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

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

on the asymptotic sizes of subset anderson rubin and lagrange multiplier tests in linear instrumental variables regression we consider tests of a simple null hypothesis on a subset of the coefficients of the exogenous and endogenous regressors in a single equation linear instrumental variables regression model with potentially weak identification existing methods of subset inference i rely on the assumption that the parameters not under test are strongly identified or ii are based on projection type arguments we show that under homoskedasticity the subset anderson and rubin 1949 test that replaces unknown parameters by limited information maximum likelihood estimates has correct asymptotic size without imposing additional identification assumptions but that the corresponding subset lagrange multiplier test is size distorted asymptotically

inference on structural parameters in instrumental variables regression with weak instruments we consider the problem of making asymptotically valid inference on structural parameters in instrumental variables regression with weak instruments using the localto zero asymptotics of staiger and stock 1997 we derive the asymptotic distributions of lr and lm type statistics for testing simple hypotheses on structural parameters based on maximum likelihood and generalized method of moments estimation methods in contrast to the nonstandard limiting behavior of wald statistics the limiting distributions of certain lm and lr statistics are bounded by a chi square distribution with degrees of freedom given by the number of instruments further we show how to construct asymptotically valid confidence sets for structural parameters by inverting these statistics

magic squares and matrix models of lie algebras this paper is concerned with the description of exceptional simple lie algebras as octonionic analogues of the classical matrix lie algebras we review the tits freudenthal construction of the magic square which includes the exceptional lie algebras as the octonionic case of a construction in terms of a jordan algebra of hermitian 3 3 matrices tits or various plane and other geometries freudenthal we present alternative constructions of the magic square which explain its symmetry and show explicitly how the use of split composition algebras leads to analogues of the matrix lie algebras su 3 sl 3 and sp 6 we adapt the magic square construction to include analogues of su 2 sl 2 and sp 4 for all real division algebras

evaluation of the distribution function of the limited information maximum likelihood estimator the distributions of the limited information maximum likelihood estimator for the coefficient of one endogenous variable are evaluated numerically tables are given for enough values of the parameters to cover all cases of interests comparisons are made with the two stage least squares estimator

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

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

the nonexistence of 100 1 alpha confidence sets of finite expected diameter in errors in variables and related models on montre que pour certains modeles il est impossible de construire des intervalles de confiance pour les parametres cles qui aient a la fois une confiance positive et une longueur esperee finie

gradings on tensor products of composition algebras and on the smirnov algebra abstract we give classifications of group gradings up to equivalence and up to isomorphism on the tensor product of a cayley algebra c and a hurwitz algebra over a field of characteristic different from 2 we also prove that the automorphism group schemes of c n and c n are isomorphic on the other hand we prove that the automorphism group schemes of a smirnov algebra t c a 35 dimensional simple exceptional structurable algebra constructed from a cayley algebra c and c are isomorphic this is used to obtain classifications up to equivalence and up to isomorphism of the group gradings on smirnov algebras

positive solutions for singular second order neumann boundary value problems via a cone fixed point theorem in this paper we consider the existence of positive solutions for the singular second order neumann boundary value problem x k 2x f t g t x 0

the tetrahedron algebra the onsager algebra and the sl2 loop algebra let k denote a field with characteristic 0 and let t denote an indeterminate we give a presentation for the three point loop algebra sl2 k t t 1 t 1 1 via generators and relations this presentation displays s4 symmetry using this presentation we obtain a decomposition of the above loop algebra into a direct sum of three subalgebras each of which is isomorphic to the onsager algebra

triality exceptional lie algebras and deligne dimension formulas abstract we give a computer free proof of the deligne cohen and de man formulas for the dimensions of the irreducible g modules appearing in g k k 4 where g ranges over the exceptional complex simple lie algebras we give additional dimension formulas for the exceptional series as well as uniform dimension formulas for other representations distinguished by freudenthal along the rows of his magic chart our proofs use the triality model of the magic square which we review and present a simplified proof of its validity we conclude with some general remarks about obtaining series of lie algebras in the spirit of deligne and vogel

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

gel statistics under weak identification abstract the central concern of this paper is the provision in a time series moment condition framework of practical recommendations of confidence regions for parameters whose coverage probabilities are robust to the strength or weakness of identification to this end we develop pearson type test statistics based on gel implied probabilities formed from general kernel smoothed versions of the moment indicators we also modify the statistics suggested in guggenberger and smith 2008 for a general kernel smoothing function importantly for our conclusions we provide gel time series counterparts to gmm and gel conditional likelihood ratio statistics given in kleibergen 2005 and smith 2007 our analysis not only demonstrates that these statistics are asymptotically conditionally pivotal under both classical asymptotic theory and weak instrument asymptotics of stock and wright 2000 but also provides asymptotic power results in the weakly identified time series context consequently the empirical null rejection probabilities of the associated tests and thereby the coverage probabilities of the corresponding confidence regions should not be affected greatly by the strength or otherwise of identification a comprehensive monte carlo study indicates that a number of the tests proposed here represent very competitive choices in comparison with those suggested elsewhere in the literature

symmetric composition algebras abstract a construction of all the okubo algebras over fields of characteristic 3 is provided and the complete classification of the composition algebras with associative norm over fields is finished

a maximum likelihood method for the incidental parameter problem this paper uses the invariance principle to solve the incidental parameter problem of 35 we seek group actions that pre serve the structural parameter and yield a maximal invariant in the parameter space with flxed dimension m estimation from the likeli hood of the maximal invariant statistic yields the maximum invariant likelihood estimator mile consistency of mile for cases in which the likelihood of the maximal invariant is the product of marginal likelihoods is straightforward we illustrate this result with a sta tionary autoregressive model with flxed efiects and an agent speciflc monotonic transformation model asymptotic properties of mile when the likelihood of the maxi mal invariant does not factorize remain an open question we are able to provide consistency asymptotic normality and e ciency results of mile when invariance yields wishart distributions two examples are an instrumental variable iv model and a dynamic panel data model with flxed efiects

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

symmetric triality relations and structurable algebras in this paper some properties of algebras satisfying local symmetric triality relations have been studied including both generalized symmetric composition algebras and the conjugate algebras of any structurable algebras we also discuss a general method of constructing lie algebras from such a system which can permit a construction of the magic square

a conditional likelihood ratio test for structural models this paper develops a general method for constructing exactly similar tests based on the conditional distribution of nonpivotal statistics in a simultaneous equations model with normal errors and known reduced form covariance matrix these tests are shown to be similar under weak instrument asymptotics when the reduced form covariance matrix is estimated and the errors are non normal the conditional test based on the likelihood ratio statistic is particularly simple and has good power properties like the score test it is optimal under the usual local to null asymptotics but it has better power when identification is weak

alternative approximations of the bias and mse of the iv estimator under weak identification with an application to bias correction abstract we provide analytical formulae for the asymptotic bias abias and mean squared error amse of the iv estimator and obtain approximations thereof based on an asymptotic scheme which essentially requires the expectation of the first stage f statistic to converge to a finite possibly small positive limit as the number of instruments approaches infinity our analytical formulae can be viewed as generalizing the bias and mse results of richardson and wu 1971 a note on the comparison of ordinary and two stage least squares estimators econometrica 39 973 982 to the case with nonnormal errors and stochastic instruments our approximations are shown to compare favorably with approximations due to morimune 1983 approximate distributions of k class estimators when the degree of overidentifiability is large compared with the sample size econometrica 51 821 841 and donald and newey 2001 choosing the number of instruments econometrica 69 1161 1191 particularly when the instruments are weak we also construct consistent estimators for the abias and amse and we use these to further construct a number of bias corrected ols and iv estimators the properties of which are examined both analytically and via a series of monte carlo experiments

bounding the difference between true and nominal rejection probabilities in tests of hypotheses about instrumental variables models abstract this paper presents a simple method for carrying out inference in a wide variety of possibly nonlinear iv models under weak assumptions the method provides a finite sample bound on the difference between the true and nominal probabilities of rejecting a correct null hypothesis the method is a non studentized version of the s test of stock and wright 2000 but is implemented and analyzed differently it does not require restrictive distributional assumptions linearity of the estimated model simultaneous equations or information about whether the instruments are strong or weak it can be applied to quantile iv models that may be nonlinear and can be used to test a parametric iv model against a nonparametric alternative it provides information about the relation between the degree of weakness of the instruments and the power of the test the bound presented here holds in finite samples regardless of the strength of the instruments

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

derivations of the cheng kac jordan superalgebras abstract the derivations of the cheng kac jordan superalgebras are studied it is shown that assuming 1 is a square in the ground field the lie superalgebra of derivations of a cheng kac jordan superalgebra is isomorphic to the lie superalgebra obtained from a simpler jordan superalgebra a kantor double superalgebra of vector type by means of the tits kantor koecher construction this is done by exploiting the s 4 symmetry of the cheng kac jordan superalgebra

structurable superalgebras of cartan type abstract we describe the simple lie superalgebras arising from the unital structurable superalgebras of characteristic 0 and construct four series of the unital simple structurable superalgebras of cartan type we give a classification of simple structurable superalgebras of cartan type over an algebraically closed field f of characteristic 0 together with the faulkner theorem on the classification of classical such superalgebras it gives a classification of the simple structurable superalgebras over f

triality of structurable and pre structurable algebras abstract we introduce a notion of pre structurable algebras based on the structurable algebra of allison as well as lie algebras that satisfy triality

pivotal statistics for testing structural parameters in instrumental variables regression we propose a novel statistic for testing the structural parameters in instrumental variables regression the statistic is straightforward to compute and has a limiting distribution that is pivotal with a degrees of freedom parameter that is equal to the number of tested parameters it therefore differs from the anderson rubin statistic whose limiting distribution is pivotal but has a degrees of freedom parameter that is equal to the number of instruments and the likelihood based wald likelihood ratio and lagrange multiplier statistics whose limiting distributions are not pivotal we analyze the relationship between the statistic and the concentrated likelihood of the structural parameters and show that its limiting distribution is not affected by weak instruments we discuss examples of the non standard shapes of the asymptotically pivotal confidence sets that can be constructed using the statistic and investigate its power properties to show its importance for practical purposes we apply the statistic to the angrist krueger 1991 data and find similar results as in staiger and stock 1997 this discussion paper has resulted in a publication in econometrica 2002 70 5 1781 1803

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

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

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

controlling the size of autocorrelation robust tests autocorrelation robust tests are notorious for suffering from size distortions and power problems we investigate under which conditions the size of autocorrelation robust tests can be controlled by an appropriate choice of critical value

lie algebras with s4 action and structurable algebras abstract the normal symmetric triality algebras sta s and the normal lie related triple algebras lrta s have been recently introduced by the second author in connection with the principle of triality it turns out that the unital normal lrta s are precisely the structurable algebras extensively studied by allison it will be shown that the normal sta s respectively lrta s are the algebras that coordinatize those lie algebras whose automorphism group contains a copy of the alternating respectively symmetric group of degree 4

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

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

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

local triality and some related algebras abstract eight dimensional symmetric composition algebras give very nice formulae for triality and local triality these latter formulae will be extended to any algebra in a suitable variety the simple algebras in this variety will be shown to be precisely the symmetric composition algebras over fields while the prime algebras are central orders in the simple ones