the mm and mmk estimators in terms of nearness to the true value in the sense of pitman

further results on projection based inference in iv regressions with weak collinear or missing instruments we study a general family of anderson rubin type procedures allowing for arbitrary collinearity among the instruments and endogenous variables using finite sample distributional theory we show that the proposed procedures besides being robust to weak instruments are also robust to the exclusion of relevant instruments and to the distribution of endogenous regressors a solution to the problem of computing linear projections from general possibly singular quadric surfaces is derived and used to build finite sample confidence sets for individual structural parameters the importance of robustness to excluded instruments is studied by simulation applications to the trade growth relationship and to education returns are presented

model implied instrumental variable generalized method of moments miiv gmm estimators for latent variable models the common maximum likelihood ml estimator for structural equation models sems has optimal asymptotic properties under ideal conditions e g correct structure no excess kurtosis etc that are rarely met in practice this paper proposes model implied instrumental variable generalized method of moments miiv gmm estimators for latent variable sems that are more robust than ml to violations of both the model structure and distributional assumptions under less demanding assumptions the miiv gmm estimators are consistent asymptotically unbiased asymptotically normal and have an asymptotic covariance matrix they are distribution free robust to heteroscedasticity and have overidentification goodness of fit j tests with asymptotic chi square distributions in addition miiv gmm estimators are scalable in that they can estimate and test the full model or any subset of equations and hence allow better pinpointing of those parts of the model that fit and do not fit the data an empirical example illustrates miiv gmm estimators two simulation studies explore their finite sample properties and find that they perform well across a range of sample sizes

problems with instrumental variables estimation when the correlation between the instruments and the endogenous explanatory variable is weak abstract we draw attention to two problems associated with the use of instrumental variables iv the importance of which for empirical work has not been fully appreciated first the use of instruments that explain little of the variation in the endogenous explanatory variables can lead to large inconsistencies in the iv estimates even if only a weak relationship exists between the instruments and the error in the structural equation second in finite samples iv estimates are biased in the same direction as ordinary least squares ols estimates the magnitude of the bias of iv estimates approaches that of ols estimates as the r 2 between the instruments and the endogenous explanatory variable approaches 0 to illustrate these problems we reexamine the results of a recent paper by angrist and krueger who used large samples from the u s census to estimate wage equations in which quarter of birth is used as an instrument for educational attainment we find evidence that despite huge sample sizes th

on the exact small sample distribution of the instrumental variable estimator the purpose of this note is to point out that the bimodality is a consequence of the special case considered by n s the exact finite sample distribution of the iv estimator has been derived in the economic literature earlier but its tabulation or plotting involves some assumptions about the nuisance parameters in this note we plot the densities of the exact finite sample distribution of the iv estimator for several parameter values and show that the bimodality feature noted by n s is a consequence of a singular covariance matrix and not of a poor instrument

exactly distribution free inference in instrumental variables regression with possibly weak instruments abstract this paper introduces a rank based test for the instrumental variables regression model that dominates the anderson rubin test in terms of finite sample size and asymptotic power in certain circumstances the test has correct size for any distribution of the errors with weak or strong instruments the test has noticeably higher power than the anderson rubin test when the error distribution has thick tails and comparable power otherwise like the anderson rubin test the rank tests considered here perform best relative to other available tests in exactly identified models

conditional moment models under semi strong identification we consider conditional moment models under semi strong identification identification strength is directly defined through the conditional moments that flatten as the sample size increases our new minimum distance estimator is consistent asymptotically normal robust to semi strong identification and does not rely on the choice of a user chosen parameter such as the number of instruments or some smoothing parameter heteroskedasticity robust inference is possible through wald testing without prior knowledge of the identification pattern simulations show that our estimator is competitive with alternative estimators based on many instruments being well centered with better coverage rates for confidence intervals

exponentially tilted likelihood inference on growing dimensional unconditional moment models abstract growing dimensional data with likelihood function unavailable are often encountered in various fields this paper presents a penalized exponentially tilted pet likelihood for variable selection and parameter estimation for growing dimensional unconditional moment models in the presence of correlation among variables and model misspecification under some regularity conditions we investigate the consistent and oracle properties of the pet estimators of parameters and show that the constrained pet likelihood ratio statistic for testing contrast hypothesis asymptotically follows the chi squared distribution theoretical results reveal that the pet likelihood approach is robust to model misspecification we study high order asymptotic properties of the proposed pet estimators simulation studies are conducted to investigate the finite performance of the proposed methodologies an example from the boston housing study is illustrated

testing with many weak instruments abstract this paper establishes the asymptotic distributions of the likelihood ratio lr anderson rubin ar and lagrange multiplier lm test statistics under many weak iv asymptotics these asymptotics are relevant when the number of ivs is large and the coefficients on the ivs are relatively small the asymptotic results hold under the null and under suitable alternatives hence power comparisons can be made provided k 3 n 0 as n where n is the sample size and k is the number of instruments these tests have correct asymptotic size this holds no matter how weak the instruments are hence the tests are robust to the strength of the instruments the asymptotic power results show that the conditional lr test is more powerful asymptotically than the ar and lm tests under many weak iv asymptotics

confidence sets for partially identified parameters that satisfy a finite number of moment inequalities this paper proposes a computationally simple way to construct confidence sets for a parameter of interest in models comprised of moment inequalities building on results from the literature on multivariate one sided tests i show how to test the hypothesis that any particular parameter value is logically consistent with the maintained moment inequalities the associated test statistic has an asymptotic chi bar square distribution and can be inverted to construct an asymptotic confidence set for the parameter of interest even if that parameter is only partially identified critical values for the test are easily computed and a monte carlo study demonstrates implementation and finite sample performance

on the normalization of structural equations properties of direction estimators in the general structural equation model only the direction of the vector of coefficients of the endogenous variables is determined the traditional normalization rule defines the coefficients that are of interest but should not be embodied in the estimation procedure we show that the properties of the traditionally defined ordinary least squares and two stage least squares estimators are distorted by their dependence on the normalization rule symmetrically normalized analogues of these estimators are defined and are shown to have essentially similar properties to those of the limited information maximum likelihood estimator exact distribution theory for the classical structural equation model in econometrics is notoriously complex and results for even the simplest cases have seemed too complicated to yield interesting analytical conclusions about the relative merits of different estimators thus anderson and sawa 1973 and anderson kunimoto and sawa 1982 for instance have resorted to extensive numerical tabulations of the exact densities to extract such information for the case of an equation with just two endogenous variables these tabulations suggest that in several respects the limited information maximum likelihood liml estimator is superior to the ordinary least squares ols and two stage least squares tsls estimators among others and these conclusions are supported by the higher order asymptotic results in anderson kunimoto and morimune 1986 and other work referenced there the basis for inference in this model is the joint distribution of the included endogenous variables represented by the reduced form together with the maintained hypothesis that a submatrix of the reduced form coefficient matrix has rank one less than its column dimension this condition determines the direction of a vector in rn where n 1 is the total number of endogenous variables in the equation but not its length thus some normalization rule is needed to determine the coefficient vector uniquely and the distribution theory referred to above has focused on results for the n coefficients remaining when the other is assumed to be unity in this paper we focus on the estimation of the direction of the n 1 dimensional coefficient vector in an unnormalized equation existing distribu tion results for the coefficients in a normalized equation are easily translated 1an earlier version of this paper bore the title on the interpretation of exact results for structural equation estimators my thanks to the referees and the co editor for comments on earlier versions of the paper that helped to both clarify the message and broaden the scope of the paper 1181

estimation and confidence regions for parameter sets in econometric models this paper develops a framework for performing estimation and inference in econometric models with partial identification focusing particularly on models characterized by moment inequalities and equalities applications of this framework include the analysis of game theoretic models revealed preference restrictions regressions with missing and corrupted data auction models structural quantile regressions and asset pricing models specifically we provide estimators and confidence regions for the set of minimizers i of an econometric criterion function q in applications the criterion function embodies testable restrictions on economic models a parameter value that describes an economic model satisfies these restrictions if q attains its minimum at this value interest therefore focuses on the set of minimizers called the identified set we use the inversion of the sample analog q n of the population criterion q to construct estimators and confidence regions for the identified set and develop consistency rates of convergence and inference results for these estimators and regions to derive these results we develop methods for analyzing the asymptotic properties of sample criterion functions under set identification

asymptotic and finite sample distribution theory for iv estimators and tests in partially identified structural equations abstract general formulae for the finite sample and asymptotic distributions of the instrumental variable estimators and the wald statistics in a simultaneous equation model are derived it is assumed that the coefficient vectors of both endogenous and exogenous variables are only partially identified even though the order condition for identification is satisfied this work extends previous results in phillips 1989 where the coefficient vector of the exogenous variables is partially identified and that of the endogenous variables is totally unidentified the effect of partial identification on the finite sample and asymptotic distributions of the estimators and the wald statistics is analyzed by isolating identifiable parts of the coefficient vectors using a rotation of the coordinate system developed in phillips 1989 the pdf s of the estimators and the wald statistics are illustrated using simulation and compared with their respective asymptotic distributions

optimal inference in a class of regression models we consider the problem of constructing confidence intervals cis for a linear functional of a regression function such as its value at a point the regression discontinuity parameter or a regression coefficient in a linear or partly linear regression our main assumption is that the regression function is known to lie in a convex function class which covers most smoothness and or shape assumptions used in econometrics we derive finite sample optimal cis and sharp efficiency bounds under normal errors with known variance we show that these results translate to uniform over the function class asymptotic results when the error distribution is not known when the function class is centrosymmetric these efficiency bounds imply thatminimax cis are close to efficient at smooth regression functions this implies in particular that it is impossible to form cis that are tighter using data dependent tuning parameters and maintain coverage over the whole function class we specialize our results to inference in a linear regression and inference on the regression discontinuity parameter and illustrate them in an empirical application

some further results on the exact small sample properties of the instrumental variable estimator we present new results on the exact small sample distribution of the instrumental variable estimator in particular we compare the small sample distribution to the asymptotic distribution

improving bias and coverage in instrumental variable analysis with weak instruments for continuous and binary outcomes causal estimates can be obtained by instrumental variable analysis using a two stage method however these can be biased when the instruments are weak we introduce a bayesian method which adjusts for the first stage residuals in the second stage regression and has much improved bias and coverage properties in the continuous outcome case this adjustment reduces median bias from weak instruments to close to zero in the binary outcome case bias from weak instruments is reduced and the estimand is changed from a marginal population based effect to a conditional effect the lack of distributional assumptions on the posterior distribution of the causal effect gives a better summary of uncertainty and more accurate coverage levels than methods that rely on the asymptotic distribution of the causal estimate we discuss these properties in the context of mendelian randomization

the classical principles of testing using instrumental variables estimates abstract the wald likelihood ratio and lagrange multiplier tests of a set of linear restrictions on the structural parameters of an equation with stochastic regressors are defined and they are generalized for the k class instrumental variables estimation the approximation of their local power function indicates that in small samples all these tests are biased the problem being more severe for the wald test that is calculated from the tsls estimator moreover the comparison of the local power of the size corrected tests shows that under certain conditions the likelihood ratio test is preferable in the class of all feasible classical tests whereas the wald test with statistic calculated from the liml estimator is preferable in the class of all wald tests