estimation of the false discovery proportion with unknown dependence summary large scale multiple testing with correlated test statistics arises frequently in much scientific research incorporating correlation information in approximating the false discovery proportion fdp has attracted increasing attention in recent years when the covariance matrix of test statistics is known fan and his colleagues provided an accurate approximation of the fdp under arbitrary dependence structure and some sparsity assumption however the covariance matrix is often unknown in many applications and such dependence information must be estimated before approximating the fdp the estimation accuracy can greatly affect the fdp approximation in the current paper we study theoretically the effect of unknown dependence on the testing procedure and establish a general framework such that the fdp can be well approximated the effects of unknown dependence on approximating the fdp are in the following two major aspects through estimating eigenvalues or eigenvectors and through estimating marginal variances to address the challenges in these two aspects we firstly develop general requirements on estimates of eigenvalues and eigenvectors for a good approximation of the fdp we then give conditions on the structures of covariance matrices that satisfy such requirements such dependence structures include banded or sparse covariance matrices and conditional sparse precision matrices within this framework we also consider a special example to illustrate our method where data are sampled from an approximate factor model which encompasses most practical situations we provide a good approximation of the fdp via exploiting this specific dependence structure the results are further generalized to the situation where the multivariate normality assumption is relaxed our results are demonstrated by simulation studies and some real data applications

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

testing for factor loading structural change under common breaks abstract this paper proposes a new test for factor loading structural change in dynamic factor models we first show that the leading test proposed by breitung and eickmeier 2011 exhibits nonmonotonic power essentially because the breaks are considered as spurious factors with stable factor loadings the proposed test eliminates the effects of the spurious factors by maximizing the test statistic over possible numbers of the original factors monte carlo simulations and an empirical example using u s treasury yield curve data clearly illustrate the usefulness of the proposed test

determining the number of factors when the number of factors can increase with sample size abstract correctly specifying the number of factors r is a fundamental issue for the application of factor models in this paper we develop an econometric method to estimate the number of factors in factor models of large dimensions where the number of factors is allowed to increase as the two dimensions cross section size n and time period t increase using similar information criteria as proposed by bai and ng 2002 we show that the number of factors can be consistently estimated using the criteria we propose a new procedure that avoids over estimating the number of factors while allowing for one to search for possible number of factors over a wide range of positive integers so that it also avoids underestimation of the number of factors we conduct monte carlo simulation to investigate the finite sample properties of the proposed approach

estimation of low rank high dimensional multivariate linear models for multi response data in this article we study low rank high dimensional multivariate linear models lrmlm for high dimensional multi response data we propose an intuitively appealing estimation approach and develop

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

on the number of common factors with high frequency data summaryin this paper we introduce a local principal component analysis approach to determining the number of common factors of a continuous time factor model with time varying factor loadings using high frequency data the model is approximated locally on shrinking blocks using discrete time factor models the number of common factors is estimated by minimizing the penalized aggregated mean squared residual error over all shrinking blocks while the local mean squared residual error on each block converges at rate min n 1 4 p where n is the sample size and p is the dimension the aggregated mean squared residual error converges at rate min n 1 2 p this achieves the convergence rate of the penalized criterion function of the global principal component analysis method assuming restrictive constant factor loading an estimator of the number of factors based on the local principal component analysis is consistent simulation results justify the performance of our estimator a real financial dataset is analysed

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

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

blended implicit methods for the numerical solution of dae problems recently a new approach for solving the discrete problems generated by the application of block implicit methods for the numerical solution of initial value problems for odes has been devised l brugnano blended block bvms b 3vms a family of economical implicit methods for odes j comput appl math 116 2000 41 62 l brugnano c magherini blended implementation of block implicit methods for odes appl numer math 42 2002 29 45 l brugnano d trigiante block implicit methods for odes in d trigiante ed recent trends in numerical analysis nova science publishers new york 2001 pp 81 105 this approach is based on the so called blended implementation of the methods giving corresponding blended implicit methods the latter have been implemented in the computational code bim l brugnano c magherini the bim code for the numerical solution of odes j comput appl math 164 165 2004 145 158 blended implicit methods are here extended to handle the numerical solution of dae problems resulting in a straightforward generalization of the basic approach

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

estimating latent asset pricing factors abstract we develop an estimator for latent factors in a large dimensional panel of financial data that can explain expected excess returns statistical factor analysis based on principal component analysis pca has problems identifying factors with a small variance that are important for asset pricing we generalize pca with a penalty term accounting for the pricing error in expected returns our estimator searches for factors that can explain both the expected return and covariance structure we derive the statistical properties of the new estimator and show that our estimator can find asset pricing factors which cannot be detected with pca even if a large amount of data is available applying the approach to portfolio data we find factors with sharpe ratios more than twice as large as those based on conventional pca and with smaller pricing errors

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

dynamic factor models with infinite dimensional factor space asymptotic analysis abstract factor models all particular cases of the generalized dynamic factor model gdfm introduced in forni et al 2000 have become extremely popular in the theory and practice of large panels of time series data the asymptotic properties consistency and rates of the corresponding estimators have been studied in forni et al 2004 those estimators however rely on brillinger s concept of dynamic principal components and thus involve two sided filters which leads to rather poor forecasting performances no such problem arises with estimators based on standard static principal components which have been dominant in this literature on the other hand the consistency of those static estimators requires the assumption that the space spanned by the factors has finite dimension which severely restricts their generality prohibiting for instance autoregressive factor loadings this paper derives the asymptotic properties of a semiparametric estimator of the loadings and common shocks based on one sided filters recently proposed by forni et al 2015 consistency and exact rates of convergence are obtained for this estimator under a general class of gdfms that does not require a finite dimensional factor space a monte carlo experiment and an empirical exercise on us macroeconomic data corroborate those theoretical results and demonstrate the excellent performance of those estimators in out of sample forecasting

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

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

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

robust high dimensional volatility matrix estimation for high frequency factor model high frequency financial data allow us to estimate large volatility matrices with relatively short time horizon many novel statistical methods have been introduced to address large volatility matr

on time varying factor models estimation and testing abstract conventional factor models assume that factor loadings are fixed over a long horizon of time which appears overly restrictive and unrealistic in applications in this paper we introduce a time varying factor model where factor loadings are allowed to change smoothly over time we propose a local version of the principal component method to estimate the latent factors and time varying factor loadings simultaneously we establish the limiting distributions and uniform convergence of the estimated factors and factor loadings in the standard large n and large t framework we also propose a bic type information criterion to determine the number of factors which can be used in models with either time varying or time invariant factor models based on the comparison between the estimates of the common components under the null hypothesis of no structural changes and those under the alternative we propose a consistent test for structural changes in factor loadings we establish the null distribution the asymptotic local power property and the consistency of our test simulations are conducted to evaluate both our nonparametric estimates and test statistic we also apply our test to investigate stock and watson s 2009 u s macroeconomic data set and find strong evidence of structural changes in the factor loadings

fractal tiles associated with shift radix systems shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d dimensional real vector space they generalize well known numeration systems such as beta expansions expansions with respect to rational bases and canonical number systems beta numeration and canonical number systems are known to be intimately related to fractal shapes such as the classical rauzy fractal and the twin dragon these fractals turned out to be important for studying properties of expansions in several settings in the present paper we associate a collection of fractal tiles with shift radix systems we show that for certain classes of parameters r these tiles coincide with affine copies of the well known tiles associated with beta expansions and canonical number systems on the other hand these tiles provide natural families of tiles for beta expansions with non unit pisot numbers as well as canonical number systems with non monic expanding polynomials we also prove basic properties for tiles associated with shift radix systems indeed we prove that under some algebraic conditions on the parameter r of the shift radix system these tiles provide multiple tilings and even tilings of the d dimensional real vector space these tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above such a tiling may consist of tiles having infinitely many different shapes moreover the tiles need not be self affine or graph directed self affine

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

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

sufficient direction factor model and its application to gene expression quantitative trait loci discovery rapid improvement in technology has made it relatively cheap to collect genetic data however statistical analysis of existing data is still much cheaper thus secondary analysis of single nucleotide polymorphism snp data i e reanalysing existing data in an effort to extract more information is an attractive and cost effective alternative to collecting new data we study the relationship between gene expression and snps through a combination of factor analysis and dimension reduction estimation to take advantage of the flexibility in traditional factor models where the latent factors are not required to be normal we recommend using semiparametric sufficient dimension reduction methods in the joint estimation of the combined model the resulting estimator is flexible and has superior performance relative to the existing estimator which relies on additional assumptions on the latent factors we quantify the asymptotic performance of the proposed parameter estimator and perform inference by assessing the estimation variability and by constructing confidence intervals the new results enable us to identify for the first time statistically significant snps concerning gene snp relations in lung tissue from genotype tissue expression data

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

a randomized sequential procedure to determine the number of factors this paper proposes a procedure to estimate the number of common factors k in a static approximate factor model the building block of the analysis is the fact that the first k eigenvalues of the covariance matrix of the data diverge whilst the others stay bounded on the grounds of this we propose a test for the null that the i th eigenvalue diverges using a randomised test statistic based directly on the estimated eigenvalue the test only requires minimal assumptions on the data and no assumptions are required on factors loadings or idiosyncratic errors the randomised tests are then employed in a sequential procedure to determine k

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

using principal component analysis to estimate a high dimensional factor model with high frequency data this paper constructs an estimator for the number of common factors in a setting where both the sampling frequency and the number of variables increase empirically we document that the covariance matrix of a large portfolio of us equities is well represented by a low rank common structure with sparse residual matrix when employed for out of sample portfolio allocation the proposed estimator largely outperforms the sample covariance estimator

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

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

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

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

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

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