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

delay independent stability of a special sequence of neutral difference differential equations with one delay abstract the equations of the title for the state x t r l x d a d a n x t b d b d b n x t t 0 d d dt 1 are considered in the parameter space a b b n p where p is defined by re a a is a second real point a a for a r a is the complex conjugate to a c the same notation applies for b by a tedious but elementary analysis the set s0 p for which 1 has for all t 0 only asymptotically stable solutions in the sense of liapunov is explicitly determined in the form b n a b 1 2 the stability boundary decays exponentially with respect to the multiplicity n of the zeros of the coefficient polynomials of 1 we generalize a theorem of f brauer j differential equations 69 1987 185 191 dealing with delay independent stability for characteristic equations of the form h z a z b z exp zt t 0 3 where a z b z are holomorphic in re z 0 in such a way that the neutral case can be dealt with

statistical properties of functionals of the paths of a particle diffusing in a one dimensional random potential we present a formalism for obtaining the statistical properties of functionals and inverse functionals of the paths of a particle diffusing in a one dimensional quenched random potential we demonstrate the implementation of the formalism in two specific examples 1 where the functional corresponds to the local time spent by the particle around the origin and 2 where the functional corresponds to the occupation time spent by the particle on the positive side of the origin within an observation time window of size t we compute the disorder average distributions of the local time the inverse local time the occupation time and the inverse occupation time and show that in many cases disorder modifies the behavior drastically

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

robust inference in nonlinear models with mixed identification strength abstract the paper studies inference in regression models composed of nonlinear functions with unknown transformation parameters and loading coefficients that measure the importance of each component in these models non identification and weak identification present in multiple parts of the parameter space resulting in mixed identification strength for different unknown parameters this paper proposes robust tests and confidence intervals for sub vectors and linear functions of the unknown parameters in particular the results cover applications where some nuisance parameters are non identified under the null davies 1977 1987 and some nuisance parameters are subject to a full range of identification strength to construct this robust inference procedure we develop a local limit theory that models mixed identification strength the asymptotic results involve both inconsistent estimators that depend on a localization parameter and consistent estimators with different rates of convergence a sequential argument is used to peel the criterion function based on identification strength of the parameters

convex hull of n planar brownian motions exact results and an application to ecology we compute exactly the mean perimeter and area of the convex hull of n independent planar brownian paths each of duration t both for open and closed paths we show that the mean perimeter alpha n sqrt t and the mean area beta n t for all t the prefactors alpha n and beta n computed exactly for all n increase very slowly logarithmically with increasing n this slow growth is a consequence of extreme value statistics and has interesting implications in an ecological context in estimating the home range of a herd of animals with a population size n

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

generalized arcsine laws for fractional brownian motion the three arcsine laws for brownian motion are a cornerstone of extreme value statistics for a brownian b t starting from the origin and evolving during time t one considers the following three observables i the duration t the process is positive ii the time t mathrm last the process last visits the origin and iii the time t mathrm max when it achieves its maximum or minimum all three observables have the same cumulative probability distribution expressed as an arcsine function thus the name arcsine laws we show how these laws change for fractional brownian motion x t a non markovian gaussian process indexed by the hurst exponent h it generalizes standard brownian motion i e h frac 1 2 we obtain the three probabilities using a perturbative expansion in h ensuremath frac 1 2 while all three probabilities are different this distinction can only be made at second order in our results are confirmed to high precision by extensive numerical simulations

handbook of brownian motion facts and formulae i theory i stochastic processes in general ii linear diffusions iii stochastic calculus iv brownian motion v local time as a markov process vi differential systems associated to brownian motion appendix 1 briefly on some diffusions ii tables of distributions of functionals of brownian motion and related processes 1 brownian motion 2 brownian motion with drift 3 reflecting brownian motion 4 bessel process of order 5 bessel process of order 1 2 6 bessel process of order zero 7 ornstein uhlenbeck process 8 radial ornstein uhlenbeck process 9 geometric brownian motion appendix 2 special functions appendix 3 inverse laplace transforms appendix 4 differential equations and their solutions appendix 5 formulae for n fold differentiation

new trace formulae for a quadratic pencil of the schrodinger operator this work deals with the eigenvalue problem for a quadratic pencil of the schrodinger operator on a finite closed interval with the two point boundary conditions we will obtain new regularized trace formulas for this class of differential pencil

inference when a nuisance parameter is not identified under the null hypothesis many econometric testing problems involve nuisance parameters which are not identified under the null hypotheses this paper studies the asymptotic distribution theory for such tests the asymptotic distributions of standard test statistics are described as functionals of chi square processes in general the distributions depend upon a large number of unknown parameters we show that a transformation based upon a conditional probability measure yields an asymptotic distribution free of nuisance parameters and we show that this transformation can be easily approximated via simulation the theory is applied to threshold models with special attention given to the so called self exciting threshold autoregressive model monte carlo methods are used to assess the finite sample distributions the tests are applied to u s gnp growth rates and we find that potter s 1995 threshold effect in this series can be possibly explained by sampling variation

maximum likelihood estimation and uniform inference with sporadic identification failure abstract this paper analyzes the properties of a class of estimators tests and confidence sets css when the parameters are not identified in parts of the parameter space specifically we consider estimator criterion functions that are sample averages and are smooth functions of a parameter this includes log likelihood quasi log likelihood and least squares criterion functions we determine the asymptotic distributions of estimators under lack of identification and under weak semi strong and strong identification we determine the asymptotic size in a uniform sense of standard t and quasi likelihood ratio qlr tests and css we provide methods of constructing qlr tests and css that are robust to the strength of identification the results are applied to two examples a nonlinear binary choice model and the smooth transition threshold autoregressive star model

brownian motion in wedges last passage time and the second arc sine law we consider a planar brownian motion starting from o at time t 0 and stopped at t 1 and a set f oii i 1 2 n of n semi infinite straight lines emanating from o denoting by g the last time when f is reached by the brownian motion we compute the probability law of g in particular we show that for a symmetric f and even n values this law can be expressed as a sum of arcsin or arcsin 2 functions the original result of levy is recovered as the special case n 2 a relation with the problem of reaction diffusion of a set of three particles in one dimension is discussed

kotani theory for one dimensional stochastic jacobi matrices we consider families of operators h on l2 given by h u n u n 1 u n 1 v n u n wherev is a stationary bounded ergodic sequence we prove analogs of kotani s results including that for a e ac h is the essential closure of the set ofe where e the lyaponov index vanishes and the result that ifv is non deterministic then ac is empty

the infinite period limit of the inverse formalism for periodic potentials soit l operateur de hill h p d 2 dx 2 v p x quand v p x est une fonction p periodique de x on montre que les formules pour la solution du probleme inverse ont une limite si on prend v p comme une repetition periodique d une fonction a support compact v x et que l on fait tendre p vers l infini

current fluctuations in one dimensional diffusive systems with a step initial density profile we show how to apply the macroscopic fluctuation theory mft of bertini de sole gabrielli jona lasinio and landim to study the current fluctuations of diffusive systems with a step initial condition we argue that one has to distinguish between two ways of averaging the annealed and the quenched cases depending on whether we let the initial condition fluctuate or not although the initial condition is not a steady state the distribution of the current satisfies a symmetry very reminiscent of the fluctuation theorem we show how the equations of the mft can be solved in the case of non interacting particles the symmetry of these equations can be used to deduce the distribution of the current for several other models from its knowledge derrida and gerschenfeld in j stat phys 136 1 15 2009 for the symmetric simple exclusion process in the range where the integrated current q t sim sqrt t we show that the non gaussian decay exp q t 3 t of the distribution of q t is generic

stability with respect to the delay on a paper of k l cooke and p van den driessche a proposal for the correction of the main result of the title s paper is made some accompanying comments on other points are given so a simpler central result is derived an explicit representation of the set of stable delays is achieved

absolute stability in delay equations abstract an equilibrium of a delay equation is said to be absolutely stable if it is asymptotically stable for all delays this is equivalent to a certain characteristic equation s having all its roots in the left half plane for all delays we obtain a simple necessary and sufficient condition for absolute stability of a class of delay equations

spectral functions special functions and the selberg zeta function the functional determinant of an eigenvalue sequence as defined by zeta regularization can be simply evaluated by quadratures we apply this procedure to the selberg trace formula for a compact riemann surface to find a factorization of the selberg zeta function into two functional determinants respectively related to the laplacian on the compact surface itself and on the sphere we also apply our formalism to various explicit eigenvalue sequences reproducing in a simpler way classical results about the gamma function and the barnesg function concerning the latter our method explains its connection to the selberg zeta function and evaluates the related glaisher kinkelin constanta

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

exact resolution method for general 1d polynomial schrodinger equation the stationary 1d schrodinger equation with a polynomial potential v q of degree n is reduced to a system of complex exact quantization conditions of bohr sommerfeld form they arise from bilinear wronskian functional relations pairing spectral determinants of n 2 generically distinct operators all the transforms of one quantum hamiltonian under a cyclic group of complex scalings the determinants zeros define n 2 semi infinite chains of points in the complex spectral plane and these encode the original quantum problem now each chain can be assigned an exact quantization condition which constrains it relative to its neighbours resulting in closed equilibrium conditions for the global chain system these are supplemented by the standard bohr sommerfeld quantization conditions which bind the infinite tail of each chain asymptotically this reduced problem is then probed numerically for effective solvability upon test cases mostly symmetric quartic oscillators we find that the iterative enforcement of all the quantization conditions generates discrete chain dynamics which appear to converge geometrically towards the correct eigenvalues eigenfunctions we conjecture that the exact quantization then acts by specifying reduced chain dynamics which can be stable contractive and thus reconstruct the exact quantum data as their fixed point to date this statement is verified only empirically and in a vicinity of purely quartic or sextic potentials v q

last passage time for linear diffusions and application to the emptying time of a box we study the statistics of last passage time for linear diffusions first we present an elementary derivation of the laplace transform of the probability density of the last passage time thus recovering known results from the mathematical literature we then illustrate them on several explicit examples in a second step we study the spectral properties of the schrodinger operator associated to such diffusions in an even potential u x u x unveiling the role played by the so called weyl coefficient indeed in this case our approach allows us to relate the last passage times for dual diffusions i e diffusions driven by opposite force fields and to obtain new explicit formulae for the mean last passage time we further show that for such even potentials the small time t expansion of the mean last passage time on the interval 0 t involves the korteveg de vries invariants which are well known in the theory of schrodinger operators finally we apply these results to study the emptying time of a one dimensional box of size l containing n independent brownian particles subjected to a constant drift in the scaling limit where both n rightarrow infty and l rightarrow infty keeping the density rho n l fixed we show that the limiting density of the emptying time is given by a gumbel distribution our analysis provides a new example of the applications of extreme value statistics to out of equilibrium systems

random convex hulls and extreme value statistics in this paper we study the statistical properties of convex hulls of n random points in a plane chosen according to a given distribution the points may be chosen independently or they may be correlated after a non exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem we present a unifying approach based on the notion of support function of a closed curve and the associated cauchy s formulae that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics as an example of correlated points we study here in detail the case when the points represent the vertices of n independent random walks in the continuum time limit this reduces to n independent planar brownian trajectories for which we compute exactly for all n the mean perimeter and the mean area of their global convex hull our results have relevant applications in ecology in estimating the home range of a herd of animals some of these results were announced recently in a short communication phys rev lett 103 140602 2009

random walk on the bethe lattice and hyperbolic brownian motion we give the exact solution to the problem of a random walk on the bethe lattice through a mapping on an asymmetric random walk on the half line we also study the continuous limit of this model and discuss in detail the relation between the random walk on the bethe lattice and brownian motion on a space of constant negative curvature

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

the trace formula for schrodinger operators on the line this paper discusses certain aspects of the spectral and inverse spectral problems for the schrodinger operator l q frac d 2 dx 2 q x for q x c the space of bounded continuous functions the trace formula of the title is the relation q 0 lambda 0 sum limits j 1 infty lambda 2j lambda 2j 1 2 mu j with appropriate choices of j j 0 and j j 1 which is a familiar relation in the theory of hill s equation we characterize the set subseteq c of potentials for which this holds further extensions of the theory of hill s equation are also obtained from the spectrum l q a torust q is constructed which is in general infinite dimensjional everyq x can be mapped to a continuous path ont q described by the auxiliary spectrum j j 1 under certain geometrical conditions on l q this path is the orbit of ac 1 vector field ont q and the mapping extends to one from the hull q overline q x xi xi in mathbb r to the closure of this orbit in particular q is compact these results have applications in the theory of schrodinger operators with ergodic potentials

the zero information limit condition and spurious inference in weakly identified models abstract in many econometric models the asymptotic variance of a parameter estimate depends on the value of another structural parameter in such a way that the data contain little information about the former when the latter is close to a critical value this paper introduces the zero information limit condition zilc to identify such models where weak identification leads to spurious inference we find that standard errors tend to be underestimated in these cases but the size of the asymptotic t test may either be too great the intuitive case emphasized in the weak instrument literature or too small as in two cases illustrated here

m operators a generalisation of weyl titchmarsh theory the theory of the weyl titchmarsh m function for second order ordinary differential operators is generalized and applied to partial differential operators of the form q x acting in three space dimensions weyl operators m z are defined as maps from l2 s 1 to l2 s 1 s 1 unit sphere in r3 for exterior and interior boundary value problems and for the partial differential operator acting in l2 r 3 with the standard weyl titchmarsh m function recovered in the special case that q is spherically symmetric the analysis is carried out rather explicitly allowing for the determination of precise norm bounds for m operators and for the proof of higher dimensional analogues of a number of the fundamental results of standard weyl titchmarsh theory

the stability chart for the linearized cushing equation with a discrete delay and with gamma distributed delays in a proper setting the problem of the title amounts to the determination of all pairs d t for which the function z z 1 exp zt d and the functions fn z z 1 1 ztn n d n n d t 0 have all their zeros in re z 0 of the complex z plane correct versions of results and proofs found in papers of k l cooke and j m grossman j math anal appl 86 1982 592 627 and s p blythe r m nisbeth w s c gurney and n macdonald j math anal appl 109 1985 388 396 are given the argumentation in the proofs of the present paper differs from that of the studies mentioned in contrast to blythe et al the stability behaviours of both models are seen as similar stability charts and stability conditions for some other classes of systems are also given in addition to the gamma distribution for the lags other distributions which are similar in shape but differ greatly in their stability behaviour are also considered

almost periodicity of some random potentials in this paper we study some random potentials with purely absolutely continuous spectrum in one dimension there are no examples of random potentials with absolutely continuous spectrum that are not almost periodic though it is known from kotani 3 that such potentials are deterministic hence one could conjecture that any random potential with purely absolutely continuous spectrum is almost periodic we show this result for random potentials with infinitely many bands in their spectrum such that the gaps behave well in the finite bands case concini and johnson l discussed the same problem a precise definition of the behavior of the gaps could be found in condition 2 3 we make use of the results of levitan 7 kotani 3 4 and moser 9 for doing this we also discuss the behavior of the lyapunov exponent asymptotically and its relation to the smoothness of the potentials

almost periodic schrodinger operators iii the absolutely continuous spectrum in one dimension we studyh d2 dx2 v x withv x limit periodic e g v x ancos x 2n with an we prove that for a genericv and for generican in the explicit example h is a cantor nowhere dense perfect set for a dense set the spectrum is both cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous

on the measure of the spectrum for the almost mathieu operator we obtain partial results on the conjecture that for the almost mathieu operator at irrational frequency the measure of the spectrum s 4 2 for 2 we show that if n is rational and alpha n to alpha irrational then s alpha n lambda theta to 4 2 lambda

trend cycle decomposition of regime switching processes abstract we present a new approach to trend cycle decomposition of time series that follow regime switching processes the proposed approach which we label the regime dependent steady state rdss decomposition is motivated as the appropriate generalization of the beveridge and nelson decomposition beveridge s nelson c r 1981 a new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the business cycle journal of monetary economics 7 151 174 to the setting where the reduced form dynamics of a given series can be captured by a regime switching forecasting model for processes in which the underlying trend component follows a random walk with possibly regime switching drift the rdss decomposition is optimal in a minimum mean squared error sense and is more broadly applicable than directly employing an unobserved components model

stability conditions for the general linear difference differential equation with constant coefficients and one constant delay abstract for the application low degree cases of the equations of the title are of special interest lag independent stability conditions are derived for all systems up to degree two inclusive some necessary stability conditions for large lags are expounded in the non neutral case left half planes depending on the lag and containing all zeros of the characteristic function are obtained a sufficient lag independent condition for the whole class considered is found special attention is devoted to the neutral case for that case a collection of disks covering all zeros of the characteristic function is given the result is illustrated by the example of the general neutral characteristic function with coalescing zeros finally zero free regions around the zeros of the coefficient polynomials are derived to derive the results conformal mappings are studied