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

global existence for the einstein vacuum equations in wave coordinates we prove global stability of minkowski space for the einstein vacuum equations in harmonic wave coordinate gauge for the set of restricted data coinciding with the schwarzschild solution in the neighborhood of space like infinity the result contradicts previous beliefs that wave coordinates are unstable in the large and provides an alternative approach to the stability problem originally solved for unrestricted data in a different gauge and with a precise description of the asymptotic behavior at null infinity by d christodoulou and s klainerman

on the determination of general scientific models with application to asset pricing we consider a consumption based asset pricing model that uses habit persistence to overcome the known statistical inadequacies of the classical consumption based asset pricing model we find that the habit model fits reasonably well and agrees with results reported in the literature if conditional heteroskedasticity is suppressed but that it does not fit nor do results agree if conditional heteroskedasticity well known to be present in financial market data is allowed to manifest itself we also find that it is the preference parameters of the model that are most affected by the presence or absence of conditional heteroskedasticity especially the risk aversion parameter the habit model exhibits four characteristics that are often present in models developed from scientific considerations 1 a likelihood is not available 2 prior information is available 3 a portion of the prior information is expressed in terms of functionals of the model that cannot be converted into an analytic prior on model

is there a curse of dimensionality for contraction fixed points in the worst case this paper analyzes the complexity of the contraction fixed point problem compute an e approximation to the fixed point v v of a contraction mapping r that maps a banach space b d of continuous functions of d variables into itself we focus on quasi linear contractions where is a nonlinear functional of a finite number of conditional expectation operators this class includes contractive fredholm integral equations that arise in asset pricing applications and the contractive bellman equation from dynamic programming in the absence of further restrictions on the domain of the quasi linear fixed point problem is subject to the curse of dimensionality i e in the worst case the minimal number of function evaluations and arithmetic operations required to compute an e approximation to a fixed point v e b d increases exponentially in d we show that the curse of dimensionality disappears if the domain of has additional special structure we identify a particular type of special structure for which the problem is strongly tractable even in the worst case i e the number of function evaluations and arithmetic operations needed to compute an e approximation of v is bounded by ce p where c and p are constants independent of d we present examples of economic problems that have this type of special structure including a class of rational expectations asset pricing problems for which the optimal exponent p 1 is nearly achieved

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

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

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

simulated moments estimation of markov models of asset prices this paper provides a simulated moments estimator sme of the parameters of dynamic models in which the state vector follows a time homogeneous markov process conditions are provided for both weak and strong consistency as well as asymptotic normality various tradeoffs among the regularity conditions underlying the large sample properties of the sme are discussed in the context of an asset pricing model this paper provides conditions for the consistency and asymptotic normality of a simulated moments estimator sme of the parameters of asset pricing models with time homogeneous markov representations of the stochastic forc ing process sme s for economic models have been proposed by mcfadden 1989 and pakes and pollard 1989 for i i d environments and by lee and ingram 1991 for a time series environment the sme for time series models examined in this paper is as follows the state vector yt that determines asset prices is assumed to follow a time homogeneous markov process whose transi tion function depends on an unknown parameter vector 3 0 asset prices and possibly other relevant data are observed as f yt 0 for some given function f of the underlying state and parameter vector in parallel a simulated state process y is generated analytically or numerically from the economic model and corresponding simulated observations f yj3 13 are taken for a given parameter choice f3 the parameter is chosen so as to match moments that is to minimize the distance between sample moments of the data f y 8030 and those of the simulated series f yt f3 in a sense to be made precise the proposed sme extends the generalized method of moments gmm estimator hansen 1982 to a large class of asset pricing models for which the moment restrictions of interest do not have analytic representations in terms of observable variables and the unknown parameter vector we provide conditions on the transition function of yt and the observation function f under which the sme of 030 is consistent and characterize the normalized asymptotic distribu tion of the estimator for two reasons neither the regularity conditions underly ing hansen s 1982 analysis of gmm estimators for time series models without

testing for weak identification in possibly nonlinear models in this paper we propose a chi square test for identification our proposed test statistic is based on the distance between two shrinkage extremum estimators the two estimators converge in probability to the same limit when identification is strong and their asymptotic distributions are different when identification is weak the proposed test is consistent not only for the alternative hypothesis of no identification but also for the alternative of weak identification which is confirmed by our monte carlo results we apply the proposed technique to test whether the structural parameters of a representative taylor rule monetary policy reaction function are identified

on the projections of isotropic distributions the class of distributions on rm 1 c m oo which are the m dimensional marginal distributions of orthogonally invariant distributions on rm n is characterized this result is then used to provide a partial answer to the following question given a symmetric distribution on r and an integer n 2 under what conditions will there exist a random vector x e rn such that a x has the given distribution up to a positive scale factor for all a 0 a e rm

breakdown properties of location estimates based on halfspace depth and projected outlyingness we describe multivariate generalizations of the median trimmed mean and w estimates the estimates are based on a geometric construction related to projection pursuit they are both affine equivariant coordinate free and have high breakdown point the generalization of the median has a breakdown point of at least 1 d 1 in dimension d and the breakdown point can be as high as 1 3 under symmetry in contrast various estimators based on rejecting apparent outliers and taking the mean of the remaining observations have breakdown points not larger than 1 d 1 in dimension d

quadrature based methods for obtaining approximate solutions to nonlinear asset pricing models the paper develops a discrete state space solution method for a class of nonlinear rational expectations models the method works by using numerical quadrature rules to approximate the integral operators that arise in stochastic intertemporal models the method is particularly useful for approximating asset pricing models and has potential applications in other problems as well an empirical application uses the method to study the relationship between the risk premium and the conditional variability of the equity return under an arch endowment process nonlinear dynamic rational expectations models rarely admit explicit solutions techniques like the method of undetermined coefficients or forward looking expansions which often work well for linear models rarely provide explicit solutions for nonlinear models the lack of explicit solutions compli cates the tasks of analyzing the dynamic properties of such models and generat ing simulated realizations for applied policy work and other purposes this paper develops a discrete state space approximation method for a specific class of nonlinear rational expectations models the class of models is distinguished by two features first the solution functions for the endogenous variables are functions of at most a finite number of lags of an exogenous stationary state vector second the expectational equations of the model take the form of integral equations or more precisely fredholm equations of the second type the key component of the method is a technique based on numerical quadrature for forming a discrete approximation to a general time series conditional density more specifically the technique provides a means for calibrating a markov chain with a discrete state space whose probability distribution closely approximates the distribution of a given time series the quality of the approximation can be expected to get better as the discrete state space is made sufficiently finer the term discrete is used here in reference to the range space of the random variables and not to the time index time is always discrete in our analysis the discretization technique is primarily useful for taking a discrete approxi mation to the conditional density of the strictly exogenous variables of a model the specification of this conditional density could be based on a variety of 1financial support under nsf grants ses 8520244 and ses 8810357 is acknowledged we thank the co editor and referees of earlier drafts for many many helpful comments that substantially improved the manuscript

blow up for quasi linear wave equations in three space dimensions we consider equations of the form u phi x t u u u 1a for a function u u x 1 x n t u x t here denotes the d alembertian square frac partial 2 partial t 2 vartriangle 1b and u represents the vector of first derivatives u that of second derivatives of u with respect to the x k and t on linearizing 1a shall go over into the classical wave equation u 0 this means that and its first derivatives with respect to u u u shall vanish for u u u 0

general notions of statistical depth function statistical depth functions are being formulated ad hoc with increasing popularity in nonparametric inference for multivariate data here we introduce several general structures for depth functions classify many existing examples as special cases and establish results on the possession or lack thereof of four key properties desirable for depth functions in general roughly speaking these properties may be described as affine invariance maximality at center monotonicity relative to deepest point and vanishing at infinity this provides a more systematic basis for selection of a depth function in particular from these and other considerations it is found that the halfspace depth behaves very well overall in comparison with various competitors

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

testing nonnested euler conditions with quadrature based methods of approximation abstract singleton 1985 proposed a test of the validity of an euler equation specification which incorporates information on a nonnested set of euler equations from an alternative economic model the information about the alternative model is introduced into the testing framework by constructing a sequence of local alternatives to the euler equations of the maintained model which are in the direction of the euler equations of the alternative model singleton s test is easy to apply but is designed to have power against an alternative that in many cases does not have a straightforward economic interpretation in this paper we reexamine the construction of local alternatives to euler equations we compare singleton s approach with a method based on perturbing the data generation process dgp of the maintained model in the direction of the dgp of the alternative model the second approach leads to tests which are always of interest but require the complete specification of the dgp in this paper we derive a test for discriminating between two nonnested sets of euler conditions which have been estimated us ing generalized method of moments gmm the test is based on the encompassing principle of cox 1961 and mizon and richard 1986 and uses tauchen s 1986 quadrature based nethods for approximating the expectation of nonlinear functions of stationary random variables singleton s empirical examples are also reconsidered

on the stahel donoho estimator and depth weighted means of multivariate data the depth of multivariate data can be used to construct weighted means as robust estimators of location the use of projection depth leads to the stahel donoho estimator as a special case in contrast to maximal depth estimators the depth weighted means are shown to be asymptotically normal under appropriate conditions met by depth functions commonly used in the current literature we also confirm through a finite sample study that the stahel donoho estimator achieves a desirable balance between robustness and efficiency at gaussian models

combining estimators to improve structural model estimation and inference under quadratic loss asymptotically semi parametric estimators of the parameters in linear structural models have the same sampling properties in finite samples the sampling properties of these estimators vary and large biases may result for sample sizes often found in practice with a goal of improving asymptotic risk performance and finite sample efficiency properties we investigate the idea of combining correlated structural equation estimators with different finite and asymptotic sampling characteristics based on a quadratic loss measure we present evidence that the finite sample performance of the resulting combination estimator can be notably superior to that of a leading traditional moment based estimator

on the asymptotic properties of ldu based tests of the rank of a matrix abstract gill and lewbel recently introduced a test for the rank of a matrix based on the ldu decomposition unfortunately the asymptotic distribution suggested by them is incorrect except in a very limited problem in general the asymptotic distribution is that of a highly complicated nonlinear function of a normally distributed random vector that appears to defy useful characterization the ldu decomposition can be used to produce a valid test asymptotically equivalent to the minimum x 2 test

further consequences of viewing liml as an iterated aitken estimator abstract following an approach originally suggested by pagan 1979 economics letters 3 369 372 we have explored some additional consequences of viewing liml as an iterated aitken estimator a curious result we derive in this context is that the aitken estimator based on 2sls residuals produces 2sls estimate of the structural parameters we derive a simple expression for the difference between 2sls and liml in finite samples which turns out to be very similar to an expression for the approximate bias of 2sls in small samples nagar 1959 econometrica 27 575 595 buse 1992 econometrica 60 173 180 bound et al 1995 journal of american statistical association 90 443 450 the expression pinpoints the gain from iteration our analytical formula yields many of the results on the difference between 2sls and liml that anderson and sawa 1979 econometrica 47 163 182 and anderson et al 1982 econometrica 50 1009 1027 have obtained through extensive numerical simulations in a model with two endogenous variables the robustness of 2sls and liml with respect to certain deviations from normality is also examined

a new specification test for the validity of instrumental variables we develop a new specification test for iv estimators adopting a particular second order approximation of bekker the new specification test compares the difference of the forward conventional 2sls estimator of the coefficient of the right hand side endogenous variable with the reverse 2sls estimator of the same unknown parameter when the normalization is changed under the null hypothesis that conventional first order asymptotics provide a reliable guide to inference the two estimates should be very similar our test sees whether the resulting difference in the two estimates satisfies the results of second order asymptotic theory essentially the same idea is applied to develop another new specification test using second order unbiased estimators of the type first proposed by nagar if the forward and reverse nagar type estimators are not significantly different we recommend estimation by liml which we demonstrate is the optimal linear combination of the nagar type estimators to second order we also demonstrate the high degree of similarity for k class estimators between the approach of bekker and the edgeworth expansion approach of rothenberg an empirical example and monte carlo evidence demonstrate the operation of the new specification test

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

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

moment conditions and bayesian non parametrics models phrased through moment conditions are central to much of modern inference here these moment conditions are embedded within a non parametric bayesian set up handling such a model is not probabilistically straightforward as the posterior has support on a manifold we solve the relevant issues building new probability and computational tools by using hausdorff measures to analyse them on real and simulated data these new methods which involve simulating on a manifold can be applied widely including providing bayesian analysis of quasi likelihoods linear and non linear regression missing data and hierarchical models

projection based depth functions and associated medians order n uniform consistency depth regions and contours induced from projection based depth functions are investigated structural properties of depth regions and contours and general continuity and convergence results of sample depth regions are obtained affine equivariant multivariate medians induced from projection based depth functions are probed the limiting distributions as well as the strong and order n consistency of the sample projection medians are established the finite sample performance of projection medians is compared with that of a leading depth induced median the tukey halfspace median induced from the tukey halfspace depth function it turns out that with appropriate choices of univariate location and scale estimators the projection medians have a very high finite sample breakdown point and relative efficiency much higher than those of the halfspace median based on the results obtained it is found that projection depth functions and projection medians behave very well overall compared with their competitors and consequently are good alternatives to statistical depth functions and affine equivariant multivariate location estimators respectively

near exogeneity and weak identification in generalized empirical likelihood estimators many moment asymptotics abstract this paper investigates the generalized empirical likelihood gel estimators when there are local violations of the exogeneity condition near exogeneity in the case of many weak moments we also examine the tradeoff between the degree of violation of the exogeneity and the number of nearly exogenous instruments in this respect this paper extends many weak moment asymptotics of newey and windmeijer 2009a the overidentifying restrictions test can detect both mild and large violations of exogeneity in the case of minor violations the anderson rubin 1949 and wald tests are not size distorted

dose response modeling in mental health using stein like estimators with instrumental variables a mental health trial is analyzed using a dose response model in which the number of sessions attended by the patients is deemed indicative of the dose of psychotherapeutic treatment here the parameter of interest is the difference in causal treatment effects between the subpopulations that take part in different numbers of therapy sessions for this data set interactions between random treatment allocation and prognostic baseline variables provide the requisite instrumental variables while the corresponding two stage least squares tsls estimator tends to have smaller bias than the ordinary least squares ols estimator the tsls suffers from larger variance it is therefore appealing to combine the desirable properties of the ols and tsls estimators such a trade off is achieved through an affine combination of these two estimators using mean squared error as a criterion this produces the semi parametric stein like spsl estimator as introduced by judge and mittelhammer 2004 the spsl estimator is used in conjunction with multiple imputation with chained equations to provide an estimator that can exploit all available information simulated data are also generated to illustrate the superiority of the spsl estimator over its ols and tsls counterparts a package entitled steiniv implementing these methods has been made available through the r platform 2017 the authors statistics in medicine published by john wiley sons ltd

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

asymptotic robustness of tests of overidentification and predeterminedness abstract many statistical test procedures have been proposed for identification restrictions on one or several equations and the econometric predeterminedness of one or several variables in a system of structural equations this study is devoted to unifying many test procedures in a systematic way and to deriving the asymptotic distributions of the test statistics under a set of local alternative hypotheses and very general conditions on the disturbances by making use of a new martingale central limit theorem and a martingale convergence theorem we show that the limiting distributions of test statistics are noncentral 2 distributions under the local alternative hypotheses and central 2 distributions under the null hypotheses these limiting distributions are robust in the sense that they hold for a variety of disturbance distributions and models our results show that many tests already known among econometricians can be carried out without making the usual relatively restrictive assumptions

finite sample breakdown points of projection based multivariate location and scatter statistics finite sample breakdown points are obtained for two classes of projection based multivariate location and scatter statistics the stahel donoho statistics and the maronna yohai statistics the definition of these multivariate statistics are dependent on the value of location and scale statistics for all univariate projections of the data and consequently their properties depend on the nature of the corresponding univariate location and scale statistics used on the projected data the finite sample breakdown points of the multivariate statistics though are not directly related to those of the corresponding univariate location and scale statistics a uniform finite sample breakdown point concept is needed the median and the median absolute deviation about the median m a d are one possible choice for the univariate location and scale statistics respectively for sparse data sets in high dimensions though it is not recommended that the m a d be used as the univariate scale statistic for the projected data since its uniform finite sample breakdown point is shown to be much less than optimum a simple modification to the m a d however is shown to alleviate this problem for various reasons one may wish to consider univariate location and scale statistics other than the median and the m a d respectively a very broad and natural class of univariate location and scale statistics to consider for the projected data is the simultaneous m estimates of location and scale new results on their breakdown properties are given in this paper implicit formulas for the breakdown points of monotonic simultaneous m estimates of location and scale are known and they tend to imply rather low breakdown points for smooth choices of the defining weight functions it is shown here that this phenomenon does not occur for a large class of nonmonotonic simultaneous m estimates of location and scale furthermore explicit rather than implicit expressions for the uniform finite sample breakdown points are given for these nonmonotonic m estimates

the behavior of the stahel donoho robust multivariate estimator abstract the stahel donoho estimators t v of multivariate location and scatter are defined as a weighted mean and a weighted covariance matrix with weights of the form w r where w is a weight function and r is a measure of outlyingness obtained by considering all univariate projections of the data it has a high breakdown point for all dimensions and order n consistency the asymptotic bias of v for point mass contamination for suitable weight functions is compared with that of rousseeuw s minimum volume ellipsoid mve estimator a simulation shows that for a suitable w t and v exhibit high efficiency for both normal and cauchy distributions and are better than their competitors for normal data with point mass contamination the performances of the estimators for detecting outliers are compared for both a real and a synthetic data set

testing identification strength we consider models defined by a set of moment restrictions that may be subject to weak identification following the recent literature the identification of the structural parameters is characterized by the jacobian of the moment conditions we unify several definitions of identification that have been used in the literature and show how they are linked to the consistency and asymptotic normality of gmm estimators we then develop two tests to assess the identification strength of the structural parameters in models that are i either linear or separable ii neither linear nor separable both tests are straightforward to apply and allow to test specific subvectors without assuming identification of the components not under test in simulations our tests are well behaved when compared to contenders both in terms of size and power in addition we show how pretesting specific subvectors can improve inference by delivering shorter confidence regions with comparable coverage probability finally when applied to the estimation of the elasticity of intertemporal substitution our test indicates weak instruments in a larger number of countries than tests proposed by stock and yogo 2005 and montiel olea and pflueger 2012

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

generalized reduced rank tests using the singular value decomposition we propose a novel statistic to test the rank of a matrix the rank statistic overcomes deficiencies of existing rank statistics like a kronecker covariance matrix for the canonical correlation rank statistic of anderson annals of mathematical statistics 1951 22 327 351 sensitivity to the ordering of the variables for the ldu rank statistic of cragg and donald journal of the american statistical association 1996 91 1301 1309 and gill and lewbel journal of the american statistical association 1992 87 766 776 a limiting distribution that is not a standard chi squared distribution for the rank statistic of robin and smith econometric theory 2000 16 151 175 usage of numerical optimization for the objective function statistic of cragg and donald journal of econometrics 1997 76 223 250 and ignoring the non negativity restriction on the singular values in ratsimalahelo 2002 rank test based on matrix perturbation theory unpublished working paper u f r science economique university de franche comte in the non stationary cointegration case the limiting distribution of the new rank statistic is identical to that of the johansen trace statistic

influence function and maximum bias of projection depth based estimators location estimators induced from depth functions increasingly have been pursued and studied in the literature among them are those induced from projection depth functions these projection depth based estimators have favorable properties among their competitors in particular they possess the best possible finite sample breakdown point robustness however robustness of estimators cannot be revealed by the finite sample breakdown point alone the influence function gross error sensitivity maximum bias and contamination sensitivity are also important aspects of robustness in this article we study these other robustness aspects of two types of projection depth based estimators projection medians and projection depth weighted means the latter includes the stahel donoho estimator as a special case exact maximum bias the influence function and contamination and gross error sensitivity are derived and studied for both types of estimators sharp upper bounds for the maximum bias and the influence functions are established comparisons based on these robustness criteria reveal that the projection depth based estimators enjoy desirable local as well as global robustness and are very competitive among their competitors

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

spatial medians depth functions and multivariate jensen s inequality for any given partial order in a d dimensional euclidean space under mild regularity assumptions we show that the intersection of closed generalized intervals containing more than 1 2 of the probability mass is a non empty compact interval this property is shared with common intervals on real line where the intersection is the median set of the underlying probability distribution so obtained multivariate medians with respect to a partial order can be observed as special cases of centers of distribution in the sense of type d depth functions introduced by y zuo and r serfling em ann stat bf 28 2000 461 482 we show that the halfspace depth function can be realized via compact convex sets or for example closed balls in place of halfspaces and discuss structural properties of halfspace and related depth functions and their centers among other things we prove that in general the maximal guaranteed depth is frac 1 d 1 as an application of these results we provide a jensen s type inequality for functions of several variables with medians in place of expectations which is an extension of the previous work by m merkle em stat prob letters bf 71 2005 277 281

on a notion of data depth based on random simplices for a distribution f on mathbb r p and a point x in mathbb r p the simplical depth d x is introduced which is the probability that the point x is contained inside a random simplex whose vertices are p 1 independent observations from f mathematically and heuristically it is argued that d x indeed can be viewed as a measure of depth of the point x with respect to f an empirical version of d cdot gives rise to a natural ordering of the data points from the center outward the ordering thus obtained leads to the introduction of multivariate generalizations of the univariate sample median and l statistics this generalized sample median and l statistics are affine equivariant

depth weighted scatter estimators general depth weighted scatter estimators are introduced and investigated for general depth functions we find out that these affine equivariant scatter estimators are fisher consistent and unbiased for a wide range of multivariate distributions and show that the sample scatter estimators are strong and n consistent and asymptotically normal and the influence functions of the estimators exist and are bounded in general we then concentrate on a specific case of the general depth weighted scatter estimators the projection depth weighted scatter estimators which include as a special case the well known stahel donoho scatter estimator whose limiting distribution has long been open until this paper large sample behavior including consistency and asymptotic normality and efficiency and finite sample behavior including breakdown point and relative efficiency of the sample projection depth weighted scatter estimators are thoroughly investigated the influence function and the maximum bias of the projection depth weighted scatter estimators are derived and examined unlike typical high breakdown competitors the projection depth weighted scatter estimators can integrate high breakdown point and high efficiency while enjoying a bounded influence function and a moderate maximum bias curve comparisons with leading estimators on asymptotic relative efficiency and gross error sensitivity reveal that the projection depth weighted scatter estimators behave very well overall and consequently represent very favorable choices of affine equivariant multivariate scatter estimators

projection estimates of multivariate location in this paper we study the maximum asymptotic bias of the projection estimate for multivariate location based on univariate estimates of location and dispersion in particular we study the projection estimate that uses the median and median absolute deviation about the median mad as univariate location and dispersion estimates respectively this estimator may be considered a natural affine equivariant multivariate median for spherical distributions the maximum bias of this estimate depends only on the marginal distributions and not on the dimension and is approximately twice the maximum bias of the univariate median we also show that for multivariate normal distributions its maximum bias compares favorably with those of the donoho stahel minimum volume ellipsoid and minimum covariance determinant estimates in all these cases the maximum bias increases with the dimension p

multidimensional trimming based on projection depth as estimators of location parameters univariate trimmed means are well known for their robustness and efficiency they can serve as robust alternatives to the sample mean while possessing high efficiencies at normal as well as heavy tailed models this paper introduces multidimensional trimmed means based on projection depth induced regions robustness of these depth trimmed means is investigated in terms of the influence function and finite sample breakdown point the influence function captures the local robustness whereas the breakdown point measures the global robustness of estimators it is found that the projection depth trimmed means are highly robust locally as well as globally asymptotics of the depth trimmed means are investigated via those of the directional radius of the depth induced regions the strong consistency asymptotic representation and limiting distribution of the depth trimmed means are obtained relative to the mean and other leading competitors the depth trimmed means are highly efficient at normal or symmetric models and overwhelmingly more efficient when these models are contaminated simulation studies confirm the validity of the asymptotic efficiency results at finite samples

the small data solutions of general 3 d quasilinear wave equations ii abstract this paper is a continuation of the work in 8 where the authors established the global existence of smooth small data solutions to the general 3 d quasilinear wave equation i j 0 3 g i j u u i j 2 u 0 when the weak null condition holds in the present paper we show that the smooth small data solutions of equation i j 0 3 g i j u u i j 2 u 0 will blow up in finite time when the weak null condition does not hold and a generic nondegenerate condition on the initial data is satisfied moreover a precise blowup time is completely determined therefore collecting the main results in this paper and 8 we have given a basically complete study on the blowup or global existence of small data solutions to the 3 d quasilinear wave equation i j 0 3 g i j u u i j 2 u 0

a regularization approach to the many instruments problem abstract this paper focuses on the estimation of a finite dimensional parameter in a linear model where the number of instruments is very large or infinite in order to improve the small sample properties of standard instrumental variable iv estimators we propose three modified iv estimators based on three different ways of inverting the covariance matrix of the instruments these inverses involve a regularization or smoothing parameter it should be stressed that no restriction on the number of instruments is needed and that all the instruments are used in the estimation we show that the three estimators are asymptotically normal and attain the semiparametric efficiency bound higher order analysis of the mse reveals that the bias of the modified estimators does not depend on the number of instruments finally we suggest a data driven method for selecting the regularization parameter interestingly our regularization techniques lead to a consistent nonparametric estimation of the optimal instrument

lower bounds for contamination bias globally minimax versus locally linear estimation we study how robust estimators can be in parametric families obtaining a lower bound on the contamination bias of an estimator that holds for a wide class of parametric families this lower bound includes as a special case the bound used to establish that the median is bias minimax among location equivariant estimators and it is tight or nearly tight in a variety of other settings such as scale estimation discrete exponential families and multiple linear regression the minimum variation distance estimator has contamination bias within a dimension free factor of this bound a second lower bound applies to locally linear estimates and implies that such estimates cannot be bias minimax among all fisher consistent estimates in higher dimensions in linear regression this class of estimates includes the familiar m estimates gm estimates and s estimates in discrete exponential families yet another lower bound implies that the proportion of zeros estimate has minimax bias if the median of the distribution is zero a common situation in some fields this bound also implies that the information standardized sensitivity of every fisher consistent estimate of the poisson mean and of the binomial proportion is unbounded

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

generalized method of moments specification testing abstract this paper analyzes the asymptotic power properties of specification tests which are based on a finite set of moment conditions it shows that any such test may fail against general misspecification that causes estimator inconsistency the mutual asymptotic equivalence of maximal degree of freedom tests is shown and the form of optimal tests against specific forms of misspecification is derived applications to testing for exogeneity of a set of instrumental variables are presented

the influence function and maximum bias of tukey s median tukey s median is among one of the earliest known high breakdown point multivariate location statistics aside from its breakdown point though little else appears to be known about its robustness properties in this paper we investigate other aspects of tukey s median and in particular we derive and study its influence function and its maximum contamination bias function when judged by these other robustness criteria tukey s median again proves to be highly robust

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

information relaxation and dual formulation of controlled markov diffusions information relaxation and duality in markov decision processes have been studied recently by several researchers with the goal to derive dual bounds on the value function in this paper we extend this dual formulation to controlled markov diffusions in a similar way we relax the constraint that the decision should be made based on the current information and impose a penalty to punish the access to the information in advance we establish the weak duality strong duality and complementary slackness results in a parallel way as those in markov decision processes we further explore the structure of the optimal penalties and expose the connection between the optimal penalties for markov decision processes and controlled markov diffusions we demonstrate the use of this dual representation in a classic dynamic portfolio choice problem through a new class of penalties which require little extra computation and produce small duality gap on the optimal value

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

small data solutions of 3 d quasilinear wave equations under null condition abstract for the 3 d quasilinear wave equation i j 0 3 g i j u i j u 0 a global existence result has been shown by h lindblad in this paper we consider this 3 d quasilinear wave equation i j 0 3 g i j u i j u i j 0 3 f i j i u j u with small initial data and f i j satisfying null condition through deriving decay estimates of low derivatives and energy estimates for high derivatives combined with the properties of null condition a global existence solution is also established by continuous induction

min max bias robust regression abstract this paper considers the problem of minimizing the maximum asymptotic bias of regression estimates over epsilon contamination neighborhoods for the joint distribution of the response and carriers two classes of estimates are treated 1 m estimates with bounded function rho applied to the scaled residuals using a very general class of scale estimates and 2 bounded influence function type generalized m estimates estimates in the first class are obtained as the solution of a minimization problem while estimates in the second class are specified by an estimating equation the first class of m estimates is sufficiently general to include both huber proposal 2 simultaneous estimates of regression coefficients and residuals scale and rousseeuw yohai s estimates of regression it is shown than an s estimate based on a jump function type rho solves the min max bias problem for the class of m estimates with very general scale this estimate is obtained by the minimization of the alpha quantile of the squared residuals where alpha epsilon depends on the fraction of contamination epsilon when epsilon approaches limit of 5 alpha epsilon approaches limit of 5 and the min max estimator approaches the least median of squared residuals estimator introduced by rousseeuw for the bounded influence class of gm estimates it is shown the a sign type nonlinearity yields the min max estimate this estimate coincides with the minimum gross error sensitivity gm estimate

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

discretizing distributions with exact moments error estimate and convergence analysis the maximum entropy principle is a powerful tool for solving underdetermined inverse problems this paper considers the problem of discretizing a continuous distribution which arises in various applied fields we obtain the approximating distribution by minimizing the kullback leibler information relative entropy of the unknown discrete distribution relative to an initial discretization based on a quadrature formula subject to some moment constraints we study the theoretical error bound and the convergence of this approximation method as the number of discrete points increases we prove that i the theoretical error bound of the approximate expectation of any bounded continuous function has at most the same order as the quadrature formula we start with and ii the approximate discrete distribution weakly converges to the given continuous distribution moreover we present some numerical examples that show the advantage of the method and apply to numerically solving an optimal portfolio problem

tests of overidentification and predeterminedness in simultaneous equation models abstract a set of simultaneous equations that is a part of a larger complete system is considered one hypothesis concerning this set of equations is that it is overidentified as a block by a block of zero restrictions another hypothesis is that a subset of the current endogenous variables is uncorrelated with the disturbances in the set of simultaneous equations econometric predeterminedness likelihood ratio lagrange multiplier and wald tests are derived for the overidentification hypothesis against unrestricted alternatives for the predeterminedness hypothesis against unrestricted alternatives and for the predeterminedness hypothesis against overidentified alternatives the tests are derived on the basis of normality but are asymptotically valid under very general conditions this study generalizes and unifies many test procedures proposed previously some new tests are developed

testing the rank and definiteness of estimated matrices with applications to factor state space and arma models abstract consider any consistent asymptotically normal estimate of an arbitrary rectangular or square matrix a this article derives an explicit test for the rank of a and a related test of semi definiteness of a potential applications include testing for identification of structural models testing for the number of state variables in state space models including tests for the order of autoregressive moving average arma processes consumer demand analysis applications and testing for the number of factors in factor analysis and related procedures the test is based on the gaussian elimination lower diagonal upper triangular ldu decomposition the test is illustrated with an empirical application to testing the order of arma processes

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

some impossibility theorems in econometrics with applications to structural and dynamic models general characterizations of valid confidence sets and tests in problems which involve locally almost unidentified lau parameters are provided and applied to several econo metric models two types of inference problems are studied i inference about parame ters which are not identifiable on certain subsets of the parameter space and ii inference about parameter transformations with discontinuities when a lau parameter or parametric function has an unbounded range it is shown under general regularity conditions that any valid confidence set with level 1 a for this parameter must be unbounded with probability close to 1 a in the neighborhood of nonidentification subsets and will have a nonzero probability of being unbounded under any distribution compatible with the model no valid confidence set which is almost surely bounded does exist these properties hold even if identifying restrictions are imposed similar results also obtain for parameters with bounded ranges consequently a confidence set which does not satisfy this characterization has zero coverage probability level this will be the case in particular for wald type confidence intervals based on asymptotic standard errors furthermore wald type statistics for testing given values of a lau parameter cannot be pivotal functions i e they have distributions which depend on unknown nuisance param eters and even cannot be usefully bounded over the space of the nuisance parameters these results are applied to several econometric problems inference in simultaneous equations instrumental variables iv regressions linear regressions with autoregressive errors inference about long run multipliers and cointegrating vectors for example it is shown that standard asymptotically justified confidence intervals based on iv estimators such as two stage least squares and the associated standard errors have zero coverage probability and the corresponding t statistics have distributions which cannot be bounded by any finite set of distribution functions a result of interest for interpreting iv regressions with weak instruments furthermore expansion methods e g edgeworth expansions and bootstrap techniques cannot solve these difficulties finally in a number of cases where wald type methods are fundamentally flawed e g iv regressions with poor instruments it is observed that likelihood based methods e g likelihood ratio tests and confidence sets combined with projection techniques can easily yield valid tests and confidence sets

higher order properties of approximate estimators many modern estimation methods in econometrics approximate an objective function for instance through simulation or discretization these approximations typically affect both bias and variance of the resulting estimator we first provide a higher order expansion of such approximate estimators that takes into account the errors due to the use of approximations we show how a newton raphson adjustment can reduce the impact of approximations then we use our expansions to develop inferential tools that take into account approximation errors we propose adjustments of the approximate estimator that remove its first order bias and adjust its standard errors these corrections apply to a class of approximate estimators that includes all known simulation based procedures a monte carlo simulation on the mixed logit model shows that our proposed adjustments can yield significant improvements at a low computational cost

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

the global stability of minkowski space time in harmonic gauge we give a new proof of the global stability of minkowski space originally established in the vacuum case by christodoulou and klainerman the new approach which relies on the classical harmonic gauge shows that the einstein vacuum and the einstein scalar field equations with asymptotically flat initial data satisfying a global smallness condition produce global causally geodesically complete solutions asymptotically convergent to the minkowski space time

global solutions of quasilinear wave equations we show global small data existence for a class of quasilinear wave equations related to einstein s equations in harmonic coordinates these equations do not satisfy the classical null condition and the asymptotic behavior of solutions is not free but the light cones bend at infinity

global solutions of nonlinear hyperbolic equations for small initial data on considere des systemes quasilineaires d equations hyperboliques d ordre 2 qui sont des deformations non lineaires de l equation d onde on etudie l existence globale et le comportement asymptotique des solutions du probleme de cauchy quand la donnee initiale est suffisamment petite

inferring the rank of a matrix abstract this paper considers methods of inference concerning the rank of matrix a based on an asymptotically normal estimate of and some identifiable specification for one such specification is 0 in which case one is interested in the rank of we first propose and examine the properties of a test of the hypothesis that the rank is of a given size against the alternative that the rank is larger we then look at the problem of estimating the rank of this matrix using model selection procedures and sequential hypothesis testing conditions for consistency of such procedures are obtained some economic applications are discussed and the various methods are compared in a monte carlo experiment

using randomization to break the curse of dimensionality this paper introduces random versions of successive approximations and multigrid algorithms for computing approximate solutions to a class of finite and infinite horizon markovian decision problems mdps we prove that these algorithms succeed in breaking the curse of dimensionality for a subclass of mdps known as discrete decision processes ddps