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

on finite sample properties of alternative estimators of coefficients in a structural equation with many instruments abstract we compare four different estimation methods for the coefficients of a linear structural equation with instrumental variables as the classical methods we consider the limited information maximum likelihood liml estimator and the two stage least squares tsls estimator and as the semi parametric estimation methods we consider the maximum empirical likelihood mel estimator and the generalized method of moments gmm or the estimating equation estimator tables and figures of the distribution functions of four estimators are given for enough values of the parameters to cover most linear models of interest and we include some heteroscedastic cases and nonlinear cases we have found that the liml estimator has good performance in terms of the bounded loss functions and probabilities when the number of instruments is large that is the micro econometric models with many instruments in the terminology of recent econometric literature

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

some properties of the liml estimator in a dynamic panel structural equation abstract we investigate the finite sample and asymptotic properties of the within groups wg the random effects quasi maximum likelihood rqml the generalized method of moment gmm and the limited information maximum likelihood liml estimators for a panel autoregressive structural equation model with random effects when both t time dimension and n cross section dimension are large when we use the forward filtering due to alvarez and arellano 2003 the wg the rqml and gmm estimators are significantly biased when both t and n are large while t n is different from zero the liml estimator gives desirable asymptotic properties when t n converges to a constant

topological properties of self similar fractals with one parameter abstract in this paper we study two classes of planar self similar fractals t e with a shifting parameter e the first one is a class of self similar tiles by shifting x coordinates of some digits we give a detailed discussion on the disk likeness i e the property of being a topological disk in terms of e we also prove that t e determines a quasi periodic tiling if and only if e is rational the second one is a class of self similar sets by shifting diagonal digits we give a necessary and sufficient condition for t e to be connected

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

disk like tiles and self affine curves with noncollinear digits let a m n be an expanding matrix d c n a digit set and t t a d the associated self affine set it has been asked by grochenig and haas 1994 that given any expanding matrix a m 2 z whether there exists a digit set such that t is a connected or disk like i e homeomorphic to the closed unit disk tile with regard to this question collinear digit sets have been studied in the literature in this paper we consider noncollinear digit sets and show the existence of a noncollinear digit set corresponding to each expanding matrix such that t is a connected tile moreover for such digit sets we give necessary and sufficient conditions for t to be a disk like tile

t test in a structural equation properties of t ratios associated with the liml tsls and ols estimators in a structural form estimation are studied the existence of moments of these t ratios including the liml form is proved first second monte carlo simulations are performed to find out real sizes of the t test and the likelihood ratio test third asymptotic expansions of the distributions of t ratios are derived under the null hypothesis to find out deviations of real sizes from nominal sizes theoretically the asymptotic power functions are also derived

disklikeness of planar self affine tiles we consider the disklikeness of the planar self affine tile t generated by an integral expanding matrix a and a consecutive collinear digit set v 0 v 2v q 1 v z 2 let f x x 2 px q be the characteristic polynomial of a we show that the tile t is disklike if and only if 2 p g 2 moreover t is a hexagonal tile for all the cases except when p 0 in which case t is a square tile the proof depends on certain special devices to count the numbers of nodal points and neighbors of t and a criterion of bandt and wang 2001 on disklikeness

a characterization of connected self affine fractals arising from collinear digits abstract let a be an expanding integer matrix with characteristic polynomial f x x 2 p x q and let d 0 1 q 2 q m v be a collinear digit set where m 0 v z 2 it is well known that there exists a unique self affine fractal t satisfying a t t d in this paper we give a complete characterization of connectedness of t that generalizes the previous result for q 3

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

spurious inference in reduced rank asset pricing models this note studies some seemingly anomalous results that arise in possibly misspecified reduced rank linear asset pricing models estimated by the continuously updated generalized method of moments when a spurious factor that is a factor that is uncorrelated with the returns on the test assets is present the test for correct model specification has asymptotic power that is equal to the nominal size in other words applied researchers will erroneously conclude that the model is correctly specified even when the degree of misspecification is arbitrarily large the rejection probability of the test for overidentifying restrictions typically decreases further in underidentified models where the dimension of the null space is larger than 1

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

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

further 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

some exact distribution theory for maximum likelihood estimators of cointegrating coefficients in error correction models the author derives some exact finite sample disbibutions and characterizes the tail behavior of maximum likelihood estimators of the cointegrating coefficients in error correction models the reduced rank regression estimator has a distribution with cauchy like tails and no finite moments of integer order the maximum likelihood estimator of the coefficients in a particular triangular system representation has matrix t distribution tails with finite integer moments to order t n r where t is the sample size n is the total number of variables and r is the dimension of cointegration space this helps explain some recent simulation studies where extreme outliers occur more frequently for the reduced rank regression estimator than for alternative asymptotically efficient procedures based on triangular representation copyright 1994 by the econometric society

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

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

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

radiographic image sequence coding using two stage adaptive vector quantization a two stage adaptive vector quantization scheme for radiographic image sequence coding is introduced each frame in the sequence is first decomposed into a set of vectors corresponding to nonoverlapping spatially contiguous block of pixels a codebook is generated using a training set of vectors drawn from the sequence each vector is then encoded by the label representing the closest codeword of the codebook and the label values in a frame label map memory at both ends of the communication channel the changes occurring in the radiographic image sequences can be categorized into two types those due to body motion and those due to the injected contrast dye material in the second scheme proposed encoding is performed in two stages in the first stage the labels of corresponding vectors from consecutive frames are compared and the frame label map memory is replenished updated this stage is sufficient to tack the changes caused by patient motions but not due to the injected contrast dye material the resulting residual error vectors after the first stage coding are calculated for the latter changes and are further encoded by a second codebook which is updated on a frame to frame basis

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

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 time series and cross section asymptotics of dynamic panel data estimators in this paper we derive the asymptotic properties of within groups wg gmm and liml estimators for an autoregressive model with random effects when both t and n tend to infinity gmm and liml are consistent and asymptotically equivalent to the wg estimator when t n 0 the fixed t results for gmm and liml remain valid but wg although consistent has an asymptotic bias in its asymptotic distribution when t n tends to a positive constant the wg gmm and liml estimators exhibit negative asymptotic biases of order 1 t 1 n and 1 2n t respectively in addition the crude gmm estimator that neglects the autocorrelation in first differenced errors is inconsistent as t n c 0 despite being consistent for fixed t finally we discuss the properties of a random effects pseudo mle with unrestricted initial conditions when both t and n tend to infinity

testing instrument admissibility some refined asymptotic results this paper is concerned with the refined asymptotic properties of several tests for the admissibility of a subset of overidentifying instrumental variables it derives maximum likelihood and linearized maximum likelihood tests and calculates size corrections to the order 1 t the local power function of the size corrected tests is the same to the order 1 t irrespectively of the form of the test statistic or the limited information estimator used in its computation futher it compares these tests with two previously proposed tests the size and the power of the original and the size corrected tests are compared by monte carlo experiments copyright 1994 by the econometric society

connectedness of planar self affine sets associated with non consecutive collinear digit sets abstract in the paper we focus on the connectedness of planar self affine sets t a d generated by an integer expanding matrix a with det a 3 and a collinear digit set d 0 1 b v where b 1 and v r 2 such that v a v is linearly independent we discuss the domain of the digit b to determine the connectedness of t a d especially a complete characterization is obtained when we restrict b to be an integer some results on the general case of det a 3 are obtained as well

finite sample properties of single equation estimators under structural change abstract finite sample properties of single equation estimators under structural change are investigated when two endogenous variables appear in the equation of interest the single equation estimators obtained in barten and bronsard 1970 and hodoshima 1988 are derived for the model with structural change and are asymptotically efficient but in finite samples they may perform poorly in comparison with the two stage least squares estimator and the ordinary least squares estimator based on expressions of the estimators in terms of wishart matrices numerical comparisons of the estimators finite sample properties are presented

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

self affine tiles in ℝn abstract a self affine tile in r n is a set t of positive measure with a t d d t d where a is an expanding n n real matrix with det a m an integer and d d d 2 d m r n is a set of m digits it is known that self affine tiles always give tilings of r n by translation this paper extends known characterizations of digit sets d yielding self affine tiles it proves several results about the structure of tilings of r n possible using such tiles and gives examples showing the possible relations between self replicating tilings and general tilings which clarify results of kenyon on self replicating tilings

formulation and estimation of dynamic models using panel data abstract this paper presents a statistical analysis of time series regression models for longitudinal data with and without lagged dependent variables under a variety of assumptions about the initial conditions of the processes being analyzed the analysis demonstrates how the asymptotic properties of estimators of longitudinal models are critically dependent on the manner in which samples become large by expanding the number of observations per person holding the number of people fixed or by expanding the number of persons holding the number of observations per person fixed the paper demonstrates which parameters can and cannot be identified from data produced by different sampling plans

frame adaptive vector quantization for image sequence coding an adaptive technique for image sequence coding that is based on vector quantization is described each frame in the sequence is first decomposed into a set of vectors a codebook is generated using the vectors of the first frame as the training sequence and a label map is created by quantizing the vectors the vectors of the second frame are then used to generate a new codebook starting with the first codebook as seeds the updated codebook is then transmitted at the same time the label map is replenished by coding the position and the new values of the labels that have changed from one frame to the other the process is repeated for subsequent frames experimental results for a test sequences demonstrate that the technique can track the changes and maintain a nearly constant distortion over the entire sequence

image compression using adaptive vector quantization the paper demonstrates the application of adaptive vector quantization techniques to the coding of monochromatic and color pictures codebooks of representative vectors are generated for different portions of the image by means of extensive simulations the performance of the coder for monochrome images is estimated as a function of the various coding parameters vector dimensionality number of representative vectors and degree of adaptivity it is shown that for a vector size of dimension 4 there is a zone of operation ranging from 1 0 to 1 5 bits pixel where adaptive vector quantization is advantageous despite the simplicity of the decoder the performance of the adaptive vector quantizer is found to be comparable with block transform coding schemes for both monochrome and color pictures

another look at the instrumental variable estimation of error components models this article develops a framework for efficient iv estimators of random effects models with information in levels which can accommodate predetermined variables our formulation clarifies the relationship between the existing estimators and the role of transformations in panel data models we characterize the valid transformations for relevant models and show that optimal estimators are invariant to the transformation used to remove individual effects we present an alternative transformation for models with predetermined instruments which preserves the orthogonality among the errors finally we consider models with predetermined variables that have constant correlation with the effects and illustrate their importance with simulations

alternative size corrections for some gls test statistics the case of the ar 1 model abstract alternative size corrections are developed for the t and f tests in the ar 1 normal linear model edgeworth corrected critical values are obtained from normal student t chi square and f distributions alternatively we may use cornish fisher corrected test statistics to avoid the problem of negative tail probabilities of an edgeworth distribution the use of the exact distributions student t f results in approximations that are locally exact i e they reduce to the exact formulae for a sufficient simplification of the model monte carlo findings support the theoretical considerations in favour of the locally exact cornish fisher corrections

image sequence coding using vector quantization in this paper a new interframe coding technique based upon vector quantization is presented this algorithm has as its basis twodimensional block vector quantization at the frame level onto which is grafted the concept of adaptive codebook replenishment and frame replenishment this algorithm includes three strategies label replenishment label replenishment with mean shift and label replenishment with mean shift and selective codeword replacement the last two strategies efficiently update the codebook to track the changes in local statistics on a frame basis compared with three dimensional block vector quantization 16 for bit rates between 0 5 and 0 65 the normalized mean squared error nmse is reduced by a factor ranging between 2 and 2 5 by the use of replenishment

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

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

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

robust confidence sets in the presence of weak instruments abstract this paper considers instrumental variable regression with a single endogenous variable and the potential presence of weak instruments i construct confidence sets for the coefficient on the single endogenous regressor by inverting tests robust to weak instruments i suggest a numerically simple algorithm for finding the conditional likelihood ratio clr confidence sets full descriptions of possible forms of the clr anderson rubin ar and lagrange multiplier lm confidence sets are given i show that the clr confidence sets have nearly the shortest expected arc length among similar symmetric invariant confidence sets in a circular model i also prove that the clr confidence set is asymptotically valid in a model with non normal errors

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

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

asymptotic expansions of the distributions of the structural variance estimators in a simultaneous equations system abstract asymptotic expansions of three alternative classes of structural variance estimators associated with the k class estimators of structural coefficients are derived for two parameter sequences a sequence in which the non centrality parameter increases while the sample size stays fixed called large or small disturbance sequence and that in which the number of observations increases the accuracy of approximations to small sample distributions are numerically examined with help of monte carlo studies properties of the sum of squared residuals of an estimated structural equation are also found from our study

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

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

generalized threshold replenishment an adaptive vector quantization algorithm for the coding of nonstationary sources in this paper we describe a new adaptive vector quantization avq algorithm designed for the coding of non stationary sources this new algorithm generalized threshold replenishment gtr differs from prior avq algorithms in that it features an explicit online consideration of both rate and distortion because of its online nature gtr is more amenable to real time hardware and software implementation than are many prior avq algorithms that rely on traditional batch training methods additionally as rate distortion cost criteria are used in both the determination of nearest neighbor codewords and the decision to update the codebook gtr achieves rate distortion performance superior to that of other avq algorithms particularly for low rate coding results are presented that illustrate low rate performance surpassing that of other avq algorithms for the coding of both an image sequence and an artificial non stationary random process for the image sequence it is shown that 1 most avq algorithms achieve distortion much lower than that of nonadaptive vq for the same rate about 1 5 b pixel and 2 gtr achieves performance substantially superior to that of the other avq algorithms for low rate coding being the only algorithm to achieve a rate below 1 0 b pixel

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

estimating linear statistical relationships this paper on estimating linear statistical relationships includes three lectures on linear functional and structural relationships factor analysis and simultaneous equations models the emphasis is on relating the several models by a general approach and on the similarity of maximum likelihood estimators under normality in the different models in the first two lectures the observable vector is decomposed into a systematic part and a random error the systematic part satisfies the linear relationships estimators are derived for several cases and some of their properties given estimation of the coefficients of a single equation in a simultaneous equations model is shown to be equivalent to estimation of linear functional relationships

many ivs estimation of dynamic panel regression models with measurement error abstract in this paper we investigate a dynamic linear panel regression model with measurement error we consider the panel data estimation whose time dimension t is not small and comparable to the cross sectional dimension n first we show that the 2sls estimator suffers from the bias problem due to many instrumental variables using the alternative asymptotics where t 3 n goes to a constant as n t we characterize its asymptotic bias due to many ivs as a bias reduction method we investigate the jive and derive its limiting distribution under the alternative asymptotics

a reinterpretation of the tests of overidentifying restrictions abstract the anderson rubin sargan basmann tests are usually considered as testing the validity of overidentifying structural restrictions in this paper we show that a more eligible interpretation is that they test the validity of overidentifying orthogonality conditions the tests are second order unbiased against the latter alternative while under the first alternative there is a subset of the parameter space where the tests are biased the third order local power under the alternative of false orthogonality conditions is derived and is used to find the implicit null hypothesis of the tests and to propose a more informative interpretation several methods of size correction are compared by monte carlo experiments

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

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

connectedness of a class of planar self affine tiles we consider a class of planar self affine tiles t that are generated by the lower triangular expanding matrices and the product form digit sets we give necessary and sufficient conditions for t to be connected and disk like also for the disconnect case we give a condition that enumerates the number of connected components of t

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

estimation of dynamic models with error components abstract observations on n cross section units at t time points are used to estimate a simple statistical model involving an autoregressive process with an additive term specific to the unit different assumptions about the initial conditions are a initial state fixed b initial state random c the unobserved individual effect independent of the unobserved dynamic process with the initial value fixed and d the unobserved individual effect independent of the unobserved dynamic process with initial value random asymptotic properties of the maximum likelihood and covariance estimators are obtained when t and when n the relationship between the pseudo and conditional maximum likelihood estimators is clarified a simple consistent estimator that is independent of the initial conditions and the way in which t or n is also suggested

adaptive vector quantization for picture coding using neural networks the paper presents applications of neural network algorithms to the design of an adaptive vector quantizer vector quantization has been applied to the problem of displaying natural images with a reduced set of colors colormap and to the interframe coding of image sequences the first step was to test classical linde buzo gray lgb self organizing feature maps sofm and competitive learning cl algorithms for the codebook design the best results for the reconstructed quality image and the computational time are obtained using a cl algorithm with a new initialization strategy that solves the problem of underutilized nodes an adaptive vector quantization algorithm is proposed and tested in a motion compensated image coder the results of the simulations are very promising in fact the coder performance compared with that using a fixed vq is considerably improved and the subjective quality of the coded images is much better than that obtained using standard vector quantization especially when rapid motion is present in the scene

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

asymptotic probability concentrations and finite sample properties of modified liml estimators for equations with more than two endogenous variables this paper investigates the distributional properties of a class of modified limited information maximum likelihood liml estimators it is shown that the asymptotic distributions of these estimators are more concentrated than those of the modified liml estimators suggested by fuller additionally the results of an extensive monte carlo investigation of the finite sample properties of the proposed estimators show that when the equation of interest has more than two endogenous variables the liml estimator is often highly inefficient so that substantial gains in precision are realized by using the modified estimators in place of the liml estimator

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

symmetry based inference in an instrumental variable setting we describe exact inference based on group invariance assumptions that specify various forms of symmetry in the distribution of a disturbance vector in a general nonlinear model it is shown that such mild assumptions can be equivalently formulated in terms of exact confidence sets for the parameters of the functional form when applied to the linear model this exact inference provides a unified approach to a variety of parametric and distribution free tests in particular we consider exact instrumental variable inference based on symmetry assumptions the unboundedness of exact confidence sets is related to the power to reject a hypothesis of underidentification in a multivariate instrumental variables context generalizations of anderson rubin confidence sets are considered

statistical inference for panel dynamic simultaneous equations models abstract we study the identification and estimation of panel dynamic simultaneous equations models we show that the presence of time persistent individual specific effects does not lead to changes in the identification conditions of traditional cowles commission dynamic simultaneous equations models however the limiting properties of the estimators depend on the way the cross section dimension n or the time series dimension t goes to infinity we propose three limited information estimator panel simple instrumental variables piv panel generalized two stage least squares pg2sls and panel limited information maximum likelihood estimation pliml we show that they are all asymptotically unbiased independent of the way of how n or t tends to infinity monte carlo studies are conducted to compare the performance of the pliml piv pg2sls the arellano bond type generalized method of moments and the akashi kunitomo least variance ratio estimator we demonstrate that the reliability of statistical inference depends critically on whether an estimator is asymptotically unbiased or not

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

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

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

the numerical values of some key parameters in econometric models abstract in the case of two endogenous variables exogenous predetermined variables and normally distributed disturbances the distributions of the two stage least squares tsls and limited information maximum likelihood liml estimators can be compared on the basis of three key parameters the non centrality parameter a standardization of the structural coefficient and the number of excluded exogenous variables in this paper the values of these parameters are estimated in eleven structural equations from various actual econometric models the distribution functions of the normalized tsls and liml estimators are given for the first two key parameters set at approximately their trimmed means and the third at its median

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

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