an exact test for multiple inequality and equality constraints in the linear regression model abstract in this article we consider the linear regression model y x e where e is n 0 2i in this context we derive exact tests of the form h r r versus k rk for the case in which 2 is unknown we extend these results to consider hypothesis tests of the form h r1 r1 and r2 r2 versus k rk for each of these hypotheses tests we derive several equivalent forms of the test statistics using the duality theory of the quadratic programming for both tests we derive their exact distribution as a weighted sum of snedecor s f distributions normalized by the numerator degrees of freedom of each f distribution of the sum a methodology for computing critical values as well as probability values for the tests is discussed the relationship between this testing framework and the multivariate one sided hypothesis testing literature is also discussed in this context we show that for any size of the hypothesis test h 0 versus k rk the test statistic and critical value obtained a

specification tests based on artificial regressions abstract many specification tests can be computed with artificial linear regressions designed to be used as calculating devices to obtain test statistics and other quantities of interest this article discusses the general principles that underlie all artificial regressions and the use of such regressions to compute lagrange multiplier and other specification tests based on estimates under the null hypothesis the generality and power of artificial regressions as a means of computing test statistics is demonstrated how durbin wu hausman conditional moment and other tests that are not explicitly lagrange multiplier tests may be computed is shown and several special cases that illustrate the general results and can be useful in practice are discussed these include tests of parameter restrictions in nonlinear regression models and tests of binary choice models such as the logit and probit models

likelihood ration test wald test and kuhn tucker test in linear models with inequality constraints on the regression parameters likelihood ratio test wald test and kuhn tucker test in linear models with inequality constraints on the regression parameters this paper considers the problem of testing statistical hypotheses in linear regression models with inequality constraints on the regression coefficients the kuhn tucker multiplier test statistic is defined and its relationships with the likelihood ratio test and the wald test are examined it is shown in particular that these relationships are the same as in the equality constrained case it is emphasized however that their common asymptotic distribution is a mixture of chi square distributions under the null hypothesis

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

a general approach to lagrange multiplier model diagnostics abstract a diagnostic test can be formulated by taking the current model as the null hypothesis and a particular form of misspecification as the alternative the general approach of this paper is to calculate the score for some alternative and base a test on its distribution under the null this distribution is derived in two theorems corresponding to different models such a test is a langrange multiplier test and is asymptotically optimal these tests can be formulated as ways of looking at the residuals for particular types of non randomness the tests are simple to calculate even when the alternative may be very complicated examples are given of tests for omitted variables and block recursive structures

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

the spherical fourier transform of rapidly decreasing functions a simple proof of a characterization due to harish chandra helgason trombi and varadarajan abstract harish chandra s helgason p c trombi and v s varadarajan have described the image of the l p schwartz spaces 0 p 2 under the spherical fourier transform we give a short proof of their results based on the paley wiener theorem

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

on the performance of tests by wu and by hausman for detecting the ordinary least squares bias problem abstract we first consider the performance of the wu 1973 hausman 1978 w h specification error test as a test for the existence of ordinary least squares ols bias we discuss power properties of the test under alternative null hypotheses one of which has not previously been considered we next consider how the w h test performs as an indicator of the extent rather than the existence of an ols bias problem since this usage of the test seems common in applied studies finally monte carlo methods are used to evaluate wu s two step estimation procedure involving the w h test as a pretest

limited information estimators and exogeneity tests for simultaneous probit models a two step maximum likelihood procedure is proposed for estimating simultaneous probit models and is compared to alternative limited information estimators conditions under which each estimator attains the cramer rao lower bound are obtained simple tests for exogeneity based on the new two step estimator are proposed and are shown to be asymptotically equivalent to one another and to have the same local asymptotic power as classical tests based on the limited information maximum likelihood estimator finite sample comparisons between the new and alternative estimators are presented based on some monte carlo evidence the performance of the proposed tests for exogeneity is also assessed

a conditional composite likelihood ratio test with boundary constraints summary composite likelihood has been widely used in applications the asymptotic distribution of the composite likelihood ratio statistic at the boundary of the parameter space is a complicated mixture of weighted chi 2 distributions in this paper we propose a conditional test with data dependent degrees of freedom we consider a modification of the composite likelihood which satisfies the second order bartlett identity we show that the modified composite likelihood ratio statistic given the number of estimated parameters lying on the boundary converges to a simple chi 2 distribution this conditional testing procedure is validated through simulation studies

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

a unified approach to estimation and orthogonality tests in linear single equation econometric models abstract maximum likelihood estimation is considered for a generalisation of the model of anderson and rubin 1949 in which the exogenous variables in the structural equation may not be included in the reduced form equations classical and specification tests are derived for orthogonality hypotheses a necessary and sufficient condition for their equivalence is presented the classical tests are compared using bahadur s asymptotic relative efficiency criterion it is shown that a generalisation of the durbin wu hausman t 2 statistics is asymptotically bahadur efficient

hahn hausman test as a specification test abstract this paper develops a modified version of the sargan sargan j d 1958 the estimation of economic relationships using instrumental variables econometrica 26 3 393 415 restrictions and shows that it is numerically equivalent to the test statistic of hahn and hausman hahn j hausman j 2002 a new specification test for the validity of instrumental variables econometrica 70 1 163 189 up to a sign the modified sargan test is constructed such that its asymptotic distribution under the null hypothesis of correct specification is standard normal when the number of instruments increases with the sample size the equivalence result is useful in understanding what the hahn hausman test detects and its power properties

chaotic dynamics of the heat semigroup on riemannian symmetric spaces abstract we show that the heat semigroup generated by certain perturbations of the laplace beltrami operator on the riemannian symmetric spaces of noncompact type is chaotic on their l p spaces when 2 p both the range of p and the range of chaos inducing perturbation are sharp this extends a result of ji and weber 17 where it was shown that under identical conditions the heat operator is subspace chaotic on these spaces

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

likelihood ratio tests with boundary constraints using data dependent degrees of freedom when the null hypothesis constrains parameters to the boundary of the parameter space the asymptotic null distribution of the likelihood ratio statistic is often a mixture of chi squared distributions giving rise to the so called chi bar test where weights can depend on the true unknown parameter and be difficult to calculate we consider the test that conditions on the observed number of null hypothesis parameters in the interior of the parameter space this approach uses simple chi squared thresholds yields conservative asymptotic type i error and is guaranteed to give power improvements over the naive approach of simply ignoring boundary constraints simulations validate the theoretical results illustrate application settings and find power comparable with but usually less than that of the chi bar test copyright 2013 oxford university press

monte carlo tests with nuisance parameters a general approach to finite sample inference and nonstandard asymptotics abstract the technique of monte carlo mc tests dwass 1957 annals of mathematical statistics 28 181 187 barnard 1963 journal of the royal statistical society series b 25 294 provides a simple method for building exact tests from statistics whose finite sample distribution is intractable but can be simulated when no nuisance parameter is involved we extend this method in two ways first by allowing for mc tests based on exchangeable possibly discrete test statistics second by generalizing it to statistics whose null distribution involves nuisance parameters maximized mc mmc tests simplified asymptotically justified versions of the mmc method are also proposed these provide a simple way of improving standard asymptotics and dealing with nonstandard asymptotics

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

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

adaptive critical value for constrained likelihood ratio testing summary we present a new general method for constrained likelihood ratio testing which when few constraints are violated improves upon the existing approach in the literature that compares the likelihood ratio with the quantile of a mixture of chi squared distributions the improvement is in terms of both simplicity and power the proposed method compares the constrained likelihood ratio statistic against the quantile of only one chi squared random variable with data dependent degrees of freedom the new test is shown to have a valid exact significance level alpha it also has more power than the classical approach against alternatives for which the number of violations is not large we provide more details for testing a simple order mu 1 leqslant cdots leqslant mu p against all alternatives using the proposed approach and give clear guidelines as to when the new method would be advantageous a simulation study suggests that for testing a simple order the new approach is more powerful in many scenarios than the existing method that uses a mixture of chi squared variables we illustrate the results of our adaptive procedure using real data on the liquidity preference hypothesis

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

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

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