long memory continuous time models this paper presents a new family of long memory models the continuous time moving average fractional process the continuous time framework allows to reconcile two competitive types of modelling fractional integration of arma processes and fractional brownian motion a comparison with usual discrete time arfima models is lead some well known empirical evidence on macroeconomic and financial time series such as variability of forward rates aggregation of responses across heterogeneous agents are well captured by this continuous time modelling moreover the usual statistical tools for long memory series and for stochastic differential equations can be jointly applied in this setting

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

inference theory for volatility functional dependencies we develop inference theory for models involving possibly nonlinear transforms of the elements of the spot covariance matrix of a multivariate continuous time process observed at high frequency the framework can be used to study the relationship among the elements of the latent spot covariance matrix and processes defined on the basis of it such as systematic and idiosyncratic variances factor betas and correlations on a fixed interval of time the estimation is based on matching model implied moment conditions under the occupation measure induced by the spot covariance process we prove consistency and asymptotic mixed normality of our estimator of the random coefficients in the volatility model and further develop model specification tests we apply our inference methods to study variance and correlation risks in nine sector portfolios comprising the s p 500 index we document sector specific variance risks in addition to that of the market and time varying heterogeneous correlation risk among the market neutral components of the sector portfolio returns

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

the generalized dynamic factor model consistency and rates abstract a factor model generalizing those proposed by geweke in d j aigner and a s goldberger latent variables in socio economic models north holland amsterdam 1977 sargent and sims new methods in business research federal reserve bank of minneapolis minneapolis 1977 engle and watson j amer statist assoc 76 1981 774 and stock and watson j business econom statist 20 2002 147 has been introduced in forni et al rev econ statist 80 2000 540 where consistent as the number n of series and the number t of observations both tend to infinity along appropriate paths n t n estimation methods for the common component are proposed rates of convergence associated with these methods are obtained here as functions of the paths n t n along which n and t go to infinity these results show that under suitable assumptions consistency requires t n to be at least of the same order as n whereas an optimal rate of n is reached for t n of the order of n2 if convergence to the space of common components is considered consistency holds irrespective of the path t n thus can be arbitrarily slow the optimal rate is still n but only requires t n to be of the order of n

asymptotics of empirical eigenstructure for high dimensional spiked covariance we derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime which takes into account the magnitude of spiked eigenvalues sample size and dimensionality this regime allows high dimensionality and diverging eigenvalues and provides new insights into the roles that the leading eigenvalues sample size and dimensionality play in principal component analysis our results are a natural extension of those in paul 2007 to a more general setting and solve the rates of convergence problems in shen et al 2013 they also reveal the biases of estimating leading eigenvalues and eigenvectors by using principal component analysis and lead to a new covariance estimator for the approximate factor model called shrinkage principal orthogonal complement thresholding s poet that corrects the biases our results are successfully applied to outstanding problems in estimation of risks of large portfolios and false discovery proportions for dependent test statistics and are illustrated by simulation studies

econometric analysis of multivariate realised qml estimation of the covariation of equity prices under asynchronous trading estimating the covariance between assets using high frequency data is challenging due to market microstructure effects and asynchronous trading in this paper we develop a multivariate realised quasi likelihood qml approach carrying out inference as if the observations arise from an asynchronously observed vector scaled brownian model observed with error under stochastic volatility the resulting realised qml estimator is positive semi definite uses all available data is consistent and asymptotically mixed normal the quasi likelihood is computed using a kalman filter and optimised using a relatively simple em algorithm which scales well with the number of assets we derive the theoretical properties of the estimator and prove that it achieves the efficient rate of convergence we show how to make itobtain the non parametric efficiency bound for this problem the estimator is also analysed usingmonte carlo methods and applied to equity data with varying levels of liquidity

a direct formulation for sparse pca using semidefinite programming given a covariance matrix we consider the problem of maximizing the variance explained by a particular linear combination of the input variables while constraining the number of nonzero coefficients in this combination this problem arises in the decomposition of a covariance matrix into sparse factors or sparse principal component analysis pca and has wide applications ranging from biology to finance we use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix where cardinality is constrained and derive a semidefinite programming based relaxation for our problem we also discuss nesterov s smooth minimization technique applied to the semidefinite program arising in the semidefinite relaxation of the sparse pca problem the method has complexity o n 4 sqrt log n epsilon where n is the size of the underlying covariance matrix and epsilon is the desired absolute accuracy on the optimal value of the problem

a model free non parametric method for density determination with application to asset returns abstract the distribution of the returns of an asset is still an open problem despite the variety of models and methods devised to deal with it in this note we propose a model free nonparametric method for the estimation of asset returns which is applicable also to a wider set of similar problems the laplace transform of a shifted rate of return or other percent change of a positive random variables can be transformed into a fractional moment problem which can then be solved by the method of maximum entropy this is a very robust method and requires only a few values of the laplace transform to provide good reconstructions this method can be applied in multiple fields where model free density estimation of functionals like percent changes of random variables is of interest

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

optimal rates of convergence for sparse covariance matrix estimation this paper considers estimation of sparse covariance matrices and establishes the optimal rate of convergence under a range of matrix operator norm and bregman divergence losses a major focus is on the derivation of a rate sharp minimax lower bound the problem exhibits new features that are significantly different from those that occur in the conventional nonparametric function estimation problems standard techniques fail to yield good results and new tools are thus needed we first develop a lower bound technique that is particularly well suited for treating two directional problems such as estimating sparse covariance matrices the result can be viewed as a generalization of le cam s method in one direction and assouad s lemma in another this lower bound technique is of independent interest and can be used for other matrix estimation problems we then establish a rate sharp minimax lower bound for estimating sparse covariance matrices under the spectral norm by applying the general lower bound technique a thresholding estimator is shown to attain the optimal rate of convergence under the spectral norm the results are then extended to

observables on quantum structures an observable on a quantum structure is any s homomorphism of quantum structures from the borel s algebra into the quantum structure we show that our partial information on an observable known only for all intervals of the form t is sufficient to derive the whole information about the observable defined on quantum structures like s mv algebras s lattice effect algebras boolean s algebras monotone s complete effect algebras with the riesz decomposition property the effect algebra of effect operators of a hilbert space and systems of functions effect tribes

on wielandt s inequality and its application to the asymptotic distribution of the eigenvalues of a random symmetric matrix a relatively obscure eigenvalue inequality due to wielandt is used to give a simple derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix the asymptotic distributions are obtained under a fairly general setting an application of the general theory to the bootstrap distribution of the eigenvalues of the sample covariance matrix is given 1 introduction and summary the derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix arises in many papers in multivariate analysis although the main idea behind most of the derivations is quite basic i e the expansion of the sample roots about the population roots the derivations themselves are often quite involved these complications are primarily due to the mathematical rather than statistical nature of the eigenvalue problem one of the main objectives of this paper is to introduce a simple method for obtaining the asymptotic distribution of the eigenvalue of random symmetric matrices the method is based upon a relatively obscure eigenvalue inequality

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

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

convex analysis on the hermitian matrices there is growing interest in optimization problems with real symmetric matrices as variables generally the matrix functions involved are spectral they depend only on the eigenvalues of the matrix it is known that convex spectral functions can be characterized exactly as symmetric convex functions of the eigenvalues a new approach to this characterization is given via a simple fenchel conjugacy formula we then apply this formula to derive expressions for subdifferentials and to study duality relationships for convex optimization problems with positive semidefinite matrices as variables analogous results hold for hermitian matrices

self consistent asset pricing models we discuss the foundations of factor or regression models in the light of the self consistency condition that the market portfolio and more generally the risk factors is are constituted of the assets whose returns it is they are supposed to explain as already reported in several articles self consistency implies correlations between the return disturbances as a consequence the alphas and betas of the factor model are unobservable self consistency leads to renormalized betas with zero effective alphas which are observable with standard ols regressions when the conditions derived from internal consistency are not met the model is necessarily incomplete which means that some sources of risk cannot be replicated or hedged by a portfolio of stocks traded on the market even for infinite economies analytical derivations and numerical simulations show that for arbitrary choices of the proxy which are different from the true market portfolio a modified linear regression holds with a non zero value ai at the origin between an asset i s return and the proxy s return self consistency also introduces orthogonality and normality conditions linking the betas alphas as well as the residuals and the weights of the proxy portfolio two diagnostics based on these orthogonality and normality conditions are implemented on a basket of 323 assets which have been components of the s p500 in the period from january 1990 to february 2005 these two diagnostics show interesting departures from dynamical self consistency starting about 2 years before the end of the internet bubble assuming that the capm holds with the self consistency condition the ols method automatically obeys the resulting orthogonality and normality conditions and therefore provides a simple way to selfconsistently assess the parameters of the model by using proxy portfolios made only of the assets which are used in the capm regressions finally the factor decomposition with the self consistency condition derives a risk factor decomposition in the multi factor case which is identical to the principal component analysis pca thus providing a direct link between model driven and data driven constructions of risk factors this correspondence shows that pca will therefore suffer from the same limitations as the capm and its multi factor generalization namely lack of out of sample explanatory power and predictability in the multi period context the self consistency conditions force the betas to be time dependent with specific constraints r 2007 elsevier b v all rights reserved

roughness in mv algebras in this paper by considering the notion of an mv algebra we consider a relationship between rough sets and mv algebra theory we introduce the notion of rough ideal with respect to an ideal of an mv algebra which is an extended notion of ideal in an mv algebra and we give some properties of the lower and the upper approximations in an mv algebra

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

high dimensional analysis of semidefinite relaxations for sparse principal components principal component analysis pca is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix however pca is well known to behave poorly in the large p small n setting in which the problem dimension p is comparable to or larger than the sample size n this paper studies pca in this high dimensional regime but under the additional assumption that the maximal eigenvector is sparse say with at most k nonzero components we consider a spiked covariance model in which a base matrix is perturbed by adding a k sparse maximal eigenvector and we analyze two computationally tractable methods for recovering the support set of this maximal eigenvector as follows a a simple diagonal thresholding method which transitions from success to failure as a function of the rescaled sample size dia n p k n k 2 log p k and b a more sophisticated semidefinite programming sdp relaxation which succeeds once the rescaled sample size sdp n p k n klog p k is larger than a critical threshold in addition we prove that no method including the best method which has exponential time complexity can succeed in recovering the support if the order parameter sdp n p k is below a threshold our results thus highlight an interesting trade off between computational and statistical efficiency in high dimensional inference

pseudo bck algebras as partial algebras it is well known that the representation of several classes of residuated lattices involves lattice ordered groups an often applicable method to determine the representing group or groups from a residuated lattice is based on partial algebras the monoidal operation is restricted to those pairs that fulfil a certain extremality condition and else left undefined the subsequent construction applied to the partial algebra is easy transparent and leads directly to the structure needed for representation in this paper we consider subreducts of residuated lattices the monoidal and the meet operation being dropped the resulting algebras are pseudo bck semilattices assuming divisibility we can pass on to partial algebras also in this case to reconstruct the underlying group structure from this partial algebra if applicable is again straightforward we demonstrate the elegance of this method for two classes of pseudo bck semilattices semilinear divisible pseudo bck algebras and cone algebras

generalized continuous and left continuous t norms arising from algebraic semantics for fuzzy logics this paper focuses on the issue of how generalizations of continuous and left continuous t norms over linearly ordered sets should be from a logical point of view taking into account recent results in the scope of algebraic semantics for fuzzy logics over chains with a monoidal residuated operation we advocate linearly ordered bl algebras and mtl algebras as adequate generalizations of continuous and left continuous t norms respectively in both cases the underlying basic structure is that of linearly ordered residuated lattices although the residuation property is equivalent to left continuity in t norms continuous t norms have received much more attention due to their simpler structure we review their complete description in terms of ordinal sums and discuss the problem of describing the structure of their generalization to bl chains in particular we show the good behavior of bl algebras over a finite or complete chain and discuss the partial knowledge of rational bl chains then we move to the general non continuous case corresponding to left continuous t norms and mtl chains the unsolved problem of describing the structure of left continuous t norms is presented together with a fistful of construction decomposition techniques that apply to some distinguished families of t norms and finally we discuss the situation in the general study of mtl chains as a natural generalization of left continuous t norms

the standard completeness of interval valued monoidal t norm based logic in this paper we prove the strong standard completeness of interval valued monoidal t norm based logic ivmtl and some of its extensions for other extensions we show that they are not strong standard complete we also give a local deduction theorem for ivmtl and other extensions of interval valued monoidal logic similar results are obtained for interval valued fuzzy logics expanded with baaz s delta

generalized method of integrated moments for high frequency data we propose a semiparametric two step inference procedure for a finite dimensional parameter based on moment conditions constructed from high frequency data the population moment conditions take the form of temporally integrated functionals of state variable processes that include the latent stochastic volatility process of an asset in the first step we nonparametrically recover the volatility path from high frequency asset returns the nonparametric volatility estimator is then used to form sample moment functions in the second step gmm estimation which requires the correction of a high order nonlinearity bias from the first step we show that the proposed estimator is consistent and asymptotically mixed gaussian and propose a consistent estimator for the conditional asymptotic variance we also construct a bierens type consistent specification test these infill asymptotic results are based on a novel empirical process type theory for general integrated functionals of noisy semimartingale processes

on filter theory of residuated lattices the aim of this paper is to develop the filter theory of general residuated lattices first we extend some particular types of filters and fuzzy filters in bl algebras and mtl algebras naturally to general residuated lattices and further enumerate some relative results obtained in bl algebras or mtl algebras which still hold in general residuated lattices next we introduce the concepts of regular filters and fuzzy regular filters to general residuated lattices which are two new types of filters and fuzzy filters and derive some of their characterizations finally we discuss the relations between fuzzy regular filters and several other special fuzzy filters and also characterize some special classes of residuated lattices by filters or fuzzy filters

multivariate realised kernels consistent positive semi definite estimators of the covariation of equity prices with noise and non synchronous trading we propose a multivariate realised kernel to estimate the ex post covariation of log prices we show this new consistent estimator is guaranteed to be positive semi definite and is robust to measurement noise of certain types and can also handle non synchronous trading it is the first estimator which has these three properties which are all essential for empirical work in this area we derive the large sample asymptotics of this estimator and assess its accuracy using a monte carlo study we implement the estimator on some us equity data comparing our results to previous work which has used returns measured over 5 or 10 minutes intervals we show the new estimator is substantially more precise

the pasting constructions of lattice ordered effect algebras the aim of this paper is to present some ways of constructing a lattice ordered effect algebra from the given family of mv algebras the loop lemma of lattice ordered effect algebras is proved then the definitions of greechie diagrams of effect algebras are given all these results have generalized those of orthomodular lattices as applications of loop lemma and greechie diagrams some lattice ordered effect algebras without states are presented

states on finite monoidal t norm based algebras the aim of this paper is to study the existences of bosbach states and riecan states on finite monoidal t norm based algebras mtl algebras for short we give some examples to show that there exist mtl algebras having no bosbach states and riecan states the conditions under which mtl algebras have bosbach states and riecan states are investigated respectively we prove that riecan states on mtl algebras are reduced to states on imtl algebras furthermore the necessary and sufficient conditions for finite linearly ordered locally finite mtl algebras and peculiar mtl algebras having bosbach states and riecan states are obtained respectively in addition the notions of pseudo quasi equivalent and a subalgebra under pseudo quasi equivalent are proposed and some of their properties are investigated

fokker planck equation of distributions of financial returns and power laws abstract our purpose is to relate the fokker planck formalism proposed by friedrich et al phys rev lett 84 2000 5224 for the distribution of stock market returns to the empirically well established power law distribution with an exponent in the range 3 5 we show how to use friedrich et al s formalism to predict that the distribution of returns is indeed asymptotically a power law with an exponent that can be determined from the kramers moyal coefficients determined by friedrich et al however with their values determined for the u s dollar german mark exchange rates the exponent predicted from their theory is found to be around 12 in disagreement with the often quoted value between 3 and 5 this could be explained by the fact that the large asymptotic value of 12 does not apply to real data that lie still far from the stationary state of the fokker planck description another possibility is that power laws are inadequate the mechanism for the power law is based on the presence of multiplicative noise across time scales which is different from the multiplicative noise at fixed time scales implicit in the arch models developed in the finance literature

inference for continuous semimartingales observed at high frequency the econometric literature of high frequency data often relies on moment estimators which are derived from assuming local constancy of volatility and related quantities we here study this local constancy approximation as a general approach to estimation in such data we show that the technique yields asymptotic properties consistency normality that are correct subject to an ex post adjustment involving asymptotic likelihood ratios these adjustments are derived and documented several examples of estimation are provided powers of volatility leverage effect and integrated betas the first order approximations based on local constancy can be over the period of one observation or over blocks of successive observations it has the advantage of gaining in transparency in defining and analyzing estimators the theory relies heavily on the interplay between stable convergence and measure change and on asymptotic expansions for martingales

on the estimation of integrated covariance matrices of high dimensional diffusion processes we consider the estimation of integrated covariance icv matrices of high dimensional diffusion processes based on high frequency observations we start by studying the most commonly used estimator the realized covariance rcv matrix we show that in the high dimensional case when the dimension p and the observation frequency n grow in the same rate the limiting spectral distribution lsd of rcv depends on the covolatility process not only through the targeting icv but also on how the covolatility process varies in time we establish a marcenko pastur type theorem for weighted sample covariance matrices based on which we obtain a marcenko pastur type theorem for rcv for a class mathcal c of diffusion processes the results explicitly demonstrate how the time variability of the covolatility process affects the lsd of rcv we further propose an alternative estimator the time variation adjusted realized covariance tvarcv matrix we show that for processes in class mathcal c the tvarcv possesses the desirable property that its lsd depends solely on that of the targeting icv through the marcenko pastur equation and hence in particular the tvarcv can be used to recover the empirical spectral distribution of the icv by using existing algorithms

a non associative generalization of hajek s bl algebras hajek introduced basic logic bl as the logic of continuous t norms and their residua basic logic is a fuzzy logic i e it is complete with respect to linearly ordered models algebraic semantics of bl is the variety of bl algebras it was proved by cignoli esteva godo and torrens that the variety of bl algebras is generated just by the continuous t norms on the interval 0 1 of reals the main goal of the paper is to present a non associative generalization of hajek s bl logic which has a class nabl of non associative bl algebras as its algebraic semantics moreover it is shown that nabl forms a variety generated just by non associative t norms consequently the non associative bl logic is the logic of non associative t norms and their residua

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

principal component analysis of high frequency data we develop the necessary methodology to conduct principal component analysis at high frequency we construct estimators of realized eigenvalues eigenvectors and principal components and provide

pre averaging estimators of the ex post covariance matrix in noisy diffusion models with non synchronous data we show how pre averaging can be applied to the problem of measuring the ex post covariance of financial asset returns under microstructure noise and non synchronous trading a pre averaged realised covariance is proposed and we present an asymptotic theory for this new estimator which can be configured to possess an optimal convergence rate or to ensure positive semi definite covariance matrix estimates we also derive a noise robust hayashi yoshida estimator that can be implemented on the original data without prior alignment of prices we uncover the finite sample properties of our estimators with simulations and illustrate their practical use on high frequency equity data

a test for the rank of the volatility process the random perturbation approach in this paper we present a test for the maximal rank of the matrix valued volatility process in the continuous it o semimartingale framework our idea is based upon a random perturbation of the original high frequency observations of an it o semimartingale which opens the way for rank testing we develop the complete limit theory for the test statistic and apply it to various null and alternative hypotheses finally we demonstrate a homoscedasticity test for the rank process

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

asymptotic inference for eigenvectors abstract asymptotic procedures are given for testing certain hypotheses concerning eigenvectors and for constructing confidence regions for eigenvectors these asymptotic procedures are derived under fairly general conditions on the estimates of the matrix whose eigenvectors are of interest applications of the general results to principal components analysis and canonical variate analysis are given author

application of quantum master equation for long term prognosis of asset prices abstract this study combines the disciplines of behavioral finance and an extension of econophysics namely the concepts and mathematical structure of quantum physics we apply the formalism of quantum theory to model the dynamics of some correlated financial assets where the proposed model can be potentially applied for developing a long term prognosis of asset price formation at the informational level the asset price states interact with each other by the means of a financial bath the latter is composed of agents expectations about the future developments of asset prices on the finance market as well as financially important information from mass media society and politicians one of the essential behavioral factors leading to the quantum like dynamics of asset prices is the irrationality of agents expectations operating on the finance market these expectations lead to a deeper type of uncertainty concerning the future price dynamics of the assets than given by a classical probability theory e g in the framework of the classical financial mathematics which is based on the theory of stochastic processes the quantum dimension of the uncertainty in price dynamics is expressed in the form of the price states superposition and entanglement between the prices of the different financial assets in our model the resolution of this deep quantum uncertainty is mathematically captured with the aid of the quantum master equation its quantum markov approximation we illustrate our model of preparation of a future asset price prognosis by a numerical simulation involving two correlated assets their returns interact more intensively than understood by a classical statistical correlation the model predictions can be extended to more complex models to obtain price configuration for multiple assets and portfolios

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

adaptive estimation of continuous time regression models using high frequency data abstract we derive the asymptotic efficiency bound for regular estimates of the slope coefficient in a linear continuous time regression model for the continuous martingale parts of two ito semimartingales observed on a fixed time interval with asymptotically shrinking mesh of the observation grid we further construct an estimator from high frequency data that achieves this efficiency bound and indeed is adaptive to the presence of infinite dimensional nuisance components the estimator is formed by taking optimal weighted average of local nonparametric volatility estimates that are constructed over blocks of high frequency observations the asymptotic efficiency bound is derived under a markov assumption for the bivariate process while the high frequency estimator and its asymptotic properties are derived in a general ito semimartingale setting to study the asymptotic behavior of the proposed estimator we introduce a general spatial localization procedure which extends known results on the estimation of integrated volatility functionals to more general classes of functions of volatility empirically relevant numerical examples illustrate that the proposed efficient estimator provides nontrivial improvement over alternatives in the extant literature

high frequency covariance estimates with noisy and asynchronous financial data this article proposes a consistent and efficient estimator of the high frequency covariance quadratic covariation of two arbitrary assets observed asynchronously with market microstructure noise this estimator is built on the marriage of the quasi maximum likelihood estimator of the quadratic variation and the proposed generalized synchronization scheme and thus is not influenced by the epps effect moreover the estimation procedure is free of tuning parameters or bandwidths and is readily implementable monte carlo simulations show the advantage of this estimator by comparing it with a variety of estimators with specific synchronization methods the empirical studies of six foreign exchange future contracts illustrate the time varying correlations of the currencies during the 2008 global financial crisis demonstrating the similarities and differences in their roles as key currencies in the global market

vast volatility matrix estimation for high frequency financial data high frequency data observed on the prices of financial assets are commonly modeled by diffusion processes with micro structure noise and realized volatility based methods are often used to estimate integrated volatility for problems involving a large number of assets the estimation objects we face are volatility matrices of large size the existing volatility estimators work well for a small number of assets but perform poorly when the number of assets is very large in fact they are inconsistent when both the number p of the assets and the average sample size n of the price data on the p assets go to infinity this paper proposes a new type of estimators for the integrated volatility matrix and establishes asymptotic theory for the proposed estimators in the framework that allows both n and p to approach to infinity the theory shows that the proposed estimators achieve high convergence rates under a sparsity assumption on the integrated volatility matrix the numerical studies demonstrate that the proposed estimators perform well for large p and complex price and volatility models the proposed method is applied to real high frequency financial data

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

dynamic factor models with infinite dimensional factor spaces one sided representations abstract factor model methods recently have become extremely popular in the theory and practice of large panels of time series data those methods rely on various factor models which all are particular cases of the generalized dynamic factor model gdfm introduced in forniet al 2000 that paper however rests on brillinger s dynamic principal components the corresponding estimators are two sided filters whose performance at the end of the observation period or for forecasting purposes is rather poor no such problem arises with estimators based on standard principal components which have been dominant in this literature on the other hand those estimators require the assumption that the space spanned by the factors has finite dimension in the present paper we argue that such an assumption is extremely restrictive and potentially quite harmful elaborating upon recent results by anderson and deistler 2008a b on singular stationary processes with rational spectrum we obtain one sided representations for the gdfm without assuming finite dimension of the factor space construction of the corresponding estimators is also briefly outlined in a companion paper we establish consistency and rates for such estimators and provide monte carlo results further motivating our approach

the variety of modular basic algebras generated by mv chains and horizontal sums of three element chain basic algebras basic algebras or lattices with section antitone involutions turned out to be appropriate structures when axiomatizing both many valued and quantum logics see e g 11 the aim of the present paper is to describe the variety of basic algebras generated by mv chains and horizontal sums of three element chain basic algebras