a complete classification of bifurcation diagrams of a dirichlet problem with concave convex nonlinearities abstract we study the bifurcation diagrams of positive solutions of the two point boundary value problem u x f u x 0 1 u 1 u 1 0 where f u g u h u g h c 0 c2 0 and 0 is a bifurcation parameter we assume that functions g and h satisfy hypotheses h1 h3 under hypotheses h1 h3 we give a complete classification of bifurcation diagrams and we prove that on the u plane each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the left hence the problem has at most two positive solutions for each 0 more precisely we prove the exact multiplicity of positive solutions

multiple positive solutions for semilinear elliptic systems with nonlinear boundary condition abstract in this paper we study the nonlinear boundary value problem u u f u u v in v v f v u v in u n g u u v v n g v u v on where is a bounded domain in r n with smooth boundary and the pairs of f u f v and g u g v are subcritical nonlinearities with the help of the nehari manifold we show existence of at least two positive solutions when one of the pairs of f u f v and g u g v is sublinear and the other one superlinear

on consistency and sparsity for principal components analysis in high dimensions principal components analysis pca is a classic method for the reduction of dimensionality of data in the form of n observations or cases of a vector with p variables contemporary datasets often have p comparable with or even much larger than n our main assertions in such settings are a that some initial reduction in dimensionality is desirable before applying any pca type search for principal modes and b the initial reduction in dimensionality is best achieved by working in a basis in which the signals have a sparse representation we describe a simple asymptotic model in which the estimate of the leading principal component vector via standard pca is consistent if and only if p n n 0 we provide a simple algorithm for selecting a subset of coordinates with largest sample variances and show that if pca is done on the selected subset then consistency is recovered even if p n n

existence and concentration of solutions with compact support for elliptic problems with critical growth in rn abstract for a continuous and sign changing function a x we study the existence and asymptotic behavior of compactly supported solutions of the following elliptic problem in r n with n 3 u a x u q u p u 0 and u d 1 2 r n where 0 0 q 1 p n 2 n 2 and a a or a a a when is small both cases admit two solutions v and v where v is a local minimizer and v is a minimizing sequence for the sobolev embedding from d 1 2 r n to l p 1 r n as 0 in the case a a a v blows up in the l norm and concentrates at one point as 0 consequently we show that v cannot be radially symmetric when a 0 0 and is small enough

on semilinear elliptic equations involving concave convex nonlinearities and sign changing weight function abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of positive solutions for semilinear elliptic equations with a sign changing weight function with the help of the nehari manifold we prove that there are at least two positive solutions for eq e f in bounded domains

multiple positive solutions of a nonlinear boundary value problem involving a sign changing weight abstract in this paper we study the multiplicity of positive solutions for the following elliptic equation u u 0 in r n u a x u q 2 u b x u p 2 u on r n where 1 q 2 p 2 2 2 n 1 n 2 if n 3 2 if n 2 r n x x n r n 1 r x n 0 is an upper half space in r n 0 and the functions a and b are satisfying some suitable conditions using the decomposition of the nehari manifold we prove that the equation has at least two positive solutions provided is sufficiently small

cross sectional averages versus principal components in spite of the increased use of factor augmented regressions in recent years little is known regarding the relative merits of the two main approaches to estimation and inference namely the cross sectional average and principal component estimators by providing a formal comparison of the approaches the current paper fills this gap in the literature

exact multiplicity and bifurcation diagrams of positive solutions of a one dimensional multiparameter prescribed mean curvature problem abstract we study the exact multiplicity and bifurcation diagrams of positive solutions u c 2 l l c l l of the one dimensional multiparameter prescribed mean curvature problem u x 1 u x 2 u p u q l x l u l u l 0 where 0 is a bifurcation parameter l 0 the radius of the one dimensional ball l l is an evolution parameter and 0 p q are two constants we prove that the problem has at most two positive solutions for any 0 p q and l 0 in addition if 0 p q q p p 1 2 p 1 we give a classification of totally three qualitatively different bifurcation diagrams on the u plane for any l 0 for any fixed p 0 and q q p p 2 2 2 p 3 we prove that there exist positive l l such that the bifurcation diagrams on the u plane are individually qualitatively different for the cases i 0 l l or l l ii l l iii l l l

panel data models with multiple time varying individual effects abstract this paper considers a panel data model with time varying individual effects the data are assumed to contain a large number of cross sectional units repeatedly observed over a fixed number of time periods the model has a feature of the fixed effects model in that the effects are assumed to be correlated with the regressors the unobservable individual effects are assumed to have a factor structure for consistent estimation of the model it is important to estimate the true number of individual effects we propose a generalized methods of moments procedure by which both the number of individual effects and the regression coefficients can be consistently estimated some important identification issues are also discussed our simulation results indicate that the proposed methods produce reliable estimates

linear regression for panel with unknown number of factors as interactive fixed effects in this paper we study the least squares ls estimator in a linear panel regression model with unknown number of factors appearing as interactive fixed effects assuming that the number of factors used in estimation is larger than the true number of factors in the data we establish the limiting distribution of the ls estimator for the regression coefficients as the number of time periods and the number of cross sectional units jointly go to infinity the main result of the paper is that under certain assumptions the limiting distribution of the ls estimator is independent of the number of factors used in the estimation as long as this number is not underestimated the important practical implication of this result is that for inference on the regression coefficients one does not necessarily need to estimate the number of interactive fixed effects consistently

a complete classification of bifurcation diagrams of a p laplacian dirichlet problem we study the bifurcation diagrams of positive solutions of the p laplacian dirichlet problem p u x f u x 0 1 1 p y y p 2y p u is the one dimensional p laplacian and 0 is a bifurcation parameter we assume that functions g and h satisfy hypotheses h1 h3 under hypotheses h1 h3 we give a complete classification of bifurcation diagrams and we prove that on the u plane each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the right hence the problem has at most two positive solutions for each 0 more precisely we prove the exact multiplicity of positive solutions in addition for p 2 we give interesting examples which show the evolution phenomena of bifurcation diagrams

shrinkage estimation of dynamic panel data models with interactive fixed effects abstract we consider the problem of determining the number of factors and selecting the proper regressors in linear dynamic panel data models with interactive fixed effects based on the preliminary estimates of the slope parameters and factors a la bai 2009 and moon and weidner 2015 we propose a method for simultaneous selection of regressors and factors and estimation through the method of adaptive group lasso least absolute shrinkage and selection operator we show that with probability approaching one our method can correctly select all relevant regressors and factors and shrink the coefficients of irrelevant regressors and redundant factors to zero further we demonstrate that our shrinkage estimators of the nonzero slope parameters exhibit some oracle property we conduct monte carlo simulations to demonstrate the superb finite sample performance of the proposed method we apply our method to study the determinants of economic growth and find that in addition to three common unobserved factors selected by our method government consumption share has negative effects whereas investment share and lagged economic growth have positive effects on economic growth

exact number of solutions of a prescribed mean curvature equation abstract we study the exact number of positive solutions of the dirichlet problem for the one dimensional prescribed mean curvature equation u 1 u 2 u p u q u 0 u 1 0 where 0 is a parameter and p q satisfy either 1 p q or 0 p q 1

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

determining the number of factors in approximate factor models in this paper we develop some econometric theory for factor models of large dimensions the focus is the determination of the number of factors r which is an unresolved issue in the rapidly growing literature on multifactor models we first establish the convergence rate for the factor estimates that will allow for consistent estimation of r we then propose some panel criteria and show that the number of factors can be consistently estimated using the criteria the theory is developed under the framework of large cross sections n and large time dimensions t no restriction is imposed on the relation between n and t simulations show that the proposed criteria have good finite sample properties in many configurations of the panel data encountered in practice

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

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

exact multiplicity for semilinear elliptic dirichlet problems involving concave and convex nonlinearities let b be the unit ball in r n n 3 let 0 p q n 2 n 2 in 1994 ambrosetti et al found that the semilinear elliptic dirichlet problem admits at least two solutions for small 0 and no solution for large in this paper we prove that there is a critical number 0 such that this problem has exactly two solutions for 0 exactly one solution for and no solution for

existence and multiplicity for negative solutions of k hessian equations abstract in this paper we give some existence results for k hessian equations by sub sup method and discuss the multiplicity of k hessian equations especially for the radial solutions in a ball

comment on iv estimation of panels with factor residuals by d robertson and v sarafidis abstract this comment discusses two issues related to the paper by robertson and sarafidis 2014 hereafter rs the first issue is the potential incidental parameters problem that may arise when data are cross sectionally heteroskedastic the second issue is an identification problem in the estimation of panel models with weakly exogenous instrumental variables

exact multiplicity and numerical computation of solutions for two classes of non autonomous problems with concave convex nonlinearities abstract we establish the exact multiplicity of positive solutions and the global solution structure for two classes of problems on circular domains the first class involves non autonomous concave convex problems on a ball in r n and the second one deals with concave convex problems on a thin annulus in r n we illustrate our results by numerical computations using a novel algorithm which involves continuation in a global parameter

multiple positive solutions for dirichlet problems involving concave and convex nonlinearities abstract in this paper we study the multiplicity of positive solutions for the following dirichlet problems involving concave and convex nonlinearities u u f x u q 2 u h x u p 2 u in a u h 0 1 a where 1 q 2 p 2 2 2 n n 2 if n 3 2 if n 2 0 a r is an infinite strip in r n is a bounded domain in r n 1 and f x h x satisfy suitable conditions

existence of solutions to a class of nonlinear convergent chattering free sliding mode control systems sliding mode control is a nonlinear control technique which is robust against some classes of uncertainties and disturbances however this control produces chattering which can cause instability due to unmodeled dynamics and can also cause damage to actuators or the plant there are essentially two ways to counter the chattering phenomenon one way is to use higher order sliding mode and the other way is to add a boundary layer around the switching surface and use continuous control inside the boundary the problem with the first method is that the derivative of a certain state variable is not available for measurement and therefore methods have to be used to observe that variable in the second method it is important that the trajectories inside the boundary layer do not try to come outside the boundary after entering the boundary layer control laws producing chattering free sliding mode using a boundary layer have been proposed and the existence of solutions to the system using these control laws are presented

multiple positive solutions for a class of concave convex elliptic problems in rn involving sign changing weight in this paper we study the multiplicity of positive solutions for the following concave convex elliptic equation u u f x uq 1 g x up 1in rn u 0in rn u h1 rn where 1 q 2 p 2 2 2nn 2 if n 3 2 if n 1 2 and the parameters 0 we assume that f x f x f x is sign changing and g x a x b x where the functions f a and b satisfy suitable conditions

estimation and inference in large heterogeneous panels with a multifactor error structure this paper presents a new approach to estimation and inference in panel data models with a general multifactor error structure the unobserved factors and the individual specific errors are allowed to follow arbitrary stationary processes and the number of unobserved factors need not be estimated the basic idea is to filter the individual specific regressors by means of cross section averages such that asymptotically as the cross section dimension n tends to infinity the differential effects of unobserved common factors are eliminated the estimation procedure has the advantage that it can be computed by least squares applied to auxiliary regressions where the observed regressors are augmented with cross sectional averages of the dependent variable and the individual specific regressors a number of estimators referred to as common correlated effects cce estimators are proposed and their asymptotic distributions are derived the small sample properties of mean group and pooled cce estimators are investigated by monte carlo experiments showing that the cce estimators have satisfactory small sample properties even under a substantial degree of heterogeneity and dynamics and for relatively small values of n and t

iv estimation of panels with factor residuals this paper considers panel data regression models with weakly exogenous or endogenous regressors and residuals generated by a multi factor error structure in this case the standard dynamic panel estimators fail to provide consistent estimates of the parameters we propose a new estimation approach based on instrumental variables which retains the traditional attractive features of method of moments estimators one novelty of our approach is that we introduce new parameters to represent the unobserved covariances between the instruments and the factor component of the residual these parameters are typically estimable when n is large some important estimation and identification issues are studied in detail the finite sample performance of the proposed estimators is investigated using simulated data the results show that the method produces reliable estimates of the parameters over various parametrizations and is robust to large values of the autoregressive parameter and or the variance of the factor loadings

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

theory and methods of panel data models with interactive effects this paper considers the maximum likelihood estimation of the panel data models with interactive effects motivated in economics and other social sciences a notable feature of the model is that the explanatory variables are correlated with the unobserved effects the usual within group estimator is inconsistent existing methods for consistent estimation are either designed for panel data with short time periods or are less efficient the maximum likelihood estimator has desirable properties and is easy to implement as illustrated by the monte carlo simulations this paper develops the inferential theory for the maximum likelihood estimator including consistency rate of convergence and the limiting distributions we further extend the model to include time invariant regressors and common regressors cross section invariant the regression coefficients for the time invariant regressors are time varying and the coefficients for the common regressors are cross sectionally varying

multiple positive solutions for indefinite semilinear elliptic problems involving a critical sobolev exponent in this paper we study the decomposition of the nehari manifold by exploiting the combination of concave and convex nonlinearities the result is subsequently used in conjunction with the ljusternik schnirelmann category and variational methods to prove the existence and multiplicity of positive solutions for an indefinite elliptic problem involving a critical sobolev exponent

common correlated effects estimation of heterogeneous dynamic panel data models with weakly exogenous regressors abstract this paper extends the common correlated effects cce approach developed by pesaran 2006 to heterogeneous panel data models with lagged dependent variables and or weakly exogenous regressors we show that the cce mean group estimator continues to be valid but the following two conditions must be satisfied to deal with the dynamics a sufficient number of lags of cross section averages must be included in individual equations of the panel and the number of cross section averages must be at least as large as the number of unobserved common factors we establish consistency rates derive the asymptotic distribution suggest using covariates to deal with the effects of multiple unobserved common factors and consider jackknife and recursive de meaning bias correction procedures to mitigate the small sample time series bias theoretical findings are accompanied by extensive monte carlo experiments which show that the proposed estimators perform well so long as the time series dimension of the panel is sufficiently large

multiplicity results for a non local problem with concave and convex nonlinearities abstract we shall prove a multiplicity result for a non local problem with a super critical nonlinearity of the form 1 s u u u p 2 u u q 2 in u 0 on r n where n is a bounded domain 0 s 1 and 1 q 2 p as a consequence of our results for each p 2 there exists 0 such that for each 0 problem 1 has a sequence of solutions with a negative energy this result is new for super critical values of p we shall also explore the existence of symmetric solutions for symmetric domains our methodology is based on a variational principle established by one of the authors that allows one to deal with super critical problems when the standard euler lagrange functional is restricted to certain convex sets

panel models with interactive effects abstract the multiplication of individual specific effects and time specific effects provides a more general formulation than the traditionally used additive form to capture the unobserved heterogeneity in panel data modeling it is also a useful approach for dimension reduction for modeling cross section dependence however and are unobservable we explore the implications for econometric modeling under various formulations of the interactive effects models and suggest a quasi likelihood approach as a common framework to study issues of estimation and statistical inference when regressors are either strictly exogenous or predetermined and under different combinations of the data size of cross sectional dimension n and time series dimensions t we also suggest some computationally simpler estimation methods in light of the quasi likelihood approach monte carlo studies are conducted to highlight the issues involved

iv gmm or likelihood approach to estimate dynamic panel models when either n or t or both are large abstract we examine the asymptotic properties of iv gmm or mle to estimate dynamic panel data models when either n o r t or both are large we show that the anderson and hsiao 1981 1982 simple instrumental variable estimator iv or maximizing the likelihood function with initial value distribution properly treated quasi maximum likelihood estimator is asymptotically unbiased when either n or t or both tend to infinity on the other hand the qmle mistreating the initial value as fixed is asymptotically unbiased only if n is fixed and t is large if both n and t are large and n t c c 0 c as t it is asymptotically biased of order n t we also explore the source of the bias of the arellano and bond 1991 type gmm estimator we show that it is asymptotically biased of order t n if t n c c 0 c as n even if we restrict the number of instruments used monte carlo studies show that whether an estimator is asymptotically biased or not has important implications on the actual size of the conventional t test

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

gmm estimation of linear panel data models with time varying individual effects abstract this paper considers models for panel data in which the individual effects vary over time the temporal pattern of variation is arbitrary but it is the same for all individuals the model thus allows one to control for time varying unobservables that are faced by all individuals e g macro economic events and to which individuals may respond differently a generalized within estimator is consistent under strong assumptions on the errors but it is dominated by a generalized method of moments estimator this is perhaps surprising because the generalized within estimator is the mle under normality the efficiency gains from imposing second moment error assumptions are evaluated they are substantial when the regressors and effects are weakly correlated

panel data models with interactive fixed effects this paper considers large n and large t panel data models with unobservable multiple interactive eects these models are useful for both micro and macro econometric modelings in earnings studies for example workers motivation persistence and diligence combined to influence the earnings in addition to the usual argument of innate ability in macroeconomics the interactive eects represent unobservable common shocks and their heterogeneous responses over cross sections since the interactive eects are allowed to be correlated with the regressors they are treated as fixed eects parameters to be estimated along with the common slope coecients the model is estimated by the least squares method which provides the interactive eect s counterpart of the within estimator we first consider model identification and then derive the rate of convergence and the limiting distribution of the interactive eect s estimator of the common slope coecients the estimator is shown to be p nt consistent this rate is valid even in the presence of correlations and heteroskedasticities in both dimensions a striking contrast with fixed t framework in which serial correlation and heteroskedasticity imply unidentification the asymptotic distribution is not necessarily centered at zero biased corrected estimators are derived we also derive the constrained estimator and its limiting distribution imposing additivity coupled with interactive eects the problem of testing additive versus interactive eects is also studied we also derive identification conditions for models with grand mean time invariant regressors and common regressors it is shown that there exists a set of necessary and sucient identification conditions for those models given identification the rate of convergence and limiting results continue to hold

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

on the evolution and qualitative behaviors of bifurcation curves for a boundary value problem abstract we study the evolution and qualitative behaviors of bifurcation curves of positive solutions for u x u 1 sin u u p 1 x 1 u 1 u 1 0 where 0 is a bifurcation parameter and p 1 is an evolution parameter on the u plane we prove that the bifurcation curve has exactly one turning point where the curve turns to the left for p 2 it is a monotone curve for p 2 it has at least two turning points for 1 p p 1 2 cos 1 sin 1 1 2 cos 1 sin 1 1 629 and it has infinitely many turning points for p 1 hence we are able to determine the exact number of positive solutions in particular we give complete descriptions of the structure of bifurcation curves when p 2 we also give some numerical simulations of bifurcation curves for p 1

efficient estimation of models for dynamic panel data abstract in this paper we consider a dynamic model for panel data we show that under standard assumptions there are more moment conditions than are currently exploited in the literature some of these are linear but others are quadratic so that nonlinear gmm is required we also show that exogenous regressors generate a larger number of relevant moment conditions in a dynamic model than they would in a static model

multiple positive solutions for a dirichlet problem involving critical sobolev exponent abstract in this paper we study the decomposition of the nehari manifold via the combination of concave and convex nonlinearities furthermore we use this result and the ljusternik schnirelmann category to prove that the existence of multiple positive solutions for a dirichlet problem involving critical sobolev exponent

finite sample approximation results for principal component analysis a matrix perturbation approach principal component analysis pca is a standard tool for dimensional reduction of a set of n observations samples each with p variables in this paper using a matrix perturbation approach we study the nonasymptotic relation between the eigenvalues and eigenvectors of pca computed on a finite sample of size n and those of the limiting population pca as n oo as in machine learning we present a finite sample theorem which holds with high probability for the closeness between the leading eigenvalue and eigenvector of sample pca and population pca under a spiked covariance model in addition we also consider the relation between finite sample pca and the asymptotic results in the joint limit p n with p n c we present a matrix perturbation view of the phase transition phenomenon and a simple linear algebra based derivation of the eigenvalue and eigenvector overlap in this asymptotic limit moreover our analysis also applies for finite p n where we show that although there is no sharp phase transition as in the infinite case either as a function of noise level or as a function of sample size n the eigenvector of sample pca may exhibit a sharp loss of tracking suddenly losing its relation to the true eigenvector of the population pca matrix this occurs due to a crossover between the eigenvalue due to the signal and the largest eigenvalue due to noise whose eigenvector points in a random direction

a complete classification of bifurcation diagrams of classes of multiparameter p laplacian boundary value problems we study the bifurcation diagrams of positive solutions of the multiparameter p laplacian problem p u x f u x 0 1 x 1 u 1 u 1 0 where p 1 p y y p 2y p u is the one dimensional p laplacian f u g u h u and 0 and 0 are two bifurcation parameters 0 and 0 are two given real numbers assuming that functions g and h satisfy hypotheses h1 h3 and h4a resp h1 h3 and h4b for fixed 0 resp 0 we give a classification of totally eight qualitatively different bifurcation diagrams we prove that on the u plane resp u plane each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the right hence the problem has at most two positive solutions for each 0 resp 0 more precisely we prove the exact multiplicity of positive solutions in addition for all p 1 we give interesting examples which show complete evolution of bifurcation diagrams as resp varies

asymptotics of the principal components estimator of large factor models with weakly influential factors abstract this paper introduces a drifting parameter asymptotic framework to derive accurate approximations to the finite sample distribution of the principal components pc estimator in situations when the factors explanatory power does not strongly dominate the explanatory power of the cross sectionally and temporally correlated idiosyncratic terms under our asymptotics the pc estimator is inconsistent we find explicit formulae for the amount of the inconsistency and propose an estimator of the number of factors for which the pc estimator works reasonably well for the special case when the idiosyncratic terms are cross sectionally but not temporally correlated or vice versa we show that the coefficients in the ols regressions of the pc estimates of factors loadings on the true factors true loadings are asymptotically normal and find explicit formulae for the corresponding asymptotic covariance matrix we explain how to estimate the parameters of the derived asymptotic distributions our monte carlo analysis suggests that our asymptotic formulae and estimators work well even for relatively small n and t we apply our theoretical results to test a hypothesis about the factor content of the us stock return data

arbitrage factor structure and mean variance analysis on large asset markets we examine the implications of arbitrage in a market with many assets the absence of arbitrage opportunities implies that the linear functionals that give the mean and cost of a portfolio are continuous hence there exist unique portfolios that represent these functionals these portfolios span the mean variance efficient set we resolve the question of when a market with many assets permits so much diversification that risk free investment opportunities are available ross 12 14 showed that if there is a factor structure then the mean returns are approximately linear functions of factor loadings we define an approximate factor structure and show that this weaker restriction is sufficient for ross result if the covariance matrix of the asset returns has only k unbounded eigenvalues then there is an approximate factor structure and it is unique the corresponding k eigenvectors converge and play the role of factor loadings hence only a principal component analysis is needed in empirical work

the structure of the solution set of a generalized ambrosetti brezis cerami problem in one space variable abstract we study the structure of solution set of the nonlinear two point boundary value problem u x f u x 0 1 x 1 u 1 u 1 0 where 0 is a bifurcation parameter and f u i 1 m a i u q i j 1 n b j u p j satisfies a1 a4 under a1 a4 we prove that there exists 0 such that the problem has exactly two positive solutions for 0 exactly one positive solution for and no positive solution for more precisely we give a complete description of the structure of the solution set

three positive solutions for dirichlet problems involving critical sobolev exponent and sign changing weight abstract in this paper we study the decomposition of the nehari manifold via the combination of concave and convex nonlinearities furthermore we use this result and the lusternik schnirelman category to prove that a semilinear elliptic equation involving a sign changing weight function has at least three positive solutions

four positive solutions for a semilinear elliptic equation involving concave and convex nonlinearities abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of positive solutions for a semilinear elliptic equation with the help of the nehari manifold and the center mass function we prove that there are at least four positive solutions for a semilinear elliptic equation in a finite strip with a hole

multiple solutions for neumann and periodic problems with singular ϕ laplacian abstract we use the critical point theory for convex lower semicontinuous perturbations of c 1 functionals to establish existence of multiple radial solutions for some one parameter neumann problems involving the operator v div v 1 v 2 similar results for periodic problems are also provided