hamilton differential harnack inequality and w entropy for witten laplacian on riemannian manifolds abstract in this paper we prove the hamilton differential harnack inequality for positive solutions to the heat equation of the witten laplacian on complete riemannian manifolds with the c d k m condition where m n and k 0 are two constants moreover we introduce the w entropy and prove the w entropy formula for the fundamental solution of the witten laplacian on complete riemannian manifolds with the c d k m condition and on compact manifolds equipped with k m super ricci flows

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

li yau gradient bounds on compact manifolds under nearly optimal curvature conditions abstract we prove li yau type gradient bounds for the heat equation either on manifolds with fixed metric or under the ricci flow in the former case the curvature condition is r i c l p for some p n 2 or sup m m r i c 2 y d 2 n x y d y where n is the dimension of the manifold in the later case one only needs scalar curvature being bounded we will explain why the conditions are nearly optimal and give an application the li yau bound for the heat equation on manifolds with fixed metric seems to be the first one allowing ricci curvature not bounded from below

two stage least squares estimation of average causal effects in models with variable treatment intensity abstract two stage least squares tsls is widely used in econometrics to estimate parameters in systems of linear simultaneous equations and to solve problems of omitted variables bias in single equation estimation we show here that tsls can also be used to estimate the average causal effect of variable treatments such as drug dosage hours of exam preparation cigarette smoking and years of schooling the average causal effect in which we are interested is a conditional expectation of the difference between the outcomes of the treated and what these outcomes would have been in the absence of treatment given mild regularity assumptions the probability limit of tsls is a weighted average of per unit average causal effects along the length of an appropriately defined causal response function the weighting function is illustrated in an empirical example based on the relationship between schooling and earnings

remarks on differential harnack inequalities abstract in this paper we give a generalization of global and local differential harnack inequalities for heat equations obtained by li and xu j f li x j xu differential harnack inequalities on riemannian manifolds i linear heat equation adv math 226 5 2011 4456 4491 and baudoin and garofalo f baudoin n garofalo perelman s entropy and doubling property on riemannian manifolds j geom anal 21 2011 1119 1131 from this we can derive new harnack inequalities and new lower bounds for the associated heat kernel also we provide some new entropy formulas with monotonicity

the incidental parameter problem since 1948 this paper was written to mark the 50th anniversary of neyman and scott s econometrica paper defining the incidental parameter problem it surveys the history both of the paper and of the problem in the statistics and econometrics literature

the conjugate heat equation and ancient solutions of the ricci flow abstract we prove gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the ricci flow as a consequence for dimension 4 and higher we show that the backward limit of type i solutions of the ricci flow must be a non flat gradient shrinking ricci soliton this extends perelman s previous result on backward limits of solutions in dimension 3 in which case the curvature operator is nonnegative it follows from hamilton ivey curvature pinching estimate as an application this also addresses an issue left in naber 2010 23 where naber proves the interesting result that there exists a type i dilation limit that converges to a gradient shrinking ricci soliton but that soliton might be flat the gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest

geometric properties of heisenberg type groups on montre qu il y a une certaine metrique riemannienne contravariante qui induit une longueur d arc

minimum distance approach to inference with many instruments abstract i analyze a linear instrumental variables model with a single endogenous regressor and many instruments i use invariance arguments to construct a new minimum distance objective function with respect to a particular weight matrix the minimum distance estimator is equivalent to the random effects estimator of chamberlain and imbens 2004 and the estimator of the coefficient on the endogenous regressor coincides with the limited information maximum likelihood estimator this weight matrix is inefficient unless the errors are normal and i construct a new more efficient estimator based on the optimal weight matrix finally i show that when the minimum distance objective function does not impose a proportionality restriction on the reduced form coefficients the resulting estimator corresponds to a version of the bias corrected two stage least squares estimator i use the objective function to construct confidence intervals that remain valid when the proportionality restriction is violated

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

analysis and geometry on manifolds with integral ricci curvature bounds ii we extend several geometrical results for manifolds with lower ricci curvature bounds to situations where one has integral lower bounds in particular we generalize colding s volume convergence results and extend the cheeger colding splitting theorem

using a likelihood perspective to sharpen econometric discourse three examples this paper discusses a number of areas of inference where dissatisfaction by applied workers with the prescriptions of econometric high theory is strong and where a likelihood approach diverges strongly from the mainstream approach in its practical prescriptions two of the applied areas are related and have in common that they involve nonstationarity macroeconomic time series modeling and analysis of panel data in the presence of potential nonstationarity the third area is nonparametric kernel regression methods the conclusion is that in these areas a likelihood perspective leads to more useful honest and objective reporting of results and characterization of uncertainty it also leads to insights not as easily available from the usual perspective on inference

liouville theorems and gradient estimates for a nonlinear elliptic equation in this paper we establish gradient estimates for positive solutions to the equation 0 1 fup u on any smooth metric measure space whose m bakry emery curvature is bounded from below by m 1 k with k 0 these estimates imply liouville theorems for 0 1 when p 1 our main theorem reduces to the gradient estimate of wang 2010 9 as applications several harnack inequalities are obtained

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

hamilton s gradient estimates and liouville theorems for porous medium equations on noncompact riemannian manifolds abstract let m be a complete noncompact riemannian manifold of dimension n in this paper we derive a local gradient estimate for positive solutions of porous medium equation t u u 1 1 1 2 n 1 on m 0 we also obtain a theorem of liouville type for positive solutions of the porous medium equation

gradient estimates for ut δf u on manifolds and some liouville type theorems abstract in this paper we first prove a localized hamilton type gradient estimate for the positive solutions of porous media type equations u t f u with f u 0 on a complete riemannian manifold with ricci curvature bounded from below in the second part we study fast diffusion equation fde and porous media equation pme u t u p p 0 and obtain localized hamilton type gradient estimates for fde and pme in a larger range of p than that for aronson benilan estimate harnack inequalities and cauchy problems in the literature applying the localized gradient estimates for fde and pme we prove some liouville type theorems for positive global solutions of fde and pme on noncompact complete manifolds with nonnegative ricci curvature generalizing yau s celebrated liouville theorem for positive harmonic functions

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

testing overidentifying restrictions with many instruments and heteroskedasticity this paper gives a test of overidentifying restrictions that is robust to many instruments and heteroskedasticity it is based on a jackknife version of the overidentifying test statistic correct asymptotic critical values are derived for this statistic when the number of instruments grows large at a rate up to the sample size it is also shown that the test is valid when the number of instruments is fixed and there is homoskedasticity this test improves on recently proposed tests by allowing for heteroskedasticity and by avoiding assumptions on the instrument projection matrix this paper finds in monte carlo studies that the test is more accurate and less sensitive to the number of instruments than the hausman sargan or gmm tests of overidentifying restrictions

gradient estimate for the heat kernel of a complete riemannian manifold and its applications abstract we derive some higher order gradient estimates for the heat kernels on complete manifolds applying these results we show that the riesz transform on a complete manifold with nonnegative ricci curvature is of weak type 1 1 in addition we consider the boundedness of the riesz potential on a complete manifold with nonnegative ricci curvature

on eigenvalue pinching in positive ricci curvature we shall show that for manifolds with ric n 1 the radius is close to iff the n 1 st eigenvalue is close to n this extends results of cheng and croke which show that the diameter is close to iff the first eigenvalue is close to n we shall also give a new proof of an important theorem of colding to the effect that if the radius is close to then the volume is close to that of the sphere and the manifold is gromov hausdorff close to the sphere from work of cheeger and colding these conditions imply that the manifold is diffeomorphic to a sphere

optimal inference for instrumental variables regression with non gaussian errors abstract this paper is concerned with inference on the coefficient on the endogenous regressor in a linear instrumental variables model with a single endogenous regressor nonrandom exogenous regressors and instruments and i i d errors whose distribution is unknown it is shown that under mild smoothness conditions on the error distribution it is possible to develop tests which are nearly efficient in the sense of andrews et al 2006 when identification is weak and consistent and asymptotically optimal when identification is strong in addition an estimator is presented which can be used in the usual way to construct valid indeed optimal confidence intervals when identification is strong the estimator is of the two stage least squares variety and is asymptotically efficient under strong identification whether or not the errors are normal

h type groups and iwasawa decompositions since their introduction by a kaplan kpl some ten years ago generalised heisenberg groups also known as groups of heisenberg type or h type groups have provided a framework in which to construct interesting examples in geometry and analysis see for instance c2 kp2 kp3 kpr k2 rl r2 tl tv the iwasawa n groups associated to all the real rank one simple lie groups are h type so one has a convenient vehicle for studying these in a unified way many problems on these simple lie groups can be reduced to a problem on h type groups via the so called noncompact picture and often problems on h type groups can be solved on all the groups of the family in one fell swoop as in for example ch ck dr out of this approach to studying simple lie groups several problems arise such as why only some h type groups correspond to simple lie groups of real rank one in this paper we discuss various features of iwasawa n groups which distinguish them in the class of all h type groups we shall show that all h type groups which possess certain geometric properties clearly possessed by iwasawa n groups satisfy a lie algebraic condition implicit in the work of b kostant kt2 that we shall call the j condition we shall also use elementary clifford algebra to classify the

gradient estimates on the weighted p laplace heat equation in this paper by a regularization process we derive new gradient estimates for positive solutions to the weighted p laplace heat equation when the m bakry emery curvature is bounded from below by k for some constant k 0 when the potential function is constant which reduce to the gradient estimate established by ni and kotschwar for positive solutions to the p laplace heat equation on closed manifolds with nonnegative ricci curvature if k 0 and reduce to the davies hamilton and li xu s gradient estimates for positive solutions to the heat equation on closed manifolds with ricci curvature bounded from below if p 2

harnack estimates for a heat type equation under the ricci flow abstract in this paper we consider the gradient estimates for a positive solution of the nonlinear parabolic equation t u t u h u p on a riemannian manifold whose metrics evolve under the ricci flow two harnack inequalities and other interesting results are obtained

multivariate regression models for panel data abstract the paper examines the relationship between heterogeneity bias and strict exogeneity in a distributed lag regression of y on x the relationship is very strong when x is continuous weaker when x is discrete and non existent as the order of the distributed lag becomes infinite the individual specific random variables introduce nonlinearity and heteroskedasticity so the paper provides an appropriate framework for the estimation of multivariate linear predictors restrictions are imposed using a minimum distance estimator it is generally more efficient than the conventional estimators such as quasi maximum likelihood there are computationally simple generalizations of two and three stage least squares that achieve this efficiency gain some of these ideas are illustrated using the sample of young men in the national longitudinal survey the paper reports regressions on the leads and lags of variables measuring union coverage smsa and region the results indicate that the leads and lags could have been generated just by a random intercept this gives some support for analysis of covariance type estimates these estimates indicate a substantial heterogeneity bias in the union smsa and region coefficients

differential harnack inequalities on riemannian manifolds i linear heat equation abstract in the first part of this paper we get new li yau type gradient estimates for positive solutions of heat equation on riemannian manifolds with ricci m k k r as applications several parabolic harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with ricci curvature bounded from below in the second part we establish a perelman type li yau hamilton differential harnack inequality for heat kernels on manifolds with ricci m k which generalizes a result of l ni 2004 2006 20 21 as applications we obtain new harnack inequalities and heat kernel estimates on general manifolds we also obtain various entropy monotonicity formulas for all compact riemannian manifolds

jackknife instrumental variable estimation with heteroskedasticity we present a new jackknife estimator for instrumental variable inference with unknown heteroskedasticity it weighs observations such that many instruments consistency is guaranteed while the signal component in the data is maintained we show that this results in a smaller signal component in the many instruments asymptotic variance when compared to estimators that neglect a part of the signal to achieve consistency both many strong instruments and many weak instruments asymptotic distributions are derived using high level assumptions that allow for instruments with identifying power that varies between explanatory variables standard errors are formulated compactly we review briefly known estimators and show in particular that our symmetric jackknife estimator performs well when compared to the hlim and hful estimators of hausman et al in monte carlo experiments

manifolds of positive ricci curvature with almost maximal volume 10 in this note we consider complete riemannian manifolds with ricci curvature bounded from below the well known theorems of myers and bishop imply that a manifold mn with ric n 1 satisfies diam mn diam sn l vol mn vol sn l it follows from ch that equality in either of these estimates can be achieved only if mn is isometric to sn 1 the natural conjecture is that a manifold mn with almost maximal diameter or volume must be a topological equivalent to sn with respect to diameter this is true only if mn satisfies some additional assumptions see an 0 gp e with respect to volume however no extra restriction is necesary

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

gradient estimates for the heat equation under the ricci flow the paper considers a manifold m evolving under the ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on m

gradient estimates for a simple elliptic equation on complete non compact riemannian manifolds in this paper we study the local gradient estimate for the positive solution to the following equation u aulog u bu 0i nm where a 0 b are real constants m is a complete non compact riemannian manifold our result is optimal in the sense when m g is a complete non compact expanding gradient ricci soliton by definition m g is called an expanding gradient ricci soliton if for some constant c 0 it satisfies that rc cg d 2 f

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

ricci curvature and volume convergence the purpose of this paper is to give a new integral estimate of distances and angles on manifolds with a given lower ricci curvature bound this will provide us with an integral version of the toponogov comparison triangle theorem for ricci curvature and almost extreme triangles see the earlier works cl and c2 for an analog of this when the manifold has positive ricci curvature using this we prove the anderson cheeger conjecture saying that the volume is a continuous function on the space of all closed n manifolds with ricci curvature greater or equal to n 1 equipped with the gromovhausdorff metric we also prove gromov s conjecture for n 57 3 saying that an almost nonnegatively ricci curved n manifold with first betti number equal to n is a torus further we prove a conjecture of anderson cheeger saying that an open n manifold with nonnegative ricci curvature whose tangent cone at infinity is in is itself in finally we prove a conjecture of fukaya yamaguchi we will now describe these results in more detail let dgh denote the gromov hausdorff distance glp first we have the following result which was conjectured by anderson cheeger

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

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

a clifford algebra approach to real simple lie algebras ii the algebras with root system bcr the algebras with root system bcr

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

identification and estimation of local average treatment effects we investigate conditions sufficient for identification of average treatment effects using instrumental variables first we show that the existence of valid instruments is not sufficient to identify any meaningful average treatment effect we then establish that the combination of an instrument and a condition on the relation between the instrument and the participation status is sufficient for identification of a local average treatment effect for those who can be induced to change their participation status by changing the value of the instrument finally we derive the probability limit of the standard iv estimator under these conditions it is seen to be a weighted average of local average treatment effects this abstract was borrowed from another version of this item

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

harnack inequality heat kernel bounds and eigenvalue estimates under integral ricci curvature bounds abstract let m n g i j be a complete riemannian manifold we prove that for any p n 2 when k p 1 is small enough some parabolic type gradient bounds hold for the positive solutions of a nonlinear parabolic equation u t u a u log u on geodesic balls b o r in m with 0 r 1 we can also derive the gradient estimates for any solutions to the above nonlinear parabolic equation along the ricci flow on a closed manifold without any curvature conditions as its application we derive some parabolic type gradient estimates for a positive solution u x t of the heat equation u t u moreover our estimate is stronger than zhang and zhang s estimate see remark 2 in section 4 those results are generalizations of li yau hamilton li xu type gradient estimates under the integral ricci curvature bounds by utilizing the gradient estimates of the heat equation we obtain harnack inequalities the upper bound and the lower bound for the heat kernel eigenvalue estimate and the lower bound of green s function on riemannian manifolds under the integral ricci curvature bounds

heat kernel on smooth metric measure spaces and applications we derive a harnack inequality for positive solutions of the f heat equation and gaussian upper and lower bound estimates for the f heat kernel on complete smooth metric measure spaces with bakry emery ricci curvature bounded below both upper and lower bound estimates are sharp when the bakry emery ricci curvature is nonnegative the main argument is the de giorgi nash moser theory as applications we prove an l f 1 liouville theorem for f subharmonic functions and an l f 1 uniqueness theorem for f heat equations when f has at most linear growth we also obtain eigenvalues estimates and f green s function estimates for the f laplace operator

gradient estimates for doubly nonlinear diffusion equations abstract in this short note we obtain some gradient estimates for the positive solution to the doubly nonlinear diffusion equation on closed riemannian manifold with ricci curvature bounded below by a non positive constant as applications we also derive corresponding harnack inequalities

aronson benilan estimates for the fast diffusion equation under the ricci flow abstract we study the fast diffusion equation with a linear forcing term 0 1 u t div u p 1 u r u under the ricci flow on a complete manifold m such that m r 2 has bounded curvature and nonnegative isotropic curvature where 0 p 1 and r r x t is the evolving scalar curvature of m at time t we prove aronson benilan and li yau hamilton type differential harnack estimates for positive solutions of 0 1 in addition we use similar method to prove certain li yau hamilton estimates for the heat equation and conjugate heat equation which extend those obtained in cao and hamilton 2009 cao 2008 and kuang and zhang 2008 to the noncompact setting

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

random effects estimators with many instrumental variables in this paper we propose a new estimator for a model with one endogenous regressor and many instrumental variables our motivation comes from the recent literature on the poor properties of standard instrumental variables estimators when the instrumental variables are weakly correlated with the endogenous regressor our proposed estimator puts a random coefficients structure on the relation between the endogenous regressor and the instruments the variance of the random coefficients is modelled as an unknown parameter in addition to proposing a new estimator our analysis yields new insights into the properties of the standard two stage least squares tsls and limited information maximum likelihood liml estimators in the case with many weak instruments we show that in some interesting cases tsls and liml can be approximated by maximizing the random effects likelihood subject to particular constraints we show that statistics based on comparisons of the unconstrained estimates of these parameters to the implicit tsls and liml restrictions can be used to identify settings when standard large sample approximations to the distributions of tsls and liml are likely to perform poorly we also show that with many weak instruments liml confidence intervals are likely to have under coverage even though its finite sample distribution is approximately centered at the true value of the parameter in an application with real data and simulations around this data set the proposed estimator performs markedly better than tsls and liml both in terms of coverage rate and in terms of risk

local sobolev constant estimate for integral ricci curvature bounds abstract we obtain a local sobolev constant estimate for integral ricci curvature which enables us to extend several important tools such as the maximal principle the gradient estimate the heat kernel estimate and the l 2 hessian estimate to manifolds with integral ricci lower bounds without the non collapsing conditions

on complete manifolds with nonnegative ricci curvature complete open riemannian manifolds mn g with nonnegative sectional curvature are well understood the basic results are toponogov s splitting theorem and the soul theorem cg1 the splitting theorem has been extended to manifolds of nonnegative ricci curvature cg2 on the other hand the soul theorem does not extend even topologically according to recent examples in gm2 a different method to construct manifolds which carry a metric with ric 0 but no metric with nonnegative sectional curvature has been given by l berard bergery bb this leads to the question cf also y1 is there any finiteness result for complete riemannian manifolds with ric 0 the answer is certainly affirmative in the low dimensional special cases n 2 where all notions of curvature coincide and n 3 where nonnegative ricci curvature has been studied by means of stable minimal surfaces msy sy on the other hand j p sha and d g yang shy have constructed complete manifolds with strictly positive ricci curvature in higher dimensions for example they can choose the underlying space to be r4 x s3 with infinitely many copies of s3 x cp 2 attached to it by surgery cf also shy 1 it is therefore clear that any finiteness result for arbitrary dimensions requires additional assumptions the purpose of this paper is to establish the following main result

consistent estimation with a large number of weak instruments this paper analyzes the conditions under which consistent estimation can be achieved in instrumental variables iv regression when the available instruments are weak and the number of instruments k n goes to infinity with the sample size we show that consistent estimation depends importantly on the strength of the instruments as measured by r the rate of growth of the so called concentration parameter and also on k n in particular when k n the concentration parameter can grow even if each individual instrument is only weakly correlated with the endogenous explanatory variables and consistency of certain estimators can be established under weaker conditions than have previously been assumed in the literature hence the use of many weak instruments may actually improve the performance of certain point estimators more specifically we find that the limited information maximum likelihood liml estimator and the bias corrected two stage least squares b2sls estimator are consistent when k n r n 0 while the two stage least squares 2sls estimator is consistent only if k n r n 0 as n these consistency results suggest that liml and b2sls are more robust to instrument weakness than 2sls

deforming the metric on complete riemannian manifolds soit m g ij x une variete de riemann a n dimensions complete non compacte de tenseur de complexe riemannien r ijkl satisfaisant r ijkl 2 k 0 sur m ou 0 0 dependant seulement de n et de k 0 telle que l equation d evolution g ij x t t 2r ij x t sur m g ij x o g ij x x m ait une solution lisse g ij x t 0 pour un temps court 0 t t u k 0 et satisfait l estimation suivante pour un entier m 0 il existe des constantes c m 0 dependant seulement de n m et k 0 telles que sup x m v m r ijkl x t 2 c m t m 0 t t u k 0

gradient estimates and harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on riemannian manifolds abstract we derive the gradient estimates and harnack inequalities for positive solutions of nonlinear parabolic and nonlinear elliptic equations t u x t h x t u x t 0 and u b u hu 0 on riemannian manifolds we also obtain a theorem of liouville type for positive solutions of the nonlinear elliptic equation

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