bootstrap validity for the score test when instruments may be weak it is well known that size adjustments based on bootstrapping the t statistic perform poorly when instruments are weakly correlated with the endogenous explanatory variable in this paper we provide a theoretical proof that guarantees the validity of the bootstrap for the score statistic this theory does not follow from standard results since the score statistic is not a smooth function of sample means and some parameters are not consistently estimable when the instruments are uncorrelated with the explanatory variable

decision theory applied to an instrumental variables model this paper applies some general concepts in decision theory to a simple instrumental variables model there are two endogenous variables linked by a single structural equation k of the exogenous variables are excluded from this structural equation and provide the instrumental variables iv the reduced form distribution of the endogenous variables conditional on the exogenous variables corresponds to independent draws from a bivariate normal distribution with linear regression functions and a known co variance matrix a canonical form of the model has parameter vector p o w where o is the parameter of interest and is normalized to be a point on the unit circle the reduced form coefficients on the instrumental variables are split into a scalar parameter p and a parameter vector which is normalized to be a point on the k 1 dimensional unit sphere p measures the strength of the association between the endogenous variables and the instrumental variables and w is a measure of direction a prior distribution is introduced for the iv model the parameters o p and are treated as independent random variables the distribution for o is uniform on the unit circle the distribution for w is uniform on the unit sphere with dimension k 1 these choices arise from the solution of a minimax problem the prior for p is left general it turns out that given any positive value for p the bayes estimator of o does not depend on p it equals the maximum likelihood estimator this bayes estimator has constant risk because it minimizes average risk with respect to a proper prior it is minimax the same general concepts are applied to obtain confidence intervals the prior distribution is used in two ways the first way is to integrate out the nuisance parameter w in the iv model this gives an integrated likelihood function with two scalar parameters o and p inverting a likelihood ratio test based on the integrated likelihood function provides a confidence interval for o this lacks finite sample optimality but invariance arguments show that the risk function depends only on p and not on o or w the second approach to confidence sets aims for finite sample optimality by setting up a loss function that trades off coverage against the length of the interval the automatic uniform priors are used for o and but a prior is also needed for the scalar p and no guidance is offered on this choice the bayes rule is a highest posterior density set invariance arguments show that the risk function depends only on p and not on o or the optimality result combines average risk and maximum risk the confidence set minimizes the average with respect to the prior distribution for p of the maximum risk where the maximization is with respect to o and

effective nonparametric estimation in the case of severely discretized data often economic data are discretized or rounded to some extent this paper proposes a regression and a density estimator that work especially well when discretization causes conventional kernel based estimators to behave poorly the estimator proposed here is a weighted average of neighboring frequency estimators and the weights are composed of cubic b splines interestingly we show that this estimator can have both a smaller bias and variance than frequency estimators as a means to obtain asymptotic normality and rates of convergence we assume that the discreteness becomes finer as the sample size increases

gmm with many moment conditions this paper provides a first order asymptotic theory for generalized method of moments gmm estimators when the number of moment conditions is allowed to increase with the sample size and the moment conditions may be weak examples in which these asymptotics are relevant include instrumental variable iv estimation with many possibly weak or uninformed instruments and some panel data models that cover moderate time spans and have correspondingly large numbers of instruments under certain regularity conditions the gmm estimators are shown to converge in probability but not necessarily to the true parameter and conditions for consistent gmm estimation are given a general framework for the gmm limit distribution theory is developed based on epiconvergence methods some illustrations are provided including consistent gmm estimation of a panel model with time varying individual effects consistent limited information maximum likelihood estimation as a continuously updated gmm estimator and consistent iv structural estimation using large numbers of weak or irrelevant instruments some simulations are reported

a semi parametric bayesian approach to the instrumental variable problem we develop a bayesian semi parametric approach to the instrumental variable problem we assume linear structural and reduced form equations but model the error distributions non parametrically a dirichlet process prior is used for the joint distribution of structural and instrumental variable equations errors our implementation of the dirichlet process prior uses a normal distribution as a base model it can therefore be interpreted as modeling the unknown joint distribution with a mixture of normal distributions with a variable number of mixture components we demonstrate that this procedure is both feasible and sensible using actual and simulated data sampling experiments compare inferences from the non parametric bayesian procedure with those based on procedures from the recent literature on weak instrument asymptotics when errors are non normal our procedure is more efficient than standard bayesian or classical methods

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

examining bias in estimators of linear rational expectations models under misspecification most rational expectations models involve equations in which the dependent variable is a function of its lags and its expected future value we investigate the asymptotic bias of generalized method of moment gmm and maximum likelihood ml estimators in such models under misspecification we consider several misspecifications and focus more specifically on the case of omitted dynamics in the dependent variable in a stylized dgp we derive analytically the asymptotic biases of these estimators we establish that in many cases of interest the two estimators of the degree of forward lookingness are asymptotically biased in opposite direction with respect to the true value of the parameter we also propose a quasi hausman test of misspecification based on the difference between the gmm and ml estimators using monte carlo simulations we show that the ordering and direction of the estimators still hold in a more realistic new keynesian macroeconomic model in this set up misspecification is in general found to be more harmful to gmm than to ml estimators

efficient estimation and inference in linear pseudo panel data models we consider pseudo panel data models constructed from repeated cross sections in which the number of individuals per group is large relative to the number of groups and time periods first we show that when time invariant group fixed effects are neglected the ols estimator does not converge in probability to a constant but rather to a random variable second we show that while the fixed effects fe estimator is consistent the usual t statistic is not asymptotically normally distributed and we propose a new robust t statistic whose asymptotic distribution is standard normal third we propose efficient gmm estimators using the orthogonality conditions implied by grouping and we provide t tests that are valid even in the presence of time invariant group effects our monte carlo results show that the proposed gmm estimator is more precise than the fe estimator and that our new t test has good size and is powerful

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

bias testing bias correction and confounder selection using an instrumental variable model instrumental variable iv analysis can be used to address bias due to unobserved confounding when estimating the causal effect of a treatment on an outcome of interest however if a proposed iv is correlated with unmeasured confounders and or weakly correlated with the treatment the standard iv estimator may be more biased than an ordinary least squares ols estimator several methods have been proposed that compare the bias of the iv and ols estimators relying on the belief that measured covariates can be used as proxies for the unmeasured confounder despite these developments there is lack of discussion about approaches that can be used to formally test whether the iv estimator may be less biased than the ols estimator thus we have developed a testing framework to compare the bias and a criterion to select informative measured covariates for bias comparison and regression adjustment we also have developed a bias correction method which allows one to use an invalid iv to correct the bias of the ols or iv estimator numerical studies demonstrate that the proposed methods perform well with realistic sample sizes

bootstrap inference for instrumental variable models with many weak instruments this study s main contribution is to theoretically analyze the application of bootstrap methods to instrumental variable models when the available instruments may be weak and the number of instruments goes to infinity with the sample size we demonstrate that a standard residual based bootstrap procedure cannot consistently estimate the distribution of the limited information maximum likelihood estimator or fuller 1977 estimator under many many weak instrument sequence the primary reason is that the standard procedure fails to capture the instrument strength in the sample adequately in addition we consider the restricted efficient re bootstrap of davidson and mackinnon 2008 2010 2014 that generates bootstrap data under the null restricted and uses an efficient estimator of the coefficient of the reduced form equation efficient we find that the re bootstrap is also invalid however it effectively mimics more key features in the limiting distributions of interest and thus is less distorted in finite samples than the standard bootstrap procedure finally we propose modified bootstrap procedures that provide a valid distributional approximation for the two estimators with many many weak instruments a monte carlo experiment shows that hypothesis testing based on the asymptotic normal approximation can have severe size distortions in finite samples instead our modified bootstrap procedures greatly reduce these distortions

a practical two step method for testing moment inequalities this paper considers the problem of testing a finite number of moment inequalities we propose a two step approach in the first step a confidence region for the moments is constructed in the second step this set is used to provide information about which moments are negative a bonferonni type correction is used to account for the fact that with some probability the moments may not lie in the confidence region it is shown that the test controls size uniformly over a large class of distributions for the observed data an important feature of the proposal is that it remains computationally feasible even when the number of moments is large the finite sample properties of the procedure are examined via a simulation study which demonstrates among other things that the proposal remains competitive with existing procedures while being computationally more attractive

on bootstrap inconsistency and bonferroni based size correction for the subset anderson rubin test under conditional homoskedasticity abstract we focus on the linear instrumental variable model with two endogenous regressors under conditional homoskedasticity and study the subset anderson and rubin 1949 ar test when the nuisance structural parameter the unrestricted slope coefficient of endogenous regressor may be weakly identified weak identification leads to nonstandard null limiting distributions and alternative to the usual chi squared critical value is needed we first investigate the bootstrap validity for the subset ar test based on various plug in estimators and show that the bootstrap provides asymptotic refinement when the nuisance structural parameter is strongly identified but is inconsistent when it is weakly identified this is in contrast to the result of bootstrap validity in moreira et al 2009 then we propose a bonferroni based size correction method that yields correct asymptotic size for all the test statistics considered the power performance of size corrected tests can be further improved by applying the mapping between structural and endogenous parameters in the model monte carlo experiments confirm the bootstrap inconsistency and demonstrate that all the subset tests based on our correction technique control the size

weak identification robust tests in an instrumental quantile model abstract we develop a testing procedure that is robust to identification quality in an instrumental quantile model in order to reduce the computational burden a multi step approach is taken and a two step anderson rubin ar statistic is considered we then propose an orthogonal decomposition of the ar statistic where the null distribution of each component does not depend on the assumption of a full rank of the jacobian power experiments are conducted and inferences on returns to schooling using the angrist and krueger data are considered as an empirical example

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

bias in causal estimates from mendelian randomization studies with weak instruments mendelian randomization studies using genetic instrumental variables ivs are now being commonly used to estimate the causal association of a phenotype on an outcome even when the necessary underlying assumptions are valid estimates from analyses using ivs are biased in finite samples the source and nature of this bias appear poorly understood in the epidemiological field we explain why the bias is in the direction of the confounded observational association with magnitude relating to the statistical strength of association between the instrument and phenotype we comment on the size of the bias from simulated data showing that when multiple instruments are used although the variance of the iv estimator decreases the bias increases we discuss ways to analyse mendelian randomization studies to alleviate the problem of weak instrument bias

robust inference in nonlinear models with mixed identification strength abstract the paper studies inference in regression models composed of nonlinear functions with unknown transformation parameters and loading coefficients that measure the importance of each component in these models non identification and weak identification present in multiple parts of the parameter space resulting in mixed identification strength for different unknown parameters this paper proposes robust tests and confidence intervals for sub vectors and linear functions of the unknown parameters in particular the results cover applications where some nuisance parameters are non identified under the null davies 1977 1987 and some nuisance parameters are subject to a full range of identification strength to construct this robust inference procedure we develop a local limit theory that models mixed identification strength the asymptotic results involve both inconsistent estimators that depend on a localization parameter and consistent estimators with different rates of convergence a sequential argument is used to peel the criterion function based on identification strength of the parameters

on distributions of ratios a large number of exact inferential procedures in statistics and econometrics involve the sampling distribution of ratios of random variables if the denominator variable is positive then tail probabilities of the ratio can be expressed as those of a suitably defined difference of random variables if in addition the joint characteristic function of numerator and denominator is known then standard fourier inversion techniques can be used to reconstruct the distribution function from it most research in this field has been based on this correspondence but which breaks down when both numerator and denominator are supported on the entire real line the present manuscript derives inversion formulae and saddlepoint approximations that remain valid in this case and reduce to known results when the denominator is almost surely positive applications include the iv est imator of a structural parameter in a just identified equation

finite sample inference for quantile regression models abstract under minimal assumptions finite sample confidence bands for quantile regression models can be constructed these confidence bands are based on the conditional pivotal property of estimating equations that quantile regression methods solve and provide valid finite sample inference for linear and nonlinear quantile models with endogenous or exogenous covariates the confidence regions can be computed using markov chain monte carlo mcmc methods we illustrate the finite sample procedure through two empirical examples estimating a heterogeneous demand elasticity and estimating heterogeneous returns to schooling we find pronounced differences between asymptotic and finite sample confidence regions in cases where the usual asymptotics are suspect

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

conditional linear combination tests for weakly identified models we introduce the class of conditional linear combination tests which reject null hypotheses concerning model parameters when a data dependent convex combination of two identification robust statistics is large these tests control size under weak identification and have a number of optimality properties in a conditional problem we show that the conditional likelihood ratio test of moreira 2003 is a conditional linear combination test in models with one endogenous regressor and that the class of conditional linear combination tests is equivalent to a class of quasi conditional likelihood ratio tests we suggest using minimax regret conditional linear combination tests and propose a computationally tractable class of tests that plug in an estimator for a nuisance parameter these plug in tests perform well in simulation and have optimal power in many strongly identified models thus allowing powerful identification robust inference in a wide range of linear and nonlinear models without sacrificing efficiency if identification is strong

efficient two sided nonsimilar invariant tests in iv regression with weak instruments abstract as nelson and startz nelson c r startz r 1990a the distribution of the instrumental variable estimator and its t ratio when the instrument is a poor one journal of business 63 s125 s140 nelson c r startz r 1990b some further results on the exact small sample properties of the instrumental variables estimator econometrica 58 967 976 dramatically demonstrated standard hypothesis tests and confidence intervals in instrumental variables regression are invalid when instruments are weak recent work on hypothesis tests for the coefficient on a single included endogenous regressor when instruments may be weak has focused on similar tests this paper extends that work to nonsimilar tests of which similar tests are a subset the power envelope for two sided invariant to rotations of the instruments nonsimilar tests is characterized theoretically then evaluated numerically for five ivs the power envelopes for similar and nonsimilar tests differ theoretically but are found to be very close numerically the nonsimilar test power envelope is effectively achieved by the moreira moreira m j 2003 a conditional likelihood ratio test for structural models econometrica 71 1027 1048 conditional likelihood ratio test so that test is effectively uniformly most powerful invariant umpi we also provide a new nonsimilar test p which has 1 2 critical values is asymptotically efficient under strong instruments involves only elementary functions and is very nearly umpi

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

regularized liml for many instruments the use of many moment conditions improves the asymptotic efficiency of the instrumental variables estimators however in finite samples the inclusion of an excessive number of moments increases the bias to solve this problem we propose regularized versions of the limited information maximum likelihood liml based on three different regularizations tikhonov landweber fridman and principal components our estimators are consistent and asymptotically normal under heteroskedastic error moreover they reach the semiparametric efficiency bound assuming homoskedastic error we show that the regularized liml estimators possess finite moments when the sample size is large enough the higher order expansion of the mean square error mse shows the dominance of regularized liml over regularized two staged least squares estimators we devise a data driven selection of the regularization parameter based on the approximate mse a monte carlo study and two empirical applications illustrate the relevance of our estimators

the hausman test and weak instruments we consider the following problem there is a structural equation of interest that contains an explanatory variable that theory predicts is endogenous there are one or more instrumental variables that credibly are exogenous with regard to this structural equation but which have limited explanatory power for the endogenous variable further there is one or more potentially strong instruments which has much more explanatory power but which may not be exogenous hausman 1978 provided a test for the exogeneity of the second instrument when none of the instruments are weak here we focus on how the standard hausman test does in the presence of weak instruments using the staiger stock asymptotics it is natural to conjecture that the standard version of the hausman test would be invalid in the weak instrument case which we confirm however we provide a version of the hausman test that is valid even in the presence of weak iv and illustrate how to implement the test in the presence of heteroskedasticity we show that the situation we analyze occurs in several important economic examples our monte carlo experiments show that our procedure works relatively well in finite samples we should note that our test is not consistent although we believe that it is impossible to construct a consistent test with weak instruments

decomposing treatment effect variation understanding and characterizing treatment effect variation in randomized experiments has become essential for going beyond the black box of the average treatment effect nonetheless traditional

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

bayesian and classical approaches to instrumental variable regression we establish relationships between certain bayesian and classical approaches to instrumental variable regression we determine the form of priors that lead to posteriors for structural parameters that have similar properties as classical 2sls and liml and in doing so provide some new insight especially in the context of weak instruments to the small sample behavior of bayesian and classical procedures using a reduced rank restriction on a multivariate linear model we determine the priors that give rise to posteriors that are identical in functional form to the sampling densities of the classical 2sls and liml estimators

instrumental variable estimation in the presence of many moment conditions abstract this paper develops shrinkage methods for addressing the many instruments problem in the context of instrumental variable estimation it has been observed that instrumental variable estimators may behave poorly if the number of instruments is large this problem can be addressed by shrinking the influence of a subset of instrumental variables the procedure can be understood as a two step process of shrinking some of the ols coefficient estimates from the regression of the endogenous variables on the instruments then using the predicted values of the endogenous variables based on the shrunk coefficient estimates as the instruments the shrinkage parameter is chosen to minimize the asymptotic mean square error the optimal shrinkage parameter has a closed form which makes it easy to implement a monte carlo study shows that the shrinkage method works well and performs better in many situations than do existing instrument selection procedures

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

gmm estimators with improved finite sample properties using principal components of the weighting matrix with an application to the dynamic panel data model gmm estimators have poor finite sample properties in highly overidentified models with many moment conditions the optimal weighting matrix is poorly estimated we suggest using principal components of the weighting matrix this effectively drops some of the moment conditions our simulations done in the context of the dynamic panel data model show that the resulting gmm estimator has better finite sample properties than the usual two step gmm estimator in the sense of smaller bias and more reliable standard errors

optimal two sided invariant similar tests for instrumental variables regression this paper considers tests of the parameter on an endogenous variable in an instrumental variables regression model the focus is on determining tests that have some optimal power properties we start by considering a model with normally distributed errors and known error covariance matrix we consider tests that are similar and satisfy a natural rotational invariance condition we determine a two sided power envelope for invariant similar tests this allows us to assess and compare the power properties of tests such as the conditional likelihood ratio clr the lagrange multiplier and the anderson rubin tests we find that the clr test is quite close to being uniformly most powerful invariant among a class of two sided tests the finite sample results of the paper are extended to the case of unknown error covariance matrix and possibly nonnormal errors via weak instrument asymptotics strong instrument asymptotic results also are provided because we seek tests that perform well under both weak and strong instruments

semiparametric tests of conditional moment restrictions under weak or partial identification abstract we propose two new semiparametric specification tests which test whether a vector of conditional moment conditions is satisfied for any vector of parameter values 0 unlike most existing tests our tests are asymptotically valid under weak and or partial identification and can accommodate discontinuities in the conditional moment functions our tests are moreover consistent provided that identification is not too weak we do not require the availability of a consistent first step estimator like robinson robinson peter m 1987 asymptotically efficient estimation in the presence of heteroskedasticity of unknown form econometrica 55 875 891 and many others in similar problems subsequently we use k nearest neighbor knn weights instead of kernel weights the advantage of using knn weights is that local power is invariant to transformations of the instruments and that under strong point identification computation of the test statistic yields an efficient estimator of 0 as a byproduct

on asymptotic size distortions in the random coefficients logit model we show that in the random coefficients logit model standard inference procedures can suffer from asymptotic size distortions the problem arises due to boundary issues and is aggravated by the standard parameterization of the model in terms of standard deviations for example in case of a single random coefficient the asymptotic size of the nominal 95 confidence interval obtained by inverting the two sided t test for the standard deviation equals 83 65 in seeming contradiction we also show that standard error estimates for the estimator of the standard deviation can be unreasonably large this problem is alleviated if the model is reparameterized in terms of variances furthermore a numerical evaluation of a conjectured lower bound suggests that the asymptotic size of the nominal 95 confidence interval obtained by inverting the two sided t test for variances means is within 0 5 percentage points of the nominal level as long as there are no more than five four random coefficients and as long as an optimal weighting matrix is employed

testing for common conditionally heteroskedastic factors this paper proposes a test for common conditionally heteroskedastic ch features in asset returns following engle and kozicki 1993 the common ch features property is expressed in terms of testable overidentifying moment restrictions however as we show these moment conditions have a degenerate jacobian matrix at the true parameter value and therefore the standard asymptotic results of hansen 1982 do not apply we show in this context that hansen s 1982 j test statistic is asymptotically distributed as the minimum of the limit of a certain random process with a markedly nonstandard distribution if two assets are considered this asymptotic distribution is a fifty fifty mixture of and where h is the number of moment conditions as opposed to a with more than two assets this distribution lies between the and p denotes the number of parameters these results show that ignoring the lack of first order identification of the moment condition model leads to oversized tests with a possibly increasing overrejection rate with the number of assets a monte carlo study illustrates these findings proquest denotes formulae symbols omitted

endogeneity in quantile regression models a control function approach this paper considers a linear triangular simultaneous equations model with conditional quantile restrictions the paper adjusts for endogeneity by adopting a control function approach and presents a simple two step estimator that exploits the partially linear structure of the model the first step consists of estimation of the residuals of the reduced form equation for the endogenous explanatory variable the second step is series estimation of the primary equation with the reduced form residual included nonparametrically as an additional explanatory variable this paper imposes no functional form restrictions on the stochastic relationship between the reduced form residual and the disturbance term in the primary equation conditional on observable explanatory variables the paper presents regularity conditions for consistency and asymptotic normality of the two step estimator in addition the paper provides some discussions on related estimation methods in the literature and on possible extensions and limitations of the estimation approach finally the numerical performance and usefulness of the estimator are illustrated by the results of monte carlo experiments and two empirical examples demand for fish and returns to schooling

performance of conditional wald tests in iv regression with weak instruments abstract we compare the powers of five tests of the coefficient on a single endogenous regressor in instrumental variables regression following moreira 2003 a conditional likelihood ratio test for structural models econometrica 71 1027 1048 all tests are implemented using critical values that depend on a statistic which is sufficient under the null hypothesis for the unknown concentration parameter so these conditional tests are asymptotically valid under weak instrument asymptotics four of the tests are based on k class wald statistics two stage least squares liml fuller s some properties of a modification of the limited information estimator econometrica 45 939 953 and bias adjusted tsls the fifth is moreira s 2003 conditional likelihood ratio clr test the heretofore unstudied conditional wald cw tests are found to perform poorly compared to the clr test in many cases the cw tests have almost no power against a wide range of alternatives our analysis is facilitated by a new algorithm presented here for the computation of the asymptotic conditional p value of the clr test

increasing the power of specification tests abstract this paper shows how to increase the power of hausman s 1978 specification test as well as the difference test in a large class of models the idea is to impose the restrictions of the null and the alternative hypotheses when estimating the covariance matrix if the null hypothesis is true then the proposed test has the same distribution as the existing ones in large samples if the hypothesis is false then the proposed test statistic is larger with probability approaching one as the sample size increases in several important applications including testing for endogeneity in the linear model

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

empirically relevant critical values for hypothesis tests a bootstrap approach tests of statistical hypotheses can be based on either of two critical values the type i critical value or the size corrected critical value the former usually depends on unknown population parameters and cannot be evaluated exactly in applications but it can often be estimated very accurately by using the bootstrap the latter does not depend on unknown population parameters but is likely to yield a test with low power the critical values used in most monte carlo studies of the powers of tests are neither type i nor size corrected they are irrelevant to empirical research

inverting the indirect the ellipse and the boomerang visualizing the confidence intervals of the structural coefficient from two stage least squares abstract in the just identified model the exact distribution of the two stage least squares 2sls estimator of the coefficient of the endogenous regressor is a ratio of two normally distributed random variables basmann 1960 basmann 1974 used fieller s 1932 result to derive the density function of the estimator in this paper we present a novel graphical exposition of fieller s 1954 technique to approximate the confidence interval for the 2sls estimator we use this approach to examine how the degree of endogeneity and instrument relevance influences the correspondence between the fieller and traditional asymptotic confidence intervals for the estimator

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

hybrid generalized empirical likelihood estimators instrument selection with adaptive lasso in this paper we use the adaptive lasso estimator to choose the relevant instruments and eliminate the irrelevant instruments the limit theory of zou 2006 is extended from univariate iid case to heteroskedastic and non gaussian data then we use the selected instruments in generalized empirical likelihood estimators gel in this sense these are called hybrid gel it is also shown that the lasso estimators are not model selection consistent whereas the adaptive lasso can select the correct model with fixed number of instruments in simulations we show that hybrid gel estimators have smaller bias and mean squared error than the other estimators in certain cases

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

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

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

exogeneity tests incomplete models weak identification and non gaussian distributions invariance and finite sample distributional theory abstract we study the distribution of durbin wu hausman dwh and revankar hartley rh tests for exogeneity from a finite sample viewpoint under the null and alternative hypotheses we consider linear structural models with possibly non gaussian errors where structural parameters may not be identified and where reduced forms can be incompletely specified or nonparametric on level control we characterize the null distributions of all the test statistics through conditioning and invariance arguments we show that these distributions do not involve nuisance parameters in particular this applies to several test statistics for which no finite sample distributional theory is yet available such as the standard statistic proposed by hausman 1978 the distributions of the test statistics may be non standard so corrections to usual asymptotic critical values are needed but the characterizations are sufficiently explicit to yield finite sample monte carlo tests of the exogeneity hypothesis the procedures so obtained are robust to weak identification missing instruments or misspecified reduced forms and can easily be adapted to allow for parametric non gaussian error distributions we give a general invariance result block triangular invariance for exogeneity test statistics this property yields a convenient exogeneity canonical form and a parsimonious reduction of the parameters on which power depends in the extreme case where no structural parameter is identified the distributions under the alternative hypothesis and the null hypothesis are identical so the power function is flat for all the exogeneity statistics however as soon as identification does not fail completely this phenomenon typically disappears we present simulation experiments which confirm the finite sample theory the theoretical results are illustrated with an empirical example the relation between trade and economic growth

credibility of confidence sets in nonstandard econometric problems confidence intervals are commonly used to describe parameter uncertainty in nonstandard problems however their frequentist coverage property does not guarantee that they do so in a reasonable fashion for instance confidence intervals may be empty or extremely short with positive probability even if they are based on inverting powerful tests we apply a betting framework and a notion of bet proofness to formalize the reasonableness of confidence intervals as descriptions of parameter uncertainty and use it for two purposes first we quantify the violations of bet proofness for previously suggested confidence intervals in nonstandard problems second we derive alternative confidence sets that are bet proof by construction we apply our framework to several nonstandard problems involving weak instruments near unit roots and moment inequalities we find that previously suggested confidence intervals are not bet proof and numerically determine alternative bet proof confidence sets

quasi maximum likelihood estimation and testing for nonlinear models with endogenous explanatory variables abstract i propose a quasi maximum likelihood framework for estimating nonlinear models with continuous or discrete endogenous explanatory variables joint and two step estimation procedures are considered the joint procedure is a quasi limited information maximum likelihood procedure as one or both of the log likelihoods may be misspecified the two step control function approach is computationally simple and leads to straightforward tests of endogeneity in the case of discrete endogenous explanatory variables i argue that the control function approach can be applied with generalized residuals to obtain average partial effects i show how the results apply to nonlinear models for fractional and nonnegative responses

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

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

robust estimation with many instruments abstract linear instrumental variables models are widely used in empirical work but often associated with low estimator precision this paper proposes an estimator that is robust to outliers and shows that the estimator is minimax optimal in a class of estimators that includes the limited maximum likelihood estimator liml intuitively this optimal robust estimator combines liml with winsorization of the structural residuals and the winsorization leads to improved precision under thick tailed error distributions consistency and asymptotic normality of the estimator are established under many instruments asymptotics and a consistent variance estimator which allows for asymptotically valid inference is provided

structural inference from reduced forms with many instruments abstract this paper develops exact finite sample and asymptotic distributions for structural equation tests based on partially restricted reduced form estimates particular attention is given to models with large numbers of instruments wherein the use of partially restricted reduced form estimates is shown to be especially advantageous in statistical testing even in cases of uniformly weak instruments comparisons are made with methods based on unrestricted reduced forms and numerical computations showing finite sample performance of the tests are reported some new results are obtained on inequalities between noncentral chi squared distributions with different degrees of freedom that assist in analytic power comparisons

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

choosing the number of instruments properties of instrumental variable estimators are sensitive to the choice of valid instruments even in large cross section applications in this paper we address this problem by deriving simple mean square error criteria that can be minimized to choose the instrument set we develop these criteria for two stage least squares 2sls limited information maximum likelihood liml and a bias adjusted version of 2sls b2sls we give a theoretical derivation of the mean square error and show optimality in monte carlo experiments we find that the instrument choice generally yields an improvement in performance also in the angrist and krueger 1991 returns to education application when the instrument set is chosen in the way we consider it turns out that both 2sls and liml give similar large returns to education

tests based on t statistics for iv regression with weak instruments abstract this paper considers tests of the parameter of an endogenous variable in an instrumental variables regression model the focus is on one sided conditional t tests theoretical and numerical work shows that the conditional 2sls and fuller t tests perform well even when instruments are weakly correlated with the endogenous variable when the population f statistic is as small as two their power is reasonably close to the power envelopes for similar and non similar tests which are invariant to rotation transformations of the instruments this finding is surprising considering the bad performance of two sided conditional t tests found in andrews et al 2007 we show these tests have bad power because the conditional null distributions of t statistics are asymmetric when instruments are weak taking this asymmetry into account we propose two sided tests based on t statistics these novel tests are approximately unbiased and can perform as well as the conditional likelihood ratio clr test

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

projection based statistical inference in linear structural models with possibly weak instruments it is well known that standard asymptotic theory is not applicable or is very unreliable in models with identification problems or weak instruments one possible way out consists of using a variant of the anderson rubin 1949 ar procedure the latter allows one to build exact tests and confidence sets only for the full vector of the coefficients of the endogenous explanatory variables in a structural equation but not for individual coefficients this problem may in principle be overcome by using projection methods dufour 1997 dufour and jasiak 2001 at first sight however this technique requires the application of costly numerical algorithms in this paper we give a general necessary and sufficient condition that allows one to check whether an ar type confidence set is bounded furthermore we provide an analytic solution to the problem of building projection based confidence sets from ar type confidence sets the latter involves the geometric properties of quadrics and can be viewed as an extension of usual confidence intervals and ellipsoids only least squares techniques are needed to build the confidence intervals

tests with correct size when instruments can be arbitrarily weak abstract this paper applies classical exponential family statistical theory to develop a unifying framework for testing structural parameters in the simultaneous equations model under the assumption of normal errors with known reduced form variance matrix the results can be divided into the limited information and full information categories in the limited information model it is possible to characterize the entire class of similar tests in a model with only one endogenous explanatory variable in the full information framework this paper proposes a family of similar tests for subsets of endogenous variables coefficients for both limited and full information models there exist power upper bounds for unbiased tests when the model is just identified the anderson rubin score and pseudo conditional likelihood ratio tests are optimal when the model is over identified the pseudo conditional likelihood ratio test has power close to the power envelope when identification is strong

a quasi differencing approach to dynamic modelling from a time series of independent cross sections we motivate and describe a gmm method of estimating linear dynamic models from a time series of independent cross sections this involves subjecting the model to a quasi differencing transformation across pairs of individuals that belong to the same group desirable features of the model include the fact that i no aggregation is involved ii dynamic response parameters can vary across groups iii the presence of unobserved individual specific heterogeneity is explicitly allowed for and iv the gmm estimators are derived by realistically assuming group size asymptotics the monte carlo experiments conducted to assess the finite sample performances of the proposed estimators provide us with encouragement

the distributions of the j and cox non nested tests in regression models with weakly correlated regressors abstract this paper examines the asymptotic null distributions of the j and cox non nested tests in the framework of two linear regression models with nearly orthogonal non nested regressors the analysis is based on the concept of near population orthogonality npo according to which the non nested regressors in the two models are nearly uncorrelated in the population distribution from which they are drawn new distributional results emerge under npo the j and cox tests tend to two different random variables asymptotically each of which is expressible as a function of a nuisance parameter c a n 0 1 variate and a 2 q variate where q is the number of non nested regressors in the alternative model the monte carlo method is used to show the relevance of the new results in finite samples and to compute alternative critical values for the two tests under npo by plugging consistent estimates of c into the relevant asymptotic expressions an empirical example illustrates the plug in procedure

initial conditions and moment restrictions in dynamic panel data models estimation of the dynamic error components model is considered using two alternative linear estimators that are designed to improve the properties of the standard first differenced gmm estimator both estimators require restrictions on the initial conditions process asymptotic efficiency comparisons and monte carlo simulations for the simple ar 1 model demonstrate the dramatic improvement in performance of the proposed estimators compared to the usual first differenced gmm estimator and compared to non linear gmm the importance of these results is illustrated in an application to the estimation of a labour demand model using company panel data

testing parameters in gmm without assuming that they are identified we propose a generalized method of moments gmm lagrange multiplier statistic i e the k statistic that uses the jacobian at the evaluated parameter value instead of the expected jacobian to obtain its limit behavior we use a novel assumption that brings gmm closer to maximum likelihood and which is easily satisfied the usual asymptotic x2 distribution of the k statistic then holds under a wider set of circumstances like weak and many instrument asymptotics and combinations thereof than the standard full rank case for the jacobian the behavior of the k statistic can be spurious around inflexion points and the maximum of the objective function since the moment conditions are then not satisfied combinations of the k statistic with statistics that test the validity of the moment equations overcome the spurious behavior we conduct a power comparison to test for the risk aversion parameter in a stochastic discount factor model and construct its confidence set for observed consumption growth and asset return series

on the asymptotic optimality of the liml estimator with possibly many instruments we consider the estimation of the coefficients of a linear structural equation in a simultaneous equation system when there are many instrumental variables we derive some asymptotic properties of the limited information maximum likelihood liml estimator when the number of instruments is large some of these results are new as well as old and we relate them to results in some recent studies we have found that the variance of the limiting distribution of the liml estimator and its modifications often attain the asymptotic lower bound when the number of instruments is large and the disturbance terms are not necessarily normally distributed that is for the micro econometric models of some cases recently called many instruments and many weak instruments