generalized dynamic factor models and volatilities estimation and forecasting abstract in large panels of financial time series with dynamic factor structure on the levels or returns the volatilities of the common and idiosyncratic components often exhibit strong correlations indicating that both are exposed to the same market volatility shocks this suggests alongside the dynamic factor decomposition of returns a dynamic factor decomposition of volatilities or volatility proxies based on this observation barigozzi and hallin 2016 proposed an entirely non parametric and model free two step general dynamic factor approach which accounts for a joint factor structure of returns and volatilities and allows for extracting the market volatility shocks here we go one step further and show how the same two step approach naturally produces volatility forecasts for the various stocks under study in an applied exercise we consider the panel of asset returns of the constituents of the s p100 index over the period 2000 2009 numerical results show that the predictors based on our two step method outperform existing univariate and multivariate garch methods as well as static factor garch models in the prediction of daily high low range while avoiding the usual problems associated with the curse of dimensionality

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

one parameter groups of regular quasimultipliers abstract groups of unbounded operators are approached in the setting of the esterle quasimultiplier theory we introduce groups of regular quasimultipliers of growth or groups for short where is a continuous weight on the real line we study the relationship of groups with families of operators and homomorphisms such as regularized distribution and integrated groups holomorphic semigroups and functional calculi some convolution banach algebras of functions with derivatives to fractional order are needed which we construct using the weyl fractional calculus

induced i l fuzzy topological spaces abstract in this paper we introduce the notion of induced i l fuzzy topological spaces by using the i l valued lower semicontinuous mappings and investigate some of their basic properties in particular we give a necessary and sufficient condition for an i l fuzzy topological space to be induced and prove that this kind of induced space preserves the cartesian product and the n compactness as well

determining the number of factors in the general dynamic factor model this article develops an information criterion for determining the number q of common shocks in the general dynamic factor model developed by forni et al as opposed to the restricted dynamic model considered by bai and ng and by amengual and watson our criterion is based on the fact that this number q is also the number of diverging eigenvalues of the spectral density matrix of the observations as the number n of series goes to infinity we provide sufficient conditions for consistency of the criterion for large n and t where t is the series length we show how the method can be implemented and provide simulations and empirics illustrating its very good finite sample performance application to real data adds a new empirical facet to an ongoing debate on the number of factors driving the u s economy

the cauchy problem and a generalization of the hille yosida theorem on demontre une generalisation du theoreme de hille yosida en caracterisant les generateurs d une certaine classe de semi groupes d operateurs non bornes sur un espace de banach on resout le probleme de cauchy abstrait

generalized dynamic factor models and volatilities consistency rates and prediction intervals abstract volatilities in high dimensional panels of economic time series with a dynamic factor structure on the levels or returns typically also admit a dynamic factor decomposition we consider a two stage dynamic factor model method recovering the common and idiosyncratic components of both levels and log volatilities specifically in a first estimation step we extract the common and idiosyncratic shocks for the levels from which a log volatility proxy is computed in a second step we estimate a dynamic factor model which is equivalent to a multiplicative factor structure for volatilities for the log volatility panel by exploiting this two stage factor approach we build one step ahead conditional prediction intervals for large n t panels of returns those intervals are based on empirical quantiles not on conditional variances they can be either equal or unequal tailed we provide uniform consistency and consistency rates results for the proposed estimators as both n and t tend to infinity we study the finite sample properties of our estimators by means of monte carlo simulations finally we apply our methodology to a panel of asset returns belonging to the s p100 index in order to compute one step ahead conditional prediction intervals for the period 2006 2013 a comparison with the componentwise garch benchmark which does not take advantage of cross sectional information demonstrates the superiority of our approach which is genuinely multivariate and high dimensional nonparametric and model free

forecasting using principal components from a large number of predictors this article considers forecasting a single time series when there are many predictors n and time series observations t when the data follow an approximate factor model the predictors can be summarized by a small number of indexes which we estimate using principal components feasible forecasts are shown to be asymptotically efficient in the sense that the difference between the feasible forecasts and the infeasible forecasts constructed using the actual values of the factors converges in probability to 0 as both n and t grow large the estimated factors are shown to be consistent even in the presence of time variation in the factor model

abstract cauchy problems for quasi linear evolution equations in the sense of hadamard this paper is devoted to the well posedness of abstract cauchy problems for quasi linear evolution equations the notion of hadamard well posedness is considered and a new type of stability condition is introduced from the viewpoint of the theory of finite difference approximations the result obtained here generalizes not only some results on abstract cauchy problems closely related with the theory of integrated semigroups or regularized semigroups but also the kato theorem on quasi linear evolution equations an application to some quasi linear partial differential equation of weakly hyperbolic type is also given

limit of the smallest eigenvalue of a large dimensional sample covariance matrix in this paper the authors show that the smallest if p leq n or the p n 1 th smallest if p n eigenvalue of a sample covariance matrix of the form 1 n xx tends almost surely to the limit 1 sqrt y 2 as n rightarrow infty and p n rightarrow y in 0 infty where x is a p times n matrix with iid entries with mean zero variance 1 and fourth moment finite also as a by product it is shown that the almost sure limit of the largest eigenvalue is 1 sqrt y 2 a known result obtained by yin bai and krishnaiah the present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices

risks of large portfolios the risk of a large portfolio is often estimated by substituting a good estimator of the volatility matrix however the accuracy of such a risk estimator is largely unknown we study factor based risk estimators under a large amount of assets and introduce a high confidence level upper bound h club to assess the estimation the h club is constructed using the confidence interval of risk estimators with either known or unknown factors we derive the limiting distribution of the estimated risks in high dimensionality we find that when the dimension is large the factor based risk estimators have the same asymptotic variance no matter whether the factors are known or not which is slightly smaller than that of the sample covariance based estimator numerically h club outperforms the traditional crude bounds and provides an insightful risk assessment in addition our simulated results quantify the relative error in the risk estimation which is usually negligible using 3 month daily data

convex hypersurfaces and lp estimates for schrodinger equations abstract this paper is concerned with schrodinger equations whose principal operators are homogeneous elliptic when the corresponding level hypersurface is convex we show the l p l q estimate of the solution operator in the free case this estimate combined with the results of fractionally integrated groups allows us to further obtain the l p estimate of solutions for the initial data belonging to a dense subset of l p in the case of integrable potentials

high dimensional covariance matrix estimation using a factor model abstract high dimensionality comparable to sample size is common in many statistical problems we examine covariance matrix estimation in the asymptotic framework that the dimensionality p tends to as the sample size n increases motivated by the arbitrage pricing theory in finance a multi factor model is employed to reduce dimensionality and to estimate the covariance matrix the factors are observable and the number of factors k is allowed to grow with p we investigate the impact of p and k on the performance of the model based covariance matrix estimator under mild assumptions we have established convergence rates and asymptotic normality of the model based estimator its performance is compared with that of the sample covariance matrix we identify situations under which the factor approach increases performance substantially or marginally the impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied the asymptotic results are supported by a thorough simulation study

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

projected principal component analysis in factor models this paper introduces a projected principal component analysis projected pca which employees principal component analysis to the projected smoothed data matrix onto a given linear space spanned by covariates when it applies to high dimensional factor analysis the projection removes noise components we show that the unobserved latent factors can be more accurately estimated than the conventional pca if the projection is genuine or more precisely when the factor loading matrices are related to the projected linear space when the dimensionality is large the factors can be estimated accurately even when the sample size is finite we propose a flexible semi parametric factor model which decomposes the factor loading matrix into the component that can be explained by subject specific covariates and the orthogonal residual component the covariates effects on the factor loadings are further modeled by the additive model via sieve approximations by using the newly proposed projected pca the rates of convergence of the smooth factor loading matrices are obtained which are much faster than those of the conventional factor analysis the convergence is achieved even when the sample size is finite and is particularly appealing in the high dimension low sample size situation this leads us to developing nonparametric tests on whether observed covariates have explaining powers on the loadings and whether they fully explain the loadings the proposed method is illustrated by both simulated data and the returns of the components of the s p 500 index

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 rank test for the number of factors with high frequency data abstract in the literature consistency of the estimates of the number of factors for large dimensional factor models had been extensively studied recently but the second order property of the estimator has long been unsolved due to lack of limiting distribution of the estimators in this paper we propose a rank test of the number of factors using large panel high frequency data contaminated with microstructure noise the rank test is realized by forming a fixed number of portfolios which reduce the dimension to a finite number in the process of constructing portfolios the number of factors is equal to the rank of the volatility matrix of the diversified portfolios asymptotically via estimating the volatility rank of a low dimensional price dynamics of the portfolios we establish a central limit theorem of the estimated factor number we then apply the asymptotic normality to testing on the number of factors numerical experiments including the monte carlo simulations and real data analysis justify our theory

regularization of wavelet approximations in this paper we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced the approach can apply readily to many other statistical contexts various new penalty functions are proposed the hard thresholding and soft thresholding estimators of donoho and johnstone are specific members of nonlinear regularized wavelet estimators they correspond to the lower and upper envelopes of a class of the penalized least squares estimators necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions the sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax to efficiently solve penalized least squares problems nonlinear regularized sobolev interpolators nrsi are proposed as initial estimators wh

some characterization theorems on induced i l topological spaces in this paper we give some necessary and sufficient conditions for an i l topological space to be a stratified space a weakly induced space or an induced space by interior and closure operators and by the convergence of molecular nets

quasi maximum likelihood analysis of high dimensional constrained factor models abstract factor models have been widely used in practice however an undesirable feature of a high dimensional factor model is that the model has too many parameters an effective way to address this issue proposed in a seminar work by tsai and tsay 2010 is to decompose the loadings matrix by a high dimensional known matrix multiplying with a low dimensional unknown matrix which tsai and tsay 2010 name the constrained factor models this paper investigates the estimation and inferential theory of constrained factor models under large n and large t setup where n denotes the number of cross sectional units and t the time periods we propose using the quasi maximum likelihood method to estimate the model and investigate the asymptotic properties of the quasi maximum likelihood estimators including consistency rates of convergence and limiting distributions a new statistic is proposed for testing the null hypothesis of constrained factor models against the alternative of standard factor models partially constrained factor models are also investigated monte carlo simulations confirm our theoretical results and show that the quasi maximum likelihood estimators and the proposed new statistic perform well in finite samples we also consider the extension to an approximate constrained factor model where the idiosyncratic errors are allowed to be weakly dependent processes

eigenvalue ratio test for the number of factors this paper proposes two new estimators for determining the number of factors r in static approximate factor models we exploit the well known fact that the r largest eigenvalues of the variance matrix of n response variables grow unboundedly as n increases while the other eigenvalues remain bounded the new estimators are obtained simply by maximizing the ratio of two adjacent eigenvalues our simulation results provide promising evidence for the two estimators

robust covariance estimation for approximate factor models in this paper we study robust covariance estimation under the approximate factor model with observed factors we propose a novel framework to first estimate the initial joint covariance matrix of the observed data and the factors and then use it to recover the covariance matrix of the observed data we prove that once the initial matrix estimator is good enough to maintain the element wise optimal rate the whole procedure will generate an estimated covariance with desired properties for data with only bounded fourth moment we propose to use adaptive huber loss minimization to give the initial joint covariance estimation this approach is applicable to a much wider class of distributions beyond sub gaussian and elliptical distributions we also present an asymptotic result for adaptive huber s m estimator with a diverging parameter the conclusions are demonstrated by extensive simulations and real data analysis

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

some characterization theorems on lattice valued induced spaces abstract in this paper we systematically discuss topological and algebraic characteristics on induced spaces and the related weakly induced spaces and fully stratified spaces

separation properties of induced i l topological spaces the concept of induced i l topological spaces has been introduced by kubiak ph d thesis uam poznan 1985 and independently by wang kexue tongbao 34 5 1989 333 in this paper the separation properties in the sense of hutton reilly of induced i l topological spaces are investigated the main result of the paper is a characterization of l topological spaces by means of the appropriate hutton reilly separation properties of its induced i l topological space

nonlocal problems for boussinesq equations abstract in this paper the existence and uniqueness of solution of the integral boundary value problem for abstract boussinesq equations are obtained

paracompactness on l fuzzy topological spaces abstract in this paper we introduce a new kind of paracompactness called paracompactness on l fuzzy topological spaces and obtain some of its properties such as that it is a good extension hereditary with respect to closed subsets and that the cartesian product of an n compaact space and a paracompact space is a paracompact space as well as that a paracompact hausdorff space is both a regular and normal space

estimating number of factors by adjusted eigenvalues thresholding determining the number of common factors is an important and practical topic in high dimensional factor models the existing literature is mainly based on the eigenvalues of the covariance matrix

large dimensional factor modeling based on high frequency observations abstract this paper develops a statistical theory to estimate an unknown factor structure based on financial high frequency data we derive an estimator for the number of factors and consistent and asymptotically mixed normal estimators of the loadings and factors under the assumption of a large number of cross sectional and high frequency observations the estimation approach can separate factors for continuous and rare jump risk the estimators for the loadings and factors are based on the principal component analysis of the quadratic covariation matrix the estimator for the number of factors uses a perturbed eigenvalue ratio statistic in an empirical analysis of the s p 500 firms we estimate four stable continuous systematic factors which can be approximated very well by a market and industry portfolios jump factors are different from the continuous factors

exponentially bounded c semigroups and generation of semigroups this paper concerns exponentially bounded c semigroups and semigroups of operators in a banach space x recently davies and pang l introduced the notion of an exponentially bounded c semigroup and characterized the generator of an exponentially bounded c semigroup on the other hand miyadera 7 discussed the complete infinitesimal generator of an exponentially bounded c semigroup and tanaka 14 has characterized the complete infinitesimal generator of an exponentially bounded c semigroup there are many important works on semigroups of linear operators for example see 4 5 12 161 extending the notion of c semigroups oharu 9 introduced the semigroups of classes cc k 0 1 2 and da prato 2 and sobolevskii 13 introduced the semigroups of growth order c 2 0 generation theorems for semigroups of these classes have been studied in 2 9 10 131 our first purpose is to clarify the relationship between semigroups and exponentially bounded c semigroups it is shown that if t t t 2 0 is a semigroup and c is an injective bounded linear operator with dense range satisfying r t c ct t for t 0 and r c c c xex lim r o t r x x then ct t t 2 0 is an exponentially bounded c semigroup whose complete infinitesimal generator is equal to that of t t t 2 0 in particular if r t t k 0 is a semigroup of class c or of growth order 1 then ct t t 2 o becomes an exponentially bounded c semigroup with a suitable c and its complete infinitesimal generator coincides with that of t t t 2 o our second and main purpose is to study the generation of semigroups via the theory of exponentially bounded c semigroups we shall characterize the complete infinitesimal generator of a semigroup r t t 2 0

a diagnostic criterion for approximate factor structure abstract we build a simple diagnostic criterion for approximate factor structure in large panel datasets given observable factors the criterion checks whether the errors are weakly cross sectionally correlated or share at least one unobservable common factor interactive effects a general version allows to determine the number of omitted common factors also for time varying structures the empirical analysis runs on ten thousand us stocks from january 1968 to december 2011 for monthly returns we select time invariant specifications with at least four financial factors and a scaled three factor specification for quarterly returns we cannot select macroeconomic models without the market factor

on factor models with random missing em estimation inference and cross validation abstract we consider the estimation and inference in approximate factor models with random missing values we show that with the low rank structure of the common component we can estimate the factors and factor loadings consistently with the missing values replaced by zeros we establish the asymptotic distributions of the resulting estimators and those based on the em algorithm we also propose a cross validation based method to determine the number of factors in factor models with or without missing values and justify its consistency simulations demonstrate that our cross validation method is robust to fat tails in the error distribution and significantly outperforms some existing popular methods in terms of correct percentage in determining the number of factors an application to the factor augmented regression models shows that a proper treatment of the missing values can improve the out of sample forecast of some macroeconomic variables

high dimensional minimum variance portfolio estimation under statistical factor models abstract we propose a high dimensional minimum variance portfolio estimator under statistical factor models and show that our estimated portfolio enjoys sharp risk consistency our approach relies on properly integrating l 1 constraint on portfolio weights with an appropriate covariance matrix estimator in terms of covariance matrix estimation we extend the theoretical results of poet fan et al 2013 to a setting that is coherent with principal component analysis simulation and extensive empirical studies on s p 100 index constituent stocks demonstrate favorable performance of our mvp estimator compared with benchmark portfolios

distribution semigroups and abstract cauchy problems we present a new definition of distribution semigroups covering in particular non densely defined generators we show that for a closed operator a in a banach space e the following assertions are equivalent a a generates a distribution semigroup b the convolution operator i a has a fundamental solution in d l e d where d denotes the domain of a supplied with the graph norm and i denotes the inclusion d e c a generates a local integrated semigroup we also show that every generator of a distribution semigroup generates a regularized semigroup introduction distribution semigroups in banach spaces and their generators were introduced by j l lions in 16 as a generalization of c0 semigroups and their generators like the generators of c0 semigroups the generators of the distribution semigroups in 16 are densely defined it was then shown in 16 that the densely defined closed linear operators a that generate a distribution semigroup are exactly those for which the convolution operator i a where i denotes the inclusion d a e has a fundamental solution treating cauchy problems as convolution equations has turned out to be very useful see 6 12 13 and 14 but in doing so the assumption that d a is dense in e is unnecessary in most of the results on the other hand in the last years there has been an interest in dealing with operators that are not densely defined starting perhaps with 7 reasons for this interest may be that it might not be so easy to decide whether a given operator is densely defined and the difficulties that arise when passing from e to e or l e so the question how the theory of distribution semigroups introduced in 16 can be extended to cover non densely defined generators in such a way that the above characterization via fundamental solutions holds seems natural it turns out that this indeed is possible in a way that even simplifies the original definitions given in 16 moreover the local strongly continuous representations see section 4 of the distribution semigroups thus obtained are exactly the local integrated semigroups for the distribution semigroups in the sense of 16 a relation to local integrated semigroups has been obtained in 23 theorem 5 6 see also 15 theorem 7 6 or 2 theorem 7 2 so the theory presented here received by the editors october 17 1995 and in revised form february 6 1997 1991 mathematics subject classification primary 47d03 34g10 47a10 46f10 c 1999 american mathematical society

high dimensional covariance matrix estimation in approximate factor models the variance covariance matrix plays a central role in the inferential theories of high dimensional factor models in finance and economics popular regularization methods of directly exploiting sparsity are not directly applicable to many financial problems classical methods of estimating the covariance matrices are based on the strict factor models assuming independent idiosyncratic components this assumption however is restrictive in practical applications by assuming sparse error covariance matrix we allow the presence of the cross sectional correlation even after taking out common factors and it enables us to combine the merits of both methods we estimate the sparse covariance using the adaptive thresholding technique as in cai and liu j amer statist assoc 106 2011 672 684 taking into account the fact that direct observations of the idiosyncratic components are unavailable the impact of high dimensionality on the covariance matrix estimation based on the factor structure is then studied

multiplicative perturbations of c regularized semigroups we present in this note two new theorems for multiplicative perturbations of c regularized semigroups which generalize the previous related ones for c0 semigroups

adaptive thresholding for sparse covariance matrix estimation in this article we consider estimation of sparse covariance matrices and propose a thresholding procedure that is adaptive to the variability of individual entries the estimators are fully data driven and demonstrate excellent performance both theoretically and numerically it is shown that the estimators adaptively achieve the optimal rate of convergence over a large class of sparse covariance matrices under the spectral norm in contrast the commonly used universal thresholding estimators are shown to be suboptimal over the same parameter spaces support recovery is discussed as well the adaptive thresholding estimators are easy to implement the numerical performance of the estimators is studied using both simulated and real data simulation results demonstrate that the adaptive thresholding estimators uniformly outperform the universal thresholding estimators the method is also illustrated in an analysis on a dataset from a small round blue cell tumor microarray experiment a supplement to this ar

embracing the blessing of dimensionality in factor models factor modeling is an essential tool for exploring intrinsic dependence structures among high dimensional random variables much progress has been made for estimating the covariance matrix from a high dimensional factor model however the blessing of dimensionality has not yet been fully embraced in the literature much of the available data are often ignored in constructing covariance matrix estimates if our goal is to accurately estimate a covariance matrix of a set of targeted variables shall we employ additional data which are beyond the variables of interest in the estimation in this article we provide sufficient conditions for an affirmative answer and further quantify its gain in terms of fisher information and convergence rate in fact even an oracle like result as if all the factors were known can be achieved when a sufficiently large number of variables is used the idea of using data as much as possible brings computational challenges a divide and conquer algorithm is thus proposed to alleviate the computational burden and also shown not to sacrifice any statistical accuracy in comparison with a pooled analysis simulation studies further confirm our advocacy for the use of full data and demonstrate the effectiveness of the above algorithm our proposal is applied to a microarray data example that shows empirical benefits of using more data supplementary materials for this article are available online

efficient estimation of heterogeneous coefficients in panel data models with common shocks this paper investigates efficient estimation of heterogeneous coefficients in panel data models with common shocks which have been a particular focus of recent theoretical and empirical literature we propose a new two step method to estimate the heterogeneous coefficients in the first step the maximum likelihood ml method is first conducted to estimate the loadings and idiosyncratic variances the second step estimates the heterogeneous coefficients by using the structural relations implied by the model and replacing the unknown parameters with their ml estimates we establish the asymptotic theory of our estimator including consistency asymptotic representation and limiting distribution the two step estimator is asymptotically efficient in the sense that it has the same limiting distribution as the infeasible generalized least squares gls estimator intensive monte carlo simulations show that the proposed estimator performs robustly in a variety of data setups

augmented factor models with applications to validating market risk factors and forecasting bond risk premia abstract we study factor models augmented by observed covariates that have explanatory powers on the unknown factors in financial factor models the unknown factors can be reasonably well explained by a few observable proxies such as the fama french factors in diffusion index forecasts identified factors are strongly related to several directly measurable economic variables such as consumption wealth variable financial ratios and term spread with those covariates both the factors and loadings are identifiable up to a rotation matrix even only with a finite dimension to incorporate the explanatory power of these covariates we propose a smoothed or projected principal component analysis pca i regress the data onto the observed covariates and ii take the principal components of the fitted data to estimate the loadings and factors this allows us to more accurately estimate the percentage of both explained and unexplained components in factors and thus to assess the explanatory power of covariates we show that both the estimated factors and loadings can be estimated with improved rates of convergence compared to the benchmark method the degree of improvement depends on the strength of the signals representing the explanatory power of the covariates on the factors the proposed estimator is robust to possibly heavy tailed distributions we apply the model to forecast us bond risk premia and find that the observed macroeconomic characteristics contain strong explanatory powers of the factors the gain of forecast is more substantial when the characteristics are incorporated to estimate the common factors than directly used for forecasts

expandable factor analysis bayesian sparse factor models have proven useful for characterizing dependence in multivariate data but scaling computation to large numbers of samples and dimensions is problematic we propose expandable factor analysis for scalable inference in factor models when the number of factors is unknown the method relies on a continuous shrinkage prior for efficient maximum a posteriori estimation of a low rank and sparse loadings matrix the structure of the prior leads to an estimation algorithm that accommodates uncertainty in the number of factors we propose an information criterion to select the hyperparameters of the prior expandable factor analysis has better false discovery rates and true positive rates than its competitors across diverse simulation settings we apply the proposed approach to a gene expression study of ageing in mice demonstrating superior results relative to four competing methods

exact separation of eigenvalues of large dimensional sample covariance matrices let b n 1 n t n 1 2 x n x n t n 1 2 where x n is n times n with i i d complex standardized entries having finite fourth moment and t n 1 2 is a hermitian square root of the nonnegative definite hermitian matrix t n it was shown in an earlier paper by the authors that under certain conditions on the eigenvalues of t n with probability 1 no eigenvalues lie in any interval which is outside the support of the limiting empirical distribution known to exist for all large n for these n the interval corresponds to one that separates the eigenvalues of t n the aim of the present paper is to prove exact separation of eigenvalues that is with probability 1 the number of eigenvalues of b n and t n lying on one side of their respective intervals are identical for all large n

asymptotic analysis of the squared estimation error in misspecified factor models abstract in this paper we obtain asymptotic approximations to the squared error of the least squares estimator of the common component in large approximate factor models with possibly misspecified number of factors the approximations are derived under both strong and weak factors asymptotics assuming that the cross sectional and temporal dimensions of the data are comparable we develop consistent estimators of these approximations and propose to use them for model comparison and for selection of the number of factors we show that the estimators of the number of factors that minimize these loss estimators are asymptotically loss efficient in the sense of shibata 1980 li 1987 and shao 1997

l fuzzy normal spaces and tietze extension theorem abstract in this paper we prove a characterization theorem for normal l fuzzy topological spaces l is an infinitely distributive lattice with an order reversing involution in the particular case l 0 1 this theorem reduces to a known result of kat tov fund math 38 1951 85 91 and tong duke math j 19 1952 289 292 as an important application we obtain a fuzzy version of tietze extension theorem this yields an affirmative answer to a question raised in a series of papers by rodabaugh fuzzy sets and systems 11 1983 163 183

hp hq estimates for dispersive equations and related applications abstract this paper studies the h p h q estimates of the solutions for a class of dispersive equations under the assumption that principal operators are homogeneous elliptic and the corresponding level hypersurfaces are convex and of finite type these estimates allow us to show higher order schrodinger operator generates a fractionally integrated group in l p r n

estimation of large dimensional factor models with an unknown number of breaks abstract in this paper we study the estimation of a large dimensional factor model when the factor loadings exhibit an unknown number of changes over time we propose a novel three step procedure to detect the breaks if any and then identify their locations in the first step we divide the whole time span into subintervals and fit a conventional factor model on each interval in the second step we apply the adaptive fused group lasso to identify intervals containing a break in the third step we devise a grid search method to estimate the location of the break on each identified interval we show that with probability approaching one our method can identify the correct number of changes and estimate the break locations simulation studies indicate superb finite sample performance of our method we apply our method to investigate stock and watson s 2009 u s monthly macroeconomic dataset and identify five breaks in the factor loadings spanning 1959 2006

lp lq estimates for dispersive equations and related applications abstract this paper is mainly concerned with the l p l q estimates of solutions for a class of general dispersive equations under the condition h b with an index b 0 1 to the end the new pointwise decay estimates of oscillatory integrals related to the fundamental solutions are proved if b 1 then the pointwise estimates are particularly consistent with the sharp results of the nondegenerate cases moreover as an application of the l p l q estimates we also show that higher order differential operator i p d v x d generates a fractionally integrated group on some l p r n from which certain l p estimates for the solutions of generalized schrodinger equations are obtained

incomplete iterated cauchy problems abstract we treat numerous higher order incomplete abstract cauchy problems by writing them in the form 1 1 k 1 n d dt a k u t 0 t 0 2 2 ui 1 0 xi 1 i j 3 3a ui 1 t bounded 1 i n j or 4 3b limt u i 1 t 0 1 i n j the abstract cauchy problem 1 2 and 3a is wellposed in a certain sense when a1 aj generate bounded c semigroups and aj 1 an generate uniformly stable c semigroups for 1 2 and 3b we need a1 aj and aj 1 an to generate stable c semigroups

inferential theory for factor models of large dimensions this paper develops an inferential theory for factor models of large dimensions the principal components estimator is considered because it is easy to compute and is asymptotically equivalent to the maximum likelihood estimator if normality is assumed we derive the rate of convergence and the limiting distributions of the estimated factors factor loadings and common components the theory is developed within the framework of large cross sections n and a large time dimension t to which classical factor analysis does not apply we show that the estimated common components are asymptotically normal with a convergence rate equal to the minimum of the square roots of n and t the estimated factors and their loadings are generally normal although not always so the convergence rate of the estimated factors and factor loadings can be faster than that of the estimated common components these results are obtained under general conditions that allow for correlations and heteroskedasticities in both dimensions stronger results are obtained when the idiosyncratic errors are serially uncorrelated and homoskedastic a necessary and sufficient condition for consistency is derived for large n but fixed t

c semigroups and the abstract cauchy problem where by a solution u x we mean that u x e c 0 co x u t x e d u for t 2 0 and acp u x is satisfied it is well known that the acp is closely related to semigroup theory and covers a number of problems of interest to the applications see s 6 9 141 recently davies and pang 2 introduced exponentially bounded c semigroups and studied their connection with the acp other relating results are in tanaka ll later delaubenfels 3 investigated c semigroups that may not be exponentially bounded and their connection with the acp roughly speaking the results in 2 3 111 are as follows

simultaneous multiple change point and factor analysis for high dimensional time series abstract we propose the first comprehensive treatment of high dimensional time series factor models with multiple change points in their second order structure we operate under the most flexible definition of piecewise stationarity and estimate the number and locations of change points consistently as well as identifying whether they originate in the common or idiosyncratic components through the use of wavelets we transform the problem of change point detection in the second order structure of a high dimensional time series into the relatively easier problem of change point detection in the means of high dimensional panel data also our methodology circumvents the difficult issue of the accurate estimation of the true number of factors in the presence of multiple change points by adopting a screening procedure we further show that consistent factor analysis is achieved over each segment defined by the change points estimated by the proposed methodology in extensive simulation studies we observe that factor analysis prior to change point detection improves the detectability of change points and identify and describe an interesting spillover effect in which substantial breaks in the idiosyncratic components get naturally enough identified as change points in the common components which prompts us to regard the corresponding change points as also acting as a form of factors our methodology is implemented in the r package factorcpt available from cran

generalized thresholding of large covariance matrices we propose a new class of generalized thresholding operators that combine thresholding with shrinkage and study generalized thresholding of the sample covariance matrix in high dimensions generalized thresholding of the covariance matrix has good theoretical properties and carries almost no computational burden we obtain an explicit convergence rate in the operator norm that shows the tradeoff between the sparsity of the true model dimension and the sample size and shows that generalized thresholding is consistent over a large class of models as long as the dimension p and the sample size n satisfy log p n 0 in addition we show that generalized thresholding has the sparsistency property meaning it estimates true zeros as zeros with probability tending to 1 and under an additional mild condition is sign consistent for nonzero elements we show that generalized thresholding covers as special cases hard and soft thresholding smoothly clipped absolute deviation and adaptive lasso and comp

deterministic parallel analysis an improved method for selecting factors and principal components factor analysis and principal component analysis are used in many application areas the first step choosing the number of components remains a serious challenge our work proposes improved methods for this important problem one of the most popular state of the art methods is parallel analysis pa which compares the observed factor strengths with simulated strengths under a noise only model the paper proposes improvements to pa we first derandomize it proposing deterministic pa which is faster and more reproducible than pa both pa and deterministic pa are prone to a shadowing phenomenon in which a strong factor makes it difficult to detect smaller but more interesting factors we propose deflation to counter shadowing we also propose to raise the decision threshold to improve estimation accuracy we prove several consistency results for our methods and test them in simulations we also illustrate our methods on data from the human genome diversity project where they significantly improve the accuracy

a bernstein type inequality and moderate deviations for weakly dependent sequences in this paper we present a bernstein type tail inequality for the maximum of partial sums of a weakly dependent sequence of random variables that is not necessarily bounded the class considered includes geometrically and subgeometrically strongly mixing sequences the result is then used to derive asymptotic moderate deviation results applications are given for classes of markov chains iterated lipschitz models and functions of linear processes with absolutely regular innovations

perturbation of local c semigroups abstract let c be a bounded linear injective operator and a be the generator of a local c semigroup s t 0 t t on a banach space x it is proved that if b is a bounded operator such that r b r c and bcx cbx for all x d a then a b also generates a local c semigroup

on compactness of induced i l fuzzy topological spaces abstract the induced i l fuzzy topological spaces for l fuzzy topological spaces introduced by wang geping is a kind of important fuzzy topological space in this paper the author studies the fuzzy compactness of induced i l fuzzy topological spaces some available relations between molecular nets of an l fuzzy topological space and that of its induced i l fuzzy topological space are firstly presented by using these relations the author obtains that the induced i l fuzzy topological space is strong fuzzy compact resp fuzzy compact n compact if and only if the l fuzzy topological space is strong fuzzy compact resp fuzzy compact n compact the author also proves that the induced i l fuzzy topological space is paracompact if and only if the l fuzzy topological space is paracompact

sufficient forecasting using factor models we consider forecasting a single time series when there is a large number of predictors and a possible nonlinear effect the dimensionality was first reduced via a high dimensional approximate factor model implemented by the principal component analysis using the extracted factors we develop a novel forecasting method called the sufficient forecasting which provides a set of sufficient predictive indices inferred from high dimensional predictors to deliver additional predictive power the projected principal component analysis will be employed to enhance the accuracy of inferred factors when a semi parametric approximate factor model is assumed our method is also applicable to cross sectional sufficient regression using extracted factors the connection between the sufficient forecasting and the deep learning architecture is explicitly stated the sufficient forecasting correctly estimates projection indices of the underlying factors even in the presence of a nonparametric forecasting function the proposed method extends the sufficient dimension reduction to high dimensional regimes by condensing the cross sectional information through factor models we derive asymptotic properties for the estimate of the central subspace spanned by these projection directions as well as the estimates of the sufficient predictive indices we further show that the natural method of running multiple regression of target on estimated factors yields a linear estimate that actually falls into this central subspace our method and theory allow the number of predictors to be larger than the number of observations we finally demonstrate that the sufficient forecasting improves upon the linear forecasting in both simulation studies and an empirical study of forecasting macroeconomic variables

high dimensionality effects in the markowitz problem and other quadratic programs with linear constraints risk underestimation we study the properties of solutions of quadratic programs with linear equality constraints whose parameters are estimated from data in the high dimensional setting where p the number of variables in the problem is of the same order of magnitude as n the number of observations used to estimate the parameters the markowitz problem in finance is a subcase of our study assuming normality and independence of the observations we relate the ecient frontier computed empirically to the true ecient frontier our computations show that there is a separation of the errors induced by estimating the mean of the observations and estimating the covariance matrix in particular the price paid for estimating the covariance matrix is an underestimation of the variance by a factor roughly equal to 1 p n therefore the risk of the optimal population solution is underestimated when we estimate it by solving a similar quadratic program with estimated parameters we also characterize the statistical behavior of linear functionals of the empirical optimal vector and show that they are biased estimators of the corresponding population quantities we investigate the robustness of our gaussian results by extending the study to certain elliptical models and models where our n observations are correlated in time we show a lack of robustness of the gaussian results but are still able to get results concerning rst order properties of the quantities of interest even in the case of relatively heavy tailed data we require two moments risk underestimation is still present in the elliptical case and more pronounced that in the gaussian case we discuss properties of the non parametric and parametric bootstrap in this context we show several results including the interesting fact that standard applications of the bootstrap generally yields inconsistent estimates of bias finally we propose some strategies to correct these problems and practically validate them in some simulations in all the paper we will assume that p n and n p tend to innity and p n

high dimensional volatility matrix estimation via waveletsand thresholding we propose a locally stationary linear model for the evolution of high dimensional financial returns where the time varying volatility matrix is modelled as a piecewise constant function of time we introduce a new wavelet based technique for estimating the volatility matrix which 10 combines four ingredients a haar wavelet decomposition variance stabilization of the haar coefficients via the fisz transform prior to thresholding a bias correction and extra time domain thresholding soft or hard under the assumption of sparsity we demonstrate the interval wise consistency of the proposed estimators of the volatility matrix and its inverse in the operator norm with rates which adapt to the features of the target matrix we also propose a version of 15 the estimators based on the polarization identity which permits a more precise derivation of the thresholds we discuss the practicalities of the algorithm including parameter selection and how to perform it online a simulation study shows the benefits of the method which is illustrated using a stock index portfolio

matrices of operators and regularized semigroups where p is a matr ix of polynomials in n variables and ia1 ia is a family of c o m m u t i n g generators of bounded linear strongly cont inuous groups of operators intuitively one wants to take the exponent ia l of the matr ix of opera to r s p a since the spec t rum of ak is real for all k this means we want the complex matr ix pi j x to be contained in a left half plane in some sense for all x in r the strongest sense is to have the numerical range of the complex matr ix conta ined in a half plane r e z 0 xer a sense that is easier to calculate or at least estimate is to have the spectrum of p d x conta ined in a left half plane for all x r the system 1 1 is then said to be petrowsky correct see 9 the intuit ion we just described is not quite true in general wi thout some