mode interaction in rayleigh be nard convection abstract a long wave model for two dimensional thermal convection is examined this model contains both quadratic and cubic nonlinearity unless the problem is symmetrical about the horizontal mid plane in which case only the cubic nonlinearity is present in the asymmetrical case a strong interaction between modes with wavenumbers k and 2k is responsible for wavenumber gaps regions in parameter space where steady states of a given spatial period cease to exist in both the symmetrical and asymmetrical cases a 1 3 resonance generates multiple steady solutions of a given spatial period we calculate steady states and determine their stability for the asymmetrical case we also solve the initial value problem and find stable travelling waves stable modulated travelling waves and stable heteroclinic cycles as predicted by an analysis of the 1 2 resonance

estimating dynamic models of imperfect competition we describe a two step algorithm for estimating dynamic games under the assumption that behavior is consistent with markov perfect equilibrium in the first step the policy functions and the law of motion for the state variables are estimated in the second step the remaining structural parameters are estimated using the optimality conditions for equilibrium the second step estimator is a simple simulated minimum distance estimator the algorithm applies to a broad class of models including industry competition models with both discrete and continuous controls such as the ericson and pakes 1995 model we test the algorithm on a class of dynamic discrete choice models with normally distributed errors and a class of dynamic oligopoly models similar to that of pakes and mcguire 1994

nonatomic games on loeb spaces in the setting of noncooperative game theory strategic negligibility of individual agents or diffuseness of information has been modeled as a nonatomic measure space typically the unit interval endowed with lebesgue measure however recent work has shown that with uncountable action sets for example the unit interval there do not exist pure strategy nash equilibria in such nonatomic games in this brief announcement we show that there is a perfectly satisfactory existence theory for nonatomic games provided this nonatomicity is formulated on the basis of a particular class of measure spaces hyperfinite loeb spaces we also emphasize other desirable properties of games on hyperfinite loeb spaces and present a synthetic treatment embracing both large games as well as those with incomplete information

a fast resample method for parametric and semiparametric models abstract we propose a fast resample method for two step nonlinear parametric and semiparametric models which does not require recomputation of the second stage estimator during each resample iteration the fast resample method directly exploits the score function representations computed on each bootstrap sample thereby reducing computational time considerably this method is used to approximate the limit distribution of parametric and semiparametric estimators possibly simulation based that admit an asymptotic linear representation monte carlo experiments demonstrate the desirable performance and vast improvement in the numerical speed of the fast bootstrap method

on the convergence in distribution of measurable multifunctions random sets normal integrands stochastic processes and stochastic infima the concept of the distribution function of a closed valued measurable multifunction is introduced and used to study the convergence in distribution of sequences of multifunctions and the epi convergence in distribution of normal integrands and stochastic processes in particular various compactness criteria are exhibited the connections with the classical convergence theory for stochastic processes are analyzed and for purposes of illustration we apply the theory to sketch out a modified derivation of donsker s theorem brownian motion as a limit of normalized random walks we also suggest the potential application of the theory to the study of the convergence of stochastic infima

finite mixture distributions sequential likelihood and the em algorithm a popular way to account for unobserved heterogeneity is to assume that the data are drawn from a finite mixture distribution a barrier to using finite mixture models is that parameters that could previously be estimated in stages must now be estimated jointly using mixture distributions destroys any additive separability of the log likelihood function we show however that an extension of the em algorithm reintroduces additive separability thus allowing one to estimate parameters sequentially during each maximization step in establishing this result we develop a broad class of estimators for mixture models returning to the likelihood problem we show that relative to full information maximum likelihood our sequential estimator can generate large computational savings with little loss of efficiency

distributional properties of correspondences on loeb spaces abstract we present some regularity properties for the set of distributions induced by the measurable selections of a correspondence over a loeb space which include closedness convexity compactness purification and semicontinuity we also note that all the properties reported in the main theorems are not satisfied by some correspondences on the unit lebesgue interval

resonant interactions in thermosolutal convection when convection occurs in a finite box successive modes become unstable as the rayleigh number is increased giving rise to branches of nonlinear solutions that are linked by mixed mode branches emerging from hopf transcritical or pitchfork bifurcations the fundamental mode in particular gives rise to resonant interactions several specific examples of these generic features have already been described here we study resonant interactions in two dimensional thermosolutal convection with point symmetry imposed where behaviour is enriched by the presence of oscillations and of subcritical steady convection steady m roll solutions are obtained numerically and their stability properties are investigated by computing the leading real and complex eigenvalues we find that the branch of fundamental m 1 solutions for a box with aspect ratio 1 5 develops loops that touch the m 3 branch in transcritical bifurcations as is reduced these loops detach themselves from the m 1 branch and eventually join the stacked m 2 branch increasing on the other hand leads to complicated triple resonances involving m 1 m 3 and m 5 solutions these resonant interactions are associated with pattern selection and the transfer of stability from single roll to multiple roll solutions in wider boxes

secondary turing type instabilities due to strong spatial resonance we investigate the dynamics of pattern forming systems in large domains near a codimension two point corresponding to a strong spatial resonance where competing instabilities with wavenumbers in th

matrix roots of eventually positive matrices abstract eventually positive matrices are real matrices whose powers become and remain strictly positive as such eventually positive matrices are a fortiori matrix roots of positive matrices which motivates us to study the matrix roots of primitive matrices using classical matrix function theory and perron frobenius theory we characterize classify and describe in terms of the real jordan canonical form the p th roots of eventually positive matrices

estimation of semiparametric models when the criterion function is not smooth we provide easy to verify sufficient conditions for the consistency and asymptotic normality of a class of semiparametric optimization estimators where the criterion function does not obey standard smoothness conditions and simultaneously depends on some nonparametric estimators that can themselves depend on the parameters to be estimated our results extend existing theories such as those of pakes and pollard 1989 andrews 1994a and newey 1994 we also show that bootstrap provides asymptotically correct confidence regions for the finite dimensional parameters we apply our results to two examples a hit rate and a partially linear median regression with some endogenous regressors

why saturated probability spaces are necessary abstract an atomless probability space a p is said to have the saturation property for a probability measure on a product of polish spaces x y if for every random element f of x whose law is marg x there is a random element g of y such that the law of f g is a p is said to be saturated if it has the saturation property for every such we show each of a number of desirable properties holds for every saturated probability space and fails for every non saturated probability space these include distributional properties of correspondences such as convexity closedness compactness and preservation of upper semi continuity and the existence of pure strategy equilibria in games with many players we also show that any probability space which has the saturation property for just one good enough measure or which satisfies just one good enough instance of the desirable properties must already be saturated our underlying themes are 1 there are many desirable properties that hold for all saturated probability spaces but fail everywhere else 2 any probability space that out performs the lebesgue unit interval in almost any way at all is already saturated

bayesian estimation of dynamic discrete choice models we propose a new methodology for structural estimation of infinite horizon dynamic discrete choice models we combine the dynamic programming dp solution algorithm with the bayesian markov chain monte carlo algorithm into a single algorithm that solves the dp problem and estimates the parameters simultaneously as a result the computational burden of estimating a dynamic model becomes comparable to that of a static model another feature of our algorithm is that even though the number of grid points on the state variable is small per solution estimation iteration the number of effective grid points increases with the number of estimation iterations this is how we help ease the curse of dimensionality we simulate and estimate several versions of a simple model of entry and exit to illustrate our methodology we also prove that under standard conditions the parameters converge in probability to the true posterior distribution regardless of the starting values

sequential estimation of dynamic discrete games a comment recursive procedures which are based on iterating on the best response mapping have difficulties converging to all equilibria in multi player games we illustrate these difficulties by revisiting the asymptotic properties of the iterative nested pseudo maximum likelihood method for estimating dynamic games introduced by aguirregabiria and mira 2007 an example shows that the iterative method may not be consistent

solutions for nonlinear convection in the presence of a lateral boundary abstract a method is described for calculating steady state patterns in large scale nonlinear systems taking into account the effect of a lateral boundary and without the need for extensive numerical calculations the key feature is the determination of the phase shift of the nonlinear periodic form distant from the boundary as a function of wavelength this is found by analyzing the solution close to the boundary where floquet theory is used to describe the departure of the solution from its periodic form for a restricted band of wavelengths lying within the eckhaus boundary dual solutions for the phase shift are found one of which corresponds to an unstable state results are presented here for the one dimensional swift hohenberg equation in a semi infinite domain but in principle the method can be applied to more general pattern forming systems the results are compared with the predictions of weakly nonlinear theory and with nonlinear computations on a large but finite domain

weak sequential convergence in l1 μ x and an exact version of fatouʼs lemma abstract the class of nonatomic finite measure spaces with the saturation property as developed in maharam 1942 and hoover keisler 1984 is characterized by the fatou and lebesgue property of a well dominated sequence of multifunctions taking values in a banach space with multifunctions reduced to functions this fatou characterization also extends to a variant of the closure property found in optimal control theory the results are developed through a considered overview of the relevant literature on the exact and approximate fatou lemma phrased in terms of bochner integration

heteroclinic cycles and modulated travelling waves in a system with d 4 symmetry we study a system of two complex ordinary differential equations with d4 symmetry describing the effects of the symmetry breaking o 2 to d4 on an interaction between fourier modes with wavenumbers in the ratio 1 2 this change from continuous to discrete symmetry is relevant to the effect of introducing riblets on a wall to reduce boundary layer drag numerical simulations planar analysis and melnikov s method reveal the persistence under this symmetry breaking of just two of the continuous group orbit of heteroclinic cycles of the o 2 system it is further shown that one of these cycles destabilizes to regular modulated travellin waves while the other loses stability to a new type of solution modulated travelling waves which quasi periodically reverse their direction of propagation a bifurcation diagram for this behavior and the subsequent destabilization of modulated travelling waves to travelling waves is established the method of averaging is used to show that some of the quasiperiodic solutions of the o 2 system persist under the symmetry breaking while others break into sets of periodic orbits

kuramoto sivashinsky dynamics on the center unstable manifold this paper studies the dynamical behavior of solutions of the kuramoto sivashinsky partial differential equation with periodic boundary conditions on a spatial interval 0 h the length h is the bifurcation parameter and reduction is made to a two complex dimensional system on a local center unstable manifold near the second bifurcation point h 2 from the trivial solution the resulting o 2 equivariant system displays all the behavior found in high precision simulations of the partial differential equation near this bifurcation point in particular bifurcation sequences to stable traveling waves unstable modulated traveling waves and attracting heteroclinic cycles are reproduced qualitatively and quantitatively within 1 in the parameter range h 2 pm 20 a clear understanding of the global dynamical behavior in this region is thus obtained

dynamics in the 1 2 spatial resonance with broken reflection symmetry the dynamics near the 1 2 spatial resonance in systems with broken o 2 symmetry are described the system is assumed to remain so 2 invariant and attention is focused on the behavior that replaces the structurally stable heteroclinic cycles present in the o 2 symmetric system the symmetry breaking destroys these cycles and introduces a variety of new bifurcations into the system some which are associated with complex behavior the possible scenarios with increasing symmetry breaking are described and possible applications of the results are discussed

formation of patterns in the rayleigh be nard convection abstract a model equation for a convection in a thin layer with nearly insulated boundaries is studied in the one dimensional case it is proved that the equation has periodic quasiperiodic and chaotic stationary solutions the stability of the periodic solutions with respect to long wave perturbations is investigated and the global dynamics of the flow is analyzed for a wide domain of parameters it is proved that almost all small initial data lead to a specific stationary periodic solution this solution corresponds to the wave number for which the zero solution is most unstable

identifying dynamic discrete decision processes in this paper we analyze the nonparametric identification of dynamic discrete choice models our methodology is based on the insight of hotz and miller 1993 that bellman equations can be interpreted as moment conditions we consider cases with and without unobserved heterogeneity not only do we show that these models are not identified rust 1994 we are also able to determine their exact degree of underidentification we begin with the case without correlated unobserved heterogeneity using bellman equations as moment conditions we show that utility functions in each alternative cannot be nonparametrically identified as long as the following structural parameters are not set the distribution function of unobserved preference shocks the discount rate and the current and future preferences in one reference alternative we also investigate how exclusion or parametric restrictions can provide identifying restrictions as the identification proof is constructive a simple method of moment estimator can be derived and overidentifying restrictions can be tested provided that one is willing to make stronger identifying assumptions dynamic discrete choice modelling is thus little different from the continuous case bellman equations can be used to recover deep structural parameters as are euler equations we continue by exploring a case where the unobserved component of preferences is correlated over time even if the functional degree of underidentification of this model is larger we present reasonable identifying assumptions that lead to the same identification results as without unobserved heterogeneity the same methodology using moment conditions is applied this paper expands upon the work in rust 1994 where the generic nonidentification result is stated we use a slightly different model in our case agents preferences have unobservable and possibly persistent components the constructive aspect of our proof allows us to interpret rust s underidentification result and to propose identifying restrictions on the technical side the insights for our identification strategy are borrowed from the works of hotz and miller 1993 hotz et al 1994 and altug and miller 1998 we

a general fatou lemma a general fatou lemma is established for a sequence of gelfand integrable functions from a vector loeb space to the dual of a separable banach space or with a weaker assumption on the sequence a banach lattice a corollary sharpens previous results in the finite dimensional setting even for the case of scalar measures counterexamples are presented to show that the results obtained here are sharp in various aspects applications include systematic generalizations of the distribution of correspondences from the case of scalar loeb spaces to the case of vector loeb spaces and a proof of the existence of a pure strategy equilibrium in games with private and public information and with compact metric action spaces

on simulated em algorithms the em algorithm is a popular and useful algorithm for finding the maximum likelihood estimator in incomplete data problems each iteration of the algorithm consists of two simple steps an e step in which a conditional expectation is calculated and an m step where the expectation is maximized in some problems however the em algorithm cannot be applied since the conditional expectation required in the e step cannot be calculated instead the expectation may be estimated by simulation we call this a simulated em algorithm the simulations can at least in principle be done in two ways either new independent random variables are drawn in each iteration or the same uniforms are re used in each iteration in this paper the properties of these two versions of the simulated em algorithm are discussed and compared

swapping the nested fixed point algorithm a class of estimators for discrete markov decision models this paper proposes a new nested algorithm npl for the estimation of a class of discrete markov decision models and studies its statistical and computational properties our method is based on a representation of the solution of the dynamic programming problem in the space of conditional choice probabilities when the npl algorithm is initialized with consistent nonparametric estimates of conditional choice probabilities successive iterations return a sequence of estimators of the structural parameters which we call k stage policy iteration estimators we show that the sequence includes as extreme cases a hotz miller estimator for k 1 and rust s nested fixed point estimator in the limit when k furthermore the asymptotic distribution of all the estimators in the sequence is the same and equal to that of the maximum likelihood estimator we illustrate the performance of our method with several examples based on rust s bus replacement model monte carlo experiments reveal a trade off between finite sample precision and computational cost in the sequence of policy iteration estimators

instabilities induced by a weak breaking of a strong spatial resonance abstract through multiple scales and symmetry arguments we derive a model set of amplitude equations describing the interaction of two steady state pattern forming instabilities in the case that the wavelengths of the instabilities are nearly in the ratio 1 2 in the case of exact 1 2 resonance the amplitude equations are odes here they are pdes we discuss the stability of spatially periodic solutions to long wavelength disturbances by including these modulational effects we are able to explore the relevance of the exact 1 2 results to spatially extended physical systems for parameter values near to this codimension two bifurcation point these new instabilities can be described in terms of reduced normal form pdes near various secondary codimension two points the robust heteroclinic cycle in the odes is destabilised by long wavelength perturbations and a stable periodic orbit is generated that lies close to the cycle an analytic expression giving the approximate period of this orbit is derived

on multiple eigenvalues of matrices depending on several parameters the eigenvalues and eigenvectors as well as their derivatives of matrices depending on one or several parameters have been studied by sun j comput math 3 1985 pp 351 364 using the implicit function theorems and others but not for the more general multiple eigenvalues in this paper the mean of the multiple eigenvalues in terms of the trace operator tr the corresponding invariant subspaces and their derivatives are studied using a generalization of sun s approach the numerical aspects concerning the computation of such derivatives by direct and iterative methods e g simultaneous and inverse simultaneous iterations will be discussed briefly the main results in this paper apply to clusters of nonmultiple eigenvalues as well the implications e g on simultaneous iteration will be explored

stabilization of unstable procedures the recursive projection method fixed point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes the recursive projection method rpm which stabilizes such procedures by computing a projection onto the unstable subspace is presented on this subspace a newton or special newton iteration is performed and the fixed point iteration is used on the complement as continuation in the parameter proceeds the projection is efficiently updated possibly increasing or decreasing the dimension of the unstable subspace the method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system convergence proofs are given and pseudo arclength continuation on the unstable subspace is introduced to allow continuation past folds examples are presented for an important application of the rpm in which a black box time integration scheme is stabilized enabling it to compute unstable steady states the rpm can also be used to accelerate iterative procedures when slow convergence is due to a few slowly decaying modes

on the theory of vector valued loeb measures and integration abstract ideas and techniques from nonstandard theories of measure spaces and banach spaces are brought together to give a further study of nonstandard vector measures an integration theory of real valued functions with respect to banach space valued measures is developed geometric properties of nonstandard hulls are also investigated some applications are given

integration of correspondences on loeb spaces we study the bochner and gel fand integration of banach space valued correspondences on a general loeb space though it is well known that the lyapunov type result on the compactness and convexity of the integral of a correspondence and the fatou type result on the preservation of upper semicontinuity by integration are in general not valid in the setting of an infinite dimensional space we show that exact versions of these two results hold in the case we study we also note that our results on a hyperfinite loeb space capture the nature of the corresponding asymptotic results for the large finite case but the unit lebesgue interval fails to provide such a framework

invading wave fronts and their oscillatory wakes are linked by a modulated travelling phase resetting wave abstract periodic wave trains are the generic solution from for oscillatory reaction diffusion equations in one space dimension it has been shown previously that invasive wavefronts generate behind them a wave train with a different speed from that of the invasion sherratt physica d 70 1994 370 382 in this paper the mechanism of wave train generation is studied in detail for systems of two reaction diffusion equations close to a supercritical hopf bifurcation in the kinetics with equal diffusion coefficients a combination of analytical and numerical evidence is presented suggesting that the invasive front and wave train are separated by a modulated travelling wave of phase gradient in which phase singularities occur periodically this calculation leads to a prediction of the amplitude and speed of the wave train generated by invasion

inference in dynamic discrete choice models with serially correlated unobserved state variables this paper develops a method for inference in dynamic discrete choice models with serially correlated unobserved state variables estimation of these models involves computing high dimensional integrals that are present in the solution to the dynamic program and in the likelihood function first the paper proposes a bayesian markov chain monte carlo estimation procedure that can handle the problem of multidimensional integration in the likelihood function second the paper presents an efficient algorithm for solving the dynamic program suitable for use in conjunction with the proposed estimation procedure

set valued functions lebesgue extensions and saturated probability spaces abstract recent advances in the theory of distributions of set valued functions have been shaped by counterexamples which hinge on the non existence of measurable selections with requisite properties these examples all based on the lebesgue interval and initially circumvented by sun in the context of loeb spaces have now led keisler and sun ks to establish a comprehensive theory of the distributions of set valued functions on saturated probability spaces introduced by hoover and keisler in contrast we show that a countably generated extension of the lebesgue interval suffices for an explicit resolution of these examples and furthermore that it does not contradict the ks necessity results we draw the fuller implications of our theorems for integration of set valued functions for lyapunov s result on the range of vector measures and for the theory of large non anonymous games

pseudo likelihood estimation and bootstrap inference for structural discrete markov decision models abstract this paper analyzes the higher order properties of the estimators based on the nested pseudo likelihood npl algorithm and the practical implementation of such estimators for parametric discrete markov decision models we derive the rate at which the npl algorithm converges to the mle and provide a theoretical explanation for the simulation results in aguirregabiria and mira aguirregabiria v mira p 2002 swapping the nested fixed point algorithm a class of estimators for discrete markov decision models econometrica 70 1519 1543 in which iterating the npl algorithm improves the accuracy of the estimator we then propose a new npl algorithm that can achieve quadratic convergence without fully solving the fixed point problem in every iteration and apply our estimation procedure to a finite mixture model we also develop one step npl bootstrap procedures for discrete markov decision models the monte carlo simulation evidence based on a machine replacement model of rust rust j 1987 optimal replacement of gmc bus engines an empirical model of harold zurcher econometrica 55 999 1033 shows that the proposed one step bootstrap test statistics and confidence intervals improve upon the first order asymptotics even with a relatively small number of iterations

nonparametric identification of finite mixture models of dynamic discrete choices in dynamic discrete choice analysis controlling for unobserved heterogeneity is an important issue and finite mixture models provide flexible ways to account for it this paper studies nonparametric identifiability of type probabilities and type specific component distributions in finite mixture models of dynamic discrete choices we derive sucient conditions for nonparametric identification for various finite mixture models of dynamic discrete choices used in applied work under dierent assumptions on the markov property stationarity and type invariance in the transition process three elements emerge as the important determinants of identification the time dimension of panel data the number of values the covariates can take and the heterogeneity of the response of dierent types to changes in the co

sequential estimation of structural models with a fixed point constraint this paper considers the estimation problem of structural models for which empirical restrictions are characterized by a fixed point constraint such as structural dynamic discrete choice models or models of dynamic games we analyze a local condition under which the nested pseudo likelihood npl algorithm converges to a consistent estimator and derive its convergence rate we find that the npl algorithm may not necessarily converge to a consistent estimator when the fixed point mapping does not have a local contraction property to address the issue of divergence we propose alternative sequential estimation procedures that can converge to a consistent estimator even when the npl algorithm does not

estimation of dynamic discrete models from time aggregated data an important component in dynamic discrete choice models and dynamic discrete games is the transition density of state variables from the current period to the next period most empirical dynamic discrete choice models identify the theoretical time interval in the behavioral model with that observed in the data set however many empirical data sets are time aggregated in this paper we show that when the time interval in the behavioral theory model differs from that in the observed data difficulties with nonparametric identification and specification arise in addition we study the properties of parametric maximum likelihood estimators and flexible semiparametric estimators of the transition density in dynamic discrete models with time aggregated data sets

nonparametric identification of dynamic models with unobserved state variables we consider the identification of a markov process wt x t when only wt a subset of the variables are observed in structural dynamic models wt includes the sequences of choice variables and observed state variables of an optimizing agent while xdenotes the sequence of serially correlated unobserved state variables in the non stationary case we show that the markov law of motion fwt x wt 1 x t 1 is identified from the observation of five periods of data wt 1 wt wt 1 wt 2 wt 3 under reason able assumptions in the stationary case only four observations wt 1 wt wt 1 wt 2 are required identification of fwt x t wt 1 x t 1 is a crucial input in methodologies for estimating markovian dynamic models based on the conditional choice probability ccp approach pioneered by hotz and miller

on the convergence of closed valued measurable multifunctions in this paper we study the convergence almost everywhere and in measure of sequences of closed valued multifunctions we first give a number of criteria for the convergence of sequences of closed subsets these results are used to obtain various characterizations for the convergence of measurable multifunctions in particular we are interested in the convergence properties of measurable selections

on the convergence in probability of random sets measurable multifunctions it is shown that the convergence in probability of random sets introduced by salinetti and wets is consistent with the more standard definitions of convergence in probability and is metric invariant

the necessity of nowhere equivalence abstract we prove some regularity properties convexity closedness compactness and preservation of upper hemicontinuity for distribution and regular conditional distribution of correspondences under the nowhere equivalence condition we show the necessity of such a condition for any of these properties to hold as an application we demonstrate that the nowhere equivalence condition is satisfied on the underlying agent space if and only if pure strategy nash equilibria exist in general large games with any fixed uncountable compact action space